2.1. Mathematical description of the problem

2.1.1. Theoretical framework

Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) ‘… put forward in particular a so-called ‘Lamb analogy’ mirroring the flow within the impacting droplet with the one induced with a flat rising expanding disk…’ (see figure 3). The equations describing the impact-induced flow within the droplet have both steady and unsteady terms but some research analyses only the latter due to its higher order of magnitude at early times (Riboux & Gordillo Reference Riboux and Gordillo2014; Philippi et al. Reference Philippi, Lagrée and Antkowiak2016; Negus et al. Reference Negus, Moore, Oliver and Cimpeanu2021). However, Lamb’s original problem (Lamb Reference Lamb1916) (below called ‘steady disk’ (SD)) describes the steady flow around a flat rising disk, which does not have an unsteady component. Philippi et al. showed that by transposing Lamb’s steady solution into rescaled lengths of $l/\sqrt {t}$ , where $l$ represents any length variable and time $t$ acts as a normalisation parameter, Lamb’s solution becomes a quasi-time-dependent expression and is an analogue to the impact-induced flow in self-similar space (below called ‘self-similar disk’ (SSD)). We build on this, but here we hypothesise that the disk radius $a$ in Lamb’s original problem is now a function of $t$ , and we will explore the full unsteady potential flow of this flat, rising, expanding disk including both steady and unsteady components (below called ‘unsteady disk’ (UD)). The UD framework has a fundamental difference from the SSD, as we shall see; and we propose the former UD as being a better framework to make an analogy with the droplet impact problem.

Figure 3. The problem of liquid droplet impact (downwards at velocity $u_0$ ) onto a solid surface (a) has been found to have analogies with the problem of a rising expanding circular disk (moving upwards at velocity $u_0$ ) in an infinite mass of liquid (b). The yellow straight arrows in (a) denote the spreading of the wet radius (or contact line before lamella ejection), while the yellow curved arrows in (b) denote the winding motions of the liquid flows around the wet radius of the disk and the blue arrows denote the radial expansion of the disk with the wet radius. Figures are made after Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016).

2.1.2. Governing equations

Lamb’s mathematical problem, the SD framework, relates to the motion of a thin circular disk of radius $a$ with constant velocity $\boldsymbol{u_0}$ normal to its plane in an infinite mass of liquid. In the inertial frame of reference moving at velocity $\boldsymbol{u_0}$ , the origin is taken at the centre of the disk for all time $t\geqslant 0$ . Lamb described the flow as an incompressible and inviscid axisymmetric ideal fluid field with the far field at rest (hence irrotational), so we can define a velocity potential $\phi (r,z)$ :

(2.1) \begin{equation} \boldsymbol{u}(r,z)=\boldsymbol{\nabla }\phi (r,z). \end{equation}

By the mass conservation law, $\phi (r,z)$ satisfies the Laplacian equation, written in cylindrical coordinates,

(2.2) \begin{equation} \boldsymbol{\nabla} ^2\phi =\frac {1}{r} \frac {\partial }{\partial r} \left ( r \frac {\partial \phi }{\partial r} \right ) + \frac {\partial ^2 \phi }{\partial z^2}=0 \end{equation}

and Bernoulli’s conservation equation for mechanical energy in an isentropic flow with negligible change in gravitational potential energy, everywhere in the liquid is

(2.3) \begin{equation} \frac {1}{2} |\boldsymbol{\nabla }\phi |^2 + \frac {p}{\rho } = \rm const. \end{equation}

where the dimensionless density $\rho$ is $1$ . The ideal fluid field in a designated control volume satisfies three boundary conditions, namely

(2.4) \begin{equation} \frac {\partial \phi }{\partial z}(r,0)=1 \quad \text{for} \quad r \leqslant a \end{equation}

due to non-permeability at the disk surface $z=0$ ; and

(2.5) \begin{equation} \phi (r,0)=0 \quad \text{for} \quad r \gt a \end{equation}

and

(2.6) \begin{equation} \boldsymbol{\nabla }\phi (r,z)\rightarrow 0 \quad \text{as} \quad r^2+z^2 \rightarrow \infty \end{equation}

because the velocity field is undisturbed far from the disk’s surface.

For the model UD framework, studied here, however, we additionally assume the disk is also expanding in radius $a(t)$ in time, so the equations are unsteady. With the new time-dependent velocity potential $\phi (r,z,t)$ :

(2.7) \begin{equation} \boldsymbol{u}(r,z,t)=\boldsymbol{\nabla }\phi (r,z,t). \end{equation}

The set of (2.2), (2.3) subject to boundary conditions (2.4)–(2.6) now read, respectively, as follows:

(2.8) \begin{equation} \boldsymbol{\nabla} ^2\phi =\frac {1}{r} \frac {\partial }{\partial r} \left ( r \frac {\partial \phi }{\partial r} \right ) + \frac {\partial ^2 \phi }{\partial z^2}=0, \end{equation}

(2.9) \begin{equation} \frac {\partial \phi }{\partial t} + \frac {1}{2} |\boldsymbol{\nabla }\phi |^2 + \frac {p}{\rho } = b(t), \end{equation}

where $b(t)$ is a function of time only, and three boundary conditions,

(2.10) \begin{equation} \frac {\partial \phi }{\partial z}(r,0,t)=1 \quad \text{for} \quad r \leqslant a(t), t \geqslant 0, \end{equation}

(2.11) \begin{equation} \phi (r,0,t)=0 \quad \text{for} \quad r \gt a(t), t \geqslant 0, \end{equation}

(2.12) \begin{equation} \boldsymbol{\nabla }\phi (r,z,t)\rightarrow 0 \quad \text{as} \quad r^2+z^2 \rightarrow \infty , t \geqslant 0. \end{equation}

Figure 4 presents the geometries of the SD and UD models, and summarises corresponding governing equations.

Figure 4. The control volume of the theoretical frameworks for a SD (a) and an UD ( $a(t)$ ) (b), including problem geometries, boundary conditions and the governing equations.

2.1.3. Analogy with an expanding water-entry problem

Figure 5. The two-dimensional potential flow theory for flow over a thin plate (a) in $\hat {z}$ plane can be derived from the potential flow past a cylinder (b) in $\hat {\zeta }$ plane using conformal mappings. By the same principle, the proposed expanding water-entry problem for an expanding cylinder (d) in (vertically flipped) $\hat {\zeta }$ plane can be mapped to the $\hat {z}$ plane (c). The dashed line in (d) denotes the free water surface in the water-entry problem (see figures 1 a and 1 b) with the intersections with the object marked by yellow crosses. The dashed line and yellow crosses are then mapped to (c) as the droplet free interface and intersections with the solid surface, respectively. Flow streamlines in (c) and (d) are continued in transparency beyond liquid free interface for comparison with (a) and (b).

When observing a side view of the disk, the rising thin flat disk in an infinite depth of flow field (in the frame of reference moving with the disk’s centre) is nothing other than the well-known flow around a thin flat plate with incidence (at an angle of attack $\pi /2$ ) in two-dimensional potential flow theory (see figure 5 a). The solution to this classic mathematical problem can be obtained by conformal mappings from the known potential flow past a cylinder (figure 5 b). In particular, the Joukovskii transformation is a conformal mapping that can achieve this goal. We use the ‘hat’ symbol to denote flow quantities in two-dimensional potential flow theory,

(2.13) \begin{equation} \hat {z}= \frac {1}{2} \left ( \hat {\zeta } + \frac {\hat {a}^2}{\hat {\zeta }} \right ) \end{equation}

where $\hat {z}$ is the complex variable in the two-dimensional plane of the plate with length $2\hat {a}$ , and $\hat {\zeta }$ is the complex variable in the two-dimensional plane of the cylinder with radius $\hat {a}$ .

By this principle, the Joukovskii transformation (2.13) relates (the side view of) the proposed expanding UD framework (figure 5 c) in the current work to an ‘expanding water-entry problem’, as in figure 5(d) with flipped vertical direction relative to figure 5(c). The latter describes a cylinder of expanding radius $a(t)$ that moves towards initially calm water. We note the analogy, as well as the difference, of this ‘expanding water-entry problem’ to the well-known water-entry problem (Wagner Reference Wagner1932; Wilson Reference Wilson1989) where the solid object has a fixed radius. In the considered analogy (i.e. expanding water-entry problem) of figure 5(d), the flow field behaves as that in figure 5(b). However, due to the presence of the water surface (denoted as dashed line in figure 5 d), only the bottom part of the flow (denoted as non-transparent streamlines) is valid in the expanding water-entry problem. These streamlines intersect the object, namely the expanding cylinder, at the yellow crosses (Wagner Reference Wagner1932; Wilson Reference Wilson1989). Now, if we apply the Joukovskii transformation (2.13) to the expanding water-entry geometry of figure 5(d), we obtain the potential flow around a thin flat plate in the $\hat {z}$ -plane in figure 5(c). It is of particular note that this $\hat {z}$ -plane flow intersects the object, the thin plate in this case, at the yellow crosses which are radially inboard of the expanding wet radius $a(t)$ . We propose that only the obtained non-transparent streamlines (in the $\hat {z}$ -plane) of the initial UD problem are applicable to droplet impact problem, and to locate this part of the flow precisely, we shall find the corresponding radial position of the yellow intersection points in the UD framework (see § 3.2).

It is worth noting that the analogy in this section is derived based on two-dimensional potential flow theory: however, the concept of the free water surface and the corresponding intersection points are applicable in three dimensions. Therefore, the obtained expanding water-entry problem in the $\hat {z}$ -plane of figure 5(c) is the analogue to the UD framework. This analogy is necessary as a correction to the rising (expanding) disk model due to the fact that, in the original droplet impact problem (figure 3 a), the flow is spreading over the surface due to the spatial restriction of the solid surface, instead of winding around the (expanding) disk (Philippi et al. Reference Philippi, Lagrée and Antkowiak2016) as in the unbounded flow domain of Lamb’s disk problem (Lamb Reference Lamb1916).

2.2. Derivations of the analytical solutions

Lamb (Reference Lamb1916) provides an implicit solution for the SD problem (i.e. Laplacian (2.2) subject to boundary conditions (2.4)–(2.6)). The implicit solution is a function of $\phi$ and $\partial \phi /\partial z$ on the side $z \geqslant 0$ assuming $\phi (r,0) = f(r)$ (we note in passing the use of $f(t)$ for the dimensionless force in other sections; we rely on the reader’s judgment to distinguish these, based on the context) at $z=0$ ,

(2.14) \begin{equation} \phi (r,z)=\int _{0}^{\infty } e^{-kz} J_0(kr) k\,\text{d}k \; \int _{0}^{\infty } f(r') J_0(kr')r' \,\text{d}r' \end{equation}

where $J_0$ is the Bessel function, and

(2.15) \begin{equation} f(r) = - C\sqrt {a^2-r^2} \end{equation}

where $C$ is an arbitrary constant.

In this work, we aim to derive the solution to the UD problem (i.e. Laplacian (2.8) subject to boundary conditions (2.10)–(2.12)) at the disk surface $z=0$ based on Lamb’s implicit solution (2.14) for the SD problem. Moreover, we will generalise the obtained solution to flow with an angle of incidence in the range from $0$ to $\pi /2$ . Here we find the velocity potential for the modified ‘unsteady Lamb disk’ derived exactly as for (2.14) but with $f(r)$ being replaced by $f(r,t)$ ,

(2.16) \begin{equation} \phi (r,z,t)=\int _{0}^{\infty } e^{-kz} J_0(kr) k\,\text{d}k \; \int _{0}^{\infty } f(r',t) J_0(kr')r' \,\text{d}r' \end{equation}

where $\phi (r,0,t) = f(r,t)$ at $z=0$ . In the UD framework, we set $f(r,t)$ as

(2.17) \begin{equation} f(r,t) = - C\sqrt {a(t)^2-r^2} \end{equation}

for $r\lt a(t)$ with $C$ an arbitrary constant. Subject to the boundary condition (2.11), we further obtain

(2.18) \begin{align} \int _{0}^{\infty } J_0(kr') \sqrt {a(t)^2-{(r')}^2} r' \,\text{d}r' &=a(t)^3 \int _{0}^{\frac {\pi }{2}} J_0(k a(t) \sin {\zeta })\sin {\zeta } \cos ^2{\zeta } \,\text{d}\zeta \nonumber \\ &=-a(t)^3 \frac {d}{ka(t) \; d(ka(t))} \frac {\sin {(ka(t))}}{ka(t)}. \end{align}

Hence, (2.16) is simplified to

(2.19) \begin{equation} \phi (r,z,t)=C\int _{0}^{\infty } e^{-kz} J_0(kr) \frac {\text{d}}{\text{d}k} \left (\frac {\sin {(ka(t))}}{k}\right ) \,\text{d}k. \end{equation}

Below, we will solve (2.19) into explicit equations of $\phi (r,z,t)$ at $z=0$ . For this, we recall important properties of Bessel’s functions, denoted as P1–P4, in Appendix A.

