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
; in the region
, the parallel plates do not permit mass transfer.
The concentration profile is:
is the velocity,
is the concentration,
is the diffusivity,
is the horizontal distance along the plates and
is the vertical distance between the plates.
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:
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 :
is the horizontal pressure gradient 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
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.