2. Problem formulation

A circular liquid domain of size $a$ and viscosity ${\it\eta}_{d}$ moves with velocity $U$ along $X$ -axis in a flat fluid membrane sheet of viscosity ${\it\eta}_{m}$ . The space above and below the domain and membrane sheet is occupied by a liquid with viscosity ${\it\mu}_{s}$ . A schematic diagram of the system is shown in figure 1. All the fluids considered are incompressible and undergo low-Reynolds-number flow.

Figure 1.

Schematic diagram of a circular liquid domain of size $a$ and thickness $h$ moving with velocity $U$ along the $X$ -axis in a membrane sheet of the same thickness. The viscosities of the domain, the surrounding membrane and the surrounding fluid are ${\it\eta}_{d}$ , ${\it\eta}_{m}$ and ${\it\mu}_{s}$ , respectively. The schematic diagram on the right is the sectional view in the $x{-}z$ plane. Both the membrane and the surrounding fluid span to infinity in their respective directions.

The governing equations for various domains are as follows. For the 3D fluid above ( $z>0$ ) and below ( $z<h$ ) the membrane sheet, including the domain, we have

(2.1a,b ) $$\begin{eqnarray}{\it\mu}_{s}{\rm\Delta}\boldsymbol{v}=\boldsymbol{{\rm\nabla}}p\quad \text{and}\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=0,\end{eqnarray}$$

where $\boldsymbol{v}(r,{\it\theta},z)=(v_{r},v_{{\it\theta}},v_{z})$ and $p(r,{\it\theta},z)$ are the velocity and pressure fields in the surrounding fluid, respectively. The ${\it\Delta}$ and $\boldsymbol{{\rm\nabla}}$ are the Laplacian and the gradient operators, respectively. For the membrane and domain liquid ( $-h\leqslant z\leqslant 0$ ), the variation of the quantities across the thickness direction is precluded due to its molecular structure. Further, because of the symmetry about the mid-plane of the membrane we assume that the velocity in the $z$ direction is zero. Thus for the in-plane motion in the domain and the surrounding membrane we have (Saffman Reference Saffman1976)

(2.2a−c ) $$\begin{eqnarray}{\it\eta}_{d}{\rm\Delta}\boldsymbol{u}+2{\it\sigma}/h=\boldsymbol{{\rm\nabla}}{\it\Pi}\quad (\text{for}~r<a),\quad {\it\eta}_{m}{\rm\Delta}\boldsymbol{u}+2{\it\sigma}/h=\boldsymbol{{\rm\nabla}}{\it\Pi}\quad (\text{for}~r>a)\quad \text{and}\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

where $\boldsymbol{u}(r,{\it\theta})=(u_{r},\;u_{{\it\theta}},\;0)$ and ${\it\Pi}(r,{\it\theta})$ are the velocity and pressure fields, respectively, and ${\it\sigma}$ is the traction (or drag) applied on the top and bottom surfaces of the membrane and domain by the surrounding fluid. No-slip boundary conditions at the membrane-surrounding liquid and domain–membrane interfaces are applied and the velocity decays to zero at infinity for both the membrane and the surrounding fluid. Note that the no-slip boundary condition at the membrane-surrounding liquid interface leads to the following relations between the velocity fields in the surrounding fluid and the domain/membrane:

(2.3a,b ) $$\begin{eqnarray}u_{r}(r,{\it\theta})=v_{r}(r,{\it\theta},z=0)\quad \text{and}\quad u_{{\it\theta}}(r,{\it\theta})=v_{{\it\theta}}(r,{\it\theta},z=0).\end{eqnarray}$$

The incompressibility conditions for the surrounding and membrane fluids are separately imposed (see equation (2.8) of Saffman (Reference Saffman1976) and equation (2.22) of Fujitani (Reference Fujitani2011)). However, it is worth mentioning that as both the domain and its surrounding are assumed to be incompressible fluids, the in-plane flow field must be divergence-free at the domain boundary. This condition has not been imposed in the present study.