2.2.1. Derivations of velocity components on the disk surface

Equation (2.19) is split into two terms by the product rule,

(2.20) \begin{equation} \phi (r,z,t)=C\int _{0}^{\infty }\left (e^{-kz} J_0(kr) \frac {\cos {\left (ka(t) \right )}}{k} a(t) + e^{-kz} J_0(kr) \frac {-\!\sin {\left (ka(t) \right )}}{k^2}\right )\,\text{d}k. \end{equation}

At $z=0$ , the vertical velocity component $u_z$ is therefore

(2.21) \begin{align} u_z(r,0,t) &=\left (\frac {\partial \phi }{\partial z}\right )_{z=0} = - k \times \phi \nonumber \\ &=C \int _{0}^{\infty } e^0 J_0(kr) \cos {(ka(t))} (-a(t))\,\text{d}k + C\int _{0}^{\infty } e^{-0} J_0(kr) \frac {\sin {(ka(t))}}{k} \,\text{d}k. \end{align}

Integrating the first integral by parts converts the first term into

(2.22) \begin{equation} C [ - J_0(kr) \sin {(ka(t))} ]_0^{\infty } + C \int _0^{\infty } r J_0'(kr) \sin {(ka(t))} \,\text{d}k. \end{equation}

Based on the property P2 and the fact $\sin {(0)}=0$ , the first term of (2.22) vanishes, and hence (2.21) becomes

(2.23) \begin{equation} u_z(r,0,t) = C \left [ r \int _{0}^{\infty } J_0'(kr) \sin {(ka(t))} \,\text{d}k + \int _{0}^{\infty } J_0(kr) \frac {\sin {(ka(t))}}{k} \,\text{d}k\right ]\!. \end{equation}

Now, the second term has the exact value given by P4 while the first integral is the derivative of the second integral with respect to $r$ . Thereby

(2.24) \begin{align} u_z(r,0,t) &= C \left [ r \frac {d\left ({\pi }/{2}\right )}{\text{d}r} + \frac {\pi }{2}\right ] = \frac {\pi C}{2} \quad \text{for} \quad r \lt a(t), \\[-12pt]\nonumber \end{align}

(2.25) \begin{align} \text{or} \quad u_z(r,0,t) &= C \left [ r \frac {d\big (\sin ^{-1} ({a(t)}/{r})\big )}{\text{d}r} + \sin ^{-1} \left ( \frac {a(t)}{r} \right ) \right ] \\[-12pt]\nonumber \\ &= C\left [\sin ^{-1} \left ( \frac {a(t)}{r} \right ) - \frac {a(t)}{\sqrt {r ^2 - a(t)^2}}\right ] \quad \text{for }\quad r \gt a(t). \end{align}

Similarly for the radial velocity component $u_r$ at $z=0$ :

(2.26) \begin{equation} u_r(r,0,t) = \left (\frac {\partial \phi }{\partial r}\right )_{z=0} = \int _0^{\infty } e^0 J_0(kr) \cos {(ka(t))} a(t)\,\text{d}k + C \int _0^{\infty } e^{-0} J_0'(kr) \frac {-\!\sin {(ka(t))}}{k} \,\text{d}k .\end{equation}

Integrating by parts the first integral, the first term becomes

(2.27) \begin{equation} C[J_0'(kr) \sin {(ka(t))}]_0^{\infty } - C r \int _0^{\infty } J_0''(kr) \sin {(ka(t))} \,\text{d}k. \end{equation}

Based on P2 and the fact $\sin {(0)}=0$ , the first term vanishes, thereby $u_r$ becomes

(2.28) \begin{equation} u_r(r,0,t) = C \left [\int _{0}^{\infty } J_0'(kr) \frac {-\!\sin {(ka(t))}}{k} \,\text{d}k - r \int _{0}^{\infty } J_0''(kr) \sin {(ka(t))} \,\text{d}k\right ]\!. \end{equation}

In contrast to (2.23), (2.28) has a second-order derivative of $J_0$ formed through integration by parts. By P3 on both terms, we simplify (2.28) to

(2.29) \begin{equation} = C \left [\int _{0}^{\infty } J_1(kr) \frac {\sin {(ka(t))}}{k} \,\text{d}k + r \int _{0}^{\infty } J_1'(kr) \sin {(ka(t))} \,\text{d}k\right ]\!. \end{equation}

Now, by P4 and its derivative with respect to $r$ , (2.29) becomes

(2.30) \begin{align} u_r(r,0,t) &= C \left [ \frac {a(t)-\sqrt {a(t)^2-r^2}}{r} + r \frac {\text{d}\left ({\left(a(t)-\sqrt {a(t)^2-r^2}\right)}/{r}\right )}{\text{d}r} \right ] \nonumber \\ &= C \frac {r}{\sqrt {a(t)^2 - r^2}} \;\;\;\;\;\;\;\; \text{for } r \lt a(t) \\[-12pt]\nonumber \end{align}

(2.31) \begin{align} \text{or} \quad u_r(r,0,t) &= C \left [ \frac {a(t)}{r}+r \frac {\text{d}\left ({a(t)}/r\right )}{\text{d}r}\right ] = 0 \quad \text{for}\quad r \gt a(t). \end{align}

Composing the obtained velocity vectors on the disk surface, we have, at $z=0$ ,

(2.32) \begin{align} &\boldsymbol{u}(r,0,t) = \boldsymbol{\nabla }\phi (r,0,t) = \left (\frac {Cr}{\sqrt {a(t)^2-r^2}}\right ) \boldsymbol{\underline {\hat {r}}} + \left (\frac {\pi C}{2}\right )\boldsymbol{\underline {\hat {z}}} \quad \text{for} \quad r \lt a(t) \\[-12pt]\nonumber\end{align}

(2.33) \begin{align} \text{and} \quad &\boldsymbol{u}(r,0,t)= (0)\boldsymbol{\underline {\hat {r}}} +\left (C\left (\sin ^{-1} \left ( \frac {a(t)}{r}\right ) - \frac {a(t)}{\sqrt {r^2-a(t)^2}}\right )\right )\boldsymbol{\underline {\hat {z}}} \quad \text{for}\quad r \gt a(t) \end{align}

where the arbitrary constant $C$ is determined to be $2/\pi$ to satisfy the non-permeability boundary condition (2.10) on the disk surface. Note that the final boundary condition (2.12) is automatically satisfied in the current solution as the exponentials in the implicit integral solution (2.19) have been chosen so as to vanish for $z \rightarrow \infty$ (Lamb Reference Lamb1916). The obtained velocity solution directly indicates the physics of the UD problem that, at any instant $t$ , within the wet radius $a(t)$ , the vertical velocity of the flow on the disk surface is unity, due to non-permeability. In contrast, while outside the wet radius, or ‘equivalently’ (see § 3.3 for details) on the free interface of the droplet in the droplet impact analogy (figure 3), the flow velocity is purely downwards.

2.2.2. Derivations of time-derivatives on the disk surface

Unique to the UD framework, the pressure distribution on the surface of the expanding disk also requires the time-derivative term $\partial \phi / \partial t$ in (2.9). Equation (2.20) allows direct time-differentiation, at $z=0$ ,

(2.34) \begin{equation} \frac {\partial \phi }{\partial t} (r,0,t)= C \int _0^{\infty }\left (J_0(kr) \frac {\partial [\sin {(ka(t)})a(t)]}{\partial t} + J_0(kr) \frac {-1}{k^2} \frac {\partial [\sin {(ka(t)})]}{\partial t}\right ) \,\text{d}k \end{equation}

where only $a(t)$ is a function of time. Based on product rule, we have

(2.35) \begin{align} \frac {1}{C \times a'(t)} \frac {\partial \phi }{\partial t} (r,0,t) = \int _0^{\infty }J_0(kr)\left (\! -\!\sin {(ka(t)}) a(t) + \frac {\cos {(ka(t))}}{k} + \frac {\!-\!\cos {(ka(t))}}{k} \,\text{d}k\right ) \end{align}

where $a'(t)$ indicates the time-derivative of $a(t)$ . The second and the third terms in (2.35) cancel. Integrating by parts on the first term then gives

(2.36) \begin{equation} =[J_0 (kr) \cos {(ka(t))}]_0^{\infty } -r \int _0^{\infty } J_0'(kr) \cos {(ka(t))} \,\text{d}k. \end{equation}

Based on the property P2 and the fact $\cos {(0)}=1$ , the first term of (2.36) equals $-1$ ; and integrating by parts again on the second term gives

(2.37) \begin{equation} =\left [J_0 '(kr) \sin {(ka(t))} \frac {1}{a(t)}\right ]_0^{\infty } -\int _0^{\infty } J_0''(kr) \sin {(ka(t))} \frac {r}{a(t)} \,\text{d}k. \end{equation}

The first term of (2.37) vanishes due to P2, and $J_0 ''(kr)=-J_1 '(kr)$ is applied to the second term. Equations (2.36) and (2.37) then combine to give

(2.38) \begin{equation} \frac {1}{C \times a'(t)} \frac {\partial \phi }{\partial t} (r,0,t)= -1 - \frac {r^2}{a(t)} \int _0^{\infty } J_1'(kr) \sin {(ka(t))} \,\text{d}k. \end{equation}

Now, using the same principle of (2.29) to (2.30), we solve for the value of the integral in (2.38),

(2.39) \begin{align} \frac {1}{C \times a'(t)} \frac {\partial \phi }{\partial t} (r,0,t)&= -1 - \frac {1}{a(t)} \left [\frac {r^2}{\sqrt {a(t)^2 - r^2}} - \big(a(t)-\sqrt {a(t)^2-r^2}\big)\right ] \nonumber \\ &=-\frac {a(t)}{\sqrt {a(t)^2 - r^2}} \quad \text{for} \quad r \lt a(t) \\[-12pt]\nonumber\end{align}

(2.40) \begin{align} \text{or} \quad \frac {1}{C \times a'(t)} \frac {\partial \phi }{\partial t} (r,0,t)&= -1 - \frac {1}{a(t)} (a(t)) = -2 \quad \text{for} \quad r \gt a(t). \end{align}

Finally, we have

(2.41) \begin{align} \frac {\partial \phi }{\partial t}(r,0,t) &=- \frac {2}{\pi } \frac {a(t)a'(t)}{\sqrt {a(t)^2-r^2}} \quad \text{for} \quad r \lt a(t) \\[-12pt]\nonumber\end{align}

(2.42) \begin{align} \text{or} \;\;\;\; \frac {\partial \phi }{\partial t}(r,0,t) &=- \frac {4}{\pi } a'(t) \quad \text{for} \quad r \gt a(t) \end{align}

with $C=\pi /2$ derived previously. It is noted that the derived solutions are applicable for Reynolds number $\textit{Re}=\rho _lU_0R_0/\mu _l\gg 1$ and Weber number $\textit{We}=\rho _lU_0^2R_0/\sigma \gg 1$ , where $\mu _l$ is the dynamic viscosity of the liquid and $\sigma$ is the liquid surface tension.

2.2.3. Generalisation to flow with incidence on the disk surface

Referring to the two-dimensional potential flow theory in § 2.1.3, for flow over an expanding thin flat plate of length $2\hat {a}(\hat {t})$ at an angle of attack $\hat {\theta }$ anticlockwise from the west (figure 6), we can directly obtain the complex potential $\hat {w}(\hat {z},\hat {t})$ from the Joukovskii transformation (2.13), as

(2.43) \begin{equation} \hat {w}(\hat {z},\hat {t},\hat {\theta })=-\hat {v}_\infty \big ( \hat {z} \; \cos (\hat {\theta }) - i \sqrt {\hat {z}-\hat {a}(\hat {t})}\sqrt {\hat {z}+\hat {a}(\hat {t})}\;\sin (\hat {\theta }) \big ) \end{equation}

where $\hat {v}_\infty =1$ is the far field flow speed and $i$ is the complex square root of $-1$ . Particularly, we are interested in the velocity potential $\hat {\phi }(\hat {x},\hat {y},\hat {t})$ on the disk surface $|\hat {x}| \leqslant \hat {a}(\hat {t})$ and $\hat {y}=0$ ,

(2.44) \begin{align} \hat {\phi }(\hat {x},0,\hat {t},\hat {\theta })=\Re \big ( \hat {w}(\hat {z},\hat {t},\hat {\theta }) \big)|_{\hat {y}=0}=-\hat {v}_\infty \big ( \hat {x} \; \cos (\hat {\theta }) + \sqrt {\hat {a}(\hat {t})^2-\hat {x}^2}\;\sin (\hat {\theta }) \big) \quad \text{for} \quad |\hat {x}| \leqslant \hat {a}(\hat {t}) \end{align}

where $\Re$ denotes the real part of the complex potential. In fundamental potential flow theory, (2.44) comes from a superposition of a uniform flow and dipole at an angle of attack $\hat {\theta }$ before the Joukovskii transformation; and the theory still holds for a three-dimensional uniform flow and dipole. The angle $\hat {\theta }$ is acting as a weight on the relative strengths of the uniform flow and dipole. Particularly, at $\hat {\theta }= ({\pi }/{2})$ (a thin plate with velocity normal to its plane in an infinite mass of liquid), the weight of the uniform flow is zero and (2.44) is nothing other than (2.17) (up to the boundary coefficient which is $\pi /2$ in three-dimensional and $1$ in two-dimensional (Tassin et al. Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2010)). It is therefore a reasonable conjecture that Lamb’s implicit solution (2.14) on the disk’s surface, and hence the time-dependent implicit solution (2.19) as well as its derived explicit solutions (2.32), (2.33), (2.41), (2.42), are merely contributed by the dipole flow. To generalise the solution into any angle of attack $\theta$ , the part of uniform flow needs to be taken into consideration subject to a weight which is a function of $\theta$ .

Figure 6. The two-dimensional potential flow theory for flow over a thin flat plate of finite length in the $\hat {z}$ plane (see figure 5) can be generalised to arbitrary incidence $\hat {\theta }$ , and the expressions can be analytically obtained by Joukovskii transformation.

