4.1 Outer equations

In the outer region, the coordinates are not stretched and the velocity components need not be rescaled. The pressures $p(x,\unicode[STIX]{x1D70E})$ and $\hat{p}(x,\unicode[STIX]{x1D70E})$ and membrane tension $\unicode[STIX]{x1D70F}(x)$ must be stretched according to (4.3) in order to match consistently with the inner solution. The radius of curvature $r(x)$ , which spans the tube cross-section, is a suitable shape function for describing the membrane shape in the outer region.

The governing equations and boundary conditions for the outer region are as follows:

(4.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}+\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}+\frac{1}{\unicode[STIX]{x1D70E}}\right)v=0, & \displaystyle\end{eqnarray}$$

(4.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\bar{p}}{\unicode[STIX]{x2202}x}-\unicode[STIX]{x1D716}^{3/2}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}^{2}}+\frac{1}{\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}\right)u=0, & \displaystyle\end{eqnarray}$$

(4.6c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\bar{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\unicode[STIX]{x1D716}^{3/2}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}^{2}}+\frac{1}{\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\frac{1}{\unicode[STIX]{x1D70E}^{2}}\right)v=0, & \displaystyle\end{eqnarray}$$

(4.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\hat{u} }{\unicode[STIX]{x2202}x}+\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}+\frac{1}{\unicode[STIX]{x1D70E}}\right)v=0, & \displaystyle\end{eqnarray}$$

(4.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\hat{\bar{p}}}{\unicode[STIX]{x2202}x}-\unicode[STIX]{x1D716}^{3/2}\unicode[STIX]{x1D705}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}^{2}}+\frac{1}{\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}\right)\hat{u} =0, & \displaystyle\end{eqnarray}$$

(4.7c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\hat{\bar{p}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\unicode[STIX]{x1D716}^{3/2}\unicode[STIX]{x1D705}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}^{2}}+\frac{1}{\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\frac{1}{\unicode[STIX]{x1D70E}^{2}}\right)\hat{v}=0, & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }x\rightarrow \pm \infty :\qquad u\rightarrow \frac{2V}{U}(1-\unicode[STIX]{x1D70E}^{2})-1,\quad v=0, & \displaystyle\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=1:\qquad u=-1,\quad v=0, & \displaystyle\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=r(x):\qquad u=0,\quad v=0, & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=0:\qquad \frac{\unicode[STIX]{x2202}\hat{u} }{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}=0,\quad \hat{v}=0, & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=r(x):\qquad \hat{u} =0,\quad \hat{v}=0, & \displaystyle\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{at }\unicode[STIX]{x1D70E}=r(x):\nonumber\\ \displaystyle & & \displaystyle \qquad \left\{-(\bar{p}-\hat{\bar{p}})+\frac{2\unicode[STIX]{x1D716}^{3/2}}{1+(\text{d}r/\text{d}x)^{2}}\left[\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\frac{\text{d}r}{\text{d}x}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\right)(v-\unicode[STIX]{x1D705}\hat{v})\right.\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad -\left.\left.\frac{\text{d}r}{\text{d}x}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\frac{\text{d}r}{\text{d}x}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\right)(u-\unicode[STIX]{x1D705}\hat{u} )\right]\right\}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad =\bar{\unicode[STIX]{x1D70F}}\left(\frac{1}{r\sqrt{1+(\text{d}r/\text{d}x)^{2}}}-\frac{\text{d}^{2}r/\text{d}x^{2}}{\sqrt{1+(\text{d}r/\text{d}x)^{2}}^{3}}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad +\,\unicode[STIX]{x1D716}\bar{\unicode[STIX]{x1D6FD}}\left\{\frac{1}{2}\left(\frac{1}{r\sqrt{1+(\text{d}r/\text{d}x)^{2}}}+\frac{\text{d}^{2}r/\text{d}x^{2}}{\sqrt{1+(\text{d}r/\text{d}x)^{2}}^{3}}\right)^{2}+\frac{1}{1+(\text{d}r/\text{d}x)^{2}}\frac{\text{d}^{2}}{\text{d}x^{2}}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad -\,\left.\frac{1}{\sqrt{1+(\text{d}r/\text{d}x)^{2}}}\frac{\text{d}r}{\text{d}x}\left(-\frac{1}{r\sqrt{1+(\text{d}r/\text{d}x)^{2}}}+\frac{\text{d}^{2}r/\text{d}x^{2}}{\sqrt{1+(\text{d}r/\text{d}x)^{2}}^{3}}\right)\frac{\text{d}}{\text{d}x}\right\}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \times \,\left(-\frac{1}{r\sqrt{1+(\text{d}r/\text{d}x)^{2}}}+\frac{\text{d}^{2}r/\text{d}x^{2}}{\sqrt{1+(\text{d}r/\text{d}x)^{2}}^{3}}\right),\end{eqnarray}$$

(4.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{at }\unicode[STIX]{x1D70E}=r(x):\nonumber\\ \displaystyle & & \displaystyle \quad \frac{1}{\sqrt{1+(\text{d}r/\text{d}x)^{2}}}\frac{\text{d}\bar{\unicode[STIX]{x1D70F}}}{\text{d}x}+\frac{2\unicode[STIX]{x1D716}^{3/2}}{1+(\text{d}r/\text{d}x)^{2}}\left(\left\{\frac{\text{d}r}{\text{d}x}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}+\left[1-\left(\frac{\text{d}r}{\text{d}x}\right)^{2}\right]\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\right\}(v-\unicode[STIX]{x1D705}\hat{v})\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \quad -\,\left.\left\{\frac{\text{d}r}{\text{d}x}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}-\left[1-\left(\frac{\text{d}r}{\text{d}x}\right)^{2}\right]\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}\right\}(u-\unicode[STIX]{x1D705}\hat{u} )\right)=0,\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }x=0:\qquad r=0,\quad \frac{\text{d}r}{\text{d}x}\rightarrow -\infty , & \displaystyle\end{eqnarray}$$

(4.16) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }x=-\ell :\qquad r=0,\quad \frac{\text{d}r}{\text{d}x}\rightarrow \infty . & \displaystyle\end{eqnarray}$$

Equations (2.14) and (2.16) are not presented for the outer problem. The leakback flux $q$ and extra pressure drop $\unicode[STIX]{x0394}p^{+}$ may be uniquely determined from the inner equations, which are presented in the next section.

4.2 Inner equations

The variables in the inner region are stretched according to the definitions in (4.2)–(4.4). The radius of curvature $r(x)$ is replaced by the scaled gap thickness $\bar{h}(\bar{x})$ , which is a more suitable choice of shape function in the inner region. Since the governing equations are autonomous, it is understood that derivatives with respect to $\bar{x}$ may equivalently be written with respect to $\bar{x}^{\ast }$ . In the development of the inner equations, only terms up to $O(\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}\unicode[STIX]{x1D705},\unicode[STIX]{x1D716}\bar{\unicode[STIX]{x1D6FD}})$ are retained.

The governing equations and boundary conditions for the inner region are as follows:

(4.17a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\bar{u}}{\unicode[STIX]{x2202}\bar{x}}+\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{y}}-\unicode[STIX]{x1D716}\right)\bar{v}=O(