path: root/content/know/concept/prandtl-equations/index.pdc

---
title: "Prandtl equations"
firstLetter: "P"
publishDate: 2021-05-29
categories:
- Physics
- Fluid mechanics
- Fluid dynamics

date: 2021-05-10T18:41:20+02:00
draft: false
markup: pandoc
---

# Prandtl equations

In fluid dynamics, the **Prandtl equations** or **boundary layer equations**
describe the movement of a [viscous](/know/concept/viscosity/) fluid
with a large [Reynolds number](/know/concept/reynolds-number/) $\mathrm{Re} \gg 1$
close to a solid surface.

Fluids with a large Reynolds number
are often approximated as having zero viscosity,
since the simpler [Euler equations](/know/concept/euler-equations)
can then be used instead of the [Navier-Stokes equations](/know/concept/navier-stokes-equations/).

However, in reality, a viscous fluid obeys the *no-slip* boundary condition:
at every solid surface the local velocity must be zero.
This implies the existence of a **boundary layer**:
a thin layer of fluid "stuck" to solid objects in the flow,
where viscosity plays an important role.
This is in contrast to the ideal flow far away from the surface.

We consider a simple theoretical case in 2D:
a large flat surface located at $y = 0$ for all $x \in \mathbb{R}$,
with a fluid *trying* to flow parallel to it at $U$.
The 2D treatment can be justified by assuming that everything is constant in the $z$-direction.
We will not solve this case,
but instead derive general equations
to describe the flow close to a flat surface.

At the wall, there is a very thin boundary layer of thickness $\delta$,
where the fluid is assumed to be completely stationary $\va{v} = 0$.
We are mainly interested in the region $\delta < y \ll L$,
where $L$ is the distance at which the fluid becomes practically ideal.
This the so-called **slip-flow** region,
in which the fluid is not stationary,
but still viscosity-dominated.

In 2D, the steady Navier-Stokes equations are as follows,
where the flow $\va{v} = (v_x, v_y)$:

$$\begin{aligned}
    v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
    &= - \frac{1}{\rho} \pdv{p}{x} + \nu \Big( \pdv[2]{v_x}{x} + \pdv[2]{v_x}{y} \Big)
    \\
    v_x \pdv{v_y}{x} + v_y \pdv{v_y}{y}
    &= - \frac{1}{\rho} \pdv{p}{y} + \nu \Big( \pdv[2]{v_y}{x} + \pdv[2]{v_y}{y} \Big)
    \\
    \pdv{v_x}{x} + \pdv{v_y}{y}
    &= 0
\end{aligned}$$

The latter represents the fluid's incompressibility.
We non-dimensionalize these equations,
and assume that changes along the $y$-axis
happen on a short scale (say, $\delta$),
and along the $x$-axis on a longer scale (say, $L$).
Let $\tilde{x}$ and $\tilde{y}$ be dimenionless variables of order $1$:

$$\begin{aligned}
    x
    = L \tilde{x}
    \qquad \quad
    y
    = \delta \tilde{x}
    \qquad \quad
    \pdv{x}
    = \frac{1}{L} \pdv{\tilde{x}}
    \qquad \quad
    \pdv{y}
    = \frac{1}{\delta} \pdv{\tilde{y}}
\end{aligned}$$

Furthermore, we choose velocity scales
to be consistent with the incompressibility condition,
and a pressure scale inspired
by [Bernoulli's theorem](/know/concept/bernoullis-theorem/):

$$\begin{aligned}
    v_x
    = U \tilde{v}_x
    \qquad \quad
    v_y
    = \frac{U \delta}{L} \tilde{v}_y
    \qquad \quad
    p
    = \rho U^2 \tilde{p}
\end{aligned}$$