Returning to the three-dimensional UD problem, we limit our attention specifically to flows on the disk surface within the wet radius, namely $\phi (r,0,t)$ for $r\lt a(t)$ . Following the insights provided by the two-dimensional potential flow over a plate with incidence in (2.44), we reset (2.17) as

(2.45) \begin{equation} f(r,\beta ,t,\theta ) = -C \big ( r \, \cos (\theta ) \cos (\beta ) + \sqrt {a(t)^2-r^2} \; \sin (\theta ) \big) \quad \text{for} \quad r \lt a(t) \end{equation}

where the first term is the potential of uniform flows in cylindrical coordinates $(r, \beta , z$ ). Note that without loss of generosity, we define $\beta =0$ as the direction of the incoming flow. By the definition of $f(r,\beta ,t,\theta )$ , this is also $\phi (r,\beta ,0,t,\theta )$ on the disk surface within the wet radius,

(2.46) \begin{equation} \phi (r,\beta ,0,t,\theta ) = -C \big ( r \, \cos (\theta ) \cos (\beta ) + \sqrt {a(t)^2-r^2} \; \sin (\theta ) \big) \quad \text{for} \quad r \lt a(t).\end{equation}

To update the explicit solution (2.32) and (2.41) regarding the flow angle of attack $\theta$ , we notice the fact that there is no time-dependency in the first term of (2.46), thereby, it is straightforward to update (2.41) as

(2.47) \begin{equation} \frac {\partial \phi }{\partial t}(r,\beta ,0,t,\theta ) =- \frac {2}{\pi } \frac {a(t)a'(t)\sin (\theta )}{\sqrt {a(t)^2-r^2}} \quad \text{for} \quad r \lt a(t). \end{equation}

Since we limit our view onto flow within the wet radius, the vertical velocity component $u_z(r,\beta ,0,t,\theta )$ has to be unity to fulfil boundary condition (2.10). In this way, it is not necessary to employ the implicit solution (2.19) again, but rather we can directly obtain the radial velocity component $u_r(r,\beta ,0,t,\theta )$ by taking the radial derivative of (2.46):

(2.48) \begin{align} u_r(r,\beta ,0,t,\theta )= \frac {\partial }{\partial r} \left ( \phi (\beta ,r,0,t) \right )= C \left ( \!-\!\cos (\theta )\cos (\beta ) + \frac {r}{\sqrt {a(t)^2-r^2}} \; \sin (\theta ) \right ) \quad \text{for} \quad r \lt a(t), \end{align}

This allows us to update the velocity solution $\boldsymbol{u}(r,\beta ,0,t,\theta )$ as

(2.49) \begin{equation} \boldsymbol{u}(r,\beta ,0,t,\theta ) = \left (\frac {2}{\pi } \left ( \!-\!\cos (\theta )\cos (\beta ) + \frac {r}{\sqrt {a(t)^2-r^2}} \; \sin (\theta ) \right )\right ) \boldsymbol{\underline {\hat {r}}} + \left (1\right )\boldsymbol{\underline {\hat {z}}} \quad \text{for} \quad r \lt a(t). \end{equation}

3.1. Contact line dynamics

Figure 7. Flows over an expanding rising disk have a stagnation point $r_1$ at the origin (in the centre of the disk; coordinate directions shown in the bottom right-hand side of each panel) for normal angle impact (a) and off-origin for generalised flow with incidence $\theta \neq \pi /2$ (b). In (a), black arrows indicate the flows at $z=0$ outside the wet radius in the (expanding) rising disk framework, which deviate from the desired interface positions marked as green arrows on the dashed line (which denotes the droplet boundary) for droplet impact problem.

The flow over a rising disk, in either the SD framework (figure 4 a) or the (expanding) UD framework (figure 4 b), has a stagnation point at the origin (figure 7 a). As might be expected, this stagnation point $r_1$ (figure 7 b) is shifted in the generalised flow with incidence (solutions (2.49) and (2.47)) in the range $0 \leqslant \theta \leqslant \pi /2$ . The stagnation point $r_1(\beta ,t,\theta )$ is derived by equating (2.48) to zero,

(3.1) \begin{equation} u_r(r,\beta ,0,t,\theta )= \frac {2}{\pi } \left ( \!-\!\cos (\theta )\cos (\beta ) + \frac {r_1(\beta ,t,\theta )}{\sqrt {a(t)^2-{r_1(\beta ,t,\theta )}^2}} \; \sin (\theta ) \right )=0 \end{equation}

with solution

(3.2) \begin{equation} r_1(\beta ,t,\theta )= \sqrt {\frac {a(t)^2}{1+\left ({\tan (\theta )}/{\cos (\beta )}\right )^2}} \quad \text{for }0 \leqslant \theta \leqslant \pi /2. \end{equation}

Equation (3.2) is not valid for $\pi /2 \leqslant \beta \lt 3\pi /2$ as there are no stagnation points in the downstream part of the flow. It is noteworthy that $r_1$ is always less than $a(t)$ for $\theta \gt 0$ (except for the trivial solution at $t=0$ ); and as $\theta$ decreases, $r_1$ shifts from the origin (flow normal to the disk at $\theta = \pi /2$ ) to the disk edge (uniform flow parallel to the disk surface at $\theta =0$ ), again as we expect.

Figure 8. Flows around an expanding rising thin flat disk with incidence $\theta =\pi /2$ in the inertial moving frame of reference at velocity $\boldsymbol{u_0}$ standing at the origin (a), in the static frame of reference standing at the far field at rest (c), and in the moving frame of reference at velocity $\boldsymbol{a'(t)}$ standing at the right-hand side of the disk edge (e). In the last frame (e), the flow loses axisymmetry and behaves as if with an angle of attack $\theta _{\!f}$ and stagnates at $r_2$ (relevant streamline not shown). The right-hand column shows similar information for flows with incidence $\theta =\pi /3$ (b,d, f). This figure is for $0\leqslant \beta \lt \pi /2$ and $3\pi /2\leqslant \beta \lt 2\pi$ , with the angle $\beta$ defined in figure 9(a).

Figure 8 shows the flow field around an expanding rising thin flat disk with incidence $\theta =\pi /2$ (figures 8 a, 8 c and 8 e) and $\theta =\pi /3$ (figures 8 b, 8 d and 8 f) from three different views. For $\theta =\pi /2$ , figure 8(a) (which shows the half-domain only due to axisymmetry) is the flow in the original inertial moving frame of reference at velocity $\boldsymbol{u_0}$ , standing at the origin, and is the same flow as in figure 7(a). The flow stagnates at the disk’s centre and sharply winds around the disk’s edge. This frame of reference describes the physics of a uniform flow falling downwards towards an expanding disk. Figure 8(c) (again, this figure shows the half-domain only, due to axisymmetry) shows the flow in the frame of reference placed in the far field, which is at rest. The flow incident above the surface $z \geqslant 0$ is diverted in the radial direction by the motion of the disk and sharply winds around at the disk edge. More interestingly, however, figure 8(e) shows another frame of reference, unique to the unsteady disk framework, which moves at the velocity $\boldsymbol{a'(t)}$ (with the magnitude $a'(t)$ towards the right) standing at (the right-hand side of) the disk edge. In this frame of reference, the flow loses its axisymmetry and the relative velocity approaches the disc at a non-normal angle, $\theta _{\!f}$ , towards a fixed disk edge. This, again, demonstrates the fact that flow over a flat disk with incidence can be composed by adding a uniform flow at a relative angle of attack. Translating to the moving frame of reference at velocity $\boldsymbol{a'(t)}$ involves adding a time-dependent uniform flow; and, accordingly, this gives a time-dependent ‘apparent’ angle of attack $\theta _{\!f}(\beta ,t)$ . Dividing (2.48) by $\sin (\theta _{\!f}(\beta ,t))$ and equating to the flow radial velocity in the moving frame, we find

(3.3) \begin{equation} \frac {2}{\pi }\left (\frac {\!-\!\cos (\beta )}{\tan \left (\theta _{\!f}(\beta ,t)\right )} +\frac {r}{\sqrt {a(t)^2-r^2}}\right ) = \frac {2}{\pi }\frac {r}{\sqrt {a(t)^2-r^2}} - a'(t). \end{equation}

Hence,

(3.4) \begin{equation} \theta _{\!f}(\beta ,t) = \tan ^{-1} \left ( \frac {2 \cos (\beta )}{\pi \times a'(t)} \right )\!. \end{equation}

Indeed, this apparent angle $\theta _{\!f}(\beta ,t)$ is a function of time in the UD framework, and as the velocity of the moving frame $a'(t)$ approaches $\infty$ , the apparent angle $\theta _{\!f}(\beta ,t)$ approaches $0$ . Nevertheless, it is noted that the angle $\theta _{\!f}(\beta ,t)$ is named ‘apparent’ because the flow in this special frame (standing at the disk edge) cannot be obtained simply by substituting the angle $\theta _{\!f}(\beta ,t)$ into the solution with incidence in the original frame (standing at the disk centre), but results into a flow with scale $1/\sin (\theta _{\!f}(\beta ,t))$ . However, this can be resolved by replacing the far-field condition $C= {2}/{\pi }$ with a time-dependent function $C_{\!f}(\beta ,t)= {2}/({\pi \times \sin (\theta _{\!f}(\beta ,t))})$ as a normalisation.

The flow field around the expanding rising thin flat disk in the three different frames of reference is generalised with incidence $\theta =\pi /3$ in figures 8(b), 8(d) and 8(f), respectively. Figure 8(b) shows the same flow as in figure 7(b) which has a stagnation point at $r_1(\beta ,t,\theta )$ (3.2) between the disk surface centre and the wet radius. Figure 8(d) is the flow in the static frame of reference standing in the stationary far field. This frame describes an expanding thin flat disk moving towards the top right-hand direction in an infinite flow domain at rest. Flow above surface $z \geqslant 0$ is, again, diverted towards the radial direction but, at the same time, is also moving towards the left-hand side, owing to the motion of the disk. This composite motion creates a stagnation point away from the disk surface in the direction of the disk motion. It is noted that, due to the extra horizontal motion, the flow winds around at the disk edge with less intensity than the normal impact case and quickly loses the effects of this winding motion away from the disk edge. Figure 8(f) shows the flow with incidence in the frame of reference moving with the edge of the disk at velocity $\boldsymbol{a'(t)}$ . Comparing with figure 8(e), the ‘apparent’ angle $\theta _{\!f}(\beta ,t,\theta )$ in figure 8(f) is greater as a combined result of the flow’s angle of attack $\theta$ in the inertial frame of reference and the effect of the moving frame of reference. Similar to (3.3), dividing (2.48) by $\sin (\theta _{\!f}(\beta ,t,\theta ))$ and multiplying by $\sin (\theta )$ , then equating to the flow radial velocity in the moving frame, we derive

(3.5) \begin{equation} \frac {2}{\pi }\left (\frac {\!-\!\cos (\beta )\sin (\theta )}{\tan (\theta _{\!f}(\beta ,t,\theta ))} +\frac {r \times \sin (\theta )}{\sqrt {a(t)^2-r^2}}\right ) = \frac {2}{\pi }\left (\!-\!\cos (\theta )\cos (\beta ) +\frac {r \times \sin (\theta )}{\sqrt {a(t)^2-r^2}}\right ) - a'(t). \end{equation}

This generalises the solution in (3.4) to

(3.6) \begin{equation} \theta _{\!f}(\beta ,t,\theta ) = \tan ^{-1} \left ( \frac {\sin (\theta )}{\cos (\theta ) + {(\pi \times a'(t)}/({2\cos (\beta ))})} \right ) \quad \text{for} \quad 0 \leqslant \theta \leqslant \pi /2. \end{equation}

Equation (3.6) recovers (3.4) at $\theta =\pi /2$ . Similarly, this ‘apparent’ angle generates a flow with scale $\sin (\theta )/\sin (\theta _{\!f}(\beta ,t,\theta ))$ , and can be normalised by the time-dependent far field condition $C_{\!f}(\beta ,t)= {(2 \times \sin (\theta )}/({\pi \times\sin (\theta _{\!f}(\beta ,t,\theta )))})$ .

