Sidewall Effects in the Smoothing of an Initial Discontinuity of Concentration

The velocity field of a binary mixture of incompressible miscible liquids is non-solenoidal when the densities of the two liquids are different. If the mixture density is linear in the volume fraction, as in the case of simple (ideal) mixtures or very nearly for glycerin and water, then the velocity can be decomposed into a solenoidal and an expansion part. In the context of this theory, we derive a new solution which describes the smoothing of an initial plane discontinuity in concentration across a channel bounded by side walls. The requirement that the velocity vanishes on the side wall introduces a different initial discontinuity not present in the solenoidal theory. The problem may be reduced to a partial differential equation in two similarity variables one for the smoothing of a concentration discontinuity without sidewalls and the other for the smoothing the velocity discontinuity at the sidewall. The similarity equations are solved explicitly in a special case.