Extension of the lubrication theory for arbitrary wall shape: An asymptotic analysis

We extend the lubrication approximation for a viscous flow in two-dimensional channels with arbitrary shape functions and moderate aspect ratio. The higher-order model is obtained following an asymptotic analysis. Velocity and pressure profiles for the approximated model are given analytically and involve the derivatives of the shape functions of the walls up to the second order. Comparisons with full-scale simulations are given and show good agreement as well as improvements from the classical standard lubrication approximation.