We emphasise our particular interest in the moving frame of reference at the velocity of the edge of the expanding disk $\boldsymbol{a'(t)}$ , as a unique feature of the UD framework which includes the interesting phenomenon of a stagnation point at a radial location $r_2$ on the disk surface which is shown in both figure 8(e) ( $\theta =\pi /2$ ) or figure 8( f) ( $0 \leqslant \theta \lt \pi /2$ ). The position of this stagnation point $r_2(\beta ,t,\theta )$ (below we call $r_2$ the ‘separation point (for the inner solution)’, to distinguish it from the stagnation point $r_1$ of conventional concept – being stationary in the laboratory frame) can be derived by setting (2.48) for the radial velocity on the surface of the disk, in the moving frame relative to the disk edge, to zero,

(3.7) \begin{align} u_r(r,\beta ,0,t,\theta )= \frac {2}{\pi } \left ( \!-\!\cos (\theta )\cos (\beta ) + \frac {r_2(\beta ,t,\theta )}{\sqrt {a(t)^2-{r_2(\beta ,t,\theta )}^2}} \; \sin (\theta ) \right ) - a'(t)=0. \end{align}

For $\pi /2 \leqslant \beta \lt 3\pi /2$ , $r_2(\beta ,t,\theta )$ decays in time from a positive value to a value of zero at time $t=t_0$ . The value of $t_0$ can be obtained from (3.7) as $\cos (\theta ) \cos (\beta )+ \pi \times a'(t_0)/2=0$ , which solves to $t_0(\beta ,\theta )=3\pi ^2/(16\cos ^2(\theta )\cos ^2(\beta ))$ . After $t=t_0$ , we set $r_2$ to zero to make the solution valid in polar coordinate. The solution is

(3.8a) \begin{align} &(\text{for }\pi /2\leqslant \beta \lt 3\pi /2) \nonumber \\ &\qquad \quad \quad r_2(\beta ,t,\theta ) = \begin{cases} \sqrt {\frac {a(t)^2}{1+\left ( {\sin (\theta )}/({\cos (\theta ) \cos (\beta )+ (\pi \times a'(t))/{2}}) \right )^2}} & 0\leqslant t \leqslant t_0 \\ 0 & t \gt t_0, \end{cases} \\[-12pt]\nonumber\end{align}

(3.8b) \begin{align} &(\text{for}\,\, 0\leqslant \beta \lt \pi /2 \text{ and} \,\,3\pi /2\leqslant \beta \lt 2\pi ) \nonumber \\ &\qquad \quad \quad r_2(\beta ,t,\theta ) =\sqrt {\frac {a(t)^2}{1+\left ( {\sin (\theta )}/(\cos (\theta ) \cos (\beta )+ (\pi \times a'(t))/{2}) \right )^2}} \; t \geqslant 0 ,\end{align}

for $0 \leqslant \theta \leqslant \pi /2$ ; or specifically for $\theta = \pi /2$ ,

(3.9) \begin{equation} r_2(\beta ,t,\pi /2)=\sqrt {\frac {a(t)^2}{1+\left ( {2}/({\pi \times a'(t)}) \right )^2}}. \end{equation}

Figure 9(a) presents the difference between the Wagner wet radius $a(t)=\sqrt {3t}$ and the separation points of (3.8) (at $\pi /4$ ) and (3.9) in polar coordinates ( $\beta$ , $r$ ) on the impact surface. In particular at $\beta =0$ or $\pi$ (the view of figure 8), solutions (3.8) and (3.9) are nothing other than substitution of the apparent angles (3.6) and (3.4) into the stagnation (3.2) in the original inertial moving frame of reference,

(3.10) \begin{align} r_2(\beta =0, t,\theta )=r_1(\beta =0,t,\theta _{\!f}) = \sqrt {\frac {a(t)^2}{1+\left ({\sin (\theta )}/({\cos (\theta ) + (\pi \times a'(t))/{2}}) \right )^2}} \quad \text{for} \quad 0 \leqslant \theta \leqslant \pi /2. \end{align}

This is due to the fact that, though the apparent angles (3.4) and (3.6) provide scaled flows of figures 8(e) and 8( f), respectively, these change neither the streamlines, nor hence the stagnation points. Thereby, by the same principle of (3.2) for $r_1(\beta ,t,\theta )$ , (3.8) and (3.9) of $r_2(\beta ,t,\theta )$ are also within the wet radius $a(t)$ (except the trivial solution at $t=0$ ). The fact illustrates the dynamics at the wet radius (figure 10) that, in the moving frame of reference at the disk edge, flows are attacking the disk at a separation point ( $r_2$ ) which is slightly inboard of the wet radius. In other words, the streamline that connects the wet radius above the disk first ‘overshoots’ to a radius less than the wet radius; but subsequently diverts away (due to the flow separation at $r_2$ ) and winds around the disk edge in an ‘S’ curve.

Figure 9. Both figures show the top views of droplet impacts with the direction of the flows at an angle $\theta$ into the page and from north to south in the plane of the page. (a) The Wagner wet radius $a(t)=\sqrt {3t}$ and the separation points $r_2$ (for normal angle impact and for flow with incidence $\theta =\pi /4$ ) are compared in polar coordinates ( $\beta$ , $r$ ) set in the impact surface of the disk. To illustrate obvious difference, we compare at the late time of $t=1$ . (b) Comparison between the predicted radius $r_2$ (superimposed as red lines) and the experimental data of water drops’ oblique impacts in García-Geijo et al. (Reference García-Geijo, Riboux and Gordillo2020). The comparisons, showing the analytical solution in the coordinate system with a moving origin at dimensionless velocity $\cos(\theta )$ from north to south, are at $\textit{We}\approx 120$ , and impact angles: $\theta =5\pi /12$ (b i); $2\pi /3$ (b ii); $\pi /4$ (b iii). The horizontal lines indicate the location of the impact point. See García-Geijo et al. (Reference García-Geijo, Riboux and Gordillo2020) for a discussion of the outer line. To be consistent, $\beta =0$ is defined as the north in both figures.

Figure 10. Flow in the moving frame of reference fixed to the disk edge. At the wet radius the flow is attacking the disk at a separation point $r_2$ , inboard of the wet radius $a(t)$ . The streamlines to the right of the separation point first ‘overshoot’ to a radius less than the wet radius, then divert away (due to the flow separation at $r_2$ ) and wind around the disk edge in an ‘S’-shaped curve. The dashed line outside the disk represents the drop’s free-interface.

Figure 9(b) superimposes the predicted separation radius $r_2$ onto the experimental data from García-Geijo et al. (Reference García-Geijo, Riboux and Gordillo2020), covering a wide range of impact angles from $45^\circ$ to $75^\circ$ , and a super-long timespan up to $t\gt 8$ . We note that the separation radius $r_2$ is the spatiotemporal boundary separating the droplet and the lamella regions (see § 3.2), and hence is not directly comparable to the maximum spreading radius in the figure, which includes lamella and rim (García-Geijo et al. Reference García-Geijo, Riboux and Gordillo2020). However, it is very difficult, if not impossible, to experimentally locate the locus between the droplet and the lamella, and the present available comparison indeed could provide some useful insights. As we see in figure 9(bi), the oblique analytical solution satisfactorily captures the first-order physics of asymmetric spreading for the large impact angle (close to $\pi /2$ ) even at late times of $t\gt 6$ . At later times or smaller impact angles, we observe that the bottom line of separation radius becomes flat and eventually bends inwards. This is because, in the moving framework of reference, as time increases or impact angle reduces, the advancing speed of the downstream separation line, here denoted as south, becomes large. As a result, the locus of the separation radius $r_2$ – where advancing speed equals the advancing speed of the wet radius $\sqrt {r/t}/2$ (Riboux & Gordillo Reference Riboux and Gordillo2014), enforced by the Wagner condition – moves upstream. This mechanism is clearer in the frame of reference of figure 9(a). It also agrees with the ‘flat-bottom’ trend of the border of the lamella predicted by García-Geijo et al. (Reference García-Geijo, Riboux and Gordillo2020) at smaller impact angles (note their impact angle is defined as $\chi =\pi /2-\theta$ ) and later times (see the bottom row in their figure 4). Particularly, we observe the downstream separation radius bends to the origin in figure 9(b) at $\theta =\pi /4$ for $t\gt 4$ . We provide two reasons for this overestimated behaviour: (i) the classical wet radius $\sqrt {3t}$ was derived under the assumption of normal impact (Wagner Reference Wagner1932; Riboux & Gordillo Reference Riboux and Gordillo2014) while a proper wet radius model considering the asymmetric spreading shape would increase the velocity threshold of the separation point; and (ii) viscosity effects are omitted in the present potential theory, which could effectively mitigate the advancing speed of the separation line, even in regimes that are often considered inviscid (Sanjay & Lohse Reference Sanjay and Lohse2025). Nevertheless, we note that even the first frames ( $t\gt 1$ ) of the compared cases in figure 9(b) fall into the late-time regime while the present analytical model is derived (mainly) for early and intermediate times ( $t\lt 0.27$ , see § 5), where viscosity and capillary effects can be omitted (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010).

3.2. Wagner condition and a solution to the droplet impact problem

From now on, we specifically consider the application of the above UD framework to the droplet impact problem (figure 3 a).

The literature associated with droplet impact problem has conventionally used, for the water-entry problem, both the intersection radius $\sqrt {2t}$ (Howison et al. Reference Howison, Ockendon and Wilson1991) and the wet radius $\sqrt {3t}$ (Wagner Reference Wagner1932; Philippi et al. Reference Philippi, Lagrée and Antkowiak2016) for $a(t)$ . Notably, the wet radius $\sqrt {3t}$ is derived from the Wagner condition for normal impact (i.e. $\theta =\pi /2$ ). In the analysis below, we will assume $a(t)=\sqrt {\lambda t}$ for general $\lambda \gt 2$ .

In § 3.1, we established the existence of a separation point $r_2$ in the UD framework, in the moving frame of reference at the disk edge; and we have derived its radial position in (3.8) and (3.9). The ratio between $r_2$ and $a$ is not constant due to the non-inertial moving frame of reference at the disk edge. We note that this dynamic ratio is a unique feature of the UD framework, and cannot be obtained from the SSD framework (Philippi et al. Reference Philippi, Lagrée and Antkowiak2016) where the ratio would be fixed in self-similar space and is equivalent to the ratio at $t=1$ of the present UD framework. In the current UD framework, the ratio between $r_2$ and $a$ varies from $1$ and $0$ . This corresponds to $r_2$ monotonically shifting from $a$ to the asymptotic limit $r_1$ in time, which is the stagnation point in the original moving frame of reference, as in the original moving frame due to the relatively negligible velocity of disk expansion at late time.

Wagner’s condition (Wagner Reference Wagner1932) is useful for linking the kinematics of the horizontal expansion of the wet radius to the vertical motion of the free surface in the context of droplet impact problem. In other words, when the downward motion of the droplet causes the free surface of the droplet to meet the solid surface, it becomes the turnover point: namely, the wet radius. See, in particular, figure $5$ of Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) as well as Riboux & Gordillo (Reference Riboux and Gordillo2014) and Negus et al. (Reference Negus, Moore, Oliver and Cimpeanu2021). This argument suggests a deduction that not only is the position of the turnover point determined by the time when a point on the curved droplet–air interface reaches the disk surface due to the interface’s predominantly vertical motion, but also that the horizontal expanding velocity at the turnover point should be determined by the radial velocity of this particular point when it reaches the disk. Therefore, the separation point $r_2$ , at which the flow has the same radial velocity as the disk’s velocity of expansion, is the particular turnover point in our expanding water-entry analogy (yellow crosses in figure 5 c). This provides us with the first reason to choose $r_2$ as the point at which we stop the solution. This leads to the solution of the expanding water-entry analogy, in the inertial moving frame of reference at the disk centre, as

(3.11) \begin{align} \boldsymbol{u}(r,\beta ,0,t,\theta ) = \left (\frac {2}{\pi } \left ( \!-\!\cos (\theta )\cos (\beta ) + \frac {r}{\sqrt {a(t)^2-r^2}} \; \sin (\theta ) \right )\right ) \boldsymbol{\underline {\hat {r}}} + \left (0\right )\boldsymbol{\underline {\hat {z}}} \quad \text{for} \quad r \leqslant r_2(\beta ,t,\theta ) \end{align}

(3.12) \begin{align} \frac {\partial \phi }{\partial t}(r,\beta ,0,t,\theta ) =- \frac {2}{\pi } \frac {a(t)a'(t)\sin (\theta )}{\sqrt {a(t)^2-r^2}} \quad \text{for} \quad r \leqslant r_2(\beta ,t,\theta ) \end{align}

evaluated up to $r_2$ ((3.8) for $0 \leqslant \theta \leqslant \pi /2$ or (3.9) for $\theta = \pi /2$ ). We propose that the set of (3.11) and (3.12) for the expanding water-entry analogy is a solution to the droplet impact problem at the solid surface.

3.3. Impact pressure on the solid surface

From (2.9), (3.11) and (3.12), we recover the unsteady form of the Bernoulli pressure on the solid surface upon droplet impact,

(3.13) \begin{align} p(r,\beta ,0,t,\theta ) &= b(t,\theta ) - \frac {{u_r}^2}{2} -\frac {\partial \phi }{\partial t} \nonumber \\ &= b(t,\theta ) - \frac {2}{\pi ^2} \left ( \cos (\theta ) \cos (\beta )- \frac {r}{\sqrt {a(t)^2-r^2}} \; \sin (\theta ) \right )^2 \nonumber \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \frac {2}{\pi } \frac {a(t)a'(t)\sin (\theta )}{\sqrt {a(t)^2-r^2}} \quad \text{for} \quad r \leqslant r_2(\beta ,t,\theta ) \end{align}

where the Bernoulli constant $b(t,\theta )$ is generalised with incidence. Also, since we neglect viscosity and surface tension in the current ideal flow field framework, by continuity of normal stress at the liquid boundary, we have $p=0$ at all free interfaces. Applying (2.9) at the droplet top interface, where the flow is moving downwards with unit (dimensionless) velocity because the bulk of the droplet is unaffected by the impact – at least at the earlier stages of impact (Riboux & Gordillo Reference Riboux and Gordillo2014), we obtain $b(t,\theta ) = 1/2$ for $t \geqslant 0$ and $0 \leqslant \theta \leqslant \pi /2$ .

