Synthetic method for boundary layer thickness

Abstract

The classical boundary layer equations are a reduced form of the Navier-Stokes equations in which some terms have been omitted on certain order considerations. Euler’s equations are also another simplified form of the Navier-Stokes equations where viscosity is completely ignored. For two-dimensional (or axisymmetrical) motion the stream function ψ satisfies a fourth order non-linear differential equation if the full Navier-Stokes equations are taken into account; a fourth order linear differential equation for slow viscous motions; a third order non-linear equation according to Prandtl’s boundary layer assumptions; and a second order linear equation if the fluid is regarded inviscid. The reduction in order of the differential equation also necessitates some sacrifice of the boundary conditions. Thus in inviscid flow only the relative normal velocity on the solid must vanish at the wall, while in viscous flow both tangential and normal velocities must be zero. Linearization of the Navier-Stokes equations, either by regarding the fluid non-viscous or the motion to be so slow that the inertia terms can be completely or partially ignored, has resulted in only limited applications of the theory. The resistance to a solid moving through a real fluid is caused by the viscosity of the fluid, and if the parameter of viscosity is small, the viscous effects are confined only to a thin region surrounding the boundary—the boundary layer region.