2.1. The tip region

The tip region has the form of a parabola $h = \sqrt {2 r_c} x^{1/2}$ , where $r_c$ is the radius of curvature. This shape, together with an appropriate velocity field, is an exact steady solution of the Stokes equation with surface tension (Hopper Reference Hopper1993). This can be shown by solving the equations using the mapping

(2.1) \begin{equation} z = 2 r_c w(\zeta ) \equiv 2r_c\left (\zeta ^2 + {\rm i}\zeta \right )\!, \end{equation}

where $\zeta$ real corresponds to the free surface. Thus the upper half of the $\zeta$ plane is mapped into the fluid domain, which is the outside of the cusp. In the Goursat representation (Jeong & Moffatt Reference Jeong and Moffatt1992), the streamfunction of the flow is written in terms of two holomorphic functions $f$ and $g$ :

(2.2) \begin{equation} \psi = {\rm Im} \left \{f(z) + \bar {z}g(z)\right \}; \end{equation}

the components of the velocity field are calculated from $u = \psi _y$ and $v = -\psi _x$ , where the subscript denotes the derivative. However, in the complex formulation the velocity field is recovered more conveniently from

(2.3) \begin{equation} u - {\rm i}v = f'(z) + \bar {z}g'(z) - \overline {g(z)}. \end{equation}

As shown in Hopper (Reference Hopper1993), the steady flow around the parabola (2.1) is given by

(2.4) \begin{equation} f = \frac {{\rm i} r_c}{2} w'(\zeta )\bar {w}(\zeta ) G(\zeta ), \quad g = -\frac {{\rm i}}{2} w'(\zeta ) G(\zeta ), \end{equation}

where

(2.5) \begin{align} G = \frac {1}{2\pi {\rm i} \sqrt {1 + 4\zeta ^2}} \ln \left (\frac {2\zeta -\sqrt {1 + 4\zeta ^2}} {2\zeta +\sqrt {1 + 4\zeta ^2}}\right ) \end{align}

and $\bar {w}(\zeta )=\overline {w(\bar {\zeta })}$ is the conjugate function.

The free surface can be parametrised by putting $\zeta = \phi /2$ , with $\phi$ real, the tangent vector to the parabola is $\boldsymbol {t} = (\phi ,1)/\sqrt {1 + \phi ^2}$ and thus the tangential velocity $u_0$ is found from $ u_0 = (u\phi + v)/\sqrt {1 + \phi ^2}$ . Using the solution (2.4), this gives

(2.6) \begin{equation} u_0(\phi ) = -\frac {1}{\pi }\ln \left (\sqrt {1 + \phi ^2}+\phi \right )\!, \quad \phi = \sqrt {2 x/r_c}, \end{equation}

in agreement with Eggers (Reference Eggers2023); the velocity near the interface can be inferred from

(2.7) \begin{align} (u,v) = \frac {(1,h^\prime )}{\sqrt {1+h^{\prime 2}}} u_0 . \end{align}

In the far field, along the interface

(2.8) \begin{equation} u_0 = -\frac {1}{2\pi }\ln \frac {8x}{r_c}, \end{equation}

which is the far field of a stokeslet, i.e. the velocity field generated by a point force of strength $2$ . This corresponds to each side of the interface pulling with force $\gamma$ (per unit length in the third direction). Since $h' = \sqrt {r_c/(2x)}$ , which vanishes for $x\rightarrow \infty$ , we have

(2.9) \begin{equation} u = -\frac {1}{2\pi }\ln \frac {8x}{r_c}, \quad v = -\frac {\sqrt {r_c}}{2^{3/2}\pi \sqrt {x}}\ln \frac {8x}{r_c}, \end{equation}

again for large $x$ .

2.2. The cusp region

In the cusp region, we can make use of slenderness and model the air gap as a cut in the complex plane along the positive $x$ axis, as illustrated in figure 2(b); this is confirmed more formally below. We are seeking solutions to the Stokes equation symmetric about the $x$ axis and with stress-free conditions along the positive $x$ axis, valid locally around the tip of the cusp, which is at the origin. This is justified by the crucial observation that the stokeslet solution does not contribute to the stress at leading order, as we see below.

