2.1 Elastic forces

The interface of a capsule is considered to be thin and elastic, as described by the Skalak model (Skalak et al. Reference Skalak, Tozeren, Zarda and Chien1973). According to the Skalak law, the non-dimensional membrane elastic tension in the principal direction $i$ (where $i$ can be $s$ or $\unicode[STIX]{x1D719}$ , see figure 1 d) is expressed as a function of the principal stretch ratios ( $\unicode[STIX]{x1D706}_{i,j}$ ) for a strain hardening membrane as

(2.3) $$\begin{eqnarray}\tilde{T}_{i}^{SK}=\frac{G^{SK}}{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{j}}[\unicode[STIX]{x1D706}_{i}^{2}(\unicode[STIX]{x1D706}_{i}^{2}-1)+C(\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{j})^{2}\{(\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{j})^{2}-1\}].\end{eqnarray}$$

Membrane tension in the principal direction $j$ can be expressed by interchanging indices in the constitutive relation (2.3). In this work, $i$ is the meridional and $j$ is the azimuthal principal direction, and the corresponding tensions are considered as $T_{s}$ and $T_{\unicode[STIX]{x1D719}}$ . The principal stretch ratios are defined as $\unicode[STIX]{x1D706}_{s}=\text{d}\tilde{s}/\text{d}\tilde{S}$ and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D719}}=\tilde{r}/\tilde{R}$ , where $s$ and $r$ are the measures of arclength and the radius of the deformed capsule, and the upper case alphabets represent their counterparts in the stress-free shape, respectively. The first and second terms on the right-hand side of equation (2.3) are the measures of the shear deformation associated with the modulus $G^{SK}$ and the area dilatation with the modulus $CG^{SK}$ , respectively (Skalak et al. Reference Skalak, Tozeren, Zarda and Chien1973). The area dilatation parameter $C$ regulates the extent of change of surface area; a large value of $C$ restricts the membrane dilatation. In the small deformation limit, $G^{SK}$ is related to the surface Young modulus, $E_{s}$  (Barthès-Biesel, Diaz & Dhenin Reference Barthès-Biesel, Diaz and Dhenin2002) as $E_{s}=2G^{SK}(1+2C)/(1+C)$ .

The membrane elastic traction (force/area), governed by the combined contribution of the elastic tensions (force/length) obtained from the constitutive law, is given by $\unicode[STIX]{x0394}\boldsymbol{f}^{el}=\unicode[STIX]{x1D70F}_{n}^{el}\boldsymbol{n}+\unicode[STIX]{x1D70F}_{t}^{el}\boldsymbol{t}$ . The components of the elastic stresses can be obtained as

(2.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{n}^{el}=-(K_{s}T_{s}+K_{\unicode[STIX]{x1D719}}T_{\unicode[STIX]{x1D719}}), & \displaystyle\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{t}^{el}=\frac{\text{d}T_{s}}{\text{d}s}+\frac{1}{r}\frac{\text{d}r}{\text{d}s}(T_{s}-T_{\unicode[STIX]{x1D719}}), & \displaystyle\end{eqnarray}$$

$K_{s}=|\text{d}\boldsymbol{t}/\text{d}s|$

$K_{\unicode[STIX]{x1D719}}=n_{r}/r$

$s$

$\unicode[STIX]{x1D719}$

$t_{y}=n_{r}=\text{d}y/\text{d}s$

$t_{r}=-n_{y}=\text{d}r/\text{d}s$

2.2 Resistance to bending

Even though a thin elastic membrane offers negligible resistance to bending, it can have a significant contribution to the restoring forces, especially in the regions of high curvature. The overall non-dimensional bending force (force/area) is given by

(2.5) $$\begin{eqnarray}\unicode[STIX]{x0394}\boldsymbol{f}^{b}=\hat{\unicode[STIX]{x1D705}}_{b}[2\unicode[STIX]{x0394}_{s}H+(2H-c_{0})(2H^{2}-2K_{G}+c_{0}H)]\boldsymbol{n},\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D705}}_{b}=\unicode[STIX]{x1D705}_{b}/a^{2}E_{s}$ is the non-dimensional bending rigidity, $H=(k_{s}+k_{\unicode[STIX]{x1D719}})/2$ is the mean curvature, $K_{G}=k_{s}k_{\unicode[STIX]{x1D719}}$ is the Gaussian curvature, and $c_{0}$ is the spontaneous curvature (Vlahovska, Podgorski & Misbah Reference Vlahovska, Podgorski and Misbah2009; Yazdani, Kalluri & Bagchi Reference Yazdani, Kalluri and Bagchi2011). In the axisymmetric cylindrical coordinate system, the Laplace–Beltrami of the mean curvature is given as (Hu, Kim & Lai Reference Hu, Kim and Lai2014)

(2.6) $$\begin{eqnarray}\unicode[STIX]{x0394}_{s}H=\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}_{s}H)=\frac{1}{r|\boldsymbol{x}_{s}|}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\left(\frac{r}{|\boldsymbol{x}_{s}|}\frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}s}\right),\end{eqnarray}$$

where $|\boldsymbol{x}_{s}|=\sqrt{(\unicode[STIX]{x2202}r/\unicode[STIX]{x2202}s)^{2}+(\unicode[STIX]{x2202}y/\unicode[STIX]{x2202}s)^{2}}$ . A spectral method is used to calculate the higher-order derivatives of mean curvature with respect to the arclength, especially for improved accuracy in estimating the bending force (Trefethen Reference Trefethen1996). For the boundary integral analyses of the deformation of a capsule and RBC in both the extensional flow and electric field, a fixed value of $\hat{\unicode[STIX]{x1D705}}_{b}=0.001$ is considered (Freund Reference Freund2014; Guckenberger & Gekle Reference Guckenberger and Gekle2017).

