text: In incompressible flow, the velocity profile is linear because the fluid temperature is constant. When the upper and lower walls are maintained at different temperatures, the velocity profile is more complicated. However, it has an exact implicit solution as shown by C. R. Illingworth in 1950.
  Consider the plane Couette flow with lower wall at rest and the upper wall in motion with constant velocity . Denote fluid properties at the lower wall with subscript  and properties at the upper wall with subscript . The properties and the pressure at the upper wall are prescribed and taken as reference quantities. Let  be the distance between the two walls. The boundary conditions are
  where  is the specific enthalpy and  is the specific heat. Conservation of mass and -momentum requires  everywhere in the flow domain. Conservation of energy and -momentum reduce to
  where  is the wall shear stress. The flow does not depend on the Reynolds number , but rather on the Prandtl number  and the Mach number , where  is the thermal conductivity,  is the speed of sound and  is the specific heat ratio. Introduce the non-dimensional variables
  In terms of these quantities, the solutions are
  where  is the heat transferred per unit time per unit area from the lower wall. Thus  are implicit functions of . One can also write the solution in terms of the recovery temperature  and recovery enthalpy  evaluated at the temperature of an insulated wall i.e., the values of  and  for which . Then the solution is
  If the specific heat is constant, then . When  and , then  and  are constant everywhere, thus recovering the incompressible Couette flow solution. Otherwise, one must know the full temperature dependence of . While there is no simple expression for  that is both accurate and general, there are several approximations for certain materials — see, e.g., temperature dependence of viscosity. When  and , the recovery quantities become unity . For air, the values  are commonly used, and the results for this case are shown in the figure.
  The effects of dissociation and ionization (i.e.,  is not constant) have also been studied; in that case the recovery temperature is reduced by the dissociation of molecules.