The initial development of a jet caused by fluid, body and free surface interaction. Part 5. Parasitic capillary waves on an initially horizontal surface

Abstract

In Part 3 of this series of papers (Needham et al., Q. J. Mech. Appl. Maths, vol. 61, 2008, pp. 581–614), we studied the free surface flow generated in a horizontal layer of inviscid fluid when a flat, rigid plate, inclined at an external angle $\unicode[STIX]{x1D6FC}$ to the horizontal, is driven into the fluid with a constant, horizontal acceleration. We found that the most interesting behaviour occurs when $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x03C0}/2$ (the plate leaning into the fluid). When $\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D6FC}_{c}$, with $\unicode[STIX]{x1D6FC}_{c}\approx 102.6^{\circ }$, we were able to find the small-time asymptotic solution structure, and solve the leading-order problem numerically. When $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{c}$, we found numerical evidence that a $120^{\circ }$ corner exists on the free surface, at leading order as $t\rightarrow 0$. For $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$, we could find no numerical solution of the leading-order problem as $t\rightarrow 0$, and hypothesised that the solution does not exist for any $t>0$ for these values of $\unicode[STIX]{x1D6FC}$. At the present time, there is no rigorous proof of this hypothesis. In this paper, we demonstrate that the likely non-existence of a solution for any $t>0$ when $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$ can be reconciled with the physics of the system by including the effect of surface tension in the model. Specifically, we find that for $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$, the solution exists for $0\leqslant t<t_{c}$, with $t_{c}\sim Bo^{-1/(3\unicode[STIX]{x1D6FE}-1)}\unicode[STIX]{x1D70F}_{c}$ and $\unicode[STIX]{x1D70F}_{c}=O(1)$ as $Bo^{-1}\rightarrow 0$, where $Bo$ is the Bond number ($Bo^{-1}$ is the square of the ratio of the capillary length to the fluid depth) and $\unicode[STIX]{x1D6FE}\equiv 1/(1-\unicode[STIX]{x03C0}/4\unicode[STIX]{x1D6FC})$. The solution does not exist for $t\geqslant t_{c}$ due to a topological transition driven by a nonlinear capillary wave, i.e. the free surface pinches off (self-intersects) when $t=t_{c}$. We are also able to compare this asymptotic solution with experimental results which show that, in an experimental case where the contact angle remains approximately constant (the modelling assumption that we make in this paper), the asymptotic solution is in good agreement. In general, the inclusion of surface tension leads to the generation of capillary waves ahead of the wavecrest, which decay as $t$ increases for $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{c}$ but dominate the flow and lead to pinch-off for $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$. These capillary waves are an unsteady analogue of the parasitic capillary waves that can be generated ahead of steadily propagating, periodic capillary–gravity waves, e.g. Lin & Rockwell (J. Fluid Mech., vol. 302, 1995, pp. 29–44).

A dam break driven by a moving source: a simple model for a powder snow avalanche

“…In the similar, single fluid dam-break flow studied in Uddin & Needham (2015), the inclusion of weak surface tension regularised the solution at small times. This was also the case in the problem studied in Billingham et al (2017), where a numerical solution showed that the oscillatory small time solution emerges after a short, surface-tensiondominated transient. Whilst we could do the same here, including surface tension is not really in the spirit of a simplified problem that is meant to be related to powder snow avalanches, for which the small scale behaviour close to the free boundary is better characterised as particle-laden, multiphase turbulence.…”