Equations (2.1) and (2.2), along with the boundary conditions, can be solved analytically for the pressure and velocity fields by exploiting the rotational symmetry about the $z$ -axis and use of Fourier and Hankel transforms (Saffman Reference Saffman1976; Fujitani Reference Fujitani2011). The contributions to the drag force on the circular liquid domain moving with a steady velocity $U$ come from the tangential components of the tractions applied to the domain by the surrounding membrane at the cylindrical interface (at $r=a$ ) and by the surrounding fluid at the top and bottom circular interfaces of radius $a$ . The expression for the magnitude of the drag force is given by the integral

(2.4) $$\begin{eqnarray}F=\lim _{\tilde{r}\rightarrow 1^{+}}\frac{{\rm\pi}}{a}\int _{0}^{\infty }\,\text{d}z\,zA(z)\text{J}_{2}(z\tilde{r}),\end{eqnarray}$$

where $\tilde{r}=r/a$ dimensionless radial distance and $A(z)$ is an as yet undetermined function. The details of the derivation of (2.4) have been omitted here to avoid repetition. They are available in Fujitani (Reference Fujitani2011). The quantity $A(z)$ is obtained, in principle, by solving the following two integral equations:

(2.5) $$\begin{eqnarray}A(z)=\frac{2acU}{{\rm\pi}z}\text{J}_{1}(z)+{\it\epsilon}\int _{0}^{\infty }\,\text{d}t\frac{t}{1+{\it\beta}t}\frac{\left[z\text{J}_{0}(t)\text{J}_{1}(z)-t\text{J}_{0}(z)\text{J}_{1}(t)\right]A(t)}{z^{2}-t^{2}}\end{eqnarray}$$

and

(2.6) $$\begin{eqnarray}2{\it\mu}_{s}Ua^{2}=\int _{0}^{\infty }\,\text{d}t\frac{A(t)\text{J}_{1}(t)}{t(1+{\it\beta}t)},\end{eqnarray}$$

where $c$ is an unknown constant, ${\it\beta}={\it\eta}_{m}/(2{\it\mu}_{s}a)$ is the Boussinesq number representing the ratio between the membrane drag and surrounding fluid drag, ${\it\epsilon}=({\it\eta}_{d}-{\it\eta}_{m})/{\it\eta}_{d}$ denotes the relative viscosity contrast between the domain and the surrounding membrane, and $\text{J}_{i}(z)$ is the Bessel function of $i$ th order and first kind. Note that, typically, $0\leqslant {\it\epsilon}\leqslant 1$ , and ${\it\epsilon}=0$ indicates that the domain and membrane have same viscosity while ${\it\epsilon}=1$ implies that the domain is rigid or the membrane viscosity is very small in comparison to the domain viscosity. Further, for ${\it\epsilon}<0$ the surrounding membrane has higher viscosity than the domain, which is not the case in general. However, it is possible to make two-phase lipid bilayer vesicles in the laboratory with appropriately chosen lipid composition such that the domain viscosity is smaller than the surrounding. A typical value of ${\it\epsilon}$ is $4/5$ and $0.2<{\it\beta}<1000$ (considering $10^{-9}<{\it\eta}_{m}<10^{-6}~\text{N}~\text{s}~\text{m}^{-1}$ and $0.5<a<2.5~{\rm\mu}\text{m}$ (Brooks et al. Reference Brooks, Fuller, Frank and Robertson1999; Cicuta et al. Reference Cicuta, Keller and Veatch2007; Aliaskarisohi et al. Reference Aliaskarisohi, Tierno, Dhar, Khattari, Blaszczynski and Fischer2010; Stanich et al. Reference Stanich, Honerkamp-Smith, Putzel, Warth, Lamprecht, Mandal, Mann, Hua and Keller2013)).

Equations (2.4)–(2.6) complete the description to obtain the drag force on the domain, and subsequently the diffusion coefficient or mobility of the domain. However, the main difficulty lies in solving the dual integral equations (2.5) and (2.6), particularly when the viscosity contrast is neither zero nor unity. For ${\it\epsilon}=0$ , the solution to the integral equation (2.5) is exact and is given by $A(z)=2acU\,\text{J}_{1}(z)/{\rm\pi}z$ , and through the subsequent use of (2.6) the constant $c$ can be evaluated. For ${\it\epsilon}\ll 1$ , Fujitani (Reference Fujitani2011) obtained a semi-analytical solution to (2.5). However, when ${\it\epsilon}$ is not very small, the dual integral equations (2.5) and (2.6) are not analytically tractable. To our knowledge, no numerical solution is available either. We develop a simple algorithm to obtain a numerical solution to (2.5) and (2.6) using the discrete collocation method (Atkinson Reference Atkinson1997) and Chebyshev polynomials. We first introduce a new variable $B(z)=zA(z)$ and rewrite (2.5) and (2.6) as

