In fluid dynamics, the Blasius boundary layer is an application of
the Prandtl equations,
which govern the flow of a fluid
at large Reynolds numberRe≫1 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 (x,z)-plane (i.e. at y=0)
and exist for all x≥0, such that its edge lies on the z-axis.
A fluid with velocity field v=Ue^x
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 x-velocity vx(x,y):
vx=Uf′(s)s≡δ(x)y
Where δ(x)≡νx/U is the boundary layer thickness
estimate that was used to derive the Prandtl equations.
Note that f′(s) is the derivative of an unknown f(s),
and that it obeys the boundary conditions f′(0)=0 and f′(∞)=1,
i.e. the fluid is stationary at the half-plane’s surface s=0,
and has velocity U far away at s→∞.
Inserting the ansatz into the incompressibility condition ∇⋅v=0 yields:
∂y∂vy=−∂x∂vx=−∂s∂vx∂x∂s=Uδδ′sf′′
Which we integrate by parts to get an expression for the y-velocity vy, namely:
vy=Uδδ′∫sf′′dy=Uδ′(sf′−∫f′ds)=Uδ′(sf′−f)
Now, consider the main Prandtl equation,
assuming the attack velocity U is constant.
Inserting our expressions for vx and vy into it gives:
We multiply by δ2/U, cancel out some terms,
and substitute δ(x)≡νx/U, leaving:
νf′′′=−Uδ′δf′′f=−U2Uνf′′f
This leads us to the Blasius equation,
which is a nonlinear ODE for f(s):
2f′′′+f′′f=0
Unfortunately, this cannot be solved analytically, only numerically.
Nevertheless, the result shows a boundary layer δ(x)
exhibiting the expected downstream thickening.
References
B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition,
CRC Press.