2.3 Electrostatics

For the analysis of the dynamics of deformation of a biconcave-discoid capsule in an externally applied DC electric field, the charge relaxation time of the outer fluid, $\tilde{t}_{e}=\unicode[STIX]{x1D716}_{e}\unicode[STIX]{x1D716}_{0}/\unicode[STIX]{x1D70E}_{e}$ , is considered as the scaling for time, where $\unicode[STIX]{x1D716}_{0}$ is the permittivity of the free space. The bulk fluids are assumed to be electroneutral on account of $\unicode[STIX]{x1D705}^{-1}\ll a$ , where $\unicode[STIX]{x1D705}^{-1}$ is the Debye screening length. Other time scales of relevance are the hydrodynamic response time, $\tilde{t}_{H}=\unicode[STIX]{x1D707}_{e}a/E_{s}$ , the Maxwell–Wagner relaxation time, $\tilde{t}_{MW}=\unicode[STIX]{x1D716}_{0}(\unicode[STIX]{x1D716}_{i}+2\unicode[STIX]{x1D716}_{e})/(\unicode[STIX]{x1D70E}_{i}+2\unicode[STIX]{x1D70E}_{e})$ , and the membrane charging time, $\tilde{t}_{cap}=aC_{m}(1/\unicode[STIX]{x1D70E}_{i}+1/2\unicode[STIX]{x1D70E}_{e})$ . The non-dimensional counterparts of these time scales are $t_{e}=1$ , $t_{H}=\tilde{t_{H}}/\tilde{t}_{e}$ , $t_{MW}=\tilde{t}_{MW}/\tilde{t}_{e}=(2+\unicode[STIX]{x1D716}_{r})/(2+\unicode[STIX]{x1D70E}_{r})$ and $t_{cap}=\tilde{t}_{cap}/\tilde{t}_{e}={\hat{C}}_{m}(1/2+1/\unicode[STIX]{x1D70E}_{r})$  (Grosse & Schwan Reference Grosse and Schwan1992). Fluid velocity ( $\tilde{\boldsymbol{u}}$ ), electric field ( $\tilde{\boldsymbol{E}}$ ), potential ( $\tilde{V}$ ), and stress are scaled by $E_{s}/\unicode[STIX]{x1D707}_{e}$ , $E_{0}$ , $E_{0}a$ and $E_{s}/a$ , respectively.

The electric potential inside and outside the biconcave-discoid capsule satisfies the Laplace equation, $\unicode[STIX]{x1D6FB}^{2}V_{i,e}=0$ . The electric potential can be obtained by solving the Laplace equation and using Green’s theorem as

(2.7a ) $$\begin{eqnarray}\displaystyle \frac{1}{2}V_{i}(\boldsymbol{x}_{0}) & = & \displaystyle \int _{s}[G^{E}(\boldsymbol{x},\boldsymbol{x}_{0})\unicode[STIX]{x1D735}V_{i}(\boldsymbol{x})\boldsymbol{\cdot }\boldsymbol{n}(\boldsymbol{x})-V_{i}(\boldsymbol{x})\boldsymbol{n}(\boldsymbol{x})\boldsymbol{\cdot }\unicode[STIX]{x1D735}G^{E}(\boldsymbol{x},\boldsymbol{x}_{0})]\,\text{d}s(\boldsymbol{x}),\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle \frac{1}{2}V_{e}(\boldsymbol{x}_{0}) & = & \displaystyle V^{\infty }(\boldsymbol{x}_{0})\nonumber\\ \displaystyle & & \displaystyle -\int _{s}[G^{E}(\boldsymbol{x},\boldsymbol{x}_{0})\unicode[STIX]{x1D735}V_{e}(\boldsymbol{x})\boldsymbol{\cdot }\boldsymbol{n}(\boldsymbol{x})-V_{e}(\boldsymbol{x})\boldsymbol{n}(\boldsymbol{x})\boldsymbol{\cdot }\unicode[STIX]{x1D735}G^{E}(\boldsymbol{x},\boldsymbol{x}_{0})]\,\text{d}s(\boldsymbol{x}),\qquad\end{eqnarray}$$

$G^{E}(\boldsymbol{x},\boldsymbol{x}_{0})=1/4\unicode[STIX]{x03C0}|\hat{\boldsymbol{x}}|$

$\hat{\boldsymbol{x}}=\boldsymbol{x}-\boldsymbol{x}_{0}$

$\boldsymbol{x}_{0}$

$\boldsymbol{x}$

The discontinuity in electrical potential at the interface is termed as the transmembrane potential ( $V_{m}=V_{i}-V_{e}$ ), and it can be calculated by solving (2.7a ) and (2.7b ) along with electrical current continuity across the membrane:

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{r}E_{n,i}+\unicode[STIX]{x1D716}_{r}\frac{\text{d}E_{n,i}}{\text{d}t}=E_{n,e}+\frac{\text{d}E_{n,e}}{\text{d}t}={\hat{C}}_{m}\frac{\text{d}V_{m}}{\text{d}t}+{\hat{G}}_{m}V_{m},\end{eqnarray}$$

where $E_{n,i,e}$ are normal electric fields inside and outside of the membrane. The current in the fluids is assumed to be Ohmic and displacement currents corresponding to the Ohmic resistance and the capacitance in the bulk to be in a parallel circuit. In this model, the thickness of the double layer is infinitesimally thin such that the interfacial free charge is given by $E_{n,e}-\unicode[STIX]{x1D716}_{r}E_{n,i}=q$ , where $q$ is the non-dimensional charge density. The non-dimensional capacitance and conductance of the membrane are ${\hat{C}}_{m}=aC_{m}/\unicode[STIX]{x1D716}_{e}\unicode[STIX]{x1D716}_{0}$ and ${\hat{G}}_{m}=aG_{m}/\unicode[STIX]{x1D70E}_{e}$ .

The Maxwell electric stress tensor in a fluid is defined as $\tilde{\unicode[STIX]{x1D749}}^{E}=\unicode[STIX]{x1D716}\unicode[STIX]{x1D716}_{0}(\tilde{\boldsymbol{E}}\tilde{\boldsymbol{E}}-(1/2)\tilde{E}^{2}\unicode[STIX]{x1D644})$ , where $\unicode[STIX]{x1D644}$ is the identity tensor. The net non-dimensional electric traction at the interface is given by $\unicode[STIX]{x0394}\boldsymbol{f}^{E}=\boldsymbol{n}\boldsymbol{\cdot }(\unicode[STIX]{x1D749}_{e}^{E}-\unicode[STIX]{x1D749}_{i}^{E})=\unicode[STIX]{x1D70F}_{n}^{E}\boldsymbol{n}+\unicode[STIX]{x1D70F}_{t}^{E}\boldsymbol{t}$ , where the components of electric stresses are

(2.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{n}^{E}={\textstyle \frac{1}{2}}[(E_{n,e}^{2}-E_{t,e}^{2})-\unicode[STIX]{x1D716}_{r}(E_{n,i}^{2}-E_{t,i}^{2})], & \displaystyle\end{eqnarray}$$