(2.7) $$\begin{eqnarray}B(z)=\frac{2acU}{{\rm\pi}}\text{J}_{1}(z)+{\it\epsilon}\int _{0}^{\infty }\,\text{d}t\,M(z,t)B(t)\end{eqnarray}$$

and

(2.8) $$\begin{eqnarray}2{\it\mu}_{s}Ua^{2}=\int _{0}^{\infty }\,\text{d}t\frac{B(t)\text{J}_{1}(t)}{t^{2}(1+{\it\beta}t)},\end{eqnarray}$$

where

(2.9) $$\begin{eqnarray}M(z,t)=\frac{z}{(1+{\it\beta}t)}\frac{[z\text{J}_{0}(t)\text{J}_{1}(z)-t\text{J}_{0}(z)\text{J}_{1}(t)]}{(z^{2}-t^{2})}.\end{eqnarray}$$

Equation (2.7) is in the form of a Fredholm equation of second kind. The drag force in terms of the new variable $B(z)$ is

(2.10) $$\begin{eqnarray}F=\lim _{\tilde{r}\rightarrow 1^{+}}\frac{{\rm\pi}}{a}\int _{0}^{\infty }\,\text{d}z\,B(z)\text{J}_{2}(z\tilde{r}).\end{eqnarray}$$

The expression (2.10) for the drag force in terms of the new variable $B(z)$ eliminates the possibility of the integrand taking prohibitively large values when $z$ is large, making the numerical computation of the integral easier. In the following, we describe the numerical solution procedure for (2.7) and (2.8).

3. Numerical solution of Fredholm integral equation

To solve (2.7), we need to map the infinite domain $[0,\infty )$ to a finite domain (see Collocation Theorem 2.6.2 of Zemyan (Reference Zemyan2012)). To achieve this, we define the change of variable $z=\tan [(X+1){\rm\pi}/4]$ , and rewrite (2.7) and (2.8) as

(3.1) $$\begin{eqnarray}B(X)=\frac{2acU}{{\rm\pi}}\text{J}_{1}(z(X))+{\it\epsilon}\frac{{\rm\pi}}{4}\int _{-1}^{1}\,\text{d}Y\sec ^{2}[(Y+1){\rm\pi}/4]M(X,Y)B(Y)\end{eqnarray}$$

and

(3.2) $$\begin{eqnarray}2{\it\mu}_{s}Ua^{2}=\frac{{\rm\pi}}{4}\int _{-1}^{1}\,\text{d}Y\frac{\sec ^{2}[(Y+1){\rm\pi}/4]B(Y)\text{J}_{1}(t(Y))}{t(Y)^{2}(1+{\it\beta}t(Y))},\end{eqnarray}$$

respectively. In the above $t(Y)=\tan [(Y+1){\rm\pi}/4]$ and the values of $X$ and $Y$ lie in the finite domain $[-1,1]$ .

Now we can approximate $B(X)$ using a set of $n$ linearly independent functions that are continuous in $[-1,1]$ (Zemyan Reference Zemyan2012). We choose Chebyshev polynomials for this purpose:

(3.3) $$\begin{eqnarray}B(X)\approx \mathop{\sum }_{i=1}^{n}B_{i}T_{i-1}(X),\end{eqnarray}$$

where $T_{i}(X)$ is the Chebyshev polynomial of first kind with degree $i$ . Note that the number of collocation points is same as the number of Chebyshev polynomials used to approximate $B(X)$ . By substituting (3.3) into (3.1), we obtain

(3.4) $$\begin{eqnarray}\mathop{\sum }_{i=1}^{n}B_{i}\left[T_{i-1}(X)-{\it\epsilon}\frac{{\rm\pi}}{4}\int _{-1}^{1}\,\text{d}Y\sec ^{2}\left[(Y+1){\rm\pi}/4\right]M(X,\,Y)T_{i-1}(Y)\right]=\frac{2acU}{{\rm\pi}}\text{J}_{1}(z).\end{eqnarray}$$