The impact pressure (3.13) has a negative singularity at the wet radius $a(t)$ for $\theta \gt 0$ . Nevertheless, the singularity is not of any concern for the solution proposed in the current paper, due to the fact that $r_2(\beta ,t,\theta )$ is always less than $a(t)$ . This provides us with a second reason to choose $r_2$ as the point at which we stop the solution: the ‘expanding rising disk’ analogue fails to describe the droplet impact problem near the wet radius. The reason can be inferred from the well-known negative pressure singularity at the disk edge of the two-dimensional potential flow over a disk/fence (figure 5 a), where the flow needs to wind over the object immediately, and hence at infinite velocity. For the current three-dimensional flow, this is illustrated in figures 8(c) and 8(d). Flow streamlines are winding around the disk edge, creating a pressure minimum at the disk edge. This is not to be expected at the wet radius in a droplet impact problem, in which the correct free interface Bernoulli condition $\partial \phi / \partial t + {(\boldsymbol{\nabla }\phi )}^2/2=0$ has been replaced by $\phi =0$ at the wet radius $a$ in the current UD framework (figure 4 b) (Korobkin & Scolan Reference Korobkin and Scolan2006). In other words, the approach of simple inviscid potential flow theory cannot lead to correct solutions to the complex phenomenon (droplet impact problem) at the wet radius (Cointe & Armand Reference Cointe and Armand1987) where, in practice, liquid viscosity and surface tension should play important roles (Gordillo et al. Reference Gordillo, Riboux and Quintero2019). In previous work, where solutions are conventionally evaluated up to the wet radius, the phenomenon of a singularity has also been noticed in several works on the water-entry problem (Wagner Reference Wagner1932; Logvinovich Reference Logvinovich1969; Faltinsen & Wu Reference Faltinsen and Wu2001). Interestingly, the unsteady term of (3.13) also has a (positive) singularity at the wet radius due to the unsteady expanding motion of the disk, which has been noted by Cointe & Armand (Reference Cointe and Armand1987), Tassin et al. (Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2010) and Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016). But the nature of the positive singularity has a smaller order of magnitude than the negative pressure singularity, and therefore the combined solution exhibits the negative singularity only as a result of the infinite flow velocity. A conventional approach to resolve the singularity was first proposed by Wagner (Reference Wagner1932) for the water-entry problem, analytically derived by Cointe & Armand (Reference Cointe and Armand1987) in two-dimensions, and Scolan (Reference Scolan2004) in three-dimensions, and adapted for the droplet impact problem by Negus et al. (Reference Negus, Moore, Oliver and Cimpeanu2021). The approach is to asymptotically compose the original singular solution with an extra splashing solution in the so-called inner region at the wet radius where the original singular solution breaks down.

A direct, and important, result from the existence of the two opposite-sign terms in (3.13) is the so-called ring-pattern pressure peak near, but inboard of, the wet radius. The pressure peak is produced by the balance between the steady and unsteady Bernoulli terms, and may explain the ring-shaped cracks in the surface observed in various experiments (Field Reference Field1966; Field et al. Reference Field, Lesser and Dear1985). Analytically, we can derive the position $r_{\textit{max}}(t,\theta )$ and the peak pressure value $p_{\textit{max}}(\beta ,t,\theta )$ by differentiating (3.13),

(3.14) \begin{equation} \frac {\partial p}{\partial r} (r_{\textit{max}},\beta ,0,t,\theta )= \frac {4}{\pi ^2} \left ( \frac {a^2\sin (\theta )\cos (\theta )\cos (\beta )}{(a^2-{r_{\textit{max}}}^2)^{3/2}} - \frac {{r_{\textit{max}}}a^2}{{(a^2-{r_{\textit{max}}}^2)}^2} \right ) + \frac {2}{\pi } \frac {aa'\sin (\theta ){r_{\textit{max}}}}{(a^2-{r_{\textit{max}}}^2)^{3/2}} = 0 \end{equation}

to solve for $0 \lt r_{\textit{max}}(\beta ,t,\theta ) \leqslant r_2(\beta ,t,\theta )$ , and hence

(3.15) \begin{equation} p_{\textit{max}}(\beta ,t,\theta )=p(r_{\textit{max}},\beta ,0,t,\theta ). \end{equation}

Regarding solutions outside the wet radius, we have derived the velocity field of (2.33) and the unsteady Bernoulli term $\partial \phi / \partial t$ of (2.42) for the expanding rising disk with flow at incidence $\pi /2$ . Also, in § 3.1, we have seen the ‘S’ shape of the streamlines outside the separation point, which, in our expanding water-entry analogy, are interpreted as the flow at the free interfaces of an impacting droplet (see figure 8 e). The nature of the separation point, $r_2$ , indicates the fact that the ‘S’ shapes on the interfaces have greater radial velocity than the expanding wet radius. However, this apparently contradicts what we find on the velocity vectors (2.33) which, outside the wet radius, have no radial component and are purely downward-directed in the original moving frame of reference at the disk centre. The apparent contradiction stems from the fact that solutions (2.33) and (2.42) describe the flows at $z=0$ in the UD framework, which is totally different from the position of the droplet interface in droplet impact problem (figure 3). The difference is illustrated in figure 7(a), where the black arrows are the solutions outside the wet radius while the green arrows (on the imaginary droplet interface) indicate the desired interface solution for droplet impact problem. Similar to the classic water-entry problem (Zhao & Faltinsen Reference Zhao and Faltinsen1993; Korobkin Reference Korobkin2004), the accuracy in approximating the desired solution on the droplet interface by the solution to the UD problem depends on the deadrise angle of the obstacle which, in the case of spherical droplet, increases from zero to normal angle as the distance from the wet radius increases. Therefore, solutions outside the wet radius are not of the concern in the current study; and it is interesting to note that the conventional method of composing a local splash solution at the wet radius is not applicable also for non-small deadrise angles, due to the order mismatch between the Bernoulli term and the splash solution in the inner region (Zhao & Faltinsen Reference Zhao and Faltinsen1993).

Figure 11. Comparisons between the present analytical pressure model with various results in the literature between $t=2\times 10^{-2}$ and $t=10^{-1}$ at half-domain (without loss of generality at $\beta =0$ ) (a–c), and full domain (without loss of generality at $\beta =0$ and $\pi$ ) (e). In the latter, the generalised present model with incidence $\pi /4$ is also superimposed. Bernoulli constants are removed from all pressure profiles for consistency. The pressure profile at $t=10^{-1}$ is zoomed-in at the wet radius (d) to illustrate the relative positions between $r_{\textit{max}}$ , $r_2$ , $r_*$ , and the wet radius $a$ at normal angle of incidence. Note, in particular, that the separation point $r_2$ is found to be very close to the radius of pressure maximum, $r_{\textit{max}}$ .

We now compare the pressure solution in (3.13) with some typical analytical pressure solutions in the literature (presented in the axisymmetric forms) in figure 11. It is noted that, for all pressure solutions in the comparison, $a(t)$ has the value of $\sqrt {3 t}$ and the Bernoulli constant $b(t,\theta )$ is removed for consistency, since the values of $a(t)$ and $b(t,\theta )$ vary in different methods and applications. Figure 11(a) compares the analytical pressure in the literature on droplet impact at normal angle at time $t=0.02{-}0.1$ with time step $\Delta t=0.02$ . The proposed solution in the current work compares well, in terms of the shapes and the magnitude of the peak pressure, especially at small $t$ , with the other two in the literature. Specifically, Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) considers only the unsteady term of the Bernoulli (2.9), and hence the pressure is monotonic at each time and rises to the positive singularity at the wet radius. Negus et al. (Reference Negus, Moore, Oliver and Cimpeanu2021) composes the splash solution at the wet radius, and correspondingly their solution shows the existence of a ‘tail’ beyond the wet radius, which expands in extent with time. The composed splash solution leads pressure to zero. Figures 11(b) and 11(c) compare also the analytical pressure given in the literature on the water-entry problem at normal angle over the same time period. The solution by Logvinovich (Reference Logvinovich1969) turns out to be ‘exactly the same’ as the derived solution in this paper, but is continued to greater radius. This is because the derived pressure (3.13) at $\theta =\pi /2$ recovers the exact pressure expression in Logvinovich (Reference Logvinovich1969); however, Logvinovich evaluates the expression to the radius $r_*$ (see (3.16) below) where the pressure is zero while (3.13) is evaluated no farther than the separation point $r_2$ . The composed solution in Zhao & Faltinsen (Reference Zhao and Faltinsen1993), Scolan (Reference Scolan2004), Wilson (Reference Wilson1989) and Howison et al. (Reference Howison, Ockendon and Wilson1991) is of the same form for flat deadrise angle, and is the precursor of the Negus et al. (Reference Negus, Moore, Oliver and Cimpeanu2021) solution. The difference between the two sets of solutions is the shift of $+6\delta /\pi$ in the radial direction, due to the difference in the location of the stagnation points in droplet impact and water-entry problems (Oliver Reference Oliver2002). Similarly, Cointe & Armand (Reference Cointe and Armand1987) and Faltinsen & Wu (Reference Faltinsen and Wu2001), instead of asymptotic analysis, use the Heaviside function to ‘peg’ the Bernoulli pressure to the splash solution, respectively, either at the radius where the unsteady Bernoulli term equals the splash pressure peak, or at the radius where the entire Bernoulli pressure equals the splash pressure peak (here as $r_{s2}$ in figure 11 c). Both pressure distributions by Cointe & Armand (Reference Cointe and Armand1987) and Faltinsen & Wu (Reference Faltinsen and Wu2001) compare well with the value of pressure peaks from (3.13), but differ in terms of the radial extent. Finally, figure 11(e) mixes the typical solutions from droplet impact and water-entry literature, and compares them with (3.13) at flow incidence $\theta = \pi /4$ . Due to loss of axisymmetry of (3.13) at non-normal angle of incidence, the pressure distributions are shown in the meridional plane; note, however, that all pressures from the literature are symmetrical in figure 11(e). For pressure (3.13) at $\theta = \pi /4$ (related to flow from right (positive $r$ -axis with $\beta =0$ ) to left (negative $r$ -axis with $\beta =\pi$ )), pressure distribution is ‘tilted’ towards the right-hand side and has a higher pressure peak near the right-hand separation $r=+r_2$ ; meanwhile, interestingly, the pressure at the centre, $r=0$ , is always lower than the centre pressure would have been at normal angle of impact (refer to the centre pressure value of Logvinovich in the same figure).

Equation (3.13) (without Bernoulli constant $b(t,\theta )$ ) reaches zero pressure at radius $r_*(t,\theta )$ where the magnitudes of the positive unsteady Bernoulli term and the negative nonlinear Bernoulli term are equal. This leads to the equation for $r_*(\beta ,t,\theta ) \leqslant a(t)$ as

(3.16) \begin{align} r_*^2\sin ^2(\theta )-\left (2 r_*\cos (\theta )\sin (\theta )+\pi a(t) a'(t) \sin (\theta )\right )\sqrt {a(t)^2-{r_*}^2}+{\cos }^2(\theta )\big (a(t)^2-{r_*}^2\big )=0. \end{align}

Equation (3.16) is a general quartic equation with a lengthy root formula; however, specifically at $\theta =\pi /2$ , it simplifies to a quadratic equation on ${r_*}^2$ ,

(3.17) \begin{equation} \frac {1}{(\pi a(t) a'(t))^2}\big ({r_*}^2\big )^2 +\big ({r_*}^2\big )^2-a(t)^2=0 \end{equation}

which can be easily solved and obtains the only reasonable positive real root,

(3.18) \begin{equation} r_*(\beta ,t,\pi /2)=\sqrt {\frac {Q}{2} \left ( \sqrt {1+\frac {4a(t)^2}{Q}}-1 \right )} \end{equation}

where $Q=(\pi a(t) a'(t))^2$ is a constant for $a(t)= O(\sqrt {t})$ . We note that Logvinovich (Reference Logvinovich1969) considered the two-dimensional problem and derived (1.8) for $x_*(t)$ , while (3.18) at $\theta =\pi /2$ is the corresponding point in the axisymmetric potential flow. Similarly, at $\theta =\pi /2$ , the quartic (3.14) is greatly simplified, and the solution can be obtained as

(3.19) \begin{equation} r_{\textit{max}}(\beta ,t,\pi /2)=\sqrt {a(t)^2-\frac {4a(t)^4}{Q}} \end{equation}

and hence the dimensionless peak pressure (without Bernoulli constant $b(t,\theta )$ ) is

(3.20) \begin{equation} p_{\textit{max}}(\beta ,t,\pi /2)=\frac {Q}{2\pi ^2a(t)^2} \left (1+ \frac {4a(t)^2}{Q}\right )\!. \end{equation}

It is not obvious whether $r_*(\beta ,t,\pi /2)$ and $r_{\textit{max}}(\beta ,t,\pi /2)$ are always inboard of the separation point $r_2(\beta ,t,\pi /2)$ of (3.9) for general $a(t)$ ; but specifically for $a(t)=\sqrt {3t}$ , $r_*(\beta ,t,\pi /2)$ is greater than $r_2(\beta ,t,\pi /2)$ at all time $t\gt 0$ . Therefore, the proposed pressure of (3.13) for the droplet impact problem is always non-negative. The relative positions between $r_{\textit{max}}(\beta ,t,\pi /2)$ , $r_2(\beta ,t,\pi /2)$ and $r_*(\beta ,t,\pi /2)$ , and the wet radius $a(t)$ are presented in figure 11(d) at a typical time $t=0.1$ for the droplet impact problem. It is worth noting the striking similarity between $r_{\textit{max}}(\beta ,t,\pi /2)$ and $r_2(\beta ,t,\pi /2)$ , and hence the conclusion that the obtained analytical pressure (3.13) at $r_2$ provides a reasonable estimation of the impact pressure maximum at early time.