We insert these scalings into the Navier-Stokes equations, yielding:

$$\begin{aligned}
    \frac{U^2}{L} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \frac{U^2}{L} \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
    &= - \frac{U^2}{L} \pdv{\tilde{p}}{\tilde{x}}
    + \nu \Big( \frac{U}{L^2} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{U}{\delta^2} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big)
    \\
    \frac{U^2 \delta}{L^2} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{U^2 \delta}{L^2} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}}
    &= - \frac{U^2}{\delta} \pdv{\tilde{p}}{\tilde{y}}
    + \nu \Big( \frac{U \delta}{L^3} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{U}{L \delta} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big)
\end{aligned}$$

For future convenience,
we multiply the former equation by $L / U^2$, and the latter by $\delta / U^2$:

$$\begin{aligned}
    \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
    &= - \pdv{\tilde{p}}{\tilde{x}}
    + \nu \Big( \frac{1}{U L} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{L}{U \delta^2} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big)
    \\
    \frac{\delta^2}{L^2} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{\delta^2}{L^2} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}}
    &= - \pdv{\tilde{p}}{\tilde{y}}
    + \nu \Big( \frac{\delta^2}{U L^3} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big)
\end{aligned}$$

We would like to estimate $\delta$.
Intuitively, we expect that higher viscosities $\nu$ give thicker layers,
and that faster velocities $U$ give thinner layers.
Furthermore, we expect *downstream thickening*:
with distance $x$, viscous stresses slow down the slip-flow,
leading to a gradual increase of $\delta(x)$.
Some dimensional analysis thus yields the following estimate:

$$\begin{aligned}
    \delta
    \approx \sqrt{\frac{\nu x}{U}}
    \sim \sqrt{\frac{\nu L}{U}}
\end{aligned}$$

We thus insert $\delta = \sqrt{\nu L / U}$ into the Navier-Stokes equations, giving us:

$$\begin{aligned}
    \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
    &= - \pdv{\tilde{p}}{\tilde{x}}
    + \nu \Big( \frac{1}{U L} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{1}{\nu} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big)
    \\
    \frac{\nu}{U L} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{\nu}{U L} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}}
    &= - \pdv{\tilde{p}}{\tilde{y}}
    + \nu \Big( \frac{\nu}{U^2 L^2} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big)
\end{aligned}$$

Here, we recognize the definition of the Reynolds number $\mathrm{Re} = U L / \nu$:

$$\begin{aligned}
    \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
    &= - \pdv{\tilde{p}}{\tilde{x}}
    + \frac{1}{\mathrm{Re}} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \pdv[2]{\tilde{v}_x}{\tilde{y}}
    \\
    \frac{1}{\mathrm{Re}} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}}
    &= - \pdv{\tilde{p}}{\tilde{y}}
    + \frac{1}{\mathrm{Re}^2} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \pdv[2]{\tilde{v}_y}{\tilde{y}}
\end{aligned}$$

Recall that we are only considering large Reynolds numbers $\mathrm{Re} \gg 1$,
in which case $\mathrm{Re}^{-1} \ll 1$,
so we can drop many terms, leaving us with these redimensionalized equations:

$$\begin{aligned}
    v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
    = - \frac{1}{\rho} \pdv{p}{x} + \nu \pdv[2]{v_x}{y}
    \qquad \quad
    \pdv{p}{y}
    = 0
\end{aligned}$$

The second one tells us that for a given $x$-value,
the pressure is the same at the surface
as in the main flow $y > L$, where the fluid is ideal.
In the latter regime, we apply Bernoulli's theorem to rewrite $p$,
using the *Bernoulli head* $H$ and the mainstream velocity $U(x)$:

$$\begin{aligned}
    p
    = \rho H - \frac{1}{2} \rho U^2
    = p_0 - \frac{1}{2} \rho U^2
\end{aligned}$$

Inserting this into the reduced Navier-Stokes equations,
we arrive at the Prandtl equations:

$$\begin{aligned}
    \boxed{
        v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
        = U \dv{U}{x} + \nu \pdv[2]{v_x}{y}
        \qquad \quad
        \pdv{v_x}{x} + \pdv{v_y}{y}
        = 0
    }
\end{aligned}$$

A notable application of these equations is
the [Blasius boundary layer](/know/concept/blasius-boundary-layer/),
where the surface in question
is a semi-infinite plane.



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