Bubbles and Fingers in Hele-Shaw Cells

The phenomena of viscous fingering in the flow of immiscible fluids in Hele-Shaw cells is discussed. Early work is reviewed and the present state of the theory is examined. It appears that the selection mechanisms by which small surface tension effects choose a particular solution from the continuum of solutions which exist when surface tension is neglected are now basically well understood. There are some uncertainties with respect to anomalously thin fingers, but there is good evidence that three-dimensional effects on the boundary conditions at the interface will resolve these questions and perturbation theories have been developed to handle these cases. Good progress has been made with regard to the stability of fingers, but questions remain with respect to the nature of finite amplitude instabilities. The theory for bubble selection and stability has been carried out but significant disagreements exist here between theory and experiment, which are probably caused by uncertainty about the boundary conditions for a retreating interface