The effect of thin film variations and transverse curvature on the shape of fingers in a Hele–Shaw cell

The steady-state shape of a finger penetrating into a viscous fluid that fills the gap between two closely spaced parallel plates is examined. Boundary conditions that take into account variations in the thickness of the thin film and both transverse curvature (across the gap) and lateral curvature (along the interface edge) are applied at the leading edge of the interface. These interface conditions, derived from local solutions in the vicinity of the interface edge, depend on μUn/T, where μ is the viscosity of the original fluid, Un is the normal velocity of the interface edge, and T is the interfacial tension. They also depend on ε/R, where ε=b/a≪1 is the ratio of gap width to cell width and aR is the lateral radius of curvature. The problem is solved by conformally mapping the domain to a circle. By expanding the solution in terms of analytic functions and satisfying the boundary conditions, the shape of the interface edge and finger width λ are determined. The agreement between numerical and experimental results is good and an improvement over previous models. The relaxation of the film thickness is also discussed.