2.1. Physical scales and intermediate range

We seek the existence of an intermediate-scale range in which a solution dominated by viscosity (see figure 2) can realistically be represented by a stable conical flow. That geometry would exist between the large global macroscopic scale and the very small scale in the region that smoothly links the conical shape with the formally infinitesimal emitted capillary jet. In that intermediate-scale range, if the solution sought existed, the flow would be self-similar. Globally stable solutions would be supported in the parametric region where this flow would be locally stable.

2.2. Conical Stokes flow and boundary conditions

In the inertia-less limit, we use the streamfunction $\varPsi _j(R,\theta )$ (see figure 1) for the Stokes flow inside the cone ( $j=0$ ) and outside ( $j=1$ ) domains (these initial domains can be extended subsequently) (Happel & Brenner Reference Happel and Brenner1973; Liu & Joseph Reference Liu and Joseph1978). Then, the Stokes equation can be brought into the form

(2.1) \begin{equation} E^2(E^2 \varPsi _j)=0, \end{equation}

where in spherical coordinates $\{R,\theta \}$ , the operator $E^2$ obeys the expression

(2.2) \begin{equation} E^2\equiv \frac {\partial ^2}{\partial R^2}+\frac {\sin \theta }{R^2}\frac {\partial }{\partial \theta }\left (\frac {1}{\sin \theta }\frac {\partial }{\partial \theta }\right ). \end{equation}

Equation (2.1) is solved assuming:

1. (i) the velocity field

   (2.3) \begin{equation} \boldsymbol{u}^{(j)}=\left \{ u_R,u_\theta \right \}^{(j)}=\left \{\frac {1}{R^2\sin \theta }\frac {\partial \varPsi _j}{\partial \theta } ,-\frac {1}{R\sin \theta }\frac {\partial \varPsi _j}{\partial R}\right \}, \end{equation}
   the components of the stress tensor $\boldsymbol{\tau }^{(j)}$ ,
   (2.4) \begin{equation} \boldsymbol{\tau }^{(j)}=\{\tau _{R,R},\tau _{R,\theta },\tau _{\theta ,\theta }\}^{(j)}=\lambda _j\left \{2\frac {\partial u_R}{\partial R}, R \frac {\partial (u_\theta /R)}{\partial R} + \frac {1}{R} \frac {\partial u_R}{\partial \theta },2 \left ( \frac {\partial u_\theta }{\partial \theta } +u_R\right )R^{-1}\right \}^{(j)} \end{equation}
   and the pressure $p_j$ at each domain $j$ satisfy the Stokes equation $\boldsymbol{\nabla }p_j= \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\tau }^{(j)}$ ;

2. (ii) the axis $\theta = 0$ and $\theta = \pi /2$ is a streamline explicitly implying

   (2.5) \begin{equation} u_{\theta }^{(0)}(R,\theta =0)=0,\, \frac {\partial u_{R}^{(0)}}{\partial \theta }(R,\theta =0)=0, \, u_{\theta }^{(1)}(R,\theta =\pi )=0,\, \frac {\partial u_{R}^{(1)}}{\partial \theta }(R,\theta =\pi )=0; \end{equation}

3. (iii) at the surface $\theta =\theta _s(R)$ , where the normal and tangential unit vectors are expressed as $\boldsymbol{n}=\{-R \theta '_s(R),1\} (1+R^2\theta '_s(R)^2 )^{-1/2}$ and $\boldsymbol{t}=\{1,R \theta '_s(R)\} (1+R^2\theta '_s(R)^2 )^{-1/2}$ , respectively – the primes indicate derivatives with respect to the variable indicated ( $R$ in this case) – the normal and tangential stress balances read

   (2.6) \begin{equation} p^{(1)}-p^{(0)}+\boldsymbol{n}\boldsymbol{\cdot }\big(\big(\boldsymbol{\tau }^{(0)}-\boldsymbol{\tau }^{(1)}\big)\boldsymbol{\cdot }\boldsymbol{n}\big) + \kappa (R)=0, \end{equation}
   and
   (2.7) \begin{equation} \boldsymbol{t}\boldsymbol{\cdot }\big(\big(\boldsymbol{\tau }^{(0)}-\boldsymbol{\tau }^{(1)}\big)\boldsymbol{\cdot }\boldsymbol{n}\big)=0 ,\end{equation}
   where the curvature $\kappa (R)$ in spherical coordinates is given in general by
   (2.8) \begin{equation} \kappa (R)= \frac {\cot (\theta _s (R))-R\!\left(r \theta ^{\prime\prime}_s(R)+\theta ^{\prime}_s(R) \left (R \theta ^{\prime}_s(R) \left (2 R \theta ^{\prime}_s(R)-\cot (\theta _s (R))\right )+3\right )\right )}{R\!\left(R^2 \theta ^{\prime}_s(R)^2+1\right )^{3/2}}, \end{equation}
   which is reduced to $\kappa ={\cot (\theta _s (R))}/{R}$ for a conical shape;

4. (iv) the continuity of tangential velocities requires

   (2.9) \begin{equation} \big(\boldsymbol{u}^{(1)}-\boldsymbol{u}^{(0)}\big) \boldsymbol{\cdot }\boldsymbol{t}=0; \end{equation}

5. (v) for a steady interface $F(R,\theta )\equiv \theta _s(R)-\theta =0$ , the general requirement ${\partial F}/{\partial t} + \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }F=0$ demands

   (2.10) \begin{equation} \boldsymbol{u}^{(0)}\boldsymbol{\cdot }\boldsymbol{n} = \boldsymbol{u}^{(1)}\boldsymbol{\cdot }\boldsymbol{n}=0; \end{equation}

6. (vi) recalling that we are using a generic length scale $l$ , and $\mu _1$ and $\gamma$ as the units of viscosity and surface tension, for any value of $R$ , the net flow rate is given by

   (2.11) \begin{equation} q=\int _0^{\theta _s(R)} 2 \pi\kern-1pt R^2 u_R^{(0)} \sin (\theta ) \, {\rm d}\theta .\end{equation}
   Alternatively, as anticipated, using the local scale $l_0= (\mu _1 Q/\gamma )^{1/2}$ , the flow rate can be set to $q=-1$ .

