The effect of temperature-dependent viscosity on the pressure drop in narrow channel flows

We investigate theoretically the effect of temperature-dependent viscosity on the pressure drop-flow rate relation in narrow pipe flows. Although seemingly a classical topic, we are not aware of previous results of this type. Different temperature boundary conditions at the wall alter the viscosity field under the same flow conditions, and we elucidate how this external heating affects the pressure drop across the pipe. We first use analytical and similarity solution methods to calculate the temperature distribution under constant temperature and constant heat flux boundary conditions, as well as assumed linear and other imposed polynomial temperature versus distance (along the flow) boundary conditions at the wall. We then employ the lubrication approximation and the Lorenz reciprocal theorem to derive an expression for the pressure drop across the channel for a temperature-dependent viscosity. Assuming a small fractional change in viscosity with temperature, we linearize the viscosity field and obtain an analytical expression for the pressure drop for a given flow rate. The results are reported as a function of the effective Peclet number for each boundary condition and the numerical results are compared with analytical predictions in the low and high Peclet number limits.