Effect of temperature-dependent viscosity on pressure drop in axisymmetric channel flows

We investigate theoretically the influence of a temperature-dependent viscosity on the pressure drop versus flow rate relationship in pipe flows for cases where the Reynolds number is small, as expected for printing and other flows of highly viscous fluids. By applying different temperature boundary conditions at the wall, the viscosity field is altered under the same flow conditions and thus we can compare how this external heating affects the pressure drop along the length of the pipe. We use analytical and similarity-solution methods to solve for 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. Also, for the momentum and energy equations we use the lubrication and boundary-layer approximations, respectively, which we expect to be typically appropriate for flows where the pipe radius is much less than the pipe length. The reciprocal theorem is used to derive an expression for the pressure drop across the channel for a viscosity field that depends on temperature and spatially varies across and along the flow. Assuming the fractional change in viscosity with temperature is small, we arrive at 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.