3.1. A first analytical conical exact self-similar solution with $q=0$

Analytical solutions of (2.1) in separable variables were known long ago. The solution part depending on the angular coordinate $\theta$ can be expressed as combinations of four independent solutions to the fourth-order differential equation derived from (2.1), which for our conical geometry we express in terms of the conical Legendre functions (Buckmaster Reference Buckmaster1972; Happel & Brenner Reference Happel and Brenner1973; Liu & Joseph Reference Liu and Joseph1978; Gañán-Calvo & Montanero Reference Gañán-Calvo and Montanero2021). The assumed existence of the intermediate region does not automatically suggest that we can set a perfectly conical flow at infinity, except in the case of the limit $q\rightarrow 0$ . In this case, there are two independent parameters of the problem: the viscosity ratio $\lambda$ and the cone angle $\alpha$ . Thus, reducing the problem to a perfectly conical self-similar flow with a meniscus of semi-angle $\theta _s(R)=\alpha$ , a solution to (2.1)–(2.11) can be written as (Buckmaster Reference Buckmaster1972; Happel & Brenner Reference Happel and Brenner1973)

(3.1) \begin{eqnarray} \varPsi _j = R^2 \big(G_{j,1}+G_{j,2} \cos (\theta ) + G_{j,3}\cos ^2(\theta ) + G_{j,4}\sin ^2(\theta ) \tanh ^{-1}(\cos (\theta ))\big)\!.\end{eqnarray}

The power $R^2$ ensures that stresses scale like $R^{-1}$ , as required by (2.6) to balance with surface tension. In this first solution, the first index of each set labels regions 0 and 1, and the second affects the amplitude of the linearly independent solutions $G_{j,i}$ . The boundary conditions lead to

(3.2) \begin{eqnarray} G_{0,1} & = & -\frac {\cos ^2(\alpha ) \cot \left (\dfrac {\alpha }{2}\right )}{4 \left (\left (\lambda -1\right ) \cos (\alpha )+\lambda +1\right )}, \nonumber \\ G_{0,2} & = & \frac {\sin (\alpha ) \cot ^2\left (\dfrac {\alpha }{2}\right )}{4 \left (\left (\lambda +1\right ) \sec (\alpha )+\lambda -1\right )}, \nonumber \\ G_{0,3} & = & -\left ({\kern-1pt}G_{0,1}+G_{0,2}\right ),\,\, G_{0,4} = 0,\nonumber \\ G_{1,1} & = & -\frac {\lambda A -B}{8 \left (\left (\lambda -1\right ) \cos (\alpha )+\lambda +1\right )}, \nonumber \\ G_{1,2} & = & -\frac {\sin ^2\left (\dfrac {\alpha }{2}\right ) \cos (\alpha ) \tan \left (\dfrac {\alpha }{2}\right ) \left (\left (\lambda -1\right ) \cos (\alpha )+\lambda \right )}{2 \left (\left (\lambda -1\right ) \cos (\alpha )+\lambda +1\right )}, \nonumber \\ G_{1,4} & = & \frac {1}{8} \sin \left (2 \alpha \right ), \,\, G_{1,3} = -\left ({\kern-1pt}G_{1,1}-G_{1,2}\right ), \end{eqnarray}

with

(3.3) \begin{eqnarray} A & = & \sin \left (2 \alpha \right ) \left (\cos (\alpha )+\left (\cos (\alpha )+1\right ) \log \left (\tan \left (\dfrac {\alpha }{2}\right )\right )\right ), \nonumber \\ B & = & \left (\sin (\alpha )-\tan \left (\dfrac {\alpha }{2}\right )\right ) \left (\cos \left (2 \alpha \right )-2 \sin ^2(\alpha ) \log \left (\tan \left (\dfrac {\alpha }{2}\right )\right )+1\right ). \end{eqnarray}

An illustration of both $u_R$ and $u_{\theta }$ as functions of $\theta$ is given in figure 3. Interestingly, the modulus of the outer velocity is nearly constant everywhere in the outer domain, except at the axis.

Figure 3.

(a) The fluid velocities $u_R^{(j)}(\theta )$ (black curves) and $u_\theta ^{(j)}(\theta )$ (blue curves) of the exact analytical solution (3.1), plotted as functions of $\theta$ . Note the singular behaviour as $\theta \rightarrow \pi$ (the region that would be occupied by the jet). (b) The pressure distributions $P^{(0)}(R)$ (black) and $P^{(1)}(R)$ (blue). Here, $\alpha =0.15$ , $\lambda =0.01$ and $q=0$ .

This velocity field is independent of $R$ and raises the inner liquid pressure towards the apex: when the inner-to-outer viscosity ratio is small, the momentum diffusion from the outer converging flow is so strong that it compresses the inner flow at the tip. In effect, the corresponding expressions for the pressure in both domains are given by

(3.4) \begin{eqnarray} p_j & = & \lambda _j \frac {2 \left (G_{j,2}-G_{j,4}\right )}{R}. \end{eqnarray}

This expression is positive for $j=0$ and negative for $j=1$ , and becomes unbounded for $R\rightarrow 0$ . Note that, in contrast with the fluid velocities, the pressure is independent of $\theta$ . Figure 3(b) illustrates the pressure distributions in both domains, which are inversely proportional to $R$ to balance the surface tension term.

For a given small viscosity ratio $\lambda$ used in figure 3, the pressure distribution is due to the intense diffusion of momentum into the inner flow from the outer domain, so that while the outer flow loses pressure only modestly, the inner flow gains it substantially. Their difference is balanced by the increase in surface tension and normal viscous forces as $R$ decreases. The increasing overpressure towards the apex in the inner domain projects the on-axis stream in the opposite direction to that of the interface towards the apex.

3.1.1. The interfacial velocity

From the solution (3.1)–(3.2) (i.e. with $q=0$ ), the velocity of the interface reads

(3.5) \begin{equation} u_R(\alpha )=-\frac {\sin \left (2 \alpha \right )}{8 \left (\left (\lambda -1\right ) \cos (\alpha )+\lambda +1\right )} .\end{equation}

This velocity can be represented as a function of the cone angle $\alpha$ and the viscosity ratio $\lambda$ , as shown in figure 4.

Figure 4.

The velocity of the interface (in the direction of the apex) according to (3.5), as a function of $\alpha$ and $\lambda$ . Iso-contours represent constant velocity values. The blue dashed line is the value of $\alpha$ that maximises the absolute value of the interfacial velocity for a given viscosity ratio $\lambda$ . Above this maximum absolute velocity for a given $\lambda$ there is no solution, while below it, each velocity gives two possible solutions and cone angles. In § 4 we show that the solutions to the right of the blue curve should not be considered.

For a given driving velocity (that can be assimilated into the interface velocity (3.5) in the self-similar problem), and a given viscosity ratio $\lambda$ , figure 4 shows that the cone angle may have two possible values, or no solution. There is one value of $\alpha$ that maximises the absolute value of the surface velocity for a given $\lambda$ , represented as a blue dashed line: it is the cone angle for which the transfer of momentum to the inner fluid is maximised. This maximising $\alpha _m$ value is given by the expression

(3.6) \begin{equation} \lambda =\frac {2 \tan ^2\left (\dfrac {\alpha _m}{2}\right ) \left (\sin ^2\left (\alpha _m\right )+\cos \left (\alpha _m\right )\right )}{2 \cos \left (\alpha _m\right )+\cos \left (2 \alpha _m\right )-1}. \end{equation}

Remarkably, for $\lambda \ll 1$ , one has that (3.6) can be approximated as

(3.7) \begin{equation} \lambda = \frac {\alpha _m^2}{4}\left (1+\frac {13}{6}\alpha _m^2\right )+O\big(\lambda ^6\big) \Longrightarrow \alpha _m= 2 \lambda ^{1/2}+O\big(\lambda ^{3/2}\big). \end{equation}

This is exactly the same relationship $\alpha _m = 2\lambda ^{1/2}$ for the critical solution (maximum strength of the external flow) found in § 4 using slender-body theory, i.e. with $\alpha _m \ll 1$ . This helps to understand why (3.5) is the first-order value of the external driving velocity, with errors $\sim \mathcal (\lambda )$ .

