Blasius boundary layer
In fluid dynamics, the Blasius boundary layer is an application of the Prandtl equations, which govern the flow of a fluid at large Reynolds number close to a surface. Specifically, the Blasius layer is the solution for a half-plane approached from the edge by a fluid.
Let the half-plane lie in the -plane (i.e. at ) and exist for all , such that its edge lies on the -axis. A fluid with velocity field approaches the half-plane’s edge head-on. To describe the fluid’s movements around the plane, we make an ansatz for the so-called slip-flow region’s -velocity :
Where is the boundary layer thickness estimate that was used to derive the Prandtl equations. Note that is the derivative of an unknown , and that it obeys the boundary conditions and , i.e. the fluid is stationary at the half-plane’s surface , and has velocity far away at .
Inserting the ansatz into the incompressibility condition yields:
Which we integrate by parts to get an expression for the -velocity , namely:
Now, consider the main Prandtl equation, assuming the attack velocity is constant. Inserting our expressions for and into it gives:
We multiply by , cancel out some terms, and substitute , leaving:
This leads us to the Blasius equation, which is a nonlinear ODE for :
Unfortunately, this cannot be solved analytically, only numerically. Nevertheless, the result shows a boundary layer exhibiting the expected downstream thickening.
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.