Fingering instability of a sheet of yield-stress fluid

We study the fingering instability that occurs at the contact line of a thin sheet of a yield-stress fluid flowing down an incline. We derive an expression for the wavelength of the finger pattern as a function of inclination angle for a Herschel-Bulkley fluid. The wavelength is predicted to diverge at a finite angle which is related to the yield stress of the fluid. We also measure the wavelength of the finger pattern with suspensions of bentonite clay in water. Our experimental results agree well with the theoretical prediction.