However, given the logarithmic singularity of the axial velocity on the outer axis ( $\theta = \pi$ ), the analytic solution presented in this section (see figure 3) does not completely solve our problem. Nevertheless, the existence of this solution suggests that such a conical flow configuration would be a good candidate as a first approach, modified as necessary to satisfy the presence of an inertialess thin jet in the vicinity of $\theta = \pi$ for small $\lambda$ values.

3.2. Modification of the analytic solution for $q \neq 0$

The modification can be accomplished in two steps.

1. (i) When $q\neq 0$ (finite constant flux), the solution (3.1)–(3.2) should be augmented as $\psi _j + \psi _{j,A}$ , where

   (3.8) \begin{equation} \psi _{j,A}=A_{j,2}\cos ^3(\theta )+ A_{j,1}\cos (\theta )+ A_{j,3} \cos (\theta ) \left (\cos (\theta )-\!\sin ^2(\theta ) \log \left (\tan \left (\frac {\theta }{2}\right )\right )\right ). \end{equation}
   Now, the scaling with $R^0$ ensures that the flux becomes independent of $R$ . The coefficients $A_{j,i}$ satisfying (2.5)–(2.11) with $q=-1$ are
   (3.9) \begin{equation} A_{j,1 } = -\frac {3 \lambda _j \cos ^2(\alpha ) \csc ^4\left (\dfrac {\alpha }{2}\right )}{16 \pi \cos (\alpha )+8 \pi }q, \quad A_{j,2} = \frac {\lambda _j \csc ^4\left (\dfrac {\alpha }{2}\right )}{16 \pi \cos (\alpha )+8 \pi }q,\quad A_{j,3} = 0, \end{equation}
   for $j=0,1$ ( $\lambda _0=1$ , $\lambda _1\equiv \lambda$ ). The coefficient $A_{j,3}$ is zero since the solution of the form
   (3.10) \begin{equation} A_{j,3} \cos (\theta ) \left (\cos (\theta )-\!\sin ^2(\theta ) \log \left (\tan \left (\frac {\theta }{2}\right )\right )\right ), \end{equation}
   which gives a strong singularity on the axis, should be excluded in both domains. On the other hand, when $R \gg 1$ , both $u_R$ and $u_{\theta }$ are independent of $R$ , as required, but they exhibit a logarithmic singularity as $\theta \rightarrow \pi$ . However, the axis belong to the region occupied by the jet and the singularity is avoided for a finite jet radius $R_j$ .

2. (ii) By forcing a regularisation around the jet of the exact solution (3.1)–(3.2) presented. This is developed next.

It is important to emphasise here that this modification not valid for $R\to 0$ .

3.2.1. Cone-jet solution: an optimal approximate analytical solution with a jet domain, excluding $R \to 0$

A possible way to tackle regularisation is to divide the space into four domains (see figure 5), where $R_J$ is the jet radius as follows:

1. (i) the inner cone (domain 0);

2. (ii) the outer flow to the cone (domain 1, now limited between the angles $\alpha$ and $\chi$ );

3. (iii) the outer jet domain (2) (or the outer flow in the jet region, between the angles $\chi$ and $\pi -R_J/R$ , for $R_J \ll R$ ); and

4. (iv) the inner jet domain (3).

Figure 5.

The four regions considered: 0 (inner cone), 1 (outer cone), 2 (outer jet region) and 3 (inner jet). The angle $\chi$ separating regions 1 and 2 is where the solutions should match. According to the procedure described, this angle is a free parameter that determines the values of the cone angle $\alpha$ and the jet radius $R_J$ for a given viscosity ratio $\lambda$ . It is related to the free parameter $\overline {{\textit{Ca}}}$ of the second solution procedure described in § 4. (a) General schematics for an arbitrary intermediate scale $l$ . The lengths $L$ and $R_{{out}}$ denote the half-length and radius of the computational box used in § 3.3. (b) The analytical solution at the local scale $l_0$ (blue lines). The excluded cone-jet transition $R\to 0$ joining cone and jet is numerically resolved (dashed line).

In this section, the presence of a jet causes the problem to lose its self-similarity in favour of a local solution to the cone-jet flow geometry, with a finite jet radius scale. By choosing $q=-1$ in the cone domain, where the apex is a sink, we implicitly use $l_0$ as the unit of length.

The analytical approach to solving domains 0 and 1 at the cone, which leads to solutions (3.1) and (3.9), is followed for domains 2 and 3, which correspond to the jet problem, by setting the boundary conditions at the cone surface. Once the cone and jet problems have been solved independently, the remaining unknowns required to solve the global problem can be determined by matching domains 1 and 2.

Focusing now on the jet surface, given by $\theta _s(R)=\pi - R_J/R$ in the limit of small $R_J$ , and similarly to the cone domain, we seek solutions of the form

(3.11) \begin{equation} \psi _j=\underbrace {\cos ^3(\theta ) A_{j,2}+\cos (\theta ) A_{j,1}}_{\text{constant flux}}+\underbrace {R^2 \big(\cos ^2(\theta ) G_{j,3}+\cos (\theta ) G_{j,2}+G_{j,1}\big)}_{\text{surface stress-balance}}, \end{equation}

with $j=2,3$ , which complies with regularity at the axis. The singular part of the solution is dropped, i.e. $G_{j,4}\big|_{j=2,3}=0$ . The conditions on the jet surface are identical to (2.6)–(2.10), except for the value of $\theta _s(R)=\pi - R_J/R$ .