(2.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{t}^{E}=E_{n,e}E_{t,e}-\unicode[STIX]{x1D716}_{r}E_{n,i}E_{t,i}. & \displaystyle\end{eqnarray}$$

2.4 Hydrodynamics

From the values of parameters in table 1, the particle Reynolds number is $Re_{p}\sim 10^{-2}$ ; therefore the flow inside and outside of the capsule can be described by the Stokes equations (Secomb Reference Secomb2017)

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D749}^{H}=0,\end{eqnarray}$$

where $\boldsymbol{u}$ is the fluid velocity and $\unicode[STIX]{x1D749}^{H}$ is the viscous stress. Viscous stresses for the inner and outer fluid media are given by

(2.11a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D749}_{i}^{H}=-P\unicode[STIX]{x1D644}+\unicode[STIX]{x1D706}(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u}^{\text{T}}), & \displaystyle\end{eqnarray}$$

(2.11b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D749}_{e}^{H}=-P\unicode[STIX]{x1D644}+(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u}^{\text{T}}), & \displaystyle\end{eqnarray}$$

$P$

Far from the capsule membrane ( $|\boldsymbol{x}|\rightarrow \infty$ ), $\boldsymbol{u}\rightarrow \boldsymbol{u}^{\infty }$ , where $\boldsymbol{u}^{\infty }$ is assumed to be the undisturbed flow velocity of the outer fluid and is given by the dimensional form of the applied velocity in (2.1). The solution of the above equations gives rise to an integral equation for the interfacial velocity Rallison & Acrivos (Reference Rallison and Acrivos1978) in a non-dimensional form as

(2.12) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x}_{0}) & = & \displaystyle \frac{2}{1+\unicode[STIX]{x1D706}}\boldsymbol{u}^{\infty }(\boldsymbol{x}_{0})-\frac{1}{1+\unicode[STIX]{x1D706}}\frac{1}{4\unicode[STIX]{x03C0}}\int _{s}\unicode[STIX]{x0394}\boldsymbol{f}(\boldsymbol{x})\boldsymbol{\cdot }\unicode[STIX]{x1D642}(\boldsymbol{x},\boldsymbol{x}_{0})\,\text{d}s(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{4\unicode[STIX]{x03C0}}\frac{1-\unicode[STIX]{x1D706}}{1+\unicode[STIX]{x1D706}}\int _{s}\boldsymbol{u}(\boldsymbol{x})\boldsymbol{\cdot }\unicode[STIX]{x1D64C}(\boldsymbol{x},\boldsymbol{x}_{0})\boldsymbol{\cdot }\boldsymbol{n}(\boldsymbol{x})\,\text{d}s(\boldsymbol{x}),\end{eqnarray}$$

where $\unicode[STIX]{x1D642}(\boldsymbol{x},\boldsymbol{x}_{0})$ and $\unicode[STIX]{x1D64C}(\boldsymbol{x},\boldsymbol{x}_{0})$ are Green’s functions for velocity and stress, respectively, and $\unicode[STIX]{x0394}\boldsymbol{f}$ is the unbalanced non-hydrodynamic traction at the interface. For an unbounded three-dimensional flow, explicit expressions for these tensors are

(2.13a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D642}(\boldsymbol{x},\unicode[STIX]{x1D66D}_{0})=\frac{\unicode[STIX]{x1D644}}{|\hat{\boldsymbol{x}}|}+\frac{\hat{\boldsymbol{x}}\hat{\boldsymbol{x}}}{|\hat{\boldsymbol{x}}|^{3}},\quad \unicode[STIX]{x1D64C}(\boldsymbol{x},\boldsymbol{x}_{0})=-6\frac{\hat{\boldsymbol{x}}\hat{\boldsymbol{x}}\hat{\boldsymbol{x}}}{|\hat{\boldsymbol{x}}|^{5}}.\end{eqnarray}$$

The general boundary integral equation (2.12) is solved for the velocity of the interface of an axisymmetric deformable capsule in an axisymmetric flow and applied electric field. For the analysis of the dynamics of a biconcave-discoid capsule and an RBC subjected to a uniform electric field, the stagnant outer fluid medium is considered; therefore $\boldsymbol{u}^{\infty }=0$ .

Finally, the shape evolves to $\boldsymbol{x}(t+\unicode[STIX]{x0394}t)$ with the kinematic condition given by

(2.14) $$\begin{eqnarray}\boldsymbol{x}(t+\unicode[STIX]{x0394}t)=\boldsymbol{x}(t)+k_{f}\boldsymbol{u}(\boldsymbol{x})\unicode[STIX]{x0394}t,\end{eqnarray}$$

where $\boldsymbol{x}(t)$ is the shape of the capsule at the current time, $t$ , and $\unicode[STIX]{x0394}t$ is the time step considered for the boundary integral simulation. In (2.14), the kinematic factor $k_{f}$ depends upon the scaling of the variables, and is described in the corresponding sections.

2.5 Pressure calculation

The inside and outside pressure of a capsule can be calculated using the boundary integral equations, given as

(2.15a ) $$\begin{eqnarray}\displaystyle P_{i}(\boldsymbol{x}_{0}) & = & \displaystyle -\frac{1}{8\unicode[STIX]{x03C0}}\int _{s}\unicode[STIX]{x0394}\boldsymbol{f}(\boldsymbol{x})\boldsymbol{\cdot }\boldsymbol{p}(\boldsymbol{x},\boldsymbol{x}_{0})\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D706}\frac{1-\unicode[STIX]{x1D706}}{8\unicode[STIX]{x03C0}}\int _{s}\boldsymbol{u}(\boldsymbol{x})\boldsymbol{\cdot }\unicode[STIX]{x1D72B}(\boldsymbol{x},\boldsymbol{x}_{0})\boldsymbol{\cdot }\boldsymbol{n}(\boldsymbol{x})\,\text{d}S(\boldsymbol{x}),\end{eqnarray}$$

(2.15b ) $$\begin{eqnarray}\displaystyle P_{e}(\boldsymbol{x}_{0}) & = & \displaystyle -\frac{1}{8\unicode[STIX]{x03C0}}\int _{s}\unicode[STIX]{x0394}\boldsymbol{f}(\boldsymbol{x})\boldsymbol{\cdot }\boldsymbol{p}(\boldsymbol{x},\boldsymbol{x}_{0})\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1-\unicode[STIX]{x1D706}}{8\unicode[STIX]{x03C0}}\int _{s}\boldsymbol{u}(\boldsymbol{x})\boldsymbol{\cdot }\unicode[STIX]{x1D72B}(\boldsymbol{x},\boldsymbol{x}_{0})\boldsymbol{\cdot }\boldsymbol{n}(\boldsymbol{x})\,\text{d}S(\boldsymbol{x}),\end{eqnarray}$$