The most convenient choice of the collocation points for (3.4) are the zeros of the $n$ th degree Chebyshev polynomial, and with this choice we obtain a system of $n$ linear equations with $n$ unknown Chebyshev coefficients:

(3.5) $$\begin{eqnarray}\left[\bar{\unicode[STIX]{x1D64F}}-{\it\epsilon}\frac{{\rm\pi}}{4}\bar{\unicode[STIX]{x1D651}}\right]\boldsymbol{b}=\boldsymbol{R},\end{eqnarray}$$

where $\bar{\unicode[STIX]{x1D651}}$ and $\bar{\unicode[STIX]{x1D64F}}$ are $n\times n$ matrices with elements $v_{i,j}=\int _{-1}^{1}\,\text{d}Y\sec ^{2}[(Y+1){\rm\pi}/4]M(z_{j},t)\times T_{i-1}(Y)$ and $t_{i,j}=T_{i-1}(X_{j})$ , respectively, and $X_{j}$ are the collocation points. The elements of the $n\times 1$ array $\boldsymbol{b}$ of Chebyshev coefficients are $b_{i}=B_{i}{\rm\pi}/2acU$ , and $r_{j}=\text{J}_{1}(z_{j})$ are the elements of the $n\times 1$ array $\boldsymbol{R}$ .

We use the Gauss–Chebyshev quadrature method (Mason & Handscomb Reference Mason and Handscomb2010) to compute the elements $v_{i,j}$ :

(3.6) $$\begin{eqnarray}v_{i,j}\approx \frac{{\rm\pi}}{n}\mathop{\sum }_{k=1}^{n}\left[\sqrt{1-Y_{k}^{2}}\sec ^{2}\left[(Y_{k}+1)\frac{{\rm\pi}}{4}\right]M(z_{j},t_{k})T_{i-1}(Y_{k})\right],\end{eqnarray}$$

where the $Y_{k}$ are the zeros of the $n$ th degree Chebyshev polynomial. Substituting the expression $v_{i,j}$ obtained from (3.6) into (3.5) and solving for $\boldsymbol{b}$ , we obtain the Chebyshev coefficients $B_{i}$ in terms of the unknown constant $c$ . To calculate $c$ , we substitute the Chebyshev polynomial approximation of function $B(X)$ , given by (3.3), into (3.2) and obtain

(3.7) $$\begin{eqnarray}c=4{\it\mu}_{s}a\left/\left\{\frac{{\rm\pi}}{n}\mathop{\sum }_{i=1}^{n}\mathop{\sum }_{k=1}^{n}\left[\frac{\displaystyle \sqrt{1-Y_{k}^{2}}\sec ^{2}\left[(Y_{k}+1)\frac{{\rm\pi}}{4}\right]b_{i}T_{i-1}(Y_{k})\text{J}_{1}(Y_{k})}{t_{k}^{2}(1+{\it\beta}t_{k})}\right]\right\}\right..\end{eqnarray}$$

Note that we use the Gauss–Chebyshev quadrature method to compute the integral appearing in (3.2). Using the results of (3.5) and (3.7), and substituting (3.3) into (2.10), we obtain the approximation to the drag force $F$ on the circular liquid domain of radius $a$ as

(3.8) $$\begin{eqnarray}F\approx \lim _{\tilde{r}\rightarrow 1^{+}}\frac{{\rm\pi}}{a}\mathop{\sum }_{i=1}^{n}\int _{0}^{\infty }\,\text{d}z\,B_{i}T_{i-1}(X)\text{J}_{2}(z\tilde{r}).\end{eqnarray}$$

We decompose the domain of integration appearing in (3.8) into two parts:

(3.9) $$\begin{eqnarray}F\approx \lim _{\tilde{r}\rightarrow 1^{+}}\frac{{\rm\pi}}{a}\mathop{\sum }_{i=1}^{n}\left[\int _{0}^{U_{L}}\,\text{d}z\,B_{i}T_{i-1}(X)\text{J}_{2}(z\tilde{r})+\int _{U_{L}}^{\infty }\,\text{d}z\,B_{i}T_{i-1}(X)\text{J}_{2}(z\tilde{r})\right].\end{eqnarray}$$