To match the flow from the cone side, i.e. condition (2.11), the flow rate condition in the jet domain for any radial position $R$ is now

(3.12) \begin{equation} q=\int _{\pi -R_J/R}^{\pi }2 \pi R\, \sin (\theta ) \,u_R^{(3)}\, R\,{\rm d}\theta = 1 ,\end{equation}

since the flow is now positive in the $R$ direction (i.e. the region $R\to 0$ is a source), where according to the values (3.9), one has for $R_J \ll R$

(3.13) \begin{equation} u_R^{(3)}= \frac {q}{\pi\!R_{J}^{2}}\left (1-\frac {3 R_J^2\lambda \cot ^2\left (\frac {\alpha _0}{2}\right )}{2 (2 \lambda -1) R^{2}\! \left (2 \cos \left (\alpha _0\right )+1\right )} \right )+ \mathcal{O}(R_J/R)^2. \end{equation}

As stated previously, the values of $A_{2,1}$ and $A_{2,2}$ can be identical to those obtained in the outside region of the cone, $A_{1,1}$ and $A_{1,2}$ . This also guarantees that the stress component $\tau _{\textit{RR}}$ , given by

(3.14) \begin{equation} \tau _{\textit{RR}}=\frac {4 \left (\cos (\theta ) \left (3 A_{1,2} \cos (\theta )-R^2 G_{1,4}\right )+A_{1,1}\right )}{R^3}, \end{equation}

is identical at both the cone and the jet regions, since no terms depending on $G_{j,i}$ appear in this expression. Moreover, to avoid tangential stresses $\tau _{R\theta }$ on the jet surface as $R \rightarrow \infty$ (i.e. the jet velocity profile should be flat), one should have $G_{j,4}= 0$ in the jet region. In contrast, $G_{j,4}\neq 0$ in cone region 1 (see expressions (3.2)), which implies a logarithmic singularity when $\theta \rightarrow \pi$ or $R\rightarrow \infty$ , since the jet surface is $\theta _s(R)=\pi -R_j/R$ , with $R_J = const.$ . Therefore, $G_{2,i}|_{i=1,2,3}$ cannot be identical to $G_{1,i}|_{i=1,2,3}$ , and a perfect match is impossible. However, the singularity on the axis diminishes with the cone angle since $G_{1,4}$ vanishes as $\alpha \rightarrow 0$ , while the other coefficients remain of order unity or even larger. This harbours the possibility of a sufficiently accurate solution for $\alpha \ll 1$ .

In the following, a way to obtain a global solution by an optimal approximate analytical match between the two solutions of (2.1) at the intermediate angle $\chi$ together with the solution on the inner side of the jet is presented in § 3.2.2.

Given a viscosity ratio $\lambda$ , a set of 11 unknowns to solve the problem is given by the following:

1. (i) eight coefficients, namely $G_{j,i}|_{j=2,3; i=1,2,3}$ and $A_{3,i}|_{i=1,2}$ ;

2. (ii) the jet diameter $R_J$ ;

3. (iii) the cone angle $\alpha$ ; and

4. (iv) the angle $\chi$ where the errors between the solutions at domains 1 and 2 are minimal (approximate matching), or zero. How to resolve the 11 described unknowns is detailed in Appendix A.

3.2.2. Approximate matching

So far, we have already obtained the best possible solutions to the surface problems of the cone (exact) and of the jet (see Appendix A). The next step is to finally solve the matching problem of solutions in regions 1 and 2 at an intermediate angle $\chi$ (see figure 5).

Recall that the outer streamfunctions $\psi _1^{(1)}$ and $\psi _1^{(2)}$ share the same part $A_{j,2}\cos ^3(\theta )+ A_{j,1}\cos (\theta )$ ( $j=1,2$ ). However, the part multiplied by $R^2$ , namely

(3.15) \begin{equation} \phi ^{(j)}=G_{j,1}+G_{j,2} \cos (\theta ) + G_{j,3}\cos ^2(\theta ) + G_{j,4}\sin ^2(\theta ) \tanh ^{-1}(\cos (\theta )), \end{equation}

is different at outside regions 1 and 2 due to the logarithmic part of the solution at region 1, which should be zero at region 2 to satisfy regularity at the axis when $R\rightarrow \infty$ . A perfect match between both solutions is impossible for simple uniqueness reasons, but one can hypothesise the existence of a certain intermediate angle $\chi$ where the velocities and stresses can be matched. This would happen if the values of $\phi ^{(j)}(\theta )\big |_{j=1,2}$ and its first and second derivatives match at $\theta =\chi$ .

Thus, the approximate matching involves only the coefficients $G_{i,j}$ , given by (A1)–(A6) since the constant flux solution (coefficients $A_{i,j}$ ) is the same in domains 1 and 2. This means that the quantity $U_J=q/(\pi\!R_{J}^{2})$ , that is, the average velocity of the jet, is the only relevant variable in the match, independently of the length scale or flow rate used. Without loss of generality, we set $q=1$ at the jet. Solving the matching problem entails to find an angle $\theta =\chi$ mediating domains 1 and 2 where, defining

(3.16) \begin{equation} \phi ^{(1)}-\phi ^{(2)}=\epsilon _0,\quad \partial _{\theta } \phi ^{(1)}-\partial _{\theta } \phi ^{(2)}=\epsilon _1,\quad \partial _{\theta }^2 \phi ^{(1)}-\partial _{\theta }^2 \phi ^{(2)}=\epsilon _2, \end{equation}

the solution to $\epsilon _i=0$ for $i=0,1,2$ would yield the exact matching of velocities and stresses at the intermediate angle $\chi$ . Theoretically, once all coefficients of $\phi ^{(j)}$ have been solved, this matching (three equations) would solve the three unknowns $R_J$ (which is the value of $R_J$ in units of the local scale $l_0$ ), the intermediate angle $\chi$ and the cone angle $\alpha$ for a given value of $\mu$ . However, this theoretical exact solution does not exist because the three sheets defined by the (3.16) in the space $ \{\alpha ,\chi ,R_J\}$ with $\epsilon _i=0$ do not meet at any non-trivial point (except at the trivial cylindrical solution $\alpha =0$ , $\chi =\pi$ for any arbitrary large value of $R_J$ ). This is due to the failure of the system (3.16), defined by successive derivatives of the first equation, to satisfy the transversality conditions of the Morse–Sard theorem (Morse Reference Morse1939; Sard Reference Sard1942) for $\epsilon _i=0$ (i.e strict independency, or topological absence of parallelism or tangency of the manifolds defined by the equations): in fact, derivation defines a linear relationship between (3.16) that violates those independency conditions.

Nevertheless, the particular nature of system (3.16), which is linear in $R_J$ and quadratic in $\lambda$ , allows one to obtain an optimal approximate solution of the matching problem. This optimal solution yields a relationship for the matching angle of the form $\chi =\chi (\alpha ;\lambda )$ , given in Appendix B, such that the matching errors $\epsilon _i$ are strictly set to zero: although this yields non-unique $R_{J}$ values (by the failure to satisfy the conditions of the Morse–Sard theorem), one can find a relationship $\chi =\chi (\alpha ;\lambda )$ that makes their differences strictly minimal. In fact, defining an error norm for their differences, the location $\xi =\chi (\alpha ;\lambda )$ is graphically visualised as a narrow ‘creek’ of that error (see Appendix B).

