Theory of bubble tips in strong viscous flows

A free surface, placed in a strong viscous flow (such that viscous forces overwhelm surface tension), often develops ends with very sharp tips. In Courrech du Pont and Eggers [Proc. Natl. Acad. U. S. A. 117, 32238 (2020)] we have shown that the axisymmetric shape of the ends, nondimensionalized by the tip curvature, is governed by a universal similarity solution. The shape of the similarity solution is close to a cone, but whose slope varies with the square root of the logarithmic distance from the tip. Here we develop the calculation of the tip similarity solution to next order, using which we demonstrate matching to previous slender-body analyses, which fail near the tip. This allows us to resolve the long-standing problem, first raised by G. I. Taylor, of finding the global solution of a bubble in a strong hyperbolic flow. We also calculate the tip curvature quantitatively, beyond the scaling behavior of the leading-order solution. Our results are shown to agree in detail with full numerical simulations of the Stokes equation.

Figure 1

On the left, a drop of low-viscosity fluid is stretched in Taylor's “four-roller” machine [3]. Very sharp tips form at the ends, while the drop remains stable. On the right, the air-liquid surface of a container draining from a whole in the bottom [4] (silicone oil, $\eta =60\mathrm{Pa}\mathrm{s}$ and flow rate $q=3.9{10}^{-3}\mathrm{ml}/\mathrm{s}$). In both experimental images the ends are in fact not strictly conical but have an opening angle which depends on the scale of observation.

Figure 2

On the top, the shape $h\left(z\right)$ of an inviscid drop (or bubble) in the flow (3) at $\mathrm{Ca}=0.4393$, as computed with the boundary integral code developed in [15]: (a). The curvature at the tip is ${\kappa }_m=2.54\times {10}^8$ in units of the unperturbed drop radius $R_d$; $z_{\mathrm{tip}}$ is the position of the left tip, and the drop half-length is $\ell =4.976$ in the same units. Panels (b)–(f) on the bottom show the corner of the drop (solid line) compared with the conical approximation (1) of Taylor's theory (dashed line). Starting from the upper left, for each panel we have zoomed in by another factor of 1/100; the local slope of the interface is seen to increase at each step.

Figure 3

Left: The slope $dHd/\zeta \left(l\right)$, rescaled in similarity form, compared with the asymptotics (76), where $l_0=0.1$ was adjusted. Right: The two components of the velocity field $v_z$ (black) and $v_r$ (red), compared with (77) and (78), respectively.

Figure 4

The drop half-length $\ell$ as function of the capillary number $\mathrm{Ca}=G{R}_d\eta /\gamma$ for for a bubble in the flow (3); see Fig. 2. The solid line is the result of our boundary integral simulation [15], the dotted line is the leading-order result [6], the dashed line is (89).

Figure 5

The natural logarithm of the tip curvature as a function of the square of the local capillary number ${\mathrm{Ca}}_{\mathrm{tip}}$ (solid line) for a bubble in the hyperbolic flow (3), as shown for a particular capillary number in Fig. 2. The dotted line is the theoretical prediction (98). For comparison, the dashed line has a constant slope of 4, corresponding to the leading-order result (49).

Figure 6

The interface slope $h^{\prime }\left(z\right)$ of an inviscid drop (or bubble) in the flow (3) at $\mathrm{Ca}=0.4369,R_d=1$ (solid line), as function of the logarithmic distance from the tip $ln((z+\ell )/R_d)$. The slope of the composite solution (99) is the dotted line, with the numerical values $\ell =4.976$ and ${\kappa }_m=1.611\times {10}^8$ as input parameters. The slope of the outer solution (83) is shown as the dashed line, and the result of the leading-order solution [6] is shown as the dot-dashed line.