3.4. Impact force on the solid surface

The impact force $f(t,\theta )$ on the solid surface upon droplet impact can be analytically obtained by integrating pressure (3.13) over the solid surface. In cylindrical coordinates ( $r, \beta , z$ ), this is

(3.21) \begin{align} f(t,\theta ) &=\int _0^{2\pi } \int _0^\infty p(r,\beta ,0,t,\theta ) r\;\text{d}r\;\text{d}\beta \nonumber \\ &= \int _0^{2\pi } \int _0^{r_2} p(r,\beta ,0,t,\theta ) r\;\text{d}r\;\text{d}\beta. \end{align}

Below, we analytically integrate the impact force for the case of normal impact $\theta =\pi /2$ where $r_2$ is independent of $\beta$ . Because of the nonlinear term in the Bernoulli equation, integration (3.21) has five terms; below we evaluate each of these separately which collectively add up to $f(t,\pi /2)$ . The Bernoulli constant $b(t,\pi /2)$ and the constant term from the nonlinear Bernoulli can be integrated, as

(3.22) \begin{align} f_1(t,\pi /2) &=2\pi \int _0^{r_2} \frac {1}{2} r\;\text{d}r=\frac {\pi {r_2}^2}{2} \\[-12pt]\nonumber \end{align}

(3.23) \begin{align} \text{and} \;\;\;\; f_2(t,\pi /2) &=\frac {-2{\cos }^2(\pi /2)}{\pi ^2} \int _0^{2\pi } \int _0^{r_2} \cos ^2(\beta )r\;\text{d}r\;\text{d}\beta =0. \end{align}

The linear term from the nonlinear part of the Bernoulli equation is integrated, as follows:

(3.24) \begin{equation} f_3(t,\pi /2) =\int _0^{2\pi } \int _0^{r_2} \frac {4r\cos (\pi /2)\sin (\pi /2)}{\pi ^2\sqrt {a^2-r^2}} \cos (\beta )r\;\text{d}r\;\text{d}\beta =0. \end{equation}

The quadratic term from the nonlinear part of the Bernoulli equation is integrated, as

(3.25) \begin{align} f_4(t,\pi /2) &=2\pi \int _0^{r_2} -\frac {2r^2{\sin }^2(\pi /2)}{\pi ^2(a^2-r^2)} r\;\text{d}r \nonumber \\ &=\frac {-4}{\pi } \int _0^{r_2} \frac {r^3}{a^2-r^2} \;\text{d}r \nonumber \\ &=\frac {2}{\pi } \Big ( \big [r^2\textrm{log}(a^2-r^2)\big ]_0^{r_2}-\big [r (\textrm{log}(r)-1)\big ]_{a^2-{r_2}^2}^{a^2} \Big ) \nonumber \\ &=\frac {2}{\pi } \big ( {r_2}^2\textrm{log}(a^2-{r_2}^2)-\big ( a^2 (\textrm{log}(a^2)-1) - (a^2-{r_2}^2) (\textrm{log}(a^2-{r_2}^2)-1) \big ) \big ) \end{align}

where $[g(x)]_{x_1}^{x_2}=g(x_2)-g(x_1)$ denotes integral evaluation. Finally, the unsteady term in the Bernoulli equation is integrated, as

(3.26) \begin{align} f_5(t,\pi /2) &=2\pi \int _0^{r_2} \frac {2aa'\sin (\pi /2)}{\pi \sqrt {a^2-r^2}} r\;\text{d}r \nonumber \\ &=4aa' \int _0^{r_2} \frac {r}{\sqrt {a^2-r^2}} \;\text{d}r \nonumber \\ &=4aa' \big (a-\sqrt {a^2-{r_2}^2} \big ). \end{align}

Therefore, returning to (3.21), we have

(3.27) \begin{equation} f(t,\pi /2)=f_1(t,\pi /2)+f_2(t,\pi /2)+f_3(t,\pi /2)+f_4(t,\pi /2)+f_5(t,\pi /2) \end{equation}

for $t \geqslant 0$ .

Figure 12. Comparisons between the present analytical impact force on the solid surface with different results in the literature. All analytical impact forces are applied to the dimensional case with parameters: $R_0=1.3\ \textrm {mm}$ and $U_0=2.67\ \textrm {m s}^{-1}$ for a water droplet. The experimental data from Zhang et al. (Reference Zhang, Zhang, Lv, Li and Guo2019) and an empirical equation for the pressure peak $F=0.84\,\rho _l\,{U_0}^2\,{(2R_0)}^2$ (Zhang et al. Reference Zhang, Li, Guo and Lv2017) are superimposed in the figure. In addition, the impact force from the generalised present model with incidence angle $\pi /4$ is also superimposed. Bernoulli constants are removed in all impact force profiles for consistency.

Figure 12 compares the analytical impact force (3.27) with an experimental measurement from the literature (Zhang et al. Reference Zhang, Zhang, Lv, Li and Guo2019) in dimensional form. The experiment entails a water droplet of diameter $2R_0=2.7\ \textrm {mm}$ impacting at velocity $U_0=2.67\ \textrm {m s}^{-1}$ normal to the solid surface. Also compared are the impact forces of all analytical pressures from § 3.3, obtained by surface integration (3.21). Bernoulli constants are removed in all impact force profiles for consistency. It is worth-noting that the unsteady Bernoulli term is integrable up to the wet radius (Cointe & Armand Reference Cointe and Armand1987; Tassin et al. Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2010; Philippi et al. Reference Philippi, Lagrée and Antkowiak2016), but the nonlinear Bernoulli term is not (Logvinovich Reference Logvinovich1969; Faltinsen & Wu Reference Faltinsen and Wu2001; Tassin et al. Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2010); nevertheless, this does not affect any of the solutions presented in figure 12 because none of these has the nonlinear Bernoulli term integrated to the wet radius. As figure 12 shows, Philippi et al.'s solution overpredicts the impact force the most because the solution lacks the nonlinear Bernoulli term which has been found to be the key component for a satisfactory prediction of the vertical impact force (Faltinsen & Wu Reference Faltinsen and Wu2001). The solutions of (3.27) and Logvinovich (Reference Logvinovich1969), by involving the nonlinear Bernoulli term, indeed predict much better impact forces. However, it is worth-noting that the impact force from (3.27) is the first of its kind to predict an impact force maximum (and at a satisfactory magnitude, see paragraphs below) while the solution by Logvinovich monotonically increases in time. Besides, solutions by Zhao & Faltinsen (Reference Zhao and Faltinsen1993), Faltinsen & Wu (Reference Faltinsen and Wu2001) and Negus et al. (Reference Negus, Moore, Oliver and Cimpeanu2021), whose pressures involve the extra splashing component, also overpredict the impact force by an increasing amount in time. Interestingly, the impact force obtained by Zhao & Faltinsen (Reference Zhao and Faltinsen1993) and Faltinsen & Wu (Reference Faltinsen and Wu2001) are surprisingly similar, though the former has used Heaviside function while the latter has used asymptotic analysis to compose the splash component. Mathematically, this indicates that the subtracted overlap pressure component (and the Bernoulli term) in the Zhao & Faltinsen (Reference Zhao and Faltinsen1993) solution is very similar, in terms of pressure magnitude, to the splash component within the wet radius. In contrast, the solution from Cointe & Armand (Reference Cointe and Armand1987), which also involves the splash solution, underpredicts the impact pressure. This is due to the fact that the splash solution is concatenated with the Bernoulli pressure at much smaller radius in Cointe & Armand (Reference Cointe and Armand1987) (see figure 11 c); moreover, Cointe & Armand’s solution is not valid after $234$ μs as the splash pressure peak decays to magnitudes less than the minimum unsteady Bernoulli pressure, and hence the pressure concatenation fails to produce a continuous pressure.

In the experimental literature (Zhang et al. Reference Zhang, Zhang, Lv, Li and Guo2019), the measured peak impact force is found to agree with the empirical formula (Zhang et al. Reference Zhang, Li, Guo and Lv2017)

(3.28) \begin{equation} F=0.84\,\rho _l {U_0}^2 {(2R_0)}^2 \end{equation}

which is also compared in figure 12 as the constant line $F(T)=0.0435\;\text{N}$ . The obtained analytical impact force (3.27), though not at the same time, predicts satisfactorily the impact force maximum in line with the empirical formula and the experiment. Interestingly, in the first attempt of the impact force problem in 1929, von Kármán (Reference von Kármán1929) had used the simple method of ‘increased inertia’ and predicted the two-dimensional force as

(3.29) \begin{equation} F=\pi \rho _l {U_0}^2 {R_0} \end{equation}

which, if we extrapolate to three-dimensions, is

(3.30) \begin{equation} F=\pi \rho _l {U_0}^2 {R_0}^2\,\approx \,0.79\,\rho _l {U_0}^2 {(2R_0)}^2. \end{equation}

Note the close similarity between (3.28) and (3.30).

In the literature which considers the analytical impact force, Cointe & Armand (Reference Cointe and Armand1987) and Korobkin (Reference Korobkin2004) have compared various analytical force distributions in time to both numerical and experimental data, however, in two-dimensions,

(3.31) \begin{equation} f(t)=2\int _{0}^{\infty } p(x,y=0,t) \,\text{d}x \end{equation}

where $p(x,y,t)$ are analytical pressures in two-dimensions. Being one dimension fewer than the axisymmetric framework studied in the present work, the ‘plane’ analytical pressures also show monotonically decreasing impact forces in time, because of the uniform pressure ‘weights’ between the centre and the edge of the wet area. Furthermore, as Cointe & Armand (Reference Cointe and Armand1987) pointed out, all the analytical solutions in two-dimensions in the comparison rise instantaneously from zero to a finite force maximum and therefore fail to be validated against experimental data at early time. Nevertheless, as evidenced by figure 12, the latter is apparently not an issue for three-dimensional forces.

4.1. Basilisk flow solver: numerical simulation configuration and validation

To provide insights to the accuracy and limitations of the developed analytical droplet impact solutions from § 3, we have also performed a numerical droplet impact simulation with the ‘free software’ (open-source) code Basilisk (Popinet Reference Popinet2003, Reference Popinet2009). Basilisk has accurate solvers for incompressible Navier–Stokes equations (and other multiphysics solvers) using the finite-volume method and a geometrical volume-of-fluid approach to represent the multiphase interfaces. The code is robust for surface-tension-driven interfacial flows due to its integrated volume-of-fluid interface representation, balanced-force continuum-surface-force surface tension formulation and height-function curvature estimation. Besides, Basilisk is appropriate for the current multiphase impact problem framework since it features an adaptive quadtree spatial discretisation (in axisymmetric configuration) which guarantees the high resolution at the fluid–fluid interfaces and impact contact surfaces while maintaining an affordable computational cost. Throughout the simulations, the schemes have a second-order convergence in space and time on regular and dynamically refined grids.

Figure 13. The square simulation domain for the axisymmetric Basilisk numerical simulation has side length of $2.1R_0$ (chosen to coincide with the Basilisk simulation in Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016)), with the initial mesh grids of $32768\times 32768$ lines. The initial configuration for the flow contains a slightly truncated droplet shape which is already touching the solid surface with penetration depth $\bar {r_0}=10^{-4}$ and ambient air elsewhere. The parameters for the numerical simulation are $U_0=1.93\ \textrm {m s}^{-1}$ at normal angle of incidence, kinematic viscosity of $20$ cSt and $\textit{Re}=106$ . The no-slip boundary condition is specified on the ‘bottom’ solid surface, and free inflow and outflows are specified on the ‘top’ and ‘left’ boundaries. The axes are x and y, and their directions are defined by Basilisk conventions for axisymmetric simulations.

The Basilisk numerical simulation in the current study consists of a two-phase flow: a liquid droplet immersed in ambient air, in an axisymmetric simulation configuration shown in figure 13. For the purpose of validation, the simulated impact conditions and parameters model an experiment in the literature (Gordillo, Sun & Cheng Reference Gordillo, Sun and Cheng2018). It is noteworthy that a different impact liquid (silicone oil droplet) than the experimental data (water droplet) used for comparisons in § 3.4 is selected deliberately to compare the analytical solutions over a wide range of impact scenarios. The liquid droplet has impact velocity $U_0=1.93\ \textrm {m s}^{-1}$ with a kinematic viscosity of $20$ cSt and Reynolds number of $106$ (Gordillo et al. Reference Gordillo, Sun and Cheng2018), and diameter $D_0=2.2\ \textrm {mm}$ by implication. The initial geometry and condition, and boundary conditions, follow the theoretical frameworks from § 2.2 and figure 4: the initial condition of the liquid phase is specified as an axisymmetric simulation domain of a spherical droplet impacting normally ( $\theta =\pi /2$ ) onto a rigid solid surface at uniform vertical velocity $U_0$ in the initially static ambient air; while for the boundary conditions, no-slip boundary condition is specified on the solid surface (denoted ‘bottom’ boundary in the simulation domain) with the ‘top’ and ‘left’ boundaries assigned as free-outflows (see figure 13). On the air–liquid interface, the interfacial acceleration due to surface tension, $\boldsymbol{a}_{\!\boldsymbol{f}}$ , is specified in Basilisk as

(4.1) \begin{equation} \boldsymbol{a}_{\!\boldsymbol{f}} = \sigma \kappa \boldsymbol{n} \delta _s / \rho _c \end{equation}

where $\sigma$ is the liquid surface tension, $\kappa$ is the interface mean curvature, $\boldsymbol{n}$ is the normal vector, $\delta _s$ is the interface Dirac function that guarantees the interfacial acceleration term is applied at cells contain interfaces only, and $\rho _c$ is the density of the interface cell. Using a continuum surface force approximation, this can be expressed through the liquid fraction  $f$ ,

(4.2) \begin{align} \boldsymbol{a}_{\!\boldsymbol{f}} = \sigma \kappa \boldsymbol{\nabla }\!f / (\rho _l(1-f)+\rho _g f) \end{align}

where $\rho _g$ is the gas (here as air) density.

Figure 14. The typical droplet morphology (a,c,e) and corresponding adaptive mesh structures (b,d, f) at, respectively, an early impact time $t=10^{-4}$ , the instant of lateral lamella ejection $t=4.2\times 10^{-2}$ , and later lamella spreading time $t=5 \times 10^{-1}$ . The finest local mesh resolution corresponds to $2^{15}$ grids per dimension and are mainly concentrated at the contact surface and at the interfaces between the two phases. The inset of (a) zooms in to the geometry of the droplet near to the surface showing the initially truncated droplet, in comparison with an auxiliary sphere (red dashed line). Note the barely visible bubbles observed on the line of $x=0$ during simulations (c,e), and see Appendix C for magnified views of the bubbles.