Under the assumption of a conical flow at infinity, $\alpha$ is a free parameter with $\chi$ a function of $\alpha$ such that $\chi _{\textit{min}} \lesssim \chi \lt \pi$ , with $\chi _{\textit{min}}\lesssim 1.22$ as shown in Appendix B. However, the slender-body theory in § 4 based on $\alpha \ll 1$ , which admits other possibilities at infinity, shows that instead of $\alpha$ as a free parameter, a local physical capillary number related to the strength of the external flow (not necessarily a conical flow) fixes $\alpha$ .

In what follows, we refer to our proposed local analytic cone-jet solution approximation as the regularised cone solution. It is completed by the numerical solution at the cone-jet transition region $R\to 0$ , which is tackled next.

3.3. The cone-jet intermediate region: numerical implementation

For a given set of $\lambda$ and $\alpha$ values, the intermediate region between the cone, a perfectly conical recirculating flow with a non-zero net flow rate (i.e. $q=1$ ), and an asymptotically cylindrical jet with a radius at infinity proportional to the local scale will be solved numerically using a local cylindrical coordinate system ( $z$ , $r$ ), with $z=R \cos \theta$ and $r=R \sin \theta$ , in a rectangular domain $[-L,L]\times [0,R_{out}]$ , where $L$ is any valid value such that $L \gg l_0$ (see figures 1, 2 and 5).

The numerical code solves the conservation of mass and a balance of linear momentum in the outer ( $j=1$ ) and inner ( $j=0$ ) subdomains, given by

(3.17) \begin{align} {\boldsymbol {\nabla }}\boldsymbol{\cdot }{\boldsymbol{v}}_j &= 0, & &(j=0,1), \end{align}

(3.18) \begin{align} {\boldsymbol {\nabla }} p_{j}&=\lambda _{j}{{\nabla} }^2 \boldsymbol {v}, & &(j=0,1), \end{align}

where $\boldsymbol {v}_j=w_j \boldsymbol{e}_z+ u_j\boldsymbol{e}_r$ is the velocity field, $p_j$ is the pressure and $\lambda _1=1$ as previously stated. We use the analytical solution as the far-field boundary conditions at the computational box boundaries.

1. (i) At right boundary, $z= L$ , (see figure 5 a) we assume that the numerical solution matches the analytical solution for the cone

   (3.19) \begin{equation} w_j=\frac {1}{r}\frac {\partial \psi _i^{\textit{cone}}}{\partial r} \quad u_{j}=-\frac {1}{r}\frac {\partial \psi _j^{\textit{cone}}}{\partial r}, \quad (i=0,1), \end{equation}
   and $F=L_{box}\tan (\alpha )$ , where $r=F(z,t)$ is the parametric representation of the interface in terms of $z$ (see magenta line in figure 5 a).

2. (ii) At the left boundary, $z=-L$ , we impose the following Neumann condition for the outer flow:

   (3.20) \begin{equation} \frac {\partial w_1}{\partial z}=\frac {\partial w_1^{\kern1pt\textit{jet}}}{\partial z},\quad \frac {\partial u_1}{\partial z}=\frac {\partial u_1^{\kern1pt\textit{jet}}}{\partial z}, \end{equation}
   where $w_1^{\kern1pt\textit{jet}}$ and $u_1^{\kern1pt\textit{jet}}$ are the velocities obtained with the analytical streamfunction of the jet solution ( $\psi _1^{\kern1pt\textit{jet}}$ ). On the other hand, for inner flow, outflow conditions are considered
   (3.21) \begin{equation} \frac {\partial w_0}{\partial z}=0,\quad u_0=0, \quad \frac {\partial F}{\partial z}=0. \end{equation}

3. (iii) At the top boundary, $r=R_{out}$ , The velocity varies continuously from the analytical solution associated with the cone to the solution associated with the jet

   (3.22) \begin{equation} z\lt -\!\sin (\chi ):\quad w_1=w_1^{\kern1pt\textit{jet}} , \quad z\gt =-\!\sin (\chi ): w_1=w_1^{\textit{cone}}\quad u_1=u_1^{\textit{cone}}. \end{equation}

Across the interface, we use the same conditions (2.6)–(2.10) already expressed for the analytical solution.

The governing equations are integrated with a variant of the numerical method proposed by Herrada & Montanero (Reference Herrada and Montanero2016), Herrada (Reference Herrada2025). The inner (0) and outer (1) domains are mapped onto rectangular domains by means of the analytical mappings

(3.23) \begin{equation} r=F(\xi ,t)\zeta _0,\quad z=\xi ,\quad [0\leqslant \zeta _0\leqslant 1]\times [-L\leqslant \xi \leqslant L], \end{equation}

for the inner domain, and

(3.24) \begin{equation} r=F(\xi ,t)+\zeta _1[R_{out}-F(\xi ,t)],\quad z=\xi ,\quad [0\leqslant \zeta _0\leqslant 1]\times [-L\leqslant \xi \leqslant L], \end{equation}

for the outer domain. These mappings are applied to the governing equations, and the resulting equations are discretised in the $\zeta$ -direction with $n_{\zeta _0}$ and $n_{\zeta _1}$ Chebyshev spectral collocation points in the inner and outer domains, respectively. Conversely, in the $\xi$ -direction we use second-order finite differences with $n_{\xi }$ points.

Steady-state solutions of the nonlinear discretised equations with all variables independent of time are obtained by solving all equations simultaneously (a so-called monolithic scheme) using a Newton–Raphson procedure.

The combined analytical–numerical cone-jet solution completed so far will be simply called the ‘cone-jet solution’. To summarise the physical framework of this solution, we have the following.

1. (i) It implies a self-similar conical flow at infinity.

2. (ii) The intermediate cone-jet solution is resolved assuming $q=1$ This means that, for numerical simplicity purposes, the intermediate scale $l$ is assumed as $l=l_0$ .

3. (iii) Therefore, from the previous points, the intermediate length $l$ should be very small compared with any other macroscopic length, including that for which a nearly conical flow is observed.

4. (iv) Thus, using $l_0$ , $m_0$ and $t_0$ as the units of length, mass and time, we also have the dimensional value of the flow rate $Q=1$ (i.e. $Q$ is the flow rate unit; or equivalently, $q=1$ in the jet domain).

5. (v) This solution has two independent variables: the viscosity ratio $\lambda$ and the cone angle $\alpha$ . The latter is not restricted except that there is a maximum velocity that the cone surface can attain, and this is obtained for a specific value of the angle given by (3.7).

An alternative solution to the problem is developed in the next section by providing a complete description of the cone-jet structure under the assumption $\alpha \ll 1$ (the slender-body approximation). Once both the cone-jet solution and the slender-body solution introduced below are obtained, their comparison will help clarify the long-standing and non-trivial issues they address. In particular, it will explain the occurrence of nearly conical tips with imperceptible ejections observed in early experiments by Taylor (Reference Taylor1934) and Rumscheidt & Mason (Reference Rumscheidt and Mason1961), as well as in more recent experimental studies (Gañán-Calvo et al. Reference Gañán-Calvo, González-Prieto, Riesco-Chueca, Herrada and Flores-Mosquera2007) and numerical investigations (Rubio et al. Reference Rubio, Montanero, Eggers and Herrada2024), among others. Nevertheless, as will be shown, a theoretical determination of the free parameter $\alpha$ remains an open problem. A possible resolution is proposed based on results that are consistent across all approaches.

