Liquid interfaces in viscous straining flows ... - Itai Cohen Group

Liquid interfaces in viscous straining flows 193Figure 11(b) shows what happens when the condition in the lower layer ischanged more drastically. In this set of calculations, a thin layer of the lowerliquid overlies a solid plate. The plate radius a 2 is used as a characteristiclength scale in the non-dimensionalization. The liquid is assumed to wet the solidsubstrate so that the plate is covered by liquid from the lower layer at all Q.The formation of a hump causes most of the liquid in the layer to drain into thehump, leaving only a very thin layer covering the rest of the solid plate. In theselective withdrawal regime, the withdrawal flow requires liquid in the lower layerto recirculate, moving along the interface towards the hump and then along thelower solid surface away from the hump. When the lower layer is thin, the viscousresistance associated with the steady-state recirculation is amplified dramatically. Inthis realization, the stress balance on the steady-state interface is strongly perturbedfrom that obtained using the simplified withdrawal model described in §§4.1 and4.2.For two-layer withdrawal with a thin layer of the lower liquid, the closed surfaceused in the boundary-integral formulation consists of the liquid interface S I andthe bottom wall of the container S b . The equation for the velocity on the interfacehas the same form as (4.3) except that the terms associated with the sidewall surfaceS side are absent. The normal stress on the solid surface σ b satisfies (4.4), withoutthe sidewall contribution.Comparing against results from the simple model and results for withdrawalfrom a finite container, we can see that reducing the thickness of the lower layerhas an effect on the steady-state shape of the interface obtained. The hump shapeis significantly more conical, and the flattening of the interface at large r resultsprimarily from volume conservation, rather than pinning at a 2 or the decay of theimposed withdrawal flow away from the sink. However, as is evident from figures 12and 13, the qualitative features are unchanged. The transition still corresponds to asaddle-node bifurcation. The evolution of the hump height h retains a logarithmiccoupling to the hump curvature κ. The main difference between withdrawal dynamicsobtained in the simplified model and the withdrawal dynamics found for a verythin layer overlying a solid plate is a shift in the slope of the rescaled κ versus hcurve.We have also conducted a series of calculations that include the effect of hydrostaticpressure explicitly. This requires that we change the form of the normal stressjump [n · σ ] + − across the liquid interface S I in the J-integral of (3.11) and (3.13)to[γ 2˜κ − ρ g · y]n, (4.5)where ρ is the density difference between the two layers, g is the acceleration due togravity and y is the location of a point on the liquid interface. Choosing the capillarylength scale l γ to be less than the pinning radius a allows the interface to flatten outat large radial distances due to stratification, as occurs in the experiment, instead ofdue to the decay of the withdrawal flow or due to surface tension effects. Results forl γ /a =0.1 andS =0.2 show that introducing stratification explicitly results in slightqualitative changes in the hump shape, but the same trend in the rescaled κ versus hcurve (figure 13).Finally, to assess the influence of details in the withdrawal flow geometry on thelogarithmic coupling (4.2), we changed the imposed withdrawal flow from a sinkflow (3.2), to that associated with a vortex ring of size a ω = a and strength ω 0 .The