Effect of inertia on pressure drop in lubrication flows through channels with varying walls

We investigate the effect of inertia on steady, incompressible Newtonian flow in confined symmetric channels with variable wall geometry. The channel rigid walls vary smoothly as a function of the distance from the inlet cross section, described by a known and predefined shape function. Using the classic lubrication theory and including the fluid inertia, we derive a non-linear partial differential equation for the streamfunction in two mapped spatial directions. The final equation is solved using three approaches: asymptotically in series form with respect to the Reynolds number and postprocessed with techniques to improve the accuracy and convergence of the series; fully spectrally in both spatial directions using appropriate orthogonal functions; and numerically, using a mixed spectral-finite difference method. All methods provide highly accurate results, with the fully spectral and numerical methods resolving the streamfunction nearly to machine precision. For linearly converging channels, hyperbolically converging channels, and periodic channels, we derive results focusing on the pressure drop required to maintain a constant flow rate. The effects of the geometrical parameter entering each shape function and the Reynolds number are studied and discussed. In all cases, the results show that the average pressure drop increases with Reynolds number, contraction ratio, or amplitude of wall variation, primarily due to the growing influence of inertia. This effect is most pronounced in the linearly converging channel and least evident in the periodic channel. For the linearly varying channel, excellent agreement is observed between the full solution(s) and the similarity solution, which exists in this case. Finally, for the periodically varying channel, the mechanical energy balance reveals that the average pressure drop increases solely due to viscous dissipation, with inertia playing no role.