A new prediction of wavelength selection in radial viscous fingering involving normal and tangential stresses

We reconsider the radial Saffman-Taylor instability when a fluid injected from a point source displaces another fluid of higher viscosity in a Hele-Shaw cell where the fluids are confined between two neighboring flat plates. The advancing fluid front is unstable and forms fingers along the circumference. The so-called Brinkman equation is used to describe the flow field which also takes into account viscous stresses in the plane of the confining plates and unlike the Darcy equation not only viscous stresses due to the confining plates. We show why in-plane stresses cannot always be neglected and how they appear naturally in the potential flow problem. The dispersion relation obtained with the Brinkman equation agrees better with the experimental results than the classical linear stability analysis of radial fingering in Hele-Shaw cells that uses Darcy's law as a model for the fluid motion.