Force Singularities in Two-Dimensional Flows.

The effects of force singularities in two-dimensional flows at low Reynolds numbers are investigated for both homogeneous and stratified fluids. The method of solution is based on Green's functions. The Navier-Stokes equations are linearized using Oseen's approximation and Boussinesq's approximation is used to describe the influence of density. Force singularities are introduced by means of Dirac's delta functions. The resulting system of equations is solved after introducing the stream function to yield fundamental solutions. These fundamental solutions are general and can be made to satisfy given boundary conditions. This process is illustrated by considering uniform flow in the infinite domain as well as flow between two moving parallel plates. The usefulness of the fundamental solutions obtained is demonstrated in the case of uniform flow in the infinite domain. In this case, fundamental solutions corresponding to a vertical force and to a horizontal force are studied in detail for both homogeneous and stratified fluids. Homogeneous fluids are obtained as a special case of stratified fluids. Results obtained by other authors are rederived as limit cases. Oseenlets, flow past a semi-infinite flat plate, viscous submerged jets and associated free boundary layers are used as illustrative examples of results that can be derived as limit cases. When the fluid is stratified, flow patterns associated with force singularities reveal an infinite series of vortex cells. It is these vortices that give rise to jets, waves, and wakes observed by other authors. The stream functions corresponding to the force singularities are shown to consist of two parts. When the force is horizontal these two parts are an inviscid sink and a viscous source. When the force is vertical, the two parts turn out to be an inviscid vortex and a viscous one in opposite directions. These results are shown for homogeneous fluids and inferred in the case of stratified fluids. Strongly stratified flow past a horizontal force singularity is obtained as a limit by the method of steepest descent corroborating Graebel's (1969) results obtained by another method. The creeping motion of a moderately stratified fluid past a cylinder is obtained in general curvilinear coordinates using matched asymptotic expansions. The procedure is essentially based on Graebel's (1969) work. Both the circular cylinder and the elliptic cylinder are treated as special cases. Finally, a summary of two-dimensional singularities is presented in the form of a table. Force singularities in stratified fluids are added to the table of two-dimensional singularities prepared by Olmstead and Gautesen (1976) for homogeneous fluids.Ph.D.MechanicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/157965/1/8025737.pd

Using Logarithms to Test the Solution of a Differential Equation in the Lab

AC2009-56