---
title: "Blasius boundary layer"
date: 2021-05-29
categories:
- Physics
- Fluid mechanics
- Fluid dynamics
layout: "concept"
---

In fluid dynamics, the **Blasius boundary layer** is an application of
the [Prandtl equations](/know/concept/prandtl-equations/),
which govern the flow of a fluid
at large Reynolds number $\mathrm{Re} \gg 1$
close to a surface.
Specifically, the Blasius layer is the solution
for a half-plane approached from the edge by a fluid.

A fluid with velocity field $\va{v} = U \vu{e}_x$ flows to the plane,
which starts at $y = 0$ and exists for $x \ge 0$.
To describe this, we make an ansatz
for the *slip-flow* region's $x$-velocity $v_x(x, y)$:

$$\begin{aligned}
    v_x
    = U f'(s)
    \qquad \quad
    s
    \equiv \frac{y}{\delta(x)}
\end{aligned}$$

Note that $f'(s)$ is the derivative of an unknown $f(s)$,
and that it obeys the boundary conditions $f'(0) = 0$ and $f'(\infty) = 1$.
Furthermore, $\delta(x)$ is the thickness of the stationary boundary layer at the surface.
To derive the Prandtl equations,
the estimate $\delta(x) = \sqrt{\nu x / U}$ was used,
which we will stick with.
For later use, it is worth writing the derivatives of $s$:

$$\begin{aligned}
    \pdv{s}{x}
    = - y \frac{\delta'}{\delta^2}
    = - s \frac{\delta'}{\delta}
    \qquad \quad
    \pdv{s}{y}
    = \frac{1}{\delta}
\end{aligned}$$

Inserting the ansatz for $v_x$ into the incompressibility condition then yields:

$$\begin{aligned}
    \pdv{v_y}{y}
    = - \pdv{v_x}{x}
    = U s f'' \frac{\delta'}{\delta}
\end{aligned}$$

Which we integrate to get an expression for the $y$-velocity $v_y$, namely:

$$\begin{aligned}
    v_y
    = U \frac{\delta'}{\delta} \int s f'' \dd{y}
    = U \delta' \: (s f' - f)
\end{aligned}$$

Now, consider the main Prandtl equation,
assuming that the attack velocity $U$ is constant:

$$\begin{aligned}
    v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
    = \nu \pdvn{2}{v_x}{y}
\end{aligned}$$

Inserting our expressions for $v_x$ and $v_y$ into this leads us to:

$$\begin{aligned}
    - U^2 \frac{\delta'}{\delta} s f'' f' + U^2 \frac{\delta'}{\delta} f'' (s f' - f)
    = \nu U \frac{1}{\delta^2} f'''
\end{aligned}$$

After  multiplying it by $\delta^2 / U$ and cancelling out some terms,
it reduces to:

$$\begin{aligned}
    \nu f''' + U \delta' \delta f'' f
    = 0
\end{aligned}$$

Then, substituting $\delta(x) = \sqrt{\nu x / U}$ and $\delta'(x) = (1/2) \sqrt{\nu / (U x)}$ yields:

$$\begin{aligned}
    \nu f''' + U \frac{\nu}{2 U} f'' f
    = 0
\end{aligned}$$

Simplifying this leads us to the **Blasius equation**,
which is a nonlinear ODE for $f(s)$:

$$\begin{aligned}
    \boxed{
        2 f''' + f'' f = 0
    }
\end{aligned}$$

Unfortunately, this cannot be solved analytically, only numerically.
Nevertheless, the result shows a boundary layer $\delta(x)$
exhibiting the expected downstream thickening.



## References
1.  B. Lautrup,
    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
    CRC Press.