$\boldsymbol{p}(\boldsymbol{x},\boldsymbol{x}_{0})$

$\unicode[STIX]{x1D72B}(\boldsymbol{x},\boldsymbol{x}_{0})$

(2.16a,b ) $$\begin{eqnarray}\boldsymbol{p}(\boldsymbol{x},\boldsymbol{x}_{0})=2\frac{\hat{\boldsymbol{x}}}{\hat{\boldsymbol{x}}^{3}},\quad \unicode[STIX]{x1D72B}(\boldsymbol{x},\boldsymbol{x}_{0})=4\left(-\frac{\unicode[STIX]{x1D644}}{|\hat{\boldsymbol{x}}|^{3}}+3\frac{\hat{\boldsymbol{x}}\hat{\boldsymbol{x}}}{|\hat{\boldsymbol{x}}|^{5}}\right).\end{eqnarray}$$

3.1 Dynamics of a capsule in extensional flow

The scaling of (2.1) gives rise to a non-dimensional quantity $Ca_{f}=\unicode[STIX]{x1D707}ea/E_{s}$ , which is termed as the flow capillary number and determines the dynamics of the deformation of a capsule. The hydrodynamic force at the interface is balanced by the elastic and bending forces, i.e. $\unicode[STIX]{x0394}\boldsymbol{f}^{H}=-(\unicode[STIX]{x0394}\boldsymbol{f}^{el}+\unicode[STIX]{x0394}\boldsymbol{f}^{b})$ . With the values of the parameters in table 1 for the simulation of a biconcave-discoid capsule in extensional flow, the kinematic factor $k_{f}=1$ . The Skalak membrane parameter $C$ is assumed to be $1$ and $10$ for two separate studies.

The analysis of the dynamics of deformation of a capsule in extensional flow is presented for a small dilatation parameter ( $C=1$ in this case). It is observed that a considerable change in area ( ${\sim}13\,\%$ ) is admitted when the capsule is subjected to a strong extensional flow. The capillary numbers are chosen in such a way that they demonstrate three possible modes of deformation. Figure 2(a–c) shows that at a small value of destabilizing force $(Ca_{f}=0.015)$ , the deformation is small and the biconcavity is maintained. The capsule slowly deforms such that the concavities at the poles are affected the most (see figure 2 a,b); subsequently, the flow reduces the depression (in a way opening them up) and eventually reaches a steady-state biconcave-cylindrical shape, as shown in figure 2(c). A capsule does not open up into a prolate spheroid until a threshold capillary number is reached (in this case, $Ca_{f}=0.017$ ). At such a threshold capillary number $(Ca_{f}=0.017)$ the capsule shows a remarkable shape transition wherein the biconcave cavities (see figure 2 d,e) open up and transform into a steady capped-cylindrical shape (a cylindrical body with convex caps at both ends) (see figure 2 f). At a still higher capillary number $(Ca_{f}=0.1)$ , the dynamics is faster, and very similar behaviour is observed (see figure 2 g–i), and the capsule attains a highly stretched prolate-spheroid shape (see figure 2 i).

Figure 2. Observed shapes during the evolution of a capsule at (a) $t=10$ , (b) $t=40$ , (c) $t=\infty$ for $Ca_{f}=0.015$ ; (d) $t=20$ , (e) $t=70$ (f) $t=\infty$ for $Ca_{f}=0.017$ ; and (g) $t=2$ , (h) $t=7$ , (i) $t=\infty$ for $Ca_{f}=0.1$ , considering the dilatation parameter $C=1$ . (j) Equilibrium displacement of the north pole from the initial position of an undeformed biconcave-discoid as a function of $Ca_{f}$ for $C=1$ (——) and $C=10$ (— —).

Figure 2(j) represents the equilibrium displacement of the north pole ( $(y_{p}-y_{p0})/y_{p0}$ , where $y_{p}$ and $y_{p0}$ are the positions of the pole in the deformed and undeformed states, respectively) from its position in the undeformed shape as a function of capillary number. Figure 2(j) suggests that the evolution of the biconcave discoid to the capped cylindrical and prolate spheroid undergoes a discontinuous shape transition at the critical capillary number, i.e. $Ca_{f}=0.017$ for $C=1$ and $Ca_{f}=0.04$ for $C=10$ . For the deformation of a Skalak capsule, a large dilatation parameter restricts the change in surface area, although at the cost of numerical stiffness. Therefore, the analysis is restricted to a maximum value of dilatation parameter $C=10$ allowing ${\sim}10\,\%$ change in surface area. The higher threshold value ( $Ca_{f}=0.04$ compared to $Ca_{f}=0.017$ at $C=1$ ) of the capillary number at $C=10$ suggests the greater extent of the membrane resistance to the deformation that leads to a biconcave-discoid to capped-cylindrical transition as well as a prolate-spheroid shape transition.

Figure 3. At $Ca_{f}=0.1$ and $C=1$ . (a–c) Pressure profile. (d–f) Variation of the meridional, $T_{s}$ (— —), azimuthal, $T_{\unicode[STIX]{x1D719}}$ (— ⋅ —) and mean, $T_{m}$ (——) elastic tensions along the arclength. (g–i) Streamlines (shown only in the right-hand half) and the velocity profile (relative magnitude of the flow is proportional to the length of the short arrows shown in the left-hand half). From left to right the columns correspond to times $t=2$ , $7$ and $\infty$ , respectively.

To understand the mechanism for the transition from a biconcave shape to a prolate spheroid, the pressure profile and elastic tension over the half arclength are plotted in figure 3(a–c) and figure 3(d–f), respectively. The applied free stream uniaxial extensional flow does not have an associated pressure profile because the velocity is linear. When a rigid spherical particle is placed in such a flow, it generates a velocity field to bring the velocity of the applied flow to zero on the particle surface. This generates stress on the particle from the equator to the poles, which the particle transmits to the fluid (from poles to equator), thereby leading to a stresslet-induced disturbance velocity field. Since, in the case of a rigid spherical particle, the particle brings the fluid to rest at its surface, the fluid has to generate a positive pressure at the equator to resist the incoming flow and a negative pressure at the poles to suck fluid into the poles. When the surface/interface is not rigid (e.g. a drop), it can give way under this pressure and build a curvature. In a drop, this can result in a dimple at the equator and a bump at the pole (the drop will try to adjust its curvature to satisfy the Young–Laplace equation at the interface) leading to a prolate-spheroid shape. Similar arguments will mean that the shape of a drop will be oblate in biaxial extensional flow.

