Heat Transfer of Couette Flow in Micro-channels: an Analytical Model of Seals

Analytical solutions of the temperature profile of a fluid in a Couette flow between two parallel plates are reported. The upper plate moves in the direction of the flow with a constant velocity and the lower plate is stationary, to simulate the conditions of a stationary seal mounted on a rotating shaft. The energy equation is solved for steady-state, laminar flow, taking into account the viscous dissipation effect of the flow due to the high shear rate of the fluid and rotational speeds being studied. An imposed temperature gradient between the two plates is used as well. The solutions use different boundary conditions of no-slip condition and a first-order slip to study the effect on the temperature profile as a result of the first-order slip condition. Brinkman number and temperature difference between the plates is also varied to see how it influences the temperature profile in the gap between the seal and the shaft. These parameters and boundary conditions influence this temperature profile. Results illustrate how slip boundary conditions at the wall show a control of the temperature rise in the annuli, resulting in reduced energy accumulation within the annuli.