Browsing by Author "Tanveer, S."
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- Analytic theory for the selection of a two-dimensional needle crystal at arbitrary Péclet numberTanveer, S. (American Physical Society, 1989-10)An accurate analytic theory is presented for the velocity selection of a two-dimensional needle crystal for arbitrary Péclet number for small values of the surface-tension parameter. The velocity selection is caused by the effect of transcendentally small terms that are determined by analytic continuation to the complex plane and analysis of nonlinear equations. The work supports the general conclusion of previous small-Péclet-number analytical results of other investigators, although there are some discrepancies in details. It also addresses questions raised by a recent investigator on the validity of selection theory owing to assumptions made on shape corrections at large distances from the tip.
- The effect of nonzero viscosity ratio on the stability of fingers and bubbles in a Hele-Shaw cellTanveer, S.; Saffman, P. G. (AIP Publishing, 1988-11)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 2 4 5, 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 3 0, 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.
- The effect of surface tension on the shape of a Hele-Shaw cell bubbleTanveer, S. (AIP Publishing, 1986-11)Numerical and asymptotic solutions are found for the steady motion of a symmetrical bubble through a parallel‐sided channel in a Hele–Shaw cell containing a viscous liquid. The degeneracy of the Taylor–Saffman zero surface‐tension solution is shown to be removed by the effect of surface tension. An apparent contradiction between numerics and perturbation arises here as it does for the finger. This contradiction is resolved analytically for small bubbles and is shown to be the result of exponentially small terms. Numerical results suggest that this is true for bubbles of arbitrary size. The limit of infinite surface tension is also analyzed.