Prandtl's Boundary-Layer Theory from the Viewpoint of a Mathematician

Abstract

The fluid flow of a continuous medium is described by the Navier-Stokes differ­ ential equations. These equations are nonlinear and-in the subsonic range-of elliptic type. Only a few explicit solutions are known; it is also quite difficult to determine approximations. An existence theory has been known for a few years, although many important questions are still unanswered. An example of a very simple unsolved problem is: does there exist a solution of the Navier-Stokes equation for the two-dimensional steady (hence laminar) flow of an incom­ pressible medium around a simply connected finite body with smooth boundary? Because of these difficulties, Ludwig Prandtl (1904) simplified the Navier­ Stokes equations by omitting nonessential terms. He thus obtained his famous boundary-layer equations. Though these equations are also nonlinear they are parabolic and hence much simpler from the theoretical and also practical point of view than the Navier-Stokes differential equations. They can be regarded as asymptotic equations of the Navier-Stokes equations in the limit of vanishing viscosity. The Prandtl boundary-layer theory very early became an invaluable device for the practical treatment of a fluid flow. Blasius (1908) first successfully computed the flow over a flat plate as an explicit solution of the Prandtl equations (but not of the Navier-Stokes equations). Many other new results were obtained in the next decades with the aid of the Prandtl concept. Viscous fluid flows were treated for twoand three-dimensional, steady and unsteady flow of an incom­ pressible or compressible medium consisting of one or more components, with or without energy addition, under the influence of magnetic forces, etc, etc. It is true that the practical success of the boundary-layer theory was over­ whelming. Therefore most people overlooked the fact that the theoretical foundation of that theory was rather vague. During the first 50 years of boundary­ layer theory the fundamental mathematical questions could not be answered. It was not possible to establish a sound mathematical connection to the Navier­ Stokes differential equations. There was no evidence of the existence, uniqueness,