The effect of a temperature-dependent viscosity on pressure drop in narrow, converging channel flows

We investigate theoretically the influence of a temperature-dependent viscosity on the pressure drop in channels with converged heated walls, for cases where the Reynolds number is small. We anticipate this to be particularly applicable wherever there are highly viscous flows through heated nozzles such as in 3D printing. We employ the Lorentz reciprocal theorem to derive an expression for pressure drop across an arbitrary geometry for a viscosity field that depends on temperature. Assuming the fractional change in viscosity with temperature is small, we linearize the viscosity field using perturbation techniques. Also, for both the momentum and energy equations we apply the lubrication approximation, which we expect to be typically appropriate for flows where the maximum channel radius is much less than the channel length. We consider linear, quadratic, and hyperbolic converging channels at different contraction ratios to elucidate how the wall shape coupled with the wall heating influence the reduction in the pressure drop. We use numerical and similarity solution methods to solve for the temperature distribution under constant temperature and constant heat flux boundary conditions for each respective geometry. We report the results as a function of the effective Peclet number for each geometry and compare the numerical results with analytical predictions in the low and high Peclet number limits.