In a capsule, the meridional and azimuthal tensions could be anisotropic and compressive, unlike the isotropic tension in a liquid–liquid interface (a drop). While a spherical drop cannot maintain its shape in extensional flow, a capsule can maintain a near-spherical shape by generating a pressure inside which is intermediate between that at the equator and the poles on the outer side. This can be attained by a compressive membrane tension (predominantly azimuthal) at the equator and a tensile membrane tension at the poles. In the case of a biconcave-discoid capsule, a similar distribution of membrane tension (compressive at the equator and tensile at the poles) can support an extensional flow, except the pressure inside is now less than that at the poles.

When the capillary number is increased beyond the threshold, at short times ( $t=2$ ), the pressure is lower at the poles, thereby generating an internal flow from the equator to the poles. This leads to the shoulders of the biconcave-discoid capsule opening up. The pressure generated at the equator is so high that the resulting inside pressure at the poles continues to drive the fluid to the poles ( $t=7$ ). This fluid flow (see figure 3 g–i) eventually leads to the opening up of the poles and transformation of the biconcave-discoid shape into a prolate spheroid (see figure 2 i).

The elastic tensions are predominantly compressive, except at the shoulders. At the poles, as the curvature is negative and the inside pressure is higher than at outside (see figure 3 a,b), the stresses have to be compressive (see figure 3 d,e). The tension at the shoulders, however, is tensile, leading to a higher pressure inside the capsule, which further assists the shoulders to disappear by pushing the fluid into the poles.

For the deformed prolate spheroid at the steady state, the inside pressure is uniform at the equator (see figure 3 c); therefore, the flow inside disappears (see figure 3 i). The pressure inside is nearly identical to the outside pressure, due to the small curvature at the equator. Thereby, the mean elastic tension at the equator disappears (see figure 3 f). At the poles, the outside pressure is lower compared to the inside pressure (see figure 3 c), and since the curvature is positive, a high tensile mean elastic tension (see figure 3 f) is generated.

3.2 Dynamics of a biconcave-discoid capsule in DC electric fields

The analysis of deformation of an elastic biconcave-discoid capsule in DC electric fields is carried out in the absence of free stream fluid velocity, $\boldsymbol{u}^{\infty }=0$ . Therefore, the hydrodynamic traction is balanced by the elastic traction, electric traction, and the traction due to bending rigidity, i.e.

(3.1) $$\begin{eqnarray}\unicode[STIX]{x0394}\boldsymbol{f}^{H}=-(\unicode[STIX]{x0394}\boldsymbol{f}^{el}+Ca_{e}\unicode[STIX]{x0394}\boldsymbol{f}^{E}+\hat{\unicode[STIX]{x1D705}}_{b}\unicode[STIX]{x0394}\boldsymbol{f}^{b}),\end{eqnarray}$$

where $Ca_{e}=\unicode[STIX]{x1D716}_{e}\unicode[STIX]{x1D716}_{0}aE_{0}^{2}/E_{s}$ is the electric capillary number. The scaling of variables results in the kinematic factor as the hydrodynamic response time, i.e. $k_{f}=1/t_{H}$ . Considering the parameters in table 1, $E_{s}\sim 0.1~\text{N}~\text{m}^{-1}$ , $\unicode[STIX]{x1D707}_{e}\sim 0.1~\text{Pa}~\text{s}$ , $\unicode[STIX]{x1D70E}_{e}\sim 0.1~\text{mS}~\text{m}^{-1}$ and $\unicode[STIX]{x1D716}_{e}\sim 80$ , for the dynamics of a capsule with an equivalent radius $a\sim 5~\unicode[STIX]{x03BC}\text{m}$ , the hydrodynamic response time can be calculated to be unity, i.e. $t_{H}=t_{e}=1$ . For this analysis, the consideration of $t_{H}=t_{e}=1$ implies that the hydrodynamic response of the capsule is over the same time scale as the evolution of the electric stresses. Also, the membrane of the capsule is considered to be purely capacitive with ${\hat{C}}_{m}=50$ and ${\hat{G}}_{m}=0$ .

3.2.1 More conducting outer fluid medium: $\unicode[STIX]{x1D70E}_{r}=0.1$

Figure 4. Observed shapes during the evolution of a biconcave-discoid capsule at (a) $t=26$ and (b) $t=\infty$ for $Ca_{e}=0.02$ and $\unicode[STIX]{x1D70E}_{r}=0.1$ . Corresponding streamlines (shown only in the right-hand half), velocity profile (relative magnitude of the flow is proportional to the length of the short arrows shown in left-hand half) and electric stress distribution (shown by short arrows from the interface) are shown in (c) and (d), respectively. Observed shapes during deformation at (e) $t=0$ and (f) $t=15$ at $Ca_{e}=0.05$ and $\unicode[STIX]{x1D70E}_{r}=0.1$ . Streamlines, velocity profile and electric stress distribution are shown in (g) corresponding to the shape shown in (e). The variation of the meridional, $T_{s}$ (— —), azimuthal, $T_{\unicode[STIX]{x1D719}}$ (— ⋅ —) and mean, $T_{m}$ (——) membrane tensions along the arclength are shown in the inset of (g).

When the outer fluid is more conducting than the inner fluid $(\unicode[STIX]{x1D70E}_{r}=0.1)$ , at $Ca_{e}=0.02$ , a biconcave-discoid capsule initially undergoes compression at the poles (see figure 4 a,c). In this case, the shapes are presented as 2D cross-sections in the plane parallel to the axis of symmetry, for better visualization in small deformation. At $t<t_{cap}$ , the polarization vector is in the direction opposite to the applied electric field; therefore, the normal electric stresses are compressive at the poles (see figure 4 c). Later, at $t>t_{cap}$ , the membrane becomes charged, and the electric stress becomes compressive, with the maxima at the equator (see figure 4 d). This drives the poles apart, and the capsule reaches a steady shape, as shown in figure 4(b). At this low capillary number, a biconcave-discoid capsule does not undergo large deformation due to the weak electric stresses at the interface.