4.1. Slender-body theory

We now show that Taylor’s theory, developed for driving by an extensional flow, can be applied to an arbitrary external flow $\boldsymbol {u}^{(\textit{ext})}(z,r),p^{(\textit{ext})}(z,r)$ , which we assume satisfies the Stokes equation in cylindrical coordinates. In the slender limit, the jet is expected to make a small correction, so we write the flow in the outer phase as

(4.1) \begin{equation} \boldsymbol {u}^{(\textit{out})} = \boldsymbol {u}^{(\textit{ext})} + \boldsymbol {u}', \end{equation}

where $\boldsymbol {u}'$ is a distribution of two-dimensional point sources along the axis, with strength $A(z)$ : $\boldsymbol {u}' = (0,A(z)/r)$ . This approximation can be verified by a systematic calculation based on an expansion in the slenderness (Buckmaster Reference Buckmaster1972), but physically it means that the jet’s only effect on the outer flow is to occupy extra volume. Now $A(z)$ follows from the kinematic boundary condition $\boldsymbol {u}\boldsymbol{\cdot }\boldsymbol {n} = 0$

(4.2) \begin{equation} A(z) = h\big[u_z^{(\textit{ext})}(z,h)h' - u_r^{(\textit{ext})}(z,h)\big]. \end{equation}

In the interior, using slenderness, we can assume a parabolic lubrication-type flow, while in the limit of small $\lambda$ the external flow is unchanged; hence we obtain

(4.3) \begin{equation} u^{(\textit{in})}_z = -\frac {1}{4\lambda \mu _0}\frac {{\rm d}p}{{\rm d}z}(h^2-r^2) + u_z^{(\textit{ext})}(z,h), \end{equation}

using continuity of the velocity. The shear balance does not need to be taken into account to leading order (Buckmaster Reference Buckmaster1972), and the normal force balance reads for a slender thread

(4.4) \begin{equation} \sigma _{\textit{rr}}^{(\textit{out})} + p^{(\textit{in})} = \frac {\gamma }{h}, \end{equation}

using the leading-order expression for the mean curvature. Calculating the stress, we obtain

(4.5) \begin{align} \sigma _{\textit{rr}}^{(\textit{out})} &= -p^{(\textit{ext})} + 2\mu _0\frac {\partial u_r^{(\textit{ext})}}{\partial r} - \frac {2\mu _0 A(z)}{h^2} \nonumber \\ &=-p^{(\textit{ext})} + 2\mu _0\frac {\partial u_r^{(\textit{ext})}}{\partial r} - \frac {2\mu _0}{h}\big[u_z^{(\textit{ext})}(z,h)h' - u_r^{(\textit{ext})}(z,h)\big]. \end{align}

Expanding in the slenderness, we have $u_z^{(\textit{ext})}(z,h) = u_0(z) + O(h^2)$ and $u_r^{(\textit{ext})}(z,h) = -u_0^{\prime}(z)h/2 + O(h^3)$ , using incompressibility. Thus to leading order we finally obtain for the inner pressure $p^{(\textit{in})}\equiv p$

(4.6) \begin{equation} p = \gamma \kappa + p_0(z) + \frac {2\mu _0}{h}(h(z)u_0(z))', \end{equation}

where $u_0(z)$ and $p_0(z)$ are the values of $u_z^{(\textit{ext})}$ and $p^{(\textit{ext})}$ on the axis, respectively. In the interior, in a steady state, the flux is constant

(4.7) \begin{equation} \frac {Q}{\pi } = -\frac {h^4}{8\lambda \mu _0}\frac {{\rm d}p}{{\rm d}z} + h^2 u_0(z). \end{equation}

Thus (4.6) and (4.7) are a closed set of equations for $h(z)$ , provided that $u_0(z),p_0(z)$ , which characterise the outer flow, are provided.

4.2. Similarity solutions of the transition region

We are interested in a local description of the entry into a thread, taken to be at $z_0$ , which is unknown, and around which we expand. As in § 2.1.1, $\sqrt {\mu _0 Q/\gamma }$ is a length scale. This suggests the similarity solution

(4.8) \begin{equation} h = \sqrt {\frac {Q\mu _0}{\pi \gamma }} \lambda ^{1/4}H(\xi ), \quad \xi = \sqrt {\frac {\pi \gamma }{Q\mu _0}}\lambda ^{1/4} (z-z_0), \quad \overline {{\textit{Ca}}} = \frac {u_0(z_0)\mu _0\lambda ^{1/2}}{\gamma }, \end{equation}

where $\overline {{\textit{Ca}}}$ is a local capillary number scaled as suggested by (3.5), (3.7). Since the local length scale vanishes in the limit of $Q\rightarrow 0$ , $u_0(z)$ and $p_0(z)$ can be expanded about $z_0$ , and only $u_0(z_0)$ matters at leading order. Moreover, for small $\lambda$ , the slope scales like $-h' =\alpha \propto \lambda ^{1/2} \ll 1$ , so the profile is flat in this limit, and the slender jet approximation can be applied.

Inserting (4.8) into (4.6) and (4.7), and retaining only the leading terms of order $Q$ , in the limit of small $\lambda$ one finds

(4.9) \begin{equation} 1 = \overline {{\textit{Ca}}} H^2 + \frac {H^2H'}{8} + \frac {\overline {{\textit{Ca}}}}{4}\left (H^2H'^2-H^3H''\right ), \end{equation}

which is the similarity description valid for $\lambda \ll 1$ . For $\xi \rightarrow \infty$ (the thread), (4.9) has a constant solution of the form

(4.10) \begin{equation} H = \frac {1}{\sqrt {\overline {{\textit{Ca}}}}}, \quad \xi \rightarrow \infty . \end{equation}

On the other hand for $\xi \rightarrow -\infty$ , (4.9) allows for two linear solutions of the form $H = -s\xi$ , where

(4.11) \begin{equation} s_{\pm } = \frac {1}{4\overline {{\textit{Ca}}}} \left (1\pm \sqrt {1-(8 \overline {{\textit{Ca}}})^2}\right ), \end{equation}

as long as $\overline { {\textit{Ca}}}\leqslant 1/8$ . Taking $\overline {{\textit{Ca}}}$ as given, there is no free parameter; thus we aim to solve (4.9), with $\overline {{\textit{Ca}}}$ as a parameter. This situation is shown on figure 8(a), for which a slope of $s_- = 1$ is approached. To do this, we reduce the order by putting $f(H) = H'$ , giving

