Thermomechanical coupling in a Couette flow between parallel plates

For planar Couette flow, the rate of entropy production of an incompressible Newtonian fluid is 

For the case of (—dP/cbc) = 0, the velocity is linear across the fluid. For a negative pressure drop the velocity is positive, and for a pressure increase, the velocity can become negative leading to backflow. At the point of reversal dv/dy = 0 aty = 0. This occurs when — dP/dx = —2 iVi/H1. The velocity gradient dv/dy from Eq. (4.25) is 

The energy equation for laminar and hydrodynamically developed flow is 

Br is the Brinkman number, which is a ratio of the viscous heating to the heat conducted through the gap of Couette device. From Eq. (4.28), we obtain the temperature gradient 

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