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Diffstat (limited to 'source/know/concept/cavitation/index.md')
| -rw-r--r-- | source/know/concept/cavitation/index.md | 36 |
1 files changed, 18 insertions, 18 deletions
diff --git a/source/know/concept/cavitation/index.md b/source/know/concept/cavitation/index.md index 346edc4..679e66f 100644 --- a/source/know/concept/cavitation/index.md +++ b/source/know/concept/cavitation/index.md @@ -20,11 +20,11 @@ the [Rayleigh-Plesset equation](/know/concept/rayleigh-plesset-equation/) for an inviscid liquid without surface tension. Note that the RP equation assumes incompressibility. -We assume that the whole liquid is at a constant pressure $p_\infty$, -and the bubble is empty, such that the interface pressure $P = 0$, -meaning $\Delta p = - p_\infty$. -At first, the radius is stationary $R'(0) = 0$, -and given by a constant $R(0) = a$. +We assume that the whole liquid is at a constant pressure $$p_\infty$$, +and the bubble is empty, such that the interface pressure $$P = 0$$, +meaning $$\Delta p = - p_\infty$$. +At first, the radius is stationary $$R'(0) = 0$$, +and given by a constant $$R(0) = a$$. The simple Rayleigh-Plesset equation is then: $$\begin{aligned} @@ -32,7 +32,7 @@ $$\begin{aligned} = - \frac{p_\infty}{\rho} \end{aligned}$$ -To solve it, we multiply both sides by $R^2 R'$ +To solve it, we multiply both sides by $$R^2 R'$$ and rewrite it in the following way: $$\begin{aligned} @@ -44,7 +44,7 @@ $$\begin{aligned} \end{aligned}$$ It is then straightforward to integrate both sides -with respect to time $\tau$, from $0$ to $t$: +with respect to time $$\tau$$, from $$0$$ to $$t$$: $$\begin{aligned} - \frac{2 p_\infty}{3 \rho} \int_0^t \dv{}{\tau}\Big( R^3 \Big) \dd{\tau} @@ -58,7 +58,7 @@ $$\begin{aligned} \end{aligned}$$ Rearranging this equation yields the following expression -for the derivative $R'$: +for the derivative $$R'$$: $$\begin{aligned} (R')^2 @@ -66,10 +66,10 @@ $$\begin{aligned} \end{aligned}$$ This equation is nasty to integrate. -The trick is to invert $R(t)$ into $t(R)$, +The trick is to invert $$R(t)$$ into $$t(R)$$, and, because we are only interested in collapse, -we just need to consider the case $R' < 0$. -The time of a given radius $R$ is then as follows, +we just need to consider the case $$R' < 0$$. +The time of a given radius $$R$$ is then as follows, where we are using slightly sloppy notation: $$\begin{aligned} @@ -79,8 +79,8 @@ $$\begin{aligned} = \int_{R}^{a} \frac{\dd{R}}{R'} \end{aligned}$$ -The minus comes from the constraint that $R' < 0$, but $t \ge 0$. -We insert the expression for $R'$: +The minus comes from the constraint that $$R' < 0$$, but $$t \ge 0$$. +We insert the expression for $$R'$$: $$\begin{aligned} t @@ -89,9 +89,9 @@ $$\begin{aligned} \end{aligned}$$ This integral needs to be looked up, -and involves the hypergeometric function ${}_2 F_1$. -However, we only care about *collapse*, which is when $R = 0$. -The time $t_0$ at which this occurs is: +and involves the hypergeometric function $${}_2 F_1$$. +However, we only care about *collapse*, which is when $$R = 0$$. +The time $$t_0$$ at which this occurs is: $$\begin{aligned} t_0 @@ -101,9 +101,9 @@ $$\begin{aligned} With our assumptions, a bubble will always collapse. However, unsurprisingly, reality turns out to be more complicated: -as $R \to 0$, the interface velocity $R' \to \infty$. +as $$R \to 0$$, the interface velocity $$R' \to \infty$$. By looking at the derivation of the Rayleigh-Plesset equation, -it can be shown that the pressure just outside the bubble diverges due to $R'$. +it can be shown that the pressure just outside the bubble diverges due to $$R'$$. This drastically changes the liquid's properties, and breaks our assumptions. |