Following the suggestions of Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016), which has successfully developed a comparable numerical droplet impact simulation using Gerris (Lagrée et al. Reference Lagrée, Staron and Popinet2011) – the predecessor of Basilisk – typical adaptive mesh refinement performed in the current study had the finest Basilisk grid refinement level (defined as $2^{\textit{level}}$ square cells per dimension, if the grid was uniform) of $12$ and as high as $15$ if needed. We note that this mesh resolution is comparable to the Basilisk simulation in (Negus et al. Reference Negus, Moore, Oliver and Cimpeanu2021) of level $13$ . This mesh resolution corresponds to between $3\times 10^{3}$ to $3\times 10^{4}$ number of grid cells per droplet diameter, which is an order of magnitude finer than $10^{3}$ per droplet diameter in calculations performed to capture droplet cavitation (Wu, Liu & Wang Reference Wu, Liu and Wang2021). Typical adaptive mesh structures during a simulation are shown in figure 14 at early impact time $t=10^{-4}$ (figures 14 a and 14 b), the lateral lamella ejection instant $t=4.2\times 10^{-2}$ (figures 14 c and 14 d), and at the later lamella spreading time $t=5 \times 10^{-1}$ (figures 14 e and 14 f). The code instructs finest levels to concentrate specifically at the contact surface and interfaces between the two phases. This is to accurately capture the pressure and force distribution on the solid surface, and the delicate evolutions of the expanding contact line, which are the main points of comparison between the performed numerical comparisons and the analytical solutions. Finally, there is the well-known, interesting phenomenon of air bubble entrapment and dimple formation when a liquid droplet undergoes an impact starting initially by being surrounded by air (Josserand & Thoroddsen Reference Josserand and Thoroddsen2016). This is also captured by numerical simulations of two-phase droplet impact in the literature (Philippi et al. Reference Philippi, Lagrée and Antkowiak2016) using the Basilisk code. Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) have successfully avoided the influence of air bubbles by setting the initial spherical droplet geometry to a slightly truncated shape already touching the solid surface. We note that Negus et al. (Reference Negus, Moore, Oliver and Cimpeanu2021) used an initial condition whereby the droplet is separated from the disk by a distance of $0.125$ and travelling with a downward vertical velocity, which is appropriate for their arrangement that involves the droplet striking a spring-supported surface. However, the remedy from Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) does not seem to work in the current simulation scenario even with the same initial sphere penetration $\bar {r_0}=10^{-4}$ being no more than the depth of one grid cell ( $\Delta x \approx 5 \times 10^{-4}$ at level $12$ ), and there are still bubbles (we decided not to artificially remove any bubbles) although these can hardly be observed in figure 14 on the line of $x=0$ (see Appendix C). This discrepancy is probably due to the different impact scenario of the current simulated impact problem, which has a lower Reynolds number $\textit{Re}=106$ , compared with the case of $\textit{Re}=5000$ in Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016), and hence a relatively lower viscosity (the way Basilisk runs is unit-less, meaning material properties of different phases are specified based on the dimensionless parameters of the main phase (here as liquid)) of the ambient air which could more easily be entrapped into bubbles. Nevertheless, this truncation approach (with the above truncation depth $\bar {r_0}$ ) is still adopted in the current numerical simulation (see figures 14 a and 14 b) as it indeed effectively reduces the number of entrapped bubbles to a reasonable and manageable size. As a result, the obtained numerical results show some degree of deviation but, overall, they are negligible and do not affect the comparison or the conclusions and the presence of bubbles does not affect the splashing or spreading process (Riboux & Gordillo Reference Riboux and Gordillo2014). The time of the simulation starts at $t=0$ with the truncated spherical droplet already touching the surface. Thus, there exists an offset of $10^{-4}$ in time definitions between the numerical simulation in § 4 and analytical solutions in §§ 2 and 3 and, from now on, we use the former as time $t$ in comparisons below.

4.1.1. Simulation grid-independence and validation tests

Figure 15. The Wagner wet radius $a$ is presented over three decades in time between $t=10^{-4}$ and $t=10^{-1}$ . The numerical results for the contact line (defined in figure 14) positions are superimposed as blue dots, and collapse onto the analytical solution up to the lamella ejection time $t_e=4.2\times 10^{-2}$ , after which contact line propagates somewhat faster than the wet radius. The analytical values of the separation points $r_2$ derived in the present study are also superimposed in the graph, which has almost identical values with Wagner’s $a$ . Nevertheless, deviations arise, with slower spreading velocity, also from time $t_e$ .

Validation of the developed Basilisk numerical simulation placed emphasis on the ability of the numerical simulation to capture the evolution of the ‘contact line’ motion during impacts, which is a key indicator of accurate flow dynamics at the impact surface. We note that the contact line is different from the wet radius after time of ejection (see figure 14 e). The former refers to the interface where three phases (liquid droplet, solid surface and ambient air) meet at the rim of the ejected lamella and is straightforward to locate, while the latter refers to the turning point of the free interface at the root of the ejected lamella. To accurately describe the results of the numerical results, here we compare the positions of contact lines for morphology validation. Figure 15 reports the obtained results of the contact line positions from the Basilisk numerical simulation as a function of time for three decades between $t=10^{-4}$ and $t=10^{-1}$ . The numerical contact line motion agrees well with the Wagner theory of $a(t)=\sqrt {3t}$ (Wagner Reference Wagner1932; Riboux & Gordillo Reference Riboux and Gordillo2014; Philippi et al. Reference Philippi, Lagrée and Antkowiak2016) up to the time of lateral lamella ejection of $t_e=0.042$ , after which the contact line spreads faster than the wet radius of the Wagner theory. The findings are in agreement with the experimental studies of Riboux & Gordillo (Reference Riboux and Gordillo2014) (see below). Readers are also referred to figure 9 of Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) (same comparison plotted with log–log axes) where very similar comparison results between the numerical contact line motion and Wagner theory were obtained. Also plotted in figure 15 is the locus of the separation position $r_2(t,\pi /2)$ , (3.9), in time, which is indistinguishably close to the Wagner theory during the duration of the plot. However, as found in § 3.1 (and can be seen from figure 11 d), the separation position $r_2(t,\pi /2)$ is always inboard of $a(t)$ , and indeed a slight deviation of $r_2(t,\pi /2)$ from $a(t)$ can be observed in figure 15 from almost the same time of $t_e$ .

Figure 16. The same results of figure 15 plotted for longer time durations, at Basilisk grid refinement level $10$ (a), level $11$ (b) and level $12$ (c). The almost identical results indicate the degree of convergence of the numerical simulation at mesh resolutions before level $12$ .

For the grid-independence test of the results, the numerically predicted contact line motions at mesh refinements of level $10$ , doubling grids of level $11$ and quadrupling grids of level $12$ are compared in figure 16 for a longer time duration until $t=0.2$ . Almost identical numerical results are obtained, and hence the numerical simulation is believed to produce grid-independent results well before the lower mesh refinement limit of level 12 used in the present study. However, note that the locus of the contact line at lower levels (here as level $10$ ) is generally found to spread slightly faster than at higher levels (here as level $12$ ), but the difference converges to nothing at higher levels above $12$ . We finally remark that the vertical lines in figure 16 represent the lateral lamella ejection time $t_e$ . All mesh levels successfully and consistently predict the correct lamella ejection (i.e. contact line spreads faster than the wet radius of the Wagner theory) at the instant of $t_e$ . Position deviations between contact lines and the wet radii were also found experimentally by Riboux & Gordillo (Reference Riboux and Gordillo2014). Readers are referred to their figure 2 for comparisons.

A further validation of the numerical simulation is reported in § 4.3, where the numerically found impact force is compared with the experimental data from Gordillo et al. (Reference Gordillo, Sun and Cheng2018). This second validation of the numerical simulation is a direct validation on the ability of the numerical simulation to develop accurate flow kinetics upon impact, particularly at the impact surface.

4.2. Impact pressure on the solid surface

4.2.1. Pressure at the centre of the solid surface

Figure 17. Analytical pressure solution at the surface centre plotted over four decades in time between $t=10^{-4}$ and $t=2.5$ . The numerical results from the Basilisk simulation are superimposed as blue dots, and collapse onto the analytical solution for three decades between $t=10^{-3}$ and $t=6.5\times 10^{-1}$ . Before $t=10^{-3}$ , there is a transition period before the numerical simulation converges to the analytical solutions. After $t=6.5\times 10^{-1}$ , the numerical pressure at the surface centre deviates from the analytical solutions. The inset shows the same data plotted with a linear y-axis.

Impact pressure at the centre (i.e. $r=0$ ) of the solid impact surface is presented in figure 17 including the analytical solution of (3.13),

(4.3) \begin{equation} p(r=0, \beta , 0, t, \pi /2) = \frac {2}{\pi } a'(t) =\frac {\sqrt {3}}{\pi \sqrt {t}} \end{equation}

and simulation results of the developed numerical simulation. Excellent agreement between the numerical simulation and the analytical solutions is obtained for over three decades between $t=10^{-3}$ to approximately $t=0.65$ . Before $t=10^{-3}$ , there is a transient period until the numerical simulation collapses on to the analytical solution, and it is found that the length of this transient period of time depends on the mesh refinement level. The finer the mesh, the shorter the period, due to the better approximation of the initial geometry to the spherical shape. This is explained by the experimental and analytical findings in the literature (Field et al. Reference Field, Lesser and Dear1985; Howison et al. Reference Howison, Ockendon and Wilson1991) that the shape, especially the curvature, of the impactor geometry plays an important role for the initial impact pressure development. After $t=0.65$ , the numerical solution deviates from the analytical solution as the latter fails to represent the surface impact pressure. Nevertheless, it is worth noting that figure 17 is presented in log–log format to highlight the comparison, while in terms of magnitude, the discrepancies between the numerical simulation and analytical solutions are small, or even negligible, at either end of the displayed time interval, due to small time period before $t=10^{-3}$ and owing to the small pressure magnitude after $t=0.65$ (see the inset of figure 17), respectively. The contributions of the latter discrepancy to impact force magnitude, when integrating the pressure over the whole surface, is elaborated on in § 4.3. The same result was also found by Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) and readers are referred to their figure $17$ for comparison. Interestingly, we note that at $r=0$ , the analytical solution of (3.13) is exactly the same as the self-similar solution derived by Philippi et al., thereby the linear lines in Philippi et al. and figure 17 of the present study are the same.

4.2.2. Radial pressure profile at the solid surface

Figure 18. Analytical radial pressure profiles plotted over four decades of time for $t=\textit {O}(10^{-4})$ (a), $t=\textit {O}(10^{-3})$ (b), $t=\textit {O}(10^{-2})$ (c) and $t=\textit {O}(10^{-1})$ (d). The numerical results from the Basilisk simulation are superimposed as blue dashed lines, which collapse onto the analytical solutions over most of the presented duration. Note, however, that in the first decade (a), the numerical simulation cannot capture the sharp pressure peaks near the wet radius; in the fourth decade (d), the numerical simulation predicts a transition from the ring pressure pattern to centre-stagnated pressure pattern, while the analytical solutions show a delay in time for this transition. In (a) and (b) from $t=5\times 10^{-4}$ to $t=3\times 10^{-3}$ the values of the peaks extend beyond the figure limits (denoted by arrows pointing to individual peak value), because the horizontal resolution of the plotting fails to capture these values. See the text for discussion of the bumps.

The radial pressure profiles on the solid surface over four decades of time, between $t=5\times 10^{-4}$ to $t=1$ , are presented in figure 18 including the analytical solution of (3.13) and the results of the numerical simulation. During the first decade ( $t=5\times 10^{-4}$ to $t=10^{-3}$ ), it is observed that there exists a large discrepancy in magnitude around the wet radius, where the impact pressure peaks, between the numerical results and the analytical solution. This is partly due to the imperfectly spherical geometry of the initial configurations in the numerical solution (see § 4.2.1), and partly due to the inviscid flow assumption of the analytical model which fails to constrain any abruptly developed high flow velocity that would not happen in the presence of liquid viscosity. Similar discrepancies were also noticed by Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016). The duration of this period matches the length of the transition period in figure 17. For the second decade ( $t=2\times 10^{-3}$ to $t=10^{-2}$ ), the match between the numerical results and the analytical solution is good for both the pressure radial and temporal evolution patterns, and for the pressure magnitudes from the impact centre to the wet radius. However, in the third decade ( $t=2\times 10^{-2}$ to $t=10^{-1}$ ), the pressure profile from the numerical simulation begins to deviate from the analytical solutions from $t=5\times 10^{-2}$ as the pressure peak at the wet radius becomes less sharp and starts shifting towards the origin with increasing time. In comparison, the analytical solution has a slower and more gradual development. Nevertheless, until the fourth decade ( $t=2\times 10^{-1}$ to $t=10^{-0}$ ), the pressure profiles from both approaches still keep the spatial ring pressure pattern until very late time, at around $t=6.5\times 10^{-1}$ . After $t=6.5\times 10^{-1}$ the numerical results show that the maximum pressure is not only located near to, or at, $y = 0$ but that the magnitude is close to $0.5$ , suggesting a quasisteady state stagnation pressure while the analytical solutions show a more or less uniform pressure radial profile, with eventually higher magnitude at all radial positions, but with farther radial extent. We note that the loss of the numerically predicted ring pressure profile starts approximately at the time of centre-pressure deviation found in figure 17 between numerical and analytical predictions.

