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diff --git a/source/know/concept/bernoullis-theorem/index.md b/source/know/concept/bernoullis-theorem/index.md index 12bd0ca..6b933d2 100644 --- a/source/know/concept/bernoullis-theorem/index.md +++ b/source/know/concept/bernoullis-theorem/index.md @@ -10,8 +10,8 @@ layout: "concept" --- For inviscid fluids, **Bernuilli's theorem** states -that an increase in flow velocity $\va{v}$ is paired -with a decrease in pressure $p$ and/or potential energy. +that an increase in flow velocity $$\va{v}$$ is paired +with a decrease in pressure $$p$$ and/or potential energy. For a qualitative argument, look no further than one of the [Euler equations](/know/concept/euler-equations/), with a [material derivative](/know/concept/material-derivative/): @@ -22,16 +22,16 @@ $$\begin{aligned} = \va{g} - \frac{\nabla p}{\rho} \end{aligned}$$ -Assuming that $\va{v}$ is constant in $t$, -it becomes clear that a higher $\va{v}$ requires a lower $p$. +Assuming that $$\va{v}$$ is constant in $$t$$, +it becomes clear that a higher $$\va{v}$$ requires a lower $$p$$. ## Simple form For an incompressible fluid -with a time-independent velocity field $\va{v}$ (i.e. **steady flow**), +with a time-independent velocity field $$\va{v}$$ (i.e. **steady flow**), Bernoulli's theorem formally states that the -**Bernoulli head** $H$ is constant along a streamline: +**Bernoulli head** $$H$$ is constant along a streamline: $$\begin{aligned} \boxed{ @@ -40,8 +40,8 @@ $$\begin{aligned} } \end{aligned}$$ -Where $\Phi$ is the gravitational potential, such that $\va{g} = - \nabla \Phi$. -To prove this theorem, we take the material derivative of $H$: +Where $$\Phi$$ is the gravitational potential, such that $$\va{g} = - \nabla \Phi$$. +To prove this theorem, we take the material derivative of $$H$$: $$\begin{aligned} \frac{\mathrm{D} H}{\mathrm{D} t} @@ -63,7 +63,7 @@ $$\begin{aligned} + \va{v} \cdot \big( \va{g} + \nabla \Phi \big) + \va{v} \cdot \Big( \frac{\nabla p}{\rho} - \frac{\nabla p}{\rho} \Big) \end{aligned}$$ -Using the fact that $\va{g} = - \nabla \Phi$, +Using the fact that $$\va{g} = - \nabla \Phi$$, we are left with the following equation: $$\begin{aligned} @@ -72,12 +72,12 @@ $$\begin{aligned} \end{aligned}$$ Assuming that the flow is steady, both derivatives vanish, -leading us to the conclusion that $H$ is conserved along the streamline. +leading us to the conclusion that $$H$$ is conserved along the streamline. In fact, there exists **Bernoulli's stronger theorem**, -which states that $H$ is constant *everywhere* in regions with -zero [vorticity](/know/concept/vorticity/) $\va{\omega} = 0$. -For a proof, see the derivation of $\va{\omega}$'s equation of motion. +which states that $$H$$ is constant *everywhere* in regions with +zero [vorticity](/know/concept/vorticity/) $$\va{\omega} = 0$$. +For a proof, see the derivation of $$\va{\omega}$$'s equation of motion. ## References |