Viscous Withdrawal of Miscible Liquid Layers

In viscous withdrawal, a converging flow imposed in an upper layer of viscous liquid entrains liquid from a lower, stably stratified layer. Using the idea that a thin tendril is entrained by a local straining flow, we propose a scaling law for the volume flux of liquid entrained from miscible liquid layers. A long-wavelength model including only local information about the withdrawal flow is degenerate, with multiple tendril solutions for one withdrawal condition. Including information about the global geometry of the withdrawal flow removes the degeneracy while introducing only a logarithmic dependence on the global flow parameters into the scaling law.

Figure 1

Axisymmetric tendrils entrained by a warmer, upwards flow in a 2-layer thermal convection experiment with miscible oils. Convection is driven by heating the bottom of the tank and cooling the top. The lower fluid is less viscous than the upper, and the spacing between tendrils is $O(10\mathrm{cm})$. The dark vertical line on the left is a thermocouple [13].

Figure 2

Sketch of surface $R(z)$ and flow streamlines for an axisymmetric withdrawal flow in the upper liquid layer entraining a thin, cylindrical tendril from a deep lower layer.

Figure 3

Rescaled tendril solutions at $S/{\ell }_z=15$, ${\mu }_0/\mu =0.1$, $c_0=3.1$ with exponents $\alpha =1.0$, 0.5, 0.25, 0.052 (top to bottom). Inset shows tendrils have different power-law divergences.

Figure 4

Rescaled tendril solutions $R(z)$ for $c_0=3.4$, 3.1, and 2.9 (solid lines, top to bottom), together with the large-scale deflection solution $R_I(z)$ (dotted line) for $S/{\ell }_z=15$ and ${\mu }_0/\mu =0.1$. Only $c_0=3.1$ allows the two shapes to merge smoothly. Inset shows $c_0$ increases logarithmically with $S/({\ell }_z\sqrt{{\mu }_0/\mu })$.