Since the asymptotic problem of a cut of vanishing thickness is lacking a characteristic length scale, we seek power-law solutions for the streamfunction of the form (Eggers & Fontelos Reference Eggers and Fontelos2015)

(2.10) \begin{equation} \psi = r^{\lambda }\left [A\sin (\lambda \varphi ) + C\sin ((\lambda -2)\varphi )\right ], \end{equation}

where $r$ is the distance from the tip and $\varphi$ the angle measured from the negative $x$ axis. This ensures that the line $\varphi =0$ in the wake of the cusp is a streamline, while the surface of the cusp is at $\varphi =\pi$ . Expansions of this form are standard in two-dimensional solid mechanics (see e.g. Muskhelishvili Reference Muskhelishvili1953; Gakhov Reference Gakhov1966). As shown in detail in Eggers & Fontelos (Reference Eggers and Fontelos2015), eliminating the pressure and demanding that the two components of the stress vanish at the free surface $\varphi =\pi$ lead to the conditions $\sin \lambda \pi = 0$ or $\cos \lambda \pi = 0$ for the scaling exponent $\lambda$ .

The first class of solutions leads to integer $\lambda$ , with vanishing transversal velocity $v=0$ along the cusp surface $x\gt 0$ . Non-singular velocity fields correspond to $\lambda = 1,2,\ldots$ , yielding $u = x^n,\;n=0,1,\ldots$ for the tangential velocity along the cusp surface. Thus with $-U$ being the leading-order velocity sweeping past the cusp, a general superposition of these solutions yields the tangential velocity

(2.11) \begin{equation} u = -U + b_1 x + b_2 x^2 + \cdots , \end{equation}

with $b_i$ being parameters. On the other hand, the second condition (and avoiding solutions too singular to match to the parabolic solution (2.4)), leads to $\lambda = 1/2,3/2,\ldots$ . Then on the positive $x$ axis, the velocity corresponding to these values is normal to the cusp, so that up to normalisation $u=0$ , $v = \pm x^{(2n-1)/2},\; n=0,1,2,\ldots$ , above and below the gap, respectively. This means a general superposition of normal velocities is of the form

(2.12) \begin{equation} u = 0, \quad v = a_0 x^{-1/2} + a_1 x^{1/2} + \cdots , \end{equation}

once more with $a_i$ as free parameters.

In addition to (2.10), another solution to the Stokes equation in the presence of a gap is a stokeslet of strength unity, pulling in the positive $x$ direction (Pozrikidis Reference Pozrikidis1992):

(2.13) \begin{equation} \psi = -\frac {y}{4\pi }\ln (r/a); \end{equation}

$a$ is a positive constant. Solution (2.13) is distinct from (2.11) and (2.12), in that it breaks scale invariance.

Once more both components of the normal stress generated by (2.13) vanish on the surface $y=0$ . Now the velocity on the positive $x$ axis corresponding to (2.13) is $u = -\ln (x/a)/(4\pi )$ and $v = 0$ , which matches (2.9) for a stokeslet of strength $2$ , and putting $a = r_c/8$ . This means that the total tangential component of the velocity is of the form

(2.14) \begin{equation} u = -\frac {1}{2\pi }\ln \left (\frac {8x}{r_c}\right ) + b_1 x + b_2 x^2 + \cdots . \end{equation}

The leading-order contribution to (2.12) for which $v$ remains finite at $x=0$ is $a_1x^{1/2}$ ; this is the one considered in Joseph et al. (Reference Joseph, Nelson, Renardy and Renardy1991) and Eggers & Fontelos (Reference Eggers and Fontelos2015). Together with a constant down-streaming $u=-U$ (which is the leading contribution to (2.11)), this produces a cusp which opens with a $3/2$ exponent. The crucial observation is that in order to match to the parabolic solution, we also have to add $a_0 x^{-1/2}$ to obtain

(2.15) \begin{equation} u = -U + b_1 x, \quad v = a_0 x^{-1/2} + a_1 x^{1/2}, \end{equation}

with higher-order terms discarded.

Then since the interface is a streamline in steady state, we obtain for the slope

(2.16) \begin{equation} h' = \frac {v}{u} = -\frac {a_0}{U}x^{-1/2} - \left (\frac {a_1}{U} - \frac {a_0 b_1}{U^2}\right ) x^{1/2} + O\bigl(x^{3/2}\bigr). \end{equation}

Thus identifying

(2.17) \begin{equation} r_c = 2\left (\frac {a_0}{U}\right )^2, \quad c = -\frac {2a_1}{3U} + \frac {a_0b_1}{U^2}, \end{equation}

and integrating (2.16), this yields

(2.18) \begin{equation} h = \sqrt {2 r_c} x^{1/2} + c x^{3/2}, \end{equation}

which is precisely the self-similar cusp profile found originally by Jeong & Moffatt (Reference Jeong and Moffatt1992). Again, higher-order terms have been discarded in order to obtain the leading-order shape valid near the cusp tip. This is the central result of this paper, in which the structure of the cusp is obtained by local analysis of the flow equations alone.

