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author | Prefetch | 2021-07-28 14:27:37 +0200 |
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committer | Prefetch | 2021-07-28 14:27:37 +0200 |
commit | e12c7ce372ecaa042d85d9fb76371a75ff518d1a (patch) | |
tree | e3658cdf571dc8107a1c0ba7c5e1efe82870dad7 /content/know/concept/rayleigh-plesset-equation | |
parent | bffb355fd906723dcf7e587ce6ad16c751ed8abe (diff) |
Expand knowledge base, fix a:visited CSS
Diffstat (limited to 'content/know/concept/rayleigh-plesset-equation')
-rw-r--r-- | content/know/concept/rayleigh-plesset-equation/index.pdc | 82 |
1 files changed, 55 insertions, 27 deletions
diff --git a/content/know/concept/rayleigh-plesset-equation/index.pdc b/content/know/concept/rayleigh-plesset-equation/index.pdc index ee8622b..9325f3f 100644 --- a/content/know/concept/rayleigh-plesset-equation/index.pdc +++ b/content/know/concept/rayleigh-plesset-equation/index.pdc @@ -19,34 +19,29 @@ describes how the radius of a spherical bubble evolves in time inside an incompressible liquid. Notably, it leads to [cavitation](/know/concept/cavitation/). - -## Simple form - -The simplest version of the Rayleigh-Plesset equation is found -in the limiting case of a liquid with zero viscosity zero surface tension. - -Consider one of the [Euler equations](/know/concept/euler-equations/) -for the velocity field $\va{v}$, -where $\rho$ is the (constant) density: +Consider the main +[Navier-Stokes equations](/know/concept/navier-stokes-equations/) +for the velocity field $\va{v}$: $$\begin{aligned} \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v} - = - \frac{\nabla p}{\rho} + = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v} \end{aligned}$$ 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 $r$. -The above Euler equation then becomes: +The above equation then becomes: $$\begin{aligned} \pdv{v}{t} + v \pdv{v}{r} = - \frac{1}{\rho} \pdv{p}{r} + + \nu \bigg( \frac{1}{r^2} \pdv{r} \Big( r^2 \pdv{v}{r} \Big) - \frac{2}{r^2} v \bigg) \end{aligned}$$ Meanwhile, the incompressibility condition -is as follows in this situation: +in [spherical coordinates](/know/concept/spherical-coordinates/) yields: $$\begin{aligned} \nabla \cdot \va{v} @@ -63,42 +58,75 @@ $$\begin{aligned} \end{aligned}$$ Where $C(t)$ is an unknown function that does not depend on $r$. -We then insert this result in the earlier Euler equation, +We then insert this result in the main Navier-Stokes equation, and isolate it for $\pdv*{p}{r}$, yielding: $$\begin{aligned} \pdv{p}{r} - = - \rho \bigg( \pdv{v}{t} + v \pdv{v}{r} \bigg) + = - \rho \bigg( \frac{1}{r^2} C' - \frac{2}{r^5} C^2 + - \nu \Big( \frac{2}{r^4} C - \frac{2}{r^4} C \Big) \bigg) = - \rho \bigg( \frac{1}{r^2} C' - \frac{2}{r^5} C^2 \bigg) \end{aligned}$$ Integrating this with respect to $r$ yields the following expression for $p$, -where $p_\infty$ is the (possibly time-dependent) pressure at $r = \infty$: +where $p_\infty(t)$ is the (possibly time-dependent) pressure at $r = \infty$: + +$$\begin{aligned} + p(r) + = p_\infty + \rho \bigg( \frac{1}{r} C' - \frac{1}{2 r^4} C^2 \bigg) +\end{aligned}$$ + +From the definition of [viscosity](/know/concept/viscosity/), +we know that the normal [stress](/know/concept/cauchy-stress-tensor/) +$\sigma_{rr}$ in the liquid is given by: + +$$\begin{aligned} + \sigma_{rr}(r) + = - p(r) + 2 \rho \nu \pdv{v(r)}{r} +\end{aligned}$$ + +We now consider a spherical bubble +with radius $R(t)$ and interior pressure $P(t)$ along its surface. +Since we know the liquid pressure $p(r)$, +we can find $P$ from $\sigma_{rr}(r)$. +Furthermore, to include the effects of surface tension, we simply add +the [Young-Laplace law](/know/concept/young-laplace-law/) to $P$: + +$$\begin{aligned} + P + = - \sigma_{rr}(R) + \alpha \frac{2}{R} + = p(R) - 2 \rho \nu \Big( \frac{-2}{R^3} C \Big) + \alpha \frac{2}{R} +\end{aligned}$$ + +We isolate this for $p(R)$, and equate it to +our expression for $p(r)$ +at the surface $r\!=\!R$: $$\begin{aligned} - p(r, t) - = p_\infty(t) + \rho \bigg( \frac{1}{r} C'(t) - \frac{1}{2 r^4} C^2(t) \bigg) + 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}$$ -We now consider a spherical bubble with radius $R(t)$ and pressure $P(t)$ along the liquid surface. -To study the liquid boundary's movement, we set $r = R$ and $p = P$, -and see that $R'(t) = v(t)$, such that $C = r^2 V = R^2 R'$. -We thus arrive at: +Isolating for $P$, +and inserting the fact that $R'(t) = v(t)$, +such that $C = r^2 v = R^2 R'$, +yields: $$\begin{aligned} P - &= p_\infty + \rho \bigg( \frac{1}{R} \dv{(R^2 R')}{t} - \frac{1}{2 R^4} (R^2 R')^2 \bigg) + &= p_\infty + \rho \bigg( \frac{1}{R} \dv{(R^2 R')}{t} - \frac{1}{2 R^4} (R^2 R')^2 + + \nu \frac{4}{R^3} (R^2 R') \bigg) + \alpha \frac{2}{R} \\ - &= p_\infty + \rho \bigg( 2 (R')^2 + R R'' - \frac{1}{2} (R')^2 \bigg) + &= 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}$$ -Rearranging this and defining $\Delta p = P - p_\infty$ -leads to the simple Rayleigh-Plesset equation: +Rearranging this and defining $\Delta p \equiv P - p_\infty$ +leads to the Rayleigh-Plesset equation: $$\begin{aligned} \boxed{ - R \dv[2]{R}{t} + \frac{3}{2} \bigg( \dv{R}{t} \bigg)^2 - = \frac{\Delta p}{\rho} + \frac{\Delta p}{\rho} + = R \dv[2]{R}{t} + \frac{3}{2} \bigg( \dv{R}{t} \bigg)^2 + \nu \frac{4}{R} \dv{R}{t} + \frac{\alpha}{\rho} \frac{2}{R} } \end{aligned}$$ |