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1 files changed, 27 insertions, 27 deletions
diff --git a/source/know/concept/prandtl-equations/index.md b/source/know/concept/prandtl-equations/index.md index 3afe405..d82657c 100644 --- a/source/know/concept/prandtl-equations/index.md +++ b/source/know/concept/prandtl-equations/index.md @@ -11,7 +11,7 @@ layout: "concept" 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/) $\mathrm{Re} \gg 1$ +with a large [Reynolds number](/know/concept/reynolds-number/) $$\mathrm{Re} \gg 1$$ close to a solid surface. Fluids with a large Reynolds number @@ -27,23 +27,23 @@ 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. +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. +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 $\va{v} = (v_x, v_y)$: +where the flow $$\va{v} = (v_x, v_y)$$: $$\begin{aligned} v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} @@ -58,10 +58,10 @@ $$\begin{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 $1$: +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 $$1$$: $$\begin{aligned} x @@ -106,7 +106,7 @@ $$\begin{aligned} \end{aligned}$$ For future convenience, -we multiply the former equation by $L / U^2$, and the latter by $\delta / U^2$: +we multiply the former equation by $$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}} @@ -118,12 +118,12 @@ $$\begin{aligned} + \nu \Big( \frac{\delta^2}{U L^3} \pdvn{2}{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdvn{2}{\tilde{v}_y}{\tilde{y}} \Big) \end{aligned}$$ -We would like to estimate $\delta$. -Intuitively, we expect that higher viscosities $\nu$ give thicker layers, -and that faster velocities $U$ give thinner layers. +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 $\delta(x)$. +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: $$\begin{aligned} @@ -132,7 +132,7 @@ $$\begin{aligned} \sim \sqrt{\frac{\nu L}{U}} \end{aligned}$$ -We thus insert $\delta = \sqrt{\nu L / U}$ into the Navier-Stokes equations, giving us: +We thus insert $$\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}} @@ -144,7 +144,7 @@ $$\begin{aligned} + \nu \Big( \frac{\nu}{U^2 L^2} \pdvn{2}{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdvn{2}{\tilde{v}_y}{\tilde{y}} \Big) \end{aligned}$$ -Here, we recognize the definition of the Reynolds number $\mathrm{Re} = U L / \nu$: +Here, we recognize the definition of the Reynolds number $$\mathrm{Re} = U L / \nu$$: $$\begin{aligned} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} @@ -156,8 +156,8 @@ $$\begin{aligned} + \frac{1}{\mathrm{Re}^2} \pdvn{2}{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \pdvn{2}{\tilde{v}_y}{\tilde{y}} \end{aligned}$$ -Recall that we are only considering large Reynolds numbers $\mathrm{Re} \gg 1$, -in which case $\mathrm{Re}^{-1} \ll 1$, +Recall that we are only considering large Reynolds numbers $$\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} @@ -168,11 +168,11 @@ $$\begin{aligned} = 0 \end{aligned}$$ -The second one tells us that for a given $x$-value, +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 $U(x)$: +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 |