Categories: Fluid dynamics, Fluid mechanics, Physics.

# 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 $\mathrm{Re} \gg 1$ close to a surface. Specifically, the Blasius layer is the solution for a half-plane approached from the edge by a fluid.

Let the half-plane lie in the $(x,z)$-plane (i.e. at $y = 0$) and exist for all $x \ge 0$, such that its edge lies on the $z$-axis. A fluid with velocity field $\va{v} = U \vu{e}_x$ 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 $v_x(x, y)$:

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

Where $\delta(x) \equiv \sqrt{\nu x / U}$ 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 $f'(\infty) = 1$, i.e. the fluid is stationary at the half-plane’s surface $s = 0$, and has velocity $U$ far away at $s \to \infty$.

Inserting the ansatz into the incompressibility condition $\nabla \cdot \va{v} = 0$ yields:

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

Which we integrate by parts 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' \bigg( s f' - \int f' \dd{s} \bigg) = U \delta' \: (s f' - f) \end{aligned}

Now, consider the main Prandtl equation, assuming the attack velocity $U$ is constant. Inserting our expressions for $v_x$ and $v_y$ into it gives:

\begin{aligned} \nu \pdvn{2}{v_x}{y} &= v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} \\ \nu \pdvn{2}{v_x}{s} \pdvn{2}{s}{y} &= v_x \pdv{v_x}{s} \pdv{s}{x} + v_y \pdv{v_x}{s} \pdv{s}{y} \\ \nu U \frac{1}{\delta^2} f''' &= - U^2 \frac{\delta'}{\delta} s f'' f' + U^2 \frac{\delta'}{\delta} f'' (s f' - f) \end{aligned}

We multiply by $\delta^2 / U$, cancel out some terms, and substitute $\delta(x) \equiv \sqrt{\nu x / U}$, leaving:

\begin{aligned} \nu f''' &= - U \delta' \delta f'' f = - U \frac{\nu}{2 U} f'' f \end{aligned}

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.

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