Exploiting the boundedness of $T_{i}(\boldsymbol{\cdot })$ and the exponentially decaying behaviour $\text{J}_{2}(z\tilde{r})$ for large $z$ , we choose a $U_{L}<\infty$ such that the second integral in (3.9) is small and can be neglected in comparison with the first integral. Finally, the expression for the drag force becomes

(3.10) $$\begin{eqnarray}F\approx \lim _{\tilde{r}\rightarrow 1^{+}}\frac{{\rm\pi}}{a}\mathop{\sum }_{i=1}^{n}\int _{0}^{U_{L}}\,\text{d}z\,B_{i}T_{i-1}(X)\text{J}_{2}(z\tilde{r}).\end{eqnarray}$$

The above integral is evaluated using Simpson’s rule.

4. Results and discussion

To compute the drag force $F$ using (3.10), we need to choose the number of Chebyshev polynomials (or collocation points) $n$ , the upper limit of integration $U_{L}$ , and the step size $h$ in Simpson’s rule. Furthermore, we need to choose an appropriate value of $\tilde{r}$ larger than but close to unity to replace the limit $\tilde{r}\rightarrow 1^{+}$ . We carry out convergence studies for the drag force $F$ with respect to the parameters. Further, we compute the error in the drag force for two limiting cases in which the analytical solution is available. They are ${\it\epsilon}=0$ (viscosity of domain and membrane are the same (Fujitani Reference Fujitani2011)) and ${\it\epsilon}=1$ (the domain is rigid (Hughes et al. Reference Hughes, Pailthorpe and White1981; Petrov & Schwille Reference Petrov and Schwille2008)).

4.1. Validation of the numerical method for a domain with the same viscosity as the surrounding membrane ( ${\it\epsilon}=0$ ) and a rigid domain ( ${\it\epsilon}=1$ )

For ${\it\epsilon}=0$ the viscosities of the domain and the surrounding membrane are the same. In this case, from (2.7) we obtain $B(X)=2acU\,\text{J}_{1}(X)/{\rm\pi}$ and the drag force on the liquid domain given by (Fujitani Reference Fujitani2011)

(4.1) $$\begin{eqnarray}F=2cU\lim _{\tilde{r}\rightarrow 1^{+}}\int _{0}^{\infty }\,\text{d}z\,\text{J}_{1}(z)\text{J}_{2}(\tilde{r}z)=2cU,\end{eqnarray}$$

where

(4.2) $$\begin{eqnarray}c={\displaystyle \frac{{\rm\pi}{\it\mu}_{s}a}{\displaystyle \int _{0}^{\infty }\,\text{d}z{\displaystyle \frac{\text{J}_{1}^{2}(z)}{(z^{2}+{\it\beta}z^{3})}}}}.\end{eqnarray}$$

In figure 2 we present the variation in the ratio $F/2cU$ computed numerically using (3.10) with respect to the parameters $U_{L}$ , $1/h$ , $\tilde{r}$ and $n$ . Note that the integral $\lim _{\tilde{r}\rightarrow 1^{+}}\int _{0}^{\infty }\,\text{d}z\,\text{J}_{1}(z)\text{J}_{2}(\tilde{r}z)=1$ , and consequently the ratio $F/2cU$ is unity and independent of ${\it\beta}$ . Upon observing the convergence and the closeness of the ratio to unity we find the following parameter values: $n=12\,000,\tilde{r}=1.009$ , $U_{L}=6000,h=0.05$ . However, we also find from figure 2(d) that the ratio is closest to unity for $n=4000$ as well. Thus to save computational time we finally choose $n=4000,\tilde{r}=1.009,U_{L}=6000$ and $h=0.05$ for calculating the drag force.

Figure 2.

Variation of $F/2cU$ for ${\it\epsilon}=0$ (a) with $U_{L}$ for $\tilde{r}=1.009$ , $n=4000,h=0.05$ , (b) with $1/h$ for $\tilde{r}=1.009,n=4000,U_{L}=6000$ , (c) with $\tilde{r}$ for $n=4000,U_{L}=6000,h=0.05$ and (d) with $n$ for $\tilde{r}=1.009,U_{L}=6000,h=0.05$ .

