In this thesis, we study the behavior of viscous flow past bodies of different shapes. In Chapter 2, we construct a boundary-fitted, numerical grid around a rigid spheroid of various aspect ratios and solve numerically the Navier-Stokes equations in steady, axisymmetric form at various Reynolds numbers. In addition, we use these steady solutions as a base flow and perform a linear stability analysis to determine the critical Reynolds numbers above which the base flow becomes unstable. We are able to confirm the results of Natarajan and Acrivos [26] and extend them to more generalized body shapes.; In Chapter 3, we solve the Navier-Stokes equations to investigate flows past an oblate ellipsoidal bubble of fixed shape, which is characterized by a free-slip boundary condition. We then compare our results with previous results by Dandy and Leal [6] and Blanco and Magnaudet [4] and use the computed steady solutions as the base flow to perform a linear stability analysis. We show that even with a free-slip boundary condition, if the body is sufficiently oblate, the flow can become unstable in a manner similar to that of flows past rigid bodies.; In Chapter 4, we develop an alternative numerical method to compute steady flows past a deforming, axisymmetric bubble. A newly developed conformal grid generation method is applied. We show that our results are in good agreement with those of Ryskin and Leal [34], [35] and then extend some of their results to higher Reynolds number.; In Chapter 5, we modify the method developed in Chapter 4 to compute steady flows past a symmetric, two-dimensional bubble. We show that the bubble deforms to an elliptical shape and that a wake can develop if the deformation of the bubble is sufficiently large.
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