Figure 1

A freely floating planar lipid bilayer membrane separating fluids with different physical (viscosity $\mu$) and electrical (permittivity $\varepsilon$ and conductivity $\sigma$) properties. A uniform electric field $E_0$ is applied far from the membrane. The time-dependent displacement of the membrane above the $x$-$y$ plane along the $z$ direction is $h(x,y,t)$.Reuse & Permissions

Figure 2

Illustration of Hill's method (see text), for $q=10$, $R=10$, $S=1$, $\omega =1$, $\delta =1$, $\mathrm{Ca}={10}^5$, $\beta =1$, $g_m=20$, $\alpha =0.1$, $\lambda =1$, and ${\xi }_0=0$. We plotted all the complex solutions $\zeta$ of the eigenvalue problem. For clarity, we kept only the first three modes of the Fourier series ($N=3$), and accordingly there are $6\times 3+3=21$ eigenvalues. The eigenvalues segregate into three distinct families of approximately equal real parts, and their imaginary parts should differ by $i\omega$ (here $\omega =0.05$). This is not the case for the family with the highest real part: too few modes were taken into account to guarantee satisfactory precision of the calculations ($N=50$ modes are used throughout this work).Reuse & Permissions

Figure 3

Growth rate $s$ as a function of wave number $q$, for $R=10$, $S=1$, $\mathrm{Ca}={10}^5$, $\beta =1$, $g_m=0$, $\alpha =0.1$, $\lambda =1$, and ${\xi }_0=0$. Dashed lines correspond to dc electric field $\delta =0$ and solid lines correspond to a sinusoidal electric field $\delta =1$ with frequency $\omega =3$. For each family of curves, $g_m$ increases from bottom to top: $g_m=$ 0 (black), 4 (red), 20, (green), and 1000 (blue). Note that, unlike what happens for a dc field, in the ac case an instability develops even for $g_m=0$. As $g_m$ increases, the ac and dc growth curves collapse (they become indistinguishable on the graph for $g_m>20$):+symbols correspond to the $g_m\gg 1$ limit, which is the same for ac and dc fields.Reuse & Permissions

Figure 4

Fastest growth rate $s_{\max }$ as a function of $\omega$ for $R=10$, $S=1$, $\delta =1$, $\mathrm{Ca}={10}^5$, $\beta =1$, $\alpha =0.1$, $\lambda =1$, ${\xi }_0=0$, and different values of $g_m$. The short horizontal lines starting at the origin of each curve represent the values calculated for the dc field $\delta =0$.Reuse & Permissions

Figure 5

Maximum growth rate $s_{\max }$ as a function of $\omega$ for $R=10$, $\delta =1$, $\mathrm{Ca}={10}^5$, $\beta =1$, $g_m=0$, $\alpha =0.1$, $\lambda =1$, $\xi =0$, and different values of $S$. The horizontal lines are calculated from the approximate solution (42).Reuse & Permissions

Figure 6

Numerical integration of the system of equations (28), (29), and (30), for $R=10$, $S=1$, $\omega =1$, $\delta =1$, $\mathrm{Ca}={10}^5$, $\beta =1$, $g_m=0$, $\alpha =0.1$, $\lambda =1$, and $\xi =0$. (a) Evolution of $h_q$ (solid lines). Dashed lines represent the normalized variation around an exponential growth, which is within $1\%$ of $h_q$. (b) In contrast, $A_{1q}$ and $A_{2q}$ vary almost sinusoidally around 0 (with a slowly growing amplitude).Reuse & Permissions

Figure 7

(a) Growth rate as a function of wave number for $R=10$, $S=1$, $\omega =1$, $\delta =1$, $\mathrm{Ca}={10}^5$, $\beta =1$, $g_m=0$, $\alpha =0.1$, $\lambda =1$, and ${\xi }_0=0$. (b) shows the largest growth rate as a function of $\omega$. In both figures, the solid line is the result of the Floquet analysis, while the dashed line represents the approximate solution (46). As expected, the latter is valid only for low frequencies ($\omega <1/\alpha$), since it does not take into account displacement currents.Reuse & Permissions

Figure 8

Properties of the instability for $R=10$, $S=1$, $\omega =1$, $\delta =1$, $\beta =1$, $g_m=0$, $\alpha =0.1$, $\lambda =1$, and $\xi =0$. (a) Growth rate $s$ as a function of $q$, for increasing values of $\mathrm{Ca}$. The growth rate converges towards a limit curve (dashed line), which is calculated numerically by setting ${\mathrm{Ca}}^{-1}=0$ in the system of equations. (b) Whereas the most unstable wave number $q_{\max }$ (and $s_{\max }$) converges towards a finite value, the highest unstable wave number $q_c$ diverges as $\mathrm{Ca}\to +\infty$.Reuse & Permissions

Figure 9

Properties of the instability for $R=10$, $S=1$, $\omega =1$, $\delta =1$, $\mathrm{Ca}={10}^5$, $g_m=0$, $\alpha =0.1$, $\lambda =1$, and $\xi =0$. (a) Growth rate $s$ as a function of $q$, for increasing values of $\beta$. The fastest growing wave number and the extent of the unstable region are almost independent of $\beta$. (b) The maximum growth rate $s_{\max }$, however, increases almost linearly with $\beta$.Reuse & Permissions