After fixing these parameters, we compute the constant $c$ using (3.7) and compare it with the exact solution (4.2) for different ${\it\beta}$ . The integral in (4.2) is evaluated using the software Mathematica. In figure 3(a,b), we present the variation in the error in $c$ and percentage error in $F/{\it\mu}_{s}aU$ with ${\it\beta}$ . We observe that up to ${\it\beta}=10^{7}$ the error in  $c$ is small and the percentage error in $F$ is below 0.05 %. Thus we claim that our method can approximate the drag force on the liquid domain accurately.

Figure 3.

Error in calculation of constant $c$ versus $\log {\it\beta}$ for ${\it\epsilon}=0$ . The parameters are $n=4000,\,U_{L}=6000$ and $h=0.05$ .

For the case of a rigid domain, Petrov & Schwille (Reference Petrov and Schwille2008) derived a simple analytical expression for the drag force for the solution developed by Hughes et al. (Reference Hughes, Pailthorpe and White1981). In figure 4 we present the variation of the error in dimensionless drag force ( $F/{\it\mu}_{s}aU$ ) for ${\it\beta}=10$ calculated using (3.10) with respect to the parameters $U_{L},\,1/h,\,\tilde{r}$ and $n$ . The drag force computed by Petrov & Schwille (Reference Petrov and Schwille2008) has been taken as the benchmark. We observe that the error is minimum when $\tilde{r}=1.014$ , $n=4000$ , $h=0.06$ and $U_{L}=6000$ and we choose these parameter values for computing the drag force on the domain. The variation of the error and the percentage error in the dimensionless drag force $F/{\it\mu}_{s}aU$ with respect to ${\it\beta}$ have been presented in figure 5(a,b), respectively. The error initially decreases with ${\it\beta}$ and then increases almost linearly. However, the percentage error asymptotically reaches a constant value of around 2.5 % (see inset of figure 5 b). Further, the percentage error is large for ${\it\beta}$ less than 5, and is around 5 % or below for ${\it\beta}$ larger than 5. Thus we say that our method is accurate for larger values of ${\it\beta}$ when ${\it\epsilon}$ is unity.

Figure 4.

Variation of the error in $F/{\it\mu}_{s}aU$ (a) with $U_{L}$ for $\tilde{r}=1.014$ , $n=4000,h=0.06$ , (b) with $1/h$ for $\tilde{r}=1.014$ , $n=4000,U_{L}=6000$ , (c) with $\tilde{r}$ for $n=4000,U_{L}=6000,h=0.06$ and (d) with $n$ for $\tilde{r}=1.014$ , $U_{L}=6000$ , $h=0.06$ . In all the above cases ${\it\beta}=10$ and ${\it\epsilon}=1$ .

Figure 5.

Variation of the error and percentage error in $F/{\it\mu}_{s}aU$ versus ${\it\beta}$ for ${\it\epsilon}=1$ . The parameters are $n=4000,\tilde{r}=1.014,U_{L}=6000,h=0.06$ .

For ${\it\epsilon}$ other than zero or unity, we do not have any analytical or existing results to compare. We carry out convergence studies for the drag force with respect to the parameters $U_{L}$ , $h$ , $\tilde{r}$ and $n$ for a few values of ${\it\epsilon}$ and ${\it\beta}$ and obtain the same parameter values at convergence. The values of ${\it\epsilon}$ and ${\it\beta}$ selected for convergence studies lie in a large range. Thus for subsequent studies we select $n=4000,\tilde{r}=1.014,U_{L}=6000$ , $h=0.06$ .

4.2. Drag on the domain for arbitrary ${\it\epsilon}$

In figure 6(a,b) we present the variation of the dimensionless drag force with the Boussinesq number ${\it\beta}$ for several values of the viscosity contrast ${\it\epsilon}$ between the domain and surrounding membrane. The values of ${\it\epsilon}$ are chosen such that the domain characteristic varies from highly viscous to one having very low viscosity with respect to the surrounding membrane. For example, ${\it\epsilon}=0.8$ implies ${\it\eta}_{d}/{\it\eta}_{m}=5$ , and for ${\it\epsilon}=-20$ we obtain ${\it\eta}_{d}/{\it\eta}_{m}=1/21$ . We have chosen ${\it\beta}$ to vary between 5 and 1000, for most cases, as this lies within the range of its possible values. Further, we obtained accuracy of 95 % or more in the computation of the drag force above ${\it\beta}=5$ in the previous subsection. From figure 6, we observe that, for any specified ${\it\epsilon}$ , the dimensionless drag force increases with ${\it\beta}$ . An increasing ${\it\beta}$ implies increasing surrounding membrane viscosity ${\it\eta}_{m}$ for fixed surrounding fluid viscosity and domain radius. Thus the drag force should increase with ${\it\beta}$ in this scenario.