(4.12) \begin{equation} f' = \frac {1}{\textit{fH}}\left (\frac {f}{2\overline {{\textit{Ca}}}}+4+f^2\right ) - \frac {4}{\overline {{\textit{Ca}}}f{\kern-1.5pt}H^{3}} = \frac {1}{\textit{fH}}\left (f+s_-\right )\left (f+s_+\right ) - \frac {4}{\overline {{\textit{Ca}}}f{\kern-1.5pt}H^{3}}. \end{equation}

To find the profile $H(\xi )$ , we solve simultaneously $\xi '(H) = 1/f$ ; the constant of integration is an arbitrary shift of the origin, which we chose as the point of maximum curvature. We solve (4.12) starting from the neighbourhood of $H = H_m = 1/\sqrt {\overline {{\textit{Ca}}}}$

(4.13) \begin{equation} H = H_m + \delta , \quad f = -\frac {-\sqrt {\overline {{\textit{Ca}}}} + \sqrt {128 \overline {{\textit{Ca}}}^3} + \overline {{\textit{Ca}}}}{4\overline {{\textit{Ca}}}}\delta ,\end{equation}

for small $\delta$ . As a result, the solution is determined uniquely.

In figure 7 we illustrate how only the smaller root $s_-\lt s_+$ is selected, it being approached by $-{{\kern-1pt}f}$ from below. Namely, the second contribution on the right of (4.12) is always positive, while the first vanishes as $s_-$ is crossed. Thus $f'$ is positive, so $-{{\kern-1pt}f}$ can only decrease, making it impossible to cross. On the other hand, for $\overline {{\textit{Ca}}} \gt 1/8$ there is no more linear solution. Instead, the leading balance in (4.12) is now $f' \approx f/H$ , because the term $f^2$ in brackets on the left dominates. Thus $f = f_0 H$ , and integrating we have $H = H_0 e^{f_0\xi }$ , the profile is increasing exponentially (see figure 7). Consequently, despite the fact that the approach to this behaviour is extremely slow, so that it looks like a linear profile with a slowly varying slope, the profile would never reach the $Q\rightarrow 0$ limit with arbitrary but finite boundary conditions, a limit that is conceptually achievable by the conical profile.

Figure 7.

Non-dimensional scaled slope $-f$ of the cone-jet profile, for different $\overline {{\textit{Ca}}}$ values and $s_-$ . The black dashed line underlines the $\overline {{\textit{Ca}}}=1/8$ case.

So far, $\overline {{\textit{Ca}}}$ is a parameter which can take any value, depending on the position of $z_0$ . Indeed, since $u(z)$ is expected to vary along the axis, $\overline {{\textit{Ca}}}$ can in principle take any value. It must adjust itself as part of matching of the local profile to the global solution. One possibility is that the solution adjusts itself such that $\overline {{\textit{Ca}}} \approx 1/8$ . Intuitively, in a regime where the flow strength is increasing with $z$ , this might be understood by the following argument: if $z_0$ moves forward, $\overline {{\textit{Ca}}}$ increases, pushing $z_0$ further forward. This will continue up to reaching a point where the similarity (conical) solution can no longer be matched, which is only possible as long as the profile is linear. This suggests that the similarity solution is tuned to its critical value. This suggests that $z_0$ is selected such that

(4.14) \begin{equation} \frac {u_0(z_0)}{v_c} \approx \frac {1}{8\sqrt {\lambda }}, \end{equation}

in the limit of small $Q$ , which implies that the local capillary number $\overline {\textit{Ca}}$ is maximised for small $\lambda$ , and in turn the local slope of the conical region is $h' =-2\sqrt {\lambda }$ , in coincidence with (3.7): it is equal to the cone angle $\alpha _m$ that maximises the local speed on the cone surface for the initial exact analytical solution (3.1)–(3.9). Since this slope is small in the limit $\lambda \rightarrow 0$ , this means that the slender-body approximation is justified in this limit.

In summary, compared with the cone-jet solution, the physical framework of this slender-body solution implies the following.

1. (i) This solution has two independent variables: the viscosity ratio $\lambda$ and the local capillary number defined in (4.8), where $u_0(z_0)$ is the velocity of the external flow on the axis at the location $z=z_0$ , in the absence of a jet as a first approximation: it measures the strength of the external flow.

2. (ii) The cone angle $\alpha$ of the cone-jet solution is related to $\overline {\textit{Ca}}$ of the slender-body solution by the expression

   (4.15) \begin{equation} \alpha =\frac {1-\sqrt {1-(8\overline {\textit{Ca}})^2}}{4\overline {\textit{Ca}}}\lambda ^{1/2}. \end{equation}

3. (iii) The latter is not restricted except for that there is a maximum value $\overline {\textit{Ca}}_{cr}=1/8$ to have a conical flow at infinity. This $\overline {\textit{Ca}}_{cr}$ leads to a slope that coincides with the angle of the cone-jet solution for which the velocity on the cone surface is maximised: $\alpha _m = 2 \lambda ^{1/2}$ .

4. (iv) It resolves the complete cone-jet intermediate region. It also provides a universal solution in terms of rescaled variables that involve the properties of the liquid, the liquid flow rate and the local strength of the external flow. Naturally, this strength can be related to the surface velocity in the cone through a universal constant of the order unity that depends on the external velocity field used.

5. (v) The flow rate $Q$ can be obtained in terms of the value of $\overline {\textit{Ca}}$ , the liquid properties and the velocity of the external flow. Like in the cone-jet solution, if one uses $l_0=\sqrt {\mu _0 Q/\gamma }$ as the length scale, $Q=1$ .

5.1. A local comparison of cone-jet and slender-body solutions

First, to illustrate how the cone-jet solution comes from a regularisation of the solution (3.1)–(3.2), figure 8 shows both the regularised and the exact analytical solution with a singularity at the axis $\theta =\pi$ , using $\lambda =0.01$ and $R=2000$ . Figure 8(a) shows the radial (black line) and angular (red line) velocities $u_R$ and $u_\theta$ for $R \gg 1$ , for $\chi = 2.5$ and $\alpha = 0.0632$ . In figure 8(b), the asymptotic behaviour of $u_R$ is represented for $\theta$ ø $\pi$ ; here, $\chi = \pi /2$ and $\alpha = 0.1481$ . To obtain these solutions, we have used the value of the cone angle $\alpha$ that minimises the differences between the values of $R_J$ obtained when the matching errors for $u_R$ and $u_\theta$ are set to zero at the matching angle $\theta =\chi$ (see Appendix B).

Figure 8.

