Blasius Equation: Some Explorations - Part 1

 

What is Blasius Equation:

Blasuis Equation describes the flow of a fluid over a flat plate.

The x-y coordinate system is chosen so that x is along the plate, and y is perpendicular to the plate. The leading edge of the plate is at x = 0, y = 0. The velocity components - u in the x-direction and v in the y-direction -- are expressed in terms of a stream function (x,y):

The fundamental equation which determines y is the x-component of the momentum equation in the boundary layer approximation. For the flat plate there is no pressure gradient, and the boundary layer approximation to the x-momentum equation takes the form

Where is the kinematic viscosity. By substituting the first eqaution into the second, we get for the partial differential equation

At the wall (y = 0) both velocity components must vanish, and far away from the plate, the horizontal velocity must approach the given free stream velocity . These conditions translate into the following conditions on :

The absence of a length scale (the plate is semi-infinite in length) suggests a similarity solution, as originally used by Blasius. The solution has the form:

The scale is comparable with the boundary layer thickness. This substitution into the equation for  leads to the following nonlinear ordinary differential equation for :

This equation is called the Blasius equation. We will solve it numerically in the next part. Once f is known, the velocity components may be computed as