The dimensionless horizontal distance is

, where

is the length of the plates.

The dimensionless vertical distance is

.

The dimensionless temperature is

.

The temperature profile of the two liquids can be obtained by solving a single partial differential equation with two different sets of parameters, one for each fluid:

,

where

is the thermal diffusivity

,

is the fluid velocity, and

,

and

are the fluid thermal conductivity, density and heat capacity, respectively.

Let

be the distance between the plates,

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 properties of the lower and upper fluid, respectively.

An 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]:

,

.

The dimensionless velocity is

.

Here

is the horizontal pressure gradient; the average (cup) temperatures of the two fluids are:

,

.

These equations are solved with the built-in Mathematica function

NDSolve, and plots of the temperature and velocity contours, as well as the average temperatures, are shown.