Inertial corrections to the Darcy law in a Hele–Shaw cell

Abstract

This note presents a derivation of the appropriate inertial corrections to the Darcy law in a Hele–Shaw cell based on a perturbative method and a polynomial approximation to the velocity field. The obtained equation is optimal in the sense that every method of weighted residuals will converge to it as the number of test functions is increased. A good agreement with the study of the shear instability in a Hele–Shaw cell at low Reynolds number is found.