Action of viscous stresses on the Young–Laplace equation in Hele-Shaw flows: A gap-averaged calculation

The Saffman–Taylor (or, viscous fingering) instability arises when a less viscous fluid displaces a more viscous one in the narrow gap of a Hele-Shaw cell. The dynamics of the fluid–fluid interface is usually described by a set of gap-averaged equations, including Darcy’s law and fluid incompressibily, supported by the pressure jump (Young–Laplace equation) and kinematic boundary conditions. Over the past two decades, various research groups have studied the influence of viscous normal stresses on the Young–Laplace equation on the development of radial fingering. However, in these works, the contribution of viscous normal stresses is included through the insertion of a legitimately two-dimensional term into the pressure jump condition. As a result, the significant variation of the fluid velocity along the direction perpendicular to the Hele-Shaw plates is neglected. In line with Hele-Shaw flow approximations, and analogous to the derivation of Darcy’s law, we introduce viscous stresses in the Young–Laplace equation through a gap-averaged calculation of the three-dimensional viscous stress tensor. We then compute the contributions of these gap-averaged viscous stresses in both the linear stability and early nonlinear analyses of the interface perturbation evolution. Our findings indicate that this approach leads to a slowdown in finger growth, and an intensification of typical tip-splitting events.