For small $x$ , (2.18) agrees with the parabolic solution (2.1). On the other hand, to leading order as $r_c\rightarrow 0$ , comparing (2.14) and (2.11) provides the down-streaming velocity $U\gt 0$ :

(2.19) \begin{equation} U = -\frac {1}{2\pi }\ln \frac {r_c}{8}, \end{equation}

which reveals an exponential dependence of the tip curvature on the externally imposed flow, as found originally by Jeong & Moffatt (Reference Jeong and Moffatt1992).

2.3. Matching

We now make sure that the solution proposed above is consistent in the limit $r_c\rightarrow 0$ , and that the different parts match. From (2.18) we see that the two terms on the right are balanced for $x = O(r_c^{1/2})$ , so that both terms are of $O(r_c^{3/4})$ . For $x$ of the order of $r_c$ , we are in the tip region; conversely for $x$ on the scale of $\sqrt {r_c}$ , we are in the cusp region, as shown in figure 2. It follows that (2.18) can be written in self-similar form as

(2.20) \begin{equation} y = r_c^{3/4} H\bigl(x/r_c^{1/2}\bigr), \quad H(\xi ) = (2\xi )^{1/2} + c\xi ^{3/2}, \end{equation}

where $c$ remains a free parameter, which depends on the particular problem at hand. The similarity solution (2.20) is precisely of the form found by Jeong & Moffatt (Reference Jeong and Moffatt1992) for a vortex dipole underneath an infinitely extended free surface, and thus for a specific value of the constant $c$ only. Our calculation now demonstrates that (2.20) is the generic structure of the cusp, and establishes $c$ as the only free parameter, once the radius of curvature $r_c$ is taken as a reference length scale.

The structure of the matching problem is illustrated in figure 2. In figure 2(a), we show the inner, tip problem, which lives on scale $r_c$ . It represents an exact solution to the Stokes equation with a free surface, which happens to be a parabola. Putting $\bar {x} = x/r_c$ and $\bar {y} = y/r_c$ , this free surface can be written as

(2.21) \begin{equation} \bar {y} = \sqrt {2\bar {x}}. \end{equation}

In figure 2(b), we illustrate the outer, cusp solution on scale $\sqrt {r_c}$ . It follows from (2.20) that in the limit $r_c\rightarrow 0$ the surface degenerates to a line, occupying the positive $x$ axis, shown as the thick red line. Moreover, we show that to leading order the boundary condition on the cut is stress-free, as assumed in deriving solutions (2.11)–(2.13). First, using the scaling (2.20), we estimate the curvature in (1.2) as $\kappa = O(r_c^{3/4}/\sqrt {r_c}^2) = O(r_c^{-1/4})$ . Second, to estimate the stress, we note that the stokeslet solution (2.13), for which $v=0$ , does not contribute to the stress. At next order, combining (2.15) and (2.17), we have $v \approx a_0 x^{-1/2} = O(\sqrt {r_c/x}\ln r_c)$ . Estimating the derivative at scale $\sqrt {r_c}$ , we find

(2.22) \begin{equation} \sigma = O\left(\sqrt {r_c/x^3}\ln r_c\right) = O\left(r_c^{-1/4}\ln r_c\right)\!, \end{equation}

which dominates the curvature by a logarithmic factor, and justifies looking for stress-free cusp solutions. This is a key observation, which explains why in spite of the complexity of the velocity field, which contains logarithmic contributions (e.g. (2.3)), finding cusp solutions reduces to finding solutions to the Stokes equation in the ’cut plane’ domain of figure 2.

Finally, we confirm the matching between the inner and outer velocity fields by comparing the outer limit of the inner solution with the inner limit of the outer solution. In the limit $r_c\rightarrow 0$ ,we find from (2.9) (the inner solution):

(2.23) \begin{equation} u = \frac {1}{2\pi }\ln \frac {r_c}{8} = -U, \quad v = \frac {\sqrt {r_c}}{2^{3/2}\pi \sqrt {x}}\ln \frac {r_c}{8} = \sqrt {\frac {r_c}{2x}} U, \end{equation}

using (2.19). If one considers the inner limit of (2.12), one obtains $v = a_1 x^{-1/2} = \sqrt {r_c/2} U x^{-1/2}$ , which is the second equation of (2.23), which demonstrates the required matching.

