Evaluation of viscous fingers width in Hele-Shaw flows

Abstract

Abstract High viscosity fluid displacement from a Hele-Shaw cell by less viscous one is considered. The cell is a pair of parallel plates separated by a small gap. While viscous fluid is displaced from a cell, the displacement front is unstable (Saffman and Taylor, 1958). The less viscous fluid penetrates through the higher viscous one forming channels called viscous fingers. It is known that the surface tension determines the finger width for the condition of immiscible fluid displacement and diffusion controls the process in the case of miscible fluids. The modified capillary number and fluid viscosity ratio are the criteria of the flows self-similarity in the first case and the Peclet number and the viscosity ratio serve as the criteria for the later one. It is shown in the paper that there is one more factor influencing the finger width. The viscous forces, acting in the flow plane, control the width of the viscous fingers if there is no surface tension at the phase interface and the fluid diffusion has no time for manifestation. It occurs under a rapid displacement of miscible fluids when Pe→∞. The criterion of self-similarity for the flows of such a kind is suggested in the paper.