At significantly high capillary number ( $Ca_{e}=0.05$ ), a biconcave-discoid capsule exhibits a pinched discoid with the poles coming in close contact (see figure 4 f). At $t<t_{cap}$ , strong compressive electric stresses at the poles and fluid flow from poles to the equator are observed (see figure 4 g, corresponding to figure 4 f). Beyond this state, the high fluid flow and electric stresses indicate possible breakup of a capsule, which manifests as a numerical singularity.

3.2.2 More conducting inner fluid medium: $\unicode[STIX]{x1D70E}_{r}=10$

Figure 5. Observed shapes during the evolution of a biconcave-discoid capsule at (a) $t=20$ , (b) $t=85$ and (c) $t=\infty$ for $Ca_{e}=0.3$ and $\unicode[STIX]{x1D70E}_{r}=10$ . Corresponding streamlines (shown only in the right-hand half), velocity profile (relative magnitude of the flow is proportional to the length of the short arrows shown in the left-hand half) and electric stress distribution (shown by short arrows from the interface) are shown in (d), (e) and (f), respectively. The variation of meridional, $T_{s}$ (— —), azimuthal, $T_{\unicode[STIX]{x1D719}}$ (— ⋅ —) and mean, $T_{m}$ (——) membrane tensions along the arclength are shown in the insets.

Figure 6. Observed shapes during the evolution of a biconcave-discoid capsule at (a) $t=10$ , (b) $t=23$ , (c) $t=29$ , (d) $t=37$ and (e) $t=\infty$ for $Ca_{e}=0.5$ and $\unicode[STIX]{x1D70E}_{r}=10$ . Corresponding streamlines (shown only in the right-hand half), velocity profile (relative magnitude of the flow is proportional to the length of the short arrows shown in the left-hand half) and electric stress distribution (shown by short arrows from the interface) are shown in (f), (g), (h), (i) and (j), respectively. The variation of meridional, $T_{s}$ (— —), azimuthal, $T_{\unicode[STIX]{x1D719}}$ (— ⋅ —) and mean, $T_{m}$ (——) membrane tensions along the arclength are shown in the insets.

Figure 7. Transmembrane potential at the interface during the shape evolution of a biconcave-discoid capsule at $(\cdots \,)$   $t=5$ , (— —) $t=20$ , (— ⋅ —) $t=35$ , (– ⋅ ⋅) $t=55$ , (– – ⋅) $t=85$ and (——) $t=\infty$ for $Ca_{e}=0.3$ (a), and at $(\cdots \,)$   $t=10$ , (— —) $t=23$ , (— ⋅ —) $t=29$ , (– ⋅ ⋅) $t=37$ , (– – ⋅) $t=50$ and (——) $t=\infty$ for $Ca_{e}=0.5$ (b).

Figure 8. Observed shapes during the deformation of a biconcave-discoid capsule for $Ca_{e}=0.4$ and $\unicode[STIX]{x1D70E}_{r}=10$ at (a) $t=0$ , (b) $20$ , (c) $42$ , (d) $52$ and (e) $\infty$ , considering $k_{f}=1$ and at (f) $t=0$ , (g) $500$ , (h) $1500$ , (i) $5000$ and (j) $\infty$ , considering $k_{f}=0.01$ .

When the inner fluid medium is more conducting, the capsule does not collapse; instead, it deforms through a series of cylindrical shapes to a steady-state shape. At a small capillary number $(Ca_{e}=0.3)$ , a biconcave-discoid capsule deforms through an intermediate cylindrical shape with concave ends (see figure 5 a). In this case, the polarization vector aligns with the applied electric field; therefore, the electric stresses are tensile near the poles and highest at the shoulders (see figure 5 d). Just after $t>t_{cap}$ , as the membrane becomes charged, the tensile electric stresses at the poles disappear, and the compressive electric stresses develop at the equator. In this case, as the normal tensile electric stresses disappear, the intermediate shape (see figure 5 b) relaxes back (because of the stabilizing elastic traction) to an equilibrium shape (see figure 5 c). This equilibrium shape is the result of the balance of the elastic traction and compressive electric stress (see figure 5 f).

At a high capillary number $(Ca_{e}=0.5)$ , the biconcave-discoid capsule at short times deforms into cylindrical shapes with diminishing concave ends near the poles (see figure 6 a,b) due to high tensile Maxwell stress at the poles and the shoulders (see figure 6 f,g). In this case, since the hydrodynamic time scale is small ( $t_{H}=t_{e}=1$ ), the deformation progresses rapidly. At $t<t_{cap}$ , due to the tensile electric stress (with the maxima near the poles, see figure 6 h) the biconcave-discoid shape evolves into a cylindrical shape (see figure 6 c). The tensile normal electric stress at the poles (see figure 6 i) continues to drive the capsule to evolve into a prolate-spheroid shape (see figure 6 d). Eventually, when the membrane becomes fully charged ( $t>t_{cap}$ ), it attains a steady-state capped-cylindrical shape (see figure 6 e), wherein the compressive electric stress at the equator (see figure 6 j) is appropriately balanced by the elastic tensions.

Thus, similar to the deformation in extensional flow (see figure 2), a biconcave-discoid capsule deforms into a steady-state capped cylinder (see figure 6), but via an intermediate prolate-spheroid shape (see figure 6 d), when the inner fluid medium is more conducting $(\unicode[STIX]{x1D70E}_{r}=10)$ . In the former case, the deformation is driven by the hydrodynamic pressure generated by the extensional flow around the biconcave-discoid capsule, whereas in the latter case, it is due to the Maxwell electric stress developed at the interface of the capsule.

During the deformation, the transmembrane potential evolves due to the charging of the membrane as well as the change in the shape of the biconcave-discoid capsule. A high transmembrane potential indicates a greater tendency to electroporate. Figure 7(a,b) shows the variation of transmembrane potential over the interface for different intermediate shapes during the deformation at $Ca_{e}=0.3$ and $Ca_{e}=0.5$ , respectively. From figure 7(a) it can be observed that the transmembrane potential is always higher at the shoulders until the membrane becomes fully charged, as the shape remains biconcave during the deformation (figure 5 a,b). For the charged membrane and the steady-state shape (figure 5 c), the transmembrane potentials from the poles to the shoulders are comparable and higher than the other locations at the interface. From figure 7(b) it can be observed that in the deformation at $Ca_{e}=0.5$ the transmembrane potential is higher at the shoulders for the cylinders with concave ends (figure 6 a–c), whereas it is higher at the poles for the prolate-spheroid (figure 6 d–e) shapes.