2.4. The outer flow

The flow defined by (2.14) has the property that it is not bounded at infinity, so one might worry that it cannot be matched to a finite outer flow. We now show that this is not a limitation of the present approach. Rather, solutions of the form (2.11) can be superimposed to produce bounded velocities. To see that, it is advantages to use the complex representation (2.2), in which solutions $u=x^n$ are described by

(2.24) \begin{equation} f(z) = \frac {z^{n+1}}{2}, \quad g(z) = \frac {z^n}{2}. \end{equation}

Indeed, the streamfunction is

(2.25) \begin{equation} \psi = \frac {1}{2}{\rm Im} \{z^{n+1}-\bar {z}z^n\} = y\,{\rm Im} \{i z^n\} = y x^n, \end{equation}

so we recover $u = \partial _y\psi = x^n$ . Similarly, the streamfunction of a stokeslet (2.13) can be written $\psi = y\,{\rm Re} \{\ln z\}$ .

Superimposing the solutions (2.25), we can write in general

(2.26) \begin{equation} \psi = y\, {\rm Re} \{\chi (z)\}, \quad \mbox{with}\; \chi (z) = d_0\ln z + \sum _{n=0}^{\infty } d_{n+1} z^n. \end{equation}

By choosing

(2.27) \begin{equation} \chi (z) = -\frac {1}{2\pi }\ln \frac {8 z}{r_c} + \frac {1}{4\pi } \sum _{n=1}^{\infty }(-1)^{n+1}\frac {{\rm e}^{-4\pi nU}(8z)^{2n}}{r_c^{2n}} = \frac {1}{4\pi }\ln \left ({\rm e}^{-4\pi U} + \frac {r_c^2}{(8z)^2}\right )\!, \end{equation}

we have constructed a solution which for $z\rightarrow 0$ behaves like (2.14). On the other hand, for $z\rightarrow \infty$ we have $\chi \approx -U$ , implying that a finite tangential velocity $u \approx -U$ is reached at infinity.

Looking at the behaviour of $\chi$ in the complex plane, and using (2.19), one obtains

(2.28) \begin{equation} \chi = \frac {1}{4\pi }\ln \left (r_c^2 + \frac {r_c^2}{(8z)^2}\right ) = \frac {1}{2\pi }\ln \frac {r_c}{8z} + \frac {1}{4\pi }\ln \left (z - {\rm i}\right ) + \frac {1}{4\pi }\ln \left (z + {\rm i}\right )\!, \end{equation}

which has logarithmic singularities at $z = \pm {\rm i}$ , or $y = \pm 1$ . According to (2.26), this means that near the singularities

(2.29) \begin{align} \psi = \pm \frac {1}{4\pi }\ln \left (z\mp {\rm i}\right )\!, \end{align}

which are counter-rotating vortices of circulation $\varGamma = \mp 1/2$ situated at $z = \pm {\rm i}$ . This is very similar to the solution considered in Jeong (Reference Jeong2010), where the flow leading to a cusp is driven by a pair of vortices. This, in turn, can be seen as an idealisation of the experimental set-ups in Joseph et al. (Reference Joseph, Nelson, Renardy and Renardy1991) and Jeong & Moffatt (Reference Jeong and Moffatt1992), where the flow is driven by two counter-propagating rollers.

The particular solution considered in (2.27) is just one of the many different ways of driving a free surface to cusp formation. However, a common feature of any such solution is the presence of singularities, of which the two vortices of (2.28) are a particular example. More generally, $\chi$ has to satisfy the requirement that $\chi \sim \ln z$ as $z\rightarrow 0$ , to represent the stokeslet generated at the tip. This implies that $z\chi '(z) \sim 1$ as $z\rightarrow 0$ : the combination $z\chi '(z)$ is free of singularities at the origin. In addition, the assumed boundedness of the velocity field at infinity implies that $z\chi '(z)$ can only grow sublinearly at infinity.

But supposing that $z\chi '(z)$ is analytic in the entire complex plane implies a contradiction: by Liouville’s theorem, such an analytic function has to grow at least linearly at infinity, which would be inconsistent with the velocity remaining bounded. Thus, the existence of singularities inside the flow, such as in (2.28), is a feature of the matching conditions we imposed. There is of course an infinite variety of ways such singularities can be imposed, corresponding to various ways in which the flow is driven.