Figure 18(c), and more clearly in figure 18(d), show that the lateral lamella is ejected with a secondary pressure ‘bump’ (marked with arrows), due to pressure between the wet radius and the contact line. This explains the small deviations in the contact line velocity found in figure 15 at $t\gt t_e$ . In particular, the location of $r_2$ from the analytical solution satisfactorily predicts the start of the pressure bump from the numerical predictions, which denotes the presence of the wet radius in the numerical calculation, up to the end of calculation at $t=1$ . Although there exist discrepancies in surface pressure profiles at both early ( $t\lt 10^{-3}$ ) and late ( $t\gt 5\times 10^{-1}$ ) time of the analytical solution, their influence on the description of the entire droplet impact process is limited because the duration of time, and the pressure magnitudes, respectively, are both small. Overall, we observe a good agreement between the analytical solutions and the numerical results in terms of radial and temporal pressure magnitudes and evolutions for the entire four decades until $t=1$ , which is enough time span to characterise the most dynamic part of droplet impact.

4.2.3. Pressure maximum on the solid surface

In § 4.2.2, we have observed that the magnitudes at the radial pressure maximum within the wet radius have the largest variations over the four decades in time. Figure 19 presents these magnitudes, and radial positions, of the pressure maximum as a function of time for a long time span, until $t=10$ , comparing the numerical results with the analytical solutions ((3.19) and (3.20)).

In figure 19(a), the development of the numerical results of the surface pressure maximum as a function of time collapses closely onto the analytical solution of (3.20) from $t=10^{-2}$ and until the late time of $t=\textit {O}(1)$ . Before $t=10^{-2}$ , the analytical solution of radial pressure maximum is above the numerical solution, and the same information was presented in figure 18, on linear axes, where the overall discrepancy is seen to be, perhaps, of small importance to the overall force exerted into the surface. A very interesting feature of figure 19(a) is the slight curvature of the numerical solutions at later time ( $t\gt 2\times 10^{-1}$ ) away from the yellow dashed auxiliary line, which is the ‘linear’ part of (3.20) in log–log axes. This behaviour is also captured by the present analytical model. This nonlinear (in log–log format) development, in time, of the analytical pressure maximum marks the key difference between the present analytical solution and the well-known self-similar theory (Philippi et al. Reference Philippi, Lagrée and Antkowiak2016) (also see paragraph below).

Figure 19. Magnitudes (a) and positions (b) of the radial pressure maximum from numerical simulation and analytical solutions, of (3.20) and (3.19), respectively, as a function of time are shown for a long time span from $t=\textit {O}(10^{-4})$ to $t=\textit {O}(10^{1})$ . The yellow dashed line in part (a) represents the linear (in log–log format) extension of the line of (3.20). Lines of the separation points $r_2$ of (3.9) and the positions of radial pressure maximum in the self-similar theory are also superimposed for comparison. Note that values in figure (b) are capped from below to $8\times 10^{-3}$ due to the log scale.

In figure 19(b), the development in time of the position of the radial pressure maximum from numerical simulation is plotted as a function of time, superimposed on the analytical solution of (3.19). Excellent agreement exists between the numerical results and the analytical solution until $t=5\times 10^{-2}$ when the radial pressure maximum from numerical simulation starts occurring ever nearer to the centre of the surface, a process which is complete by $t=7\times 10^{-1}$ . These results are consistent with our findings from figures 18(c) and 18(d). The analytical solution also reproduces the trend of the radial displacement in the location of the pressure maximum, but this occurs later than in the calculations and is complete by $t = 3\pi ^2/16 \approx 1.85$ ((3.19) has no real solutions after $t = 3\pi ^2/16$ and this marks the disappearance of the ring pattern pressure maximum for $r\gt 0$ ). Also plotted in figure 19(b) is the wet radius $a(t)$ , which denotes the position of the maximum in the radial pressure profiles in self-similar theory (Philippi et al. Reference Philippi, Lagrée and Antkowiak2016). We can see that both the present analytical solution and numerical results verify the self-similar theory at early time, but deviate from self-similarity at $t=\textit {O}(10^{-1})$ . Finally, returning to the implied question below (3.20), we also plot the separation point of (3.9) in figure 19(b) which indeed shows that $r_{\textit{max}}(t,\pi /2)$ is always at smaller radius than, i.e. inboard of, the separation point of $r_2(t,\pi /2)$ in time.

4.2.4. Temporal development of pressure along the solid surface

In this section we present the development of pressure along the surface as a function of time.

Figure 20. Comparison of temporal surface pressure between the analytical solutions and the numerical simulation at radii $0.1$ (a), $0.4$ (b), $0.6$ (c) and $0.8$ (d).

Figure 20 shows the time-series of pressure at $r=0.1$ , $0.4$ , $0.6$ and $0.8$ for durations until $t=1$ . At $r=0.1$ (figure 20 a), we observe an excellent match (except the initial sharp spike) between the numerical results and analytical solutions for the whole duration. As the radius increases, at $r=0.4$ (figure 20 b), the analytical solution has a relatively small deviation from the numerical results at large time $t\gt 0.7$ . Note that the analytical solution has an initial sharp spike from zero pressure, whereas the numerical simulation shows a gradual magnitude change in time due to the presence of air phase being ‘squeezed out’ from beneath the droplet. With increasing distance along the solid surface, this pressure magnitude deviation becomes noticeable at earlier times at $r=0.6$ (figure 20 c) from $t\gt 0.6$ , until the solutions become ‘completely detached’ at $r=0.8$ (figure 20 d), albeit nevertheless remaining quite close in magnitude. We note that the wet radius takes $t=0.8^2/3\approx 0.2$ to spread to $r=0.8$ , a time belonging to the fourth decade of figure 18(d). Therefore, the observed magnitude gap in figure 20(d) is due to the discrepancies between the ‘rounded’ numerical peak and the ‘sharp’ analytical peak around the wet radius at later time of $t=\textit {O}(10^{-1})$ , as previously observed in figure 18(d).

Figure 21. Comparison of temporal surface pressure from the analytical solutions and numerical simulation at radii (see (4.4)) $r_{0.6}$ (a), $r_{0.8}$ (b), $r_{0.9}$ (c) and $r_{0.96}$ (d). The dashed ‘auxiliary’ line represents the pressure solution of (3.13) beyond $r_2$ of (3.9). Note, all four figures limit the ordinate to pressure magnitudes of $10$ .

It is informative to compare the pressure at radial locations with fixed ratios with the wet radius,

(4.4) \begin{equation} r_k=k\times a(t) \end{equation}

where $0 \leqslant k \leqslant 1$ is constant. Figure 21(a) presents the numerical results at $r_{0.6}$ , which characterises the pressure value in the middle between the surface centre and wet radius up to $t=1$ , with the analytical solutions superimposed. Apart from the initial pressure variations in the analytical solution (due to the transition period in the numerical solution and air bubble disturbance at early time), this is in good agreement with the numerical results. For $t\gt 0.7$ , however, the same pressure discrepancies are observed again due to the same reason explained in the preceding paragraph. As the radial position ratio increases, at $r_{0.8}$ (figure 21 b), the difference in pressure magnitudes between the numerical simulation and analytical solution become apparent at earlier time, as we have already observed for fixed radial positions. However, at $r_{0.9}$ (figure 21 c), a unique ‘cliff’-feature (i.e. a discontinuity in pressure magnitude) appears in the analytical solution at $t^*=0.436$ , as $0.9\times a(t^*)=r_2(t^*,\pi /2)$ . This is due to the newly introduced separation point $r_2$ in the present analytical model, where the validity of the analytical solution as an analogue of the flow of an impacting droplet striking a surface ends. Although the pressure cliff is an artificial discontinuity, this novel feature forces the analytical solution to zero pressure directly, which effectively reduces the discrepancy gap between the numerical results and analytical solutions. For contrast, an auxiliary dashed line denotes what the discrepancy would have been without the separation point concept.

Finally, at $r_{0.96}$ (figure 21 d), which is a radial position very close to the wet radius, the ‘cliff’ now arises towards the middle of the duration in the numerical solution, the dynamics of which guarantees the analytical solution to remain a reasonable match to the time-series of the pressure profile at each radial location, even at large radius near the wet radius where the discrepancy would otherwise be substantial. To justify the significance of the artificial pressure cliff, we note that the separation point $r_2$ serves as the locus where the Wagner condition is enforced in the droplet’s flow field (Wagner Reference Wagner1932), outside of which, the thin lamella (and rim) carries the flow (see figure $1c$ in García-Geijo et al. (Reference García-Geijo, Riboux and Gordillo2020)). The lamella is a slender region with negligible pressure gradient (Gordillo et al. Reference Gordillo, Riboux and Quintero2019; García-Geijo et al. Reference García-Geijo, Riboux and Gordillo2020), and hence the pressure on the wall is essentially the atmospheric pressure, i.e. effectively zero. Thereby, guided by the underlying physics, and for convenience, we explicitly ignore the solution of the flow outside the separation point, which also makes the solution explicit and in a closed form. The flow outside the separation point, at the very edge of the lamella, has been alternatively analysed in Oliver (Reference Oliver2002) and Negus et al. (Reference Negus, Moore, Oliver and Cimpeanu2021) at the wet radius $a$ , known as ‘inner solution’. For this, we remind readers of the close relative positions (especially at early time) between $r_2$ (black line) and $a$ (yellow dash–dot line) as shown in the temporal evolutions in figure 19(b).

4.2.5. Overall temporal and spatial pressure profile on the solid surface

Figure 22. The dimensionless surface pressure contour plot upon droplet impact at normal angle as a function of time and radius between $0\lt t\lt 0.7$ and $0\lt r\lt 2$ . The sharp edge of the pressure contour denotes the locus of the separation point $r_2$ of the present analytical solution. The line of the Wagner wet radius is superimposed as the yellow dashed line for comparison.

We have so far compared the derived analytical pressure solutions on the solid surface with the numerical simulation results, ‘vertically’ at different radial positions from the centre to the wet radius as a function of time, and ‘horizontally’ at four time decades as a function of radius. In this final section, we present the overall temporal and spatial surface pressure profile as a contour plot in figure 22. Overall, the maximum impact pressure happens upon impact and decays in time. The isobars are generally slightly tilted towards ‘the north-east direction’, which is the signature of the ring pressure pattern when examining the pressure as a function of radius at any constant time. The Wagner wet radius $a(t)=\sqrt {3t}$ is also superimposed in figure 22 as a dashed line. We see that the line of $a(t)$ deviates (farther outside) from the edge of $r_2(t,\pi /2)$ in time. Hoksbergen et al. (Reference Hoksbergen, Akkerman and Baran2023) has plotted a comparable figure from a computational fluid dynamics simulation with same values on the axes (except that their droplet impact happens at $T=1\ {\unicode{x03BC}} \textrm {s}$ ) and pressure colour bars in dimensional forms. The comparison with the numerical pressure profile of Hoksbergen et al. shows that the analytical wet radius, $a(t)=\sqrt {3t}$ , not only overshoots the numerical wet radius, but also the numerical contact line of the computational fluid dynamics results. We therefore suggest the separation point, $r_2$ , to be, at least qualitatively, a better approximation to the wet radius, especially at large time.

4.3. Impact force on the solid surface

Figure 23(a) presents the time-series of the impact force from the experiments by Gordillo et al. (Reference Gordillo, Sun and Cheng2018), the developed Basilisk numerical simulation and the analytical solutions of (3.27), Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) and Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010). We note that figure 23(a) is the experimental validation of the developed Basilisk simulation in the present paper. Excellent agreement has been obtained for the entire time duration, which thereby provides confidence in the results of the numerical simulation for time span of $t=\textit {O}(1)$ .

Figure 23. (a) The impact force on the solid surface from the numerical simulation is compared with the experimental data by Gordillo et al. (Reference Gordillo, Sun and Cheng2018). Analytical predictions from (3.27) and other studies (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010; Philippi et al. Reference Philippi, Lagrée and Antkowiak2016) are also added. The experimental peak force happens at $T_p=0.16\ \textrm {ms}$ ( $0\leqslant T \leqslant T_p$ is denoted as the shaded area). (b) A sensitivity study on $a(t)\sim \sqrt {t}$ in (3.27), denoted by dashed lines. Green triangles denote the maximum forces in individual cases. The secondary $x$ -axis on the top denotes the dimensionless time $t$ , and note that the $x$ -axis range is extended in (b).

We compare the present analytical model with the self-similar theory of Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) in figure 23(a). The experimental data indeed verifies the self-similar theory for three decades in time until $T=\textit {O}(10^{-2})\ \textrm {ms}$ (equivalently $t=\textit {O}(10^{-2})$ in dimensionless form). The present analytical solution extends the consistency with Gordillo’s experiments up to the peak force time at $T\approx 0.16\ \textrm {ms}$ (Gordillo et al. Reference Gordillo, Sun and Cheng2018) (equivalently $t=\textit {O}(10^{-1})$ ). The present analytical solution’s applicability (with respect to impact force) thus extends to the fourth decade of $t=\textit {O}(10^{-1})$ . The agreement of the present analytical solution with the experimental data, and the impact force calculated by Basilisk, verifies the fact that the initial transition period observed in § 4.2 is indeed negligible as its time duration and the influenced surface area are small.

After the peak experimental force at $T=0.16\ \textrm {ms}$ , as observed in figure 12 on the experimental data of Zhang et al. (Reference Zhang, Zhang, Lv, Li and Guo2019), the impact force predicted by (3.27) continues to grow. This indicates the limitation of the present analytical model to capture the correct decay rate of the ring pressure pattern, as already seen in the pressure comparisons of § 4.2, and will be explained in § 5.