Modified linearized Burger's flow for two-dimensional viscous motion

Abstract

In this expository paper we use a complex ver- sion of the two-dimensional steady flow Navier-Stokes equa- tions, originally due to Legendre (6) and develop a new lin- earization which is similar, but different from Burger's flow. A similarity solution is found where the stream function for the potential flow is a linear combination of the linear shear flow and stagnation-point flow in two dimensions. Such a similarity solution describes a flow impinging on a plane wall at an arbi- trary angle of incidence and reduces the problem to a system of two ordinary differential equations which can be integrated numerically. The technique is similar to a method used by Ranger and Davis (8) and discussed in a nonlinear context by Dorrepaal (1). The existence of the boundary layer near the wall is interpreted from the similarity solution. A promulga- tion of the case of inflow only is carried out for the linearized model. 1. Introduction. Motion in two dimensions is characterized by the streamlines being all continuously equidistant to a fixed plane and the vorticity vector being at right angles to the plane of the motion and therefore fixed in direction. The fact that two-dimensional motion offers opportunities for special mathematical treatment and empowers us to scrutinize the nature of many phenomena which in their full three- dimensional form have so far proved intractable, is of long standing interest in the field of fluid dynamics. In the context of the fluid dynamic literature, the rareness in the exact solutions of the Navier-Stokes equations is due to the analytic nature of the nonlinear boundary-value problems. Since nonlinear problems fail to admit a superposition principle, this raises the issue of the existence of a similarity solution for certain types of problems. One such flow which admits a similarity solution to the full Navier-Stokes equations is the two-dimensional stagnation-point flow in which an incompressible viscous fluid flows steadily towards a plane wall. In such a flow a two- dimensional rigid wall occupies the entire x-axis and the fluid domain is y> 0. At a stagnation point, a streamline splits in order for flow separation to occur. Copyright c