(a) Radial and angular velocities $u_R$ and $u_\theta$ for the analytical solution. The main plot show the solution regularised at the axis, while the inset shows the ‘exact’ solution (3.1)–(3.2) with a noticeable singularity at the axis ( $\theta =\pi$ ). $\chi = 2.5$ , $\alpha = 0.0632$ (b) Asymptotic behaviour of the radial velocity as $\theta \to \pi$ , for $\chi = \pi /2$ , $\alpha = 0.1481$ , showing the initial solution (3.1)–(3.2) with a singularity at $\theta =\pi$ (dashed line: observe the non-zero slope) and the regularised solution (3.11) (continuous line). Here, $R=2000,\, \lambda =0.01$ in both (a) and (b). Here, the matching errors of $u_R$ ( $\epsilon _2=0$ , see Appendix B) and $u_\theta$ ( $\epsilon _1=0$ ) at $\theta =\chi$ are zero using expression (B1) for $R_{J,2}$ and $R_{J,1}$ respectively.

Secondly, in figure 9 we compare the numerically resolved cone-jet (where all $R$ values are allowed) and the slender-body solutions for $\overline {\textit{Ca}}\lt 1/8$ , which corresponds to $\alpha \lt \alpha _m$ . The relationship between $\overline {\textit{Ca}}$ and $\alpha$ is given by (4.15). After rescaling, the cone-jet solution according to the definitions (4.8), the agreement is nearly perfect, as expected since $\lambda \ll 1$ .

Figure 9.

Comparison of the numerical cone-jet (dashed lines) and the slender-body (continuous lines) solutions of (4.9) for two $\overline {\textit{Ca}} \lt 1/8$ values (corresponding to $\alpha \lt \alpha _m$ ); $\alpha$ and $\overline {\textit{Ca}}$ are related by (4.15). Here, $\lambda =0.0025$ . (a) Plot of the cone-jet profiles of the solutions. (b) The cone-jet slopes of the profiles.

In general, while the slender-body theory assumes that a consistent error of the order of $O(\alpha ^n)$ with $\alpha \ll 1$ and $ n \gt 1$ occurs and affects all variables everywhere, the cone-jet solution assumes that the errors are restricted to the region where two solutions of the external flow are matched, imposing exact conditions at the cone and the jet. In contrast, the slender-body theory uses a physically meaningful parameter $\overline {\textit{Ca}}$ , resolves the cone-jet region, reduces the problem to a universal scaling structure of the cone-jet geometry and indicates the maximum value of the local flow strength for which the flow can be conical for a given viscosity ratio $\lambda$ .

5.2. Comparison of the slender-body solution with a complete numerical solution

In this section, we provide evidence for the existence of a critical $\overline {\textit{Ca}}_{cr}$ , as proposed at end of § 4.2. We propose that, for increasing strengths of the external flow over a certain value which depends on the nature of that flow, the local flow is tuned to its critical value. To this end, we use the extensional flow solution used in Rubio et al. (Reference Rubio, Montanero, Eggers and Herrada2024); see figure 10 for details of the definitions used to perform the comparison. To provide the most consistent comparison, we define a local capillary

(5.1) \begin{equation} \overline {\textit{Ca}}_2=\frac {\mu _0 \lambda ^{1/2} u(z_0)}{\gamma }, \end{equation}

calculated with the velocity of the flow $u(z_0)$ at the interface of the numerical cone-jet solution, under the same extensional flow and at the same axial location $z=z_0$ (i.e. the location of the stagnation point of the inner flow) as the definition of $\overline {\textit{Ca}}$ of the slender-body solution, (4.8).

Figure 10.

Shape and streamlines of a cone-jet meniscus subject to a purely extensional flow (Rubio et al. Reference Rubio, Montanero, Eggers and Herrada2024). The definition of the local capillary number $\overline {\textit{Ca}}_2$ is indicated, where instead of the velocity of the external flow at the stagnation point $z_0$ on the axis, $u_0(z_0)$ , we use the velocity of the interface of the cone jet at the same $z_0$ , i.e. $u(z_0)$ . The magenta arrow indicates the value of the velocity of the external flow at the axial location of the stagnation point of the inner flow, in the absence of the latter, necessary to calculate $\overline {\textit{Ca}}$ as defined, while the yellow arrow indicates the value of the velocity on the interface at the axial location $z=z_0$ to calculate $\overline {\textit{Ca}}_2$ .

To perform the comparison, we have used three values of the macroscopic capillary number $C=U \mu _0/\gamma$ of the extensional flow (where $U$ is the external characteristic velocity, see Rubio et al. (Reference Rubio, Montanero, Eggers and Herrada2024)), namely $C=0.09$ , $0.11$ and $0.3$ . To find the values of $\overline {\textit{Ca}}$ used in the slender-body description, we extrapolate $\overline {\textit{Ca}}_2$ as found from full numerical solutions for successively smaller $Q$ and compute the difference $\overline {\textit{Ca}}_2-\overline {CA}$ , as shown in figure 11. In each case, $\overline {\textit{Ca}}_2$ converges like a power law toward a limiting value. According to the convergence observed, that value is precisely $\overline {\textit{Ca}}$ . However, the power-law exponent of that convergence decreases with increasing $C$ , indicating a progressively slower convergence. Given the $\overline {\textit{Ca}}$ -value thus found, the convergence of the numerical solutions to the slender-body theory for $Q\rightarrow 0$ improves as the $C$ value decreases.

Figure 11.

Convergence of the ‘external’ local capillary number $\overline {\textit{Ca}}_2$ of the numerical solution to the slender-body capillary number $\overline {\textit{Ca}}$ as $Q\rightarrow 0$ , for the three cases, $C=0.09$ , 0.11 and 0.3, analysed.

Figure 11 illustrates the convergence of the local cone-jet transition from a complete numerical solution of the flow conditions of figure 10 to the solution provided by the slender-body theory. The value of $\overline {\textit{Ca}}$ is obtained by matching the value of the rescaled profiles and their first and second derivatives at $z=z_0$ , with the numerical solution in the case of the lowest possible flow rate value $Q$ achievable by our numerical method. Intriguingly, the local solution around $z=z_0$ appears to be below $\overline {\textit{Ca}}_{cr}$ for the three values of $C$ used in this study, although the convergence is slower as $C$ increases, as shown in figure 12.

Figure 12.

Normalised cone-jet shapes $H(\xi )$ and their slopes $H'(\xi )$ for $\lambda = 0.025$ and three different $C$ values: (a) $0.09$ , (b) $0.11$ and (c) $0.3$ . Black lines correspond to the slender-body theory, while coloured lines correspond to the numerical case with actual boundary conditions and different flow rates (Rubio et al. Reference Rubio, Montanero, Eggers and Herrada2024).

Our results suggest that when the external capillary number is small, the rescaled macroscopic solution converges relatively rapidly to the slender-body solution at large scales as $Q$ decreases. However, as the external flow strength increases, the local flow appears to approach $\overline {\textit{Ca}}_{cr}=1/8$ from below, and the corresponding local angle $\alpha$ of the cone tends to $\alpha _m=2\lambda ^{1/2}$ , but the approach of the numerical solution to the macroscopic cone angle $\alpha$ eventually seems to demand extremely small values of $Q$ .