Figure 6.

Variation of $F/{\it\mu}_{s}aU$ with the Boussinesq number ${\it\beta}$ for different viscosity contrast ${\it\epsilon}$ . The parameters chosen are $n=4000,\tilde{r}=1.014$ , $U_{L}=6000$ , $h=0.06$ .

We observe from figure 6(a) that for positive ${\it\epsilon}$ the dimensionless drag force does not change significantly with ${\it\epsilon}$ . The same is true for $-1\leqslant {\it\epsilon}\leqslant 1$ . Note that when ${\it\epsilon}$ changes from 0 to unity the domain behaviour changes from having the same viscosity as the surrounding membrane to near rigid. During this drastic change in the domain behaviour the diffusion coefficient does not change. In this scenario, the domains diffuse in a membrane of low viscosity. A similar observation was also made by Cicuta et al. (Reference Cicuta, Keller and Veatch2007) in their experimental study.

For large and negative ${\it\epsilon}$ the domain viscosity is much smaller than that of the surrounding membrane. In this scenario, we observe a significant drop in the drag force for all ${\it\beta}$ (see figure 6 b). From figure 7(a), we also observe that the ratio between the drag forces on the rigid and liquid domains increases with ${\it\beta}$ initially and then decreases. The drop is most significant in the intermediate range of ${\it\beta}$ , between approximately 10 and 100. Again, from figure 7(a) we observe that the drag forces on the rigid and the liquid domains are almost the same $-1\leqslant {\it\epsilon}\leqslant 1$ (data for ${\it\epsilon}=-1$ not presented).

Figure 7.

(a) Variation of drag force on the liquid domain, relative to the drag force on a rigid domain of the same size, with the Boussinesq number ${\it\beta}$ for three different values of viscosity contrast ${\it\epsilon}$ . (b) Mobility with respect to ${\it\beta}$ for ${\it\beta}\geqslant 5$ and comparison with the results of Saffman (Reference Saffman1976) and Hughes et al. (Reference Hughes, Pailthorpe and White1981). For large and negative ${\it\epsilon}$ , significant departure from the logarithmic behaviour is observed.

Hughes et al. (Reference Hughes, Pailthorpe and White1981) have predicted that mobility varies linearly with ${\it\beta}$ for small  ${\it\beta}$ . Due to the large percentage error in our results for ${\it\beta}\leqslant 5$ , we do not attempt a quantitative comparison of mobility with the asymptotic behaviour predicted by Hughes et al. (Reference Hughes, Pailthorpe and White1981). However, we do observe that the absolute error is less for ${\it\beta}\ll 1$ and the mobility varies linearly with ${\it\beta}$ in this regime (not shown). Finally, in figure 7(b), we present the dimensionless mobility variation with ${\it\beta}$ for different ${\it\epsilon}$ . For a rigid domain ( ${\it\epsilon}=1$ ) our results agree with the logarithmic behaviour ( $U/F\approx \ln (2{\it\beta})$ ) predicted by Saffman (Reference Saffman1976) and Hughes et al. (Reference Hughes, Pailthorpe and White1981). When the domain is not rigid, but the viscosity contrast ${\it\epsilon}$ is less than unity in magnitude, the mobility for the liquid domain is close to that for the rigid domain. However, for large and negative ${\it\epsilon}$ , that is when the surrounding membrane is significantly more viscous than the domain, the mobility is significantly different and does not vary as $\ln (2{\it\beta})$ . The dimensionless mobility initially increases with ${\it\beta}$ and subsequently decreases but is always larger than the mobility for the rigid domain. To the best of our knowledge, such a behaviour has not been predicted previously. It is possible to design experiments by choosing a ternary lipid mixture of ordered and disordered lipids and cholesterol of suitable composition to obtain domains of significantly lower viscosity than the surrounding. Ternary phase diagrams presented in figures 8–10 of Veatch & Keller (Reference Veatch and Keller2005) can provide guidance to choosing such compositions.