A weak momentum source in a uniform stream

Abstract

Abstract.The interaction of a momentum source with a viscous uniform stream is studied. Taken separately, each flow is described by an exact solution of the full Navier-Stokes equations. Their combined effect can be described in terms of two nondimensional parameters related to the strength of the source and the distance from it. A perturbation solution for the case of a weak source is attempted and the initial terms are found. It is shown that the results are closely related to the Oseen expansion for viscous streaming past a spheroid and that when the momentum of the source opposes that of the stream a closed streamline of elliptical shape is formed.1. Introduction. The leading terms in the Oseen expansions for the sphere (Proudman and Pearson [4]), and the spheroid (Breach [1]), are very similar in form. This is true even when the spheroid is very slender and suggests that the mere presence of a body, rather than its particular shape, is the essential factor affecting the flow in the large. From this overall point of view, when a body is present in a uniform stream it acts as a sink of momentum. The details of how this momentum is destroyed are built into the boundary conditions on the surface of the body and are of local interest only. Therefore a fundamental problem for investigation is that of a momentum source, or sink, placed in a uniform stream. This is a simpler problem than that of streaming past a finite body in that there are no fixed boundaries present and analytical difficulties in satisfying conditions thereon do not arise.The momentum source in the absence of the uniform stream is described by a nontrivial exact solution of the full Navier-Stokes equations. The uniform stream alone corresponds to another exact solution. Hence when the two interact there is the possibility of constructing the solution by perturbations from exact solutions of the full nonlinear equations. This would seem to be more satisfactory than a perturbation starting from exact solutions of linearised equations.The problem depends on four parameters: the source strength, the viscosity, the velocity of the uniform stream, and the distance from the origin. These can be combined to give just two fundamental parameters and all the conditions that the solution must satisfy are expressible in terms of limiting values of these two.2. A solution of the Navier-Stokes equations corresponding to a momentum source. By requiring the momentum flux across any sphere around the origin to be constant and imposing axial symmetry, Landau [3] constructed an exact solution of the NavierStokes equations (see Landau and Lifshitz [4]). This was subsequently discovered independently by Squire [7). Sedov [5], by dimensional analysis, found this solution as one of a whole family of exact solutions in which the velocity components vary in-