I. The Rotation of a Gravitating Sphere in a Monatomic Gas. II. The Drag of a Body Moving Transversely in a Confined Stratified Fluid

Citation

Foster, Michael Ralph (1969) I. The Rotation of a Gravitating Sphere in a Monatomic Gas. II. The Drag of a Body Moving Transversely in a Confined Stratified Fluid. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/89KW-B656. https://resolver.caltech.edu/CaltechETD:etd-01132006-084851

Abstract

Part I - The Rotation of a Gravitating Sphere in a Monatomic Gas

The flow resulting from the steady rotation of a gravitating sphere in a monatomic gas at rest is studied in a variety of special cases. The low speed rotation problem involves the solution of non-uniform Stokes equations and exhibits the interesting property that, if the field is large enough to make a "scale height" very small compared to the sphere radius, the motion is very weak and occurs primarily in a thin boundary layer on the sphere. The asymptotic theory for the gravitational field strength very large with arbitrary rotation speed shows essentially the same boundary layer, regardless of Reynolds number; the perturbation theory presents some interesting mathematical problems as well. The high speed rotation case is finally considered, and solutions have been obtained only for a gas with small Prandtl number. Even then, the flow structure is very complex. Depending on the relative sizes of the Prandtl number and inverse Reynolds number, there are six possibilities. In every case, there is a thin Prandtl boundary layer on the surface of the sphere and an essentially incompressible jet in the equatorial plane. In some cases, a thermal layer outside the Prandtl boundary layer is required to adjust the temperature, and in every case but one, it is necessary to infer the existence of still another layer, which is inviscid but rotational, that adjusts the uniform flow into the layer required by the strong hydrostatic constraints on the outer flow to that necessary for Prandtl boundary layer entrainment. In some cases these layers are unstable to small disturbances if the temperature on the sphere is sufficiently large.

Part II - The Drag of a Body Moving Transversely in a Confined Stratified Fluid

The motion of a body through a stratified fluid bounded by vertical plates is studied in the case when the motion of the body is sufficiently slow to make the inertia of the fluid negligible. The case studied is for a very small coefficient of diffusion (for salt in water, for example). The density changes are quite large, and the drag is quite easily computed without appeal to the structure of any boundary layers or shear layers, depending only on changes of potential energy of the fluid. The solution exhibits regions where the fluid is unstably stratified, and hence mixes. Depending upon how complete the mixing process is, the body might experience a thrust!

The equations for boundary layers are given, but details of their solution are not dealt with here, because of their quasi-linear nature. The horizontal shear layers consist of a simple density adjustment layer surrounded by a thicker and quite complicated non linear dynamical layer. The more conventional Stewartson layers do not appear here, these layers, because of the non linearities, are quite complex, and details of their structure have not yet been fully worked out.

Thesis Files