When the hydrodynamic response time and the electric response time are same, $t_{H}=t_{e}=1$ , i.e. $k_{f}=1$ , at the critical capillary number ( $Ca_{e}=0.4$ ), a biconcave discoid evolves into a capped cylinder through the intermediate shapes, such as biconcave cylinder and stretched prolate spheroid, similar to the case of $Ca_{e}=0.5$ (see figure 6). At this capillary number ( $Ca_{e}=0.4$ ), the shape evolution is shown in figure 8(a–e). To demonstrate the effect of slow hydrodynamics, the shape evolutions of a biconcave-discoid capsule at $Ca_{e}=0.4$ and $\unicode[STIX]{x1D70E}_{r}=10$ are presented in figure 8(f–j) considering $k_{f}=0.01$ . In this case, a biconcave-discoid capsule does not respond instantly to the developed electric stresses at the interface. Therefore, for $t<t_{cap}$ , the intermediate stretched biconcave-cylindrical (see figure 8 b), stretched cylindrical (see figure 8 c), and highly deformed prolate-spheroid (see figure 8 d) shapes are missing in the deformation considering $k_{f}=0.01$ . Instead, the biconcave-discoid capsule undergoes a slow deformation through few intermediate shapes (see figure 8 g–i), and eventually attains a capped-cylindrical shape (see figure 8 j) at the equilibrium. Hence, for different hydrodynamic response times (here, $t_{H}=1$ and $100$ , i.e. $k_{f}=1$ and $0.01$ ), even though the final equilibrium shape, a biconcave-discoid capsule attains, is same (see figure 8 e and figure 8 j), the pathways for the shape evolution are different.

3.3 Dynamics of an RBC in AC electric fields

The electrohydrodynamic deformation of a biconcave-discoid capsule can be extended to explore its suitability for studying an RBC in an electric field. In physiological conditions, the non-dimensional parameters relevant for an RBC are: viscosity ratio $\unicode[STIX]{x1D706}=5$ , ratio of dielectric constants $\unicode[STIX]{x1D716}_{r}=0.6$ , conductivity ratio $\unicode[STIX]{x1D70E}_{r}=1.054$ , non-dimensional capacitance ${\hat{C}}_{m}=282$ and non-dimensional conductance ${\hat{G}}_{m}=6.75\times 10^{-7}$ . Considering scaling parameters as used in the case of deformation of a biconcave-discoid capsule, for the deformation of an RBC, the non-dimensional hydrodynamic time scale is very large (non-dimensional $t_{H}=0.625\times 10^{6}$ ). Therefore, the kinematic factor $k_{f}=1/t_{H}\sim O(10^{-6})$ implies that a negligible displacement (2.14) of the interface will take place in a specific numerical time step, $\unicode[STIX]{x0394}t\sim O(\tilde{t}_{e})$ , suggesting that the computation time required for reaching a steady shape could be prohibitively high.

Since the hydrodynamic response time is very high compared to the electric time scale, the electric field at the interface reaches an equilibrium value (with respect to the transmembrane potential that corresponds to the charging time of the membrane and the frequency of the applied field) even before the capsule starts responding to the electric stress developed at the interface. The electric field (2.7) is, therefore, solved with the capacitor time scale, $\tilde{t}_{cap}={\hat{C}}_{m}\,\times$ electric time scale (i.e. $\tilde{t}_{cap}={\hat{C}}_{m}\times \tilde{t}_{e}=282\times \unicode[STIX]{x1D716}_{e}\unicode[STIX]{x1D716}_{0}/\unicode[STIX]{x1D70E}_{e}$ ), and the transmembrane potential is allowed to reach a steady value at a particular shape of the RBC. Moreover, as the charge relaxation in both the inner and outer fluids are very fast ( $\unicode[STIX]{x1D716}_{e}/\unicode[STIX]{x1D70E}_{e}\ll 1$ and $\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D70E}_{i}\ll 1$ ), a simplification of the electric current continuity (2.8) across the interface is made as

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{r}E_{n,i}=E_{n,e}=\frac{\text{d}V_{m}}{\text{d}t_{e}}.\end{eqnarray}$$

Subsequently, the hydrodynamic equation for RBC deformation (2.12) is solved with the already calculated equilibrium electric stress, considering the hydrodynamic response time $(t_{H}=\unicode[STIX]{x1D707}_{e}a/E_{s})$ as the scaling parameter for the time.

All the variables but time (here $t_{e}=\tilde{t}/{\hat{C}}_{m}\times \tilde{t}_{e}$ ) for solving the electric potential (2.7) are scaled with the scaling parameters used for the solution of electrostatics in the case of a biconcave-discoid elastic capsule in a DC electric field, as discussed in the previous section. Thus, $1/({\hat{C}}_{m}\times \tilde{t}_{e})$ is used for scaling the frequency of the applied field. For solving the hydrodynamic boundary integral equation (2.12), the hydrodynamic time scale is used for scaling.

The hydrodynamic force is balanced by the non-hydrodynamic forces similar to the case of deformation of an elastic capsule in a DC electric field (3.1). In this case, time-averaged electric stress (ignoring the time-periodic stress that has instantaneous dynamics but zero average) is used in the calculation of interfacial velocity (in (2.12)). For faster numerical calculations, the viscosity ratio is considered to be unity, $\unicode[STIX]{x1D706}=1$ . The kinematic condition (2.14) for the shape evolution from the calculated interfacial velocity, because of $k_{f}=1$ , simplifies to

(3.3) $$\begin{eqnarray}\boldsymbol{x}(t+\unicode[STIX]{x0394}t)=\boldsymbol{x}(t)+\boldsymbol{u}(\boldsymbol{x})\unicode[STIX]{x0394}t.\end{eqnarray}$$

Figure 9. Transmembrane potential of the static shape of an RBC as a function of frequency. Marker point ○ represents $t_{cap}^{-1}$ .

