The effect of nonzero viscosity ratio on the stability of fingers and bubbles in a Hele–Shaw cell

The linear stability of a steadily moving bubble or a finger in a Hele–Shaw cell is considered in the case when gravity and the ratio between the viscosities of the less and more viscous fluids are nonzero. The effect of gravity is easily incorporated by a transformation of parameters introduced previously by Saffman and Taylor [Proc. R. Soc. London Ser. A 245, 312 (1958)] for the steady flow, which makes the time-dependent flows with and without gravity equivalent. For the nonzero viscosity ratio, the transformation of parameters introduced by Saffman and Taylor also makes steady finger and bubble flows with nonzero and zero viscosity ratios equivalent. However, for the unsteady case, there is no such equivalence and so a complete calculation is carried out to investigate the effect of the nonzero viscosity ratio on the stability of fingers and bubbles. The incorporation of the finite viscosity ratio is found not to qualitatively alter the linear stability features obtained in earlier work for the zero viscosity ratio, although there are quantitative differences in the growth or decay rate of various modes. For any surface tension, numerical calculation suggests that the McLean–Saffman branch of bubbles [Phys. Fluids 30, 651 (1987)] of arbitrary size is stable, whereas all the other branches are unstable. For a small bubble that is circular, the eigenvalues of the stability operator are found explicitly. The previous analytic theory for the stability of the finger in the limit of zero surface tension is extended to include the case of the finite viscosity ratio. It is found that, as in the case of bubbles, the finite viscosity ratio does not alter qualitatively any of the features obtained previously for the zero viscosity ratio