Assume that the properties of the two fluids are unchanged and that their velocity profiles are independent of horizontal location.

In the inlet region

, the liquids are in fully developed laminar flow and have different uniform concentrations

and

; in the region

, the parallel plates do not permit mass transfer.

The concentration profile is:

,

where

is the velocity,

is the concentration,

is the diffusivity,

is the horizontal distance along the plates and

is the vertical distance between the plates.

Let

be the distance between the plates and

be the fraction filled with the lower fluid, with the interface at

. Then the lower plate is at

, the upper plate is at

and the initial and boundary conditions are:

,

,

.

Here

,

where 1 and 2 represent the lower (donor) and upper (acceptor) fluid, respectively.

Analytic solution for fully developed flow of two immiscible fluids between two flat plates when the fluids have arbitrary viscosity ratios and arbitrary flow rate ratios is derived in [1]:

,

.

Here

is the horizontal pressure gradient and

and

are the viscosities of the lower and upper fluids; the average (cup) concentrations of the two fluids are:

,

.

The concentration of the transferred species at the interface is

, where

is Henry's constant. When

, the concentration is continuous across the interface; when

, the concentration of the transferred species is higher in the donor than in the acceptor fluid; and when

, the opposite is true and the donor fluid becomes the acceptor. Here we use

.

These equations are solved with the built-in Wolfram Mathematica function

NDSolve, and plots of the concentration and velocity contours are shown.