On the Fingering of Slow Immiscible, Viscous Liquid-Liquid Displacements

We assume that an Interface between two immiscible, viscous fluids of unequal mobility flowing between glass plates is fingering in the manner observed experimentally by Chuoke, van blurs and van der Poel. Chuoke, et al, analyzed this fingering effect by means of small-amplitude perturbation theory. The small-amplitude perturbation theory enables one to determine the rate of growth of small-amplitude sinusoidal fingers of any wave length. The wavelength, , with maximum rate of growth can be determined and is conjectured to be the most probable wave length to be observed in an actual physical system with fingers of large amplitude. Reasonable agreement between this conjecture and experimental observations is obtained by Chuoke, et al. The small-amplitude perturbation theory essentially exhausts itself at this stage. It cannot be used to predict the rate of growth or the shapes of large-amplitude fingers. In the present paper, we develop a different approximate theory for the growth of fingers. The approximations involved are similar to those used in first-order shallow-water theory for water waves. This approach enables us to determine the transient behavior of large amplitude fingers. Unfortunately, the approximations which we must make for mathematical expediency are not valid approximations everywhere along the fingering surface, and so the results we obtain are not quantitative. (The same is true in shallow-water theory when this theory is applied to the breaks of waves on a beach. However, this does not prevent people from applying the theory anyway because it is so difficult to get results by other, more accurate methods.)It is of interest that the finger shapes that we determine here do have some of the features observable in the experiments of Chuoke, et al.