The analysis of the deformation of an RBC in AC electric fields is conducted at a high frequency of the applied field ( $\unicode[STIX]{x1D714}=2.5$ , corresponding to ${\sim}10~\text{MHz}$ ). This is important in suggested experiments, wherein the transmembrane potential ( $V_{m}$ ) has to be restricted to a very low value to prevent the possibility of electroporation (see figure 9) (Joshi & Hu Reference Joshi and Hu2012; Das & Thaokar Reference Das and Thaokar2018a ). The electric capillary number ( $Ca_{e}$ ) remains the same as used in the case of the deformation of a biconcave-discoid capsule in a DC field. In the case of the RBC deformation, $Ca_{e}=1$ is equivalent to the applied electric field strength of $0.45~\text{kV}~\text{cm}^{-1}$ , much below the typical $1~\text{kV}~\text{cm}^{-1}$ or higher used in electroporation studies.

At $Ca_{e}=0.5$ , the small electric stresses at the interface lead to a small deformation of the RBC (see figure 10 a,b). The transmembrane potential reaches an equilibrium value before the hydrodynamic calculation begins. Since $\unicode[STIX]{x1D714}<t_{cap}^{-1}$ , the electric stress is purely compressive, with the maxima at the equator (similar to the case of a biconcave-discoid capsule in DC electric field, shown in figure 5 f). This compressive electric stress is responsible for the deformation of the RBC into a cylinder with concave ends. At larger capillary number ( $Ca_{e}=1$ ), a substantial increase in the deformation is observed (see figure 10 g,h). From figure 10(c,d,i,j), it can be observed that the azimuthal, meridional and mean elastic tensions are compressive near the equator and tensile near the poles.

Consideration of the area dilatation parameter $C=1$ allows a significant change in the area during the deformation of an RBC, i.e. 4.7 % and 13.8 % at $Ca_{e}=0.5$ and $Ca_{e}=1$ , respectively. At a higher value of the area dilatation parameter ( $C=10$ ), the constraint on the change in area (0.82 % and 1.75 % at $Ca_{e}=0.5$ and 1, respectively) restricts the deformation of an RBC. Considering $C=10$ , the shapes observed during the deformation of an RBC at $Ca_{e}=0.5$ and 1 are shown in figure 10(e,f,k,l), respectively. It can be easily visualized that the deformation of an RBC considering $C=10$ is considerably less compared to the deformation with $C=1$ .

Figure 10. Shapes observed during the deformation of an RBC at (a) $t=10$ and (b) $t=\infty$ for $Ca_{e}=0.5$ and at (g) $t=20$ and (h) $t=\infty$ for $Ca_{e}=1$ in an AC field with $\unicode[STIX]{x1D714}=2.5$ , considering $C=1$ . Variation of the meridional, $T_{s}$ (— —), azimuthal, $T_{\unicode[STIX]{x1D719}}$ (— ⋅ —) and mean, $T_{m}$ (——) elastic tensions along the arclength for the corresponding cases are shown in (c,d) and (i,j), respectively. Shapes observed during the deformation of an RBC at (e) $t=24$ and (f) $t=\infty$ for $Ca_{e}=0.5$ and at (k) $t=20$ and (l) $t=\infty$ for $Ca_{e}=1$ , considering $C=10$ .

Figure 11. At $Ca_{e}=0.5$ . (a) Transmembrane potential at the north pole in a single cycle of the applied electric field during the calculation of average electric stress at (– –) $t=10$ and (——) $t=\infty$ . (b) Variation of the transmembrane potential at the end of the particular cycle of the applied field over the arclength at (– –) $t=10$ and (——) $t=\infty$ , corresponding to figure 10(a,b), respectively.

Figure 12. At $Ca_{e}=1$ . (a) Transmembrane potential at the north pole in a single cycle of the applied electric field during the calculation of average electric stress at (– –) $t=20$ and (——) $t=\infty$ . (b) Variation of the transmembrane potential at the end of the particular cycle of the applied field over the arclength at (– –) $t=20$ and (——) $t=\infty$ , corresponding to figure 10(g,h), respectively.

Figure 13. Shapes observed during the deformation of an RBC at (a) $t=10$ , (b) $t=70$ and (c) $t=\infty$ for $Ca_{e}=0.5$ in an AC field with $\unicode[STIX]{x1D714}=2.5$ , considering $C=1$ and $\unicode[STIX]{x1D70E}_{r}=10$ .

Figure 14. At $Ca_{e}=0.5$ and $\unicode[STIX]{x1D70E}_{r}=10$ . (a) Transmembrane potential at the north pole in a single cycle of the applied electric field during the calculation of the average electric stress at (– –) $t=10$ , (— ⋅ —) $t=70$ and (——) $t=\infty$ . (b) Variation of the transmembrane potential at the end of the particular cycle of the applied field over the arclength at (– –) $t=10$ , (— ⋅ —) $t=70$ and (——) $t=\infty$ , corresponding to figure 13(a–c), respectively.

To understand the mechanical integrity of the cell membrane to the applied electric field, the analysis of the transmembrane potential, during the deformation is important. Figure 11(a) shows that at $Ca_{e}=0.5$ , the transmembrane potential of the membrane at the north pole of an RBC during the calculation of the average electric stress in a cycle of the applied electric field varies with time. Figure 11(b) represents the variation of the transmembrane potential of the interface at the end of the cycle. It can be clearly observed in figure 11(b) that the shoulders attain the maximum transmembrane potential, suggesting the possibility of electroporation in those regions. Similar behaviour of the membrane is suggested in figure 12(a,b) for the deformation of an RBC at $Ca_{e}=1$ .

In an experimental condition with a low-conducting outer fluid medium (in this case, $\unicode[STIX]{x1D70E}_{r}=10$ ), an RBC undergoes large deformation even at a low capillary number ( $Ca_{e}=0.5$ ). The strong compressive electric stress causes an RBC to deform through a cylindrical shape with biconcave ends (see figure 13 a) to a cylinder (see figure 13 b), finally resulting in a prolate spheroid (see figure 13 c). Unlike the case of a biconcave-discoid capsule, an RBC does not show the intermediate large deformation (as observed in figure 6 d), as the partial charging of the membrane results in consistent tensile electric stress at the poles and compressive electric stress at the equator. In this case, since a small value of area dilatation parameter ( $C=1$ ) is used, a significant change in the area is observed. From figure 14(a,b), it can be observed that for the intermediate cylinder with concave ends (figure 13 a) and a cylindrical shape (figure 13 b) the transmembrane potential is the highest at the shoulders. On the other hand, when the RBC evolves into a prolate spheroid at the time-averaged steady state, the transmembrane potential is maximum at the poles, thereby suggesting the possibility of electroporation at the poles.