I. Numerical Solutions of Steady Viscous Flow Past Spheres and Gas Bubbles. II. Numerical Solution of Singular Endpoint Boundary Value Problems

Citation

Brabston, Donald Campbell, Jr. (1974) I. Numerical Solutions of Steady Viscous Flow Past Spheres and Gas Bubbles. II. Numerical Solution of Singular Endpoint Boundary Value Problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4719-D351. https://resolver.caltech.edu/CaltechETD:etd-08172005-081056

Abstract

In Part I, numerical solutions of the Navier-Stokes equations are given for steady, viscous, incompressible, axisymmetric flow past a rigid and a spherical gas bubble. The problem is formulated in terms of a stream function and the vorticity which are expanded in finite Legendre series. The coefficients in these series satisfy a finite system of ordinary differential equations. A finite-difference scheme is used to solve the system with Newton's method used to solve the nonlinear difference equations. The results agree very well with low and high Reynolds number theories.

In Part II, systems of ordinary differential equations with singular points of the first kind are considered. The singular point may be at either end, at both ends, or in the interior of a finite interval. A two-point linear system of boundary conditions is imposed at the endpoints. A theory is developed stating the conditions under which a unique solution will exist A numerical method is developed for solving these problems. In this method, a series solution about the singular point is matched to a finite difference solution away from the singular point. Error estimates are developed, and numerical examples are given.

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