Collapse of a bubble in an electric field

A low-Mach-number analysis is presented of the collapse of a bubble in an electric field, which is assumed to be homogeneous, but may be unsteady. Ellipsoidal shape deformations are accounted for in the analysis, but are assumed to be small. It is shown that the presence of an electric field leads to additional terms in a modified Rayleigh-Plesset equation. This differential equation for the bubble radius and a corresponding equation for ellipsoidal shape deformations have been integrated numerically. The results indicate that a bubble can be made to collapse by instantaneously switching on an electric field. Also, nonharmonic volumetric oscillations are observed for time-dependent electric fields of sufficiently large amplitude. It is shown that the rate of a collapse driven by external pressure variations due, for instance, to acoustic forcing can be accelerated.

Figure 1

The effect of varying $W$ on the steady solutions for $R$ and $R_2$ for different ${\Pi }_o$. The solid curves from top to bottom in panels (a) and (b) have ${\Pi }_o=10$, 1, and 0.75. The dashed curves in (a) and (b) were generated with ${\Pi }_o=0.75$, $\lambda =0.1$, and $K_{\text{in}}∕K_{\text{out}}=0.5$. The dot-dashed lines in (a) and (b) represent $R=1-(3∕4W)∕(3\gamma {\Pi }_o-2)$ (with $\gamma =1$ and ${\Pi }_o=0.75$) and $R_2=3W∕16$, respectively. The rest of the parameters are $\gamma =1$ and $H=0.113$.

Figure 2

The effect of varying $\lambda$ on the steady solutions for $q$ for $K_{\text{in}}∕K=0$, 0.5, and 1. The dotted lines represent steady solutions for $q$ obtained using the limiting formula given by Eq. (52). The rest of the parameter values remain unchanged from Fig. 1.

Figure 3

The effect of varying $\Pi$ on the steady solutions for $R$ and $R_2$ for different $W$. The curves below $R=1$ in panel (a) from top to bottom have $W=0.01$, 0.05, and 0.1. In panel (b), the solid, dashed, and dot-dashed lines have $W=0.01$, 0.05, and 0.1, respectively. The rest of the parameter values remain unchanged from Fig. 1.

Figure 4

The effect of varying the initial conditions on the solution for $R\left(t\right)$ and $R_2\left(t\right)$ with $W=0.11$, $\mathrm{Oh}=0.037$, $\mathrm{Ma}=1.81\times {10}^{-3}$, ${\Pi }_o=13.7$, $\lambda =0$, and $(K,K_{\text{in}})=(4.67\times {10}^{-3},0)$. The rest of the parameters are the same as in Fig. 1.

Figure 5

The effect of varying ${\Pi }_o$ on the solution for $R\left(t\right)$ and $R_2\left(t\right)$. The rest of the parameters remain unchanged from Fig. 4.

Figure 6

The effect of varying $W$ and $\Omega$ on the solution for $R\left(t\right)$ and $R_2\left(t\right)$, shown in (a) and (c) and (b) and (d), respectively. In (a) and (c), $E\left(t\right)=\sin \left(\Omega t\right)$, ${\Pi }_o=1.39$, and $\Omega =0.62$; in (b) and (d), $E\left(t\right)=\sin \left(\Omega t\right)$ and $W=0.11$. The rest of the parameters remain unchanged from Fig. 4.

Figure 7

The effect of varying $W$ on the solution for $R\left(t\right)$ and $R_2\left(t\right)$ with time-dependent pressure and electric-field forcing. (a) and (d) $P\left(t\right)=-0.85{\Pi }_a\sin \left(\Omega t\right)$, $E\left(t\right)=(1∕2)\mathbf{(}\tanh \left[30(t-3.2)\right]-\tanh \left[30(t-4)\right]\mathbf{)}$; (b) and (e) $P\left(t\right)=-{\Pi }_a\sin \left(\Omega t\right)$, $E\left(t\right)=(1∕2)\mathbf{(}\tanh \left[30(t-3.8)\right]-\tanh \left[30(t-4.2)\right]\mathbf{)}$; (c) and (f) $P\left(t\right)=-1.5{\Pi }_a\sin \left(\Omega t\right)$, $E\left(t\right)=(1∕2)\mathbf{(}\tanh \left[30(t-5)\right]-\tanh \left[30(t-6)\right]\mathbf{)}$. $\Omega =0.62,{\Pi }_0=13.7$ and the rest of the parameters remain unchanged from Fig. 4.