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Diffstat (limited to 'source/know/concept/rayleigh-plesset-equation/index.md')
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1 files changed, 22 insertions, 22 deletions
diff --git a/source/know/concept/rayleigh-plesset-equation/index.md b/source/know/concept/rayleigh-plesset-equation/index.md index 4718683..b3ec8f5 100644 --- a/source/know/concept/rayleigh-plesset-equation/index.md +++ b/source/know/concept/rayleigh-plesset-equation/index.md @@ -16,7 +16,7 @@ Notably, it leads to [cavitation](/know/concept/cavitation/). Consider the main [Navier-Stokes equation](/know/concept/navier-stokes-equations/) -for the velocity field $\va{v}$: +for the velocity field $$\va{v}$$: $$\begin{aligned} \frac{\mathrm{D} \va{v}}{\mathrm{D} t} @@ -24,9 +24,9 @@ $$\begin{aligned} = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v} \end{aligned}$$ -We make the ansatz $\va{v} = v(r, t) \vu{e}_r$, -where $\vu{e}_r$ is the basis vector; -in other words, we demand that the only spatial variation of the flow is in $r$. +We make the ansatz $$\va{v} = v(r, t) \vu{e}_r$$, +where $$\vu{e}_r$$ is the basis vector; +in other words, we demand that the only spatial variation of the flow is in $$r$$. The above equation then becomes: $$\begin{aligned} @@ -44,17 +44,17 @@ $$\begin{aligned} = 0 \end{aligned}$$ -This is only satisfied if $r^2 v$ is constant with respect to $r$, -leading us to a solution $v(r)$ given by: +This is only satisfied if $$r^2 v$$ is constant with respect to $$r$$, +leading us to a solution $$v(r)$$ given by: $$\begin{aligned} v(r) = \frac{C(t)}{r^2} \end{aligned}$$ -Where $C(t)$ is an unknown function that does not depend on $r$. +Where $$C(t)$$ is an unknown function that does not depend on $$r$$. We then insert this result in the main Navier-Stokes equation, -and isolate it for $\ipdv{p}{r}$, yielding: +and isolate it for $$\ipdv{p}{r}$$, yielding: $$\begin{aligned} \pdv{p}{r} @@ -63,8 +63,8 @@ $$\begin{aligned} = - \rho \bigg( \frac{1}{r^2} C' - \frac{2}{r^5} C^2 \bigg) \end{aligned}$$ -Integrating this with respect to $r$ yields the following expression for $p$, -where $p_\infty(t)$ is the (possibly time-dependent) pressure at $r = \infty$: +Integrating this with respect to $$r$$ yields the following expression for $$p$$, +where $$p_\infty(t)$$ is the (possibly time-dependent) pressure at $$r = \infty$$: $$\begin{aligned} p(r) @@ -73,7 +73,7 @@ $$\begin{aligned} From the definition of [viscosity](/know/concept/viscosity/), we know that the normal [stress](/know/concept/cauchy-stress-tensor/) -$\sigma_{rr}$ in the liquid is given by: +$$\sigma_{rr}$$ in the liquid is given by: $$\begin{aligned} \sigma_{rr}(r) @@ -81,11 +81,11 @@ $$\begin{aligned} \end{aligned}$$ We now consider a spherical bubble -with radius $R(t)$ and interior pressure $P(t)$ along its surface. -Since we know the liquid pressure $p(r)$, -we can find $P$ from $\sigma_{rr}(r)$. +with radius $$R(t)$$ and interior pressure $$P(t)$$ along its surface. +Since we know the liquid pressure $$p(r)$$, +we can find $$P$$ from $$\sigma_{rr}(r)$$. Furthermore, to include the effects of surface tension, we simply add -the [Young-Laplace law](/know/concept/young-laplace-law/) to $P$: +the [Young-Laplace law](/know/concept/young-laplace-law/) to $$P$$: $$\begin{aligned} P @@ -93,18 +93,18 @@ $$\begin{aligned} = p(R) - 2 \rho \nu \Big( \frac{-2}{R^3} C \Big) + \alpha \frac{2}{R} \end{aligned}$$ -We isolate this for $p(R)$, and equate it to -our expression for $p(r)$ -at the surface $r\!=\!R$: +We isolate this for $$p(R)$$, and equate it to +our expression for $$p(r)$$ +at the surface $$r\!=\!R$$: $$\begin{aligned} P - \rho \nu \frac{4}{R^3} C - \alpha \frac{2}{R} = p_\infty + \rho \bigg( \frac{1}{R} C' - \frac{1}{2 R^4} C^2 \bigg) \end{aligned}$$ -Isolating for $P$, -and inserting the fact that $R'(t) = v(t)$, -such that $C = r^2 v = R^2 R'$, +Isolating for $$P$$, +and inserting the fact that $$R'(t) = v(t)$$, +such that $$C = r^2 v = R^2 R'$$, yields: $$\begin{aligned} @@ -115,7 +115,7 @@ $$\begin{aligned} &= p_\infty + \rho \bigg( 2 (R')^2 + R R'' - \frac{1}{2} (R')^2 + \nu \frac{4}{R} R' \bigg) + \alpha \frac{2}{R} \end{aligned}$$ -Rearranging this and defining $\Delta p \equiv P - p_\infty$ +Rearranging this and defining $$\Delta p \equiv P - p_\infty$$ leads to the Rayleigh-Plesset equation: $$\begin{aligned} |