Flow and instability of a viscous current down a slope

Abstract

If viscous fluid is released on a horizontal surface it rapidly takes up a circular plan form as it spreads. This form is observed^1,2 to be stable to any small disturbances which are initiated on the front due, for example, to irregularities in the horizontal surface or to chance perturbations. Alternatively, if some fluid is released onto a sloping surface—for example, some liquid detergent on a slanted plate—a quite different plan form occurs. One, two or more extended regions of fluid develop downslope, as shown in Fig. 1a, b. A situation intermediate between these two is now discussed. Consider a broad band of viscous fluid, uniform in depth across a slope, released so as to flow down a constant slope. By following the motion, which is initially independent of the cross-slope coordinate, the speed of advance and the depth of the flow before it breaks up into a series of waves of ever increasing amplitude can be determined. I present an expression for the wavelength of the front, which is determined by surface tension and is independent of the coefficient of viscosity.