Solution of the Laminar Boundary Layer by the Variational Method (1 st Report)

Abstract

In this paper, the frictional resistance of the flat plate, placed in the uniform stream along its direction, is calculated by the variational method. The functional J of the velocity distribution is constructed, its stationary value being equal to the shearing stress on the surface of the plate. The Euler equation for δJ=0 is same as the equation of motion. The problem is solved by Ritz's method. The obtained value of the resistance coefficient of the plate of length l is c_f=1.333√(ν/U₀l), where ν is the kinematic coefficient of viscosity and U₀ is the uniform velocity. The equations used to reduce this result are regarded as the equations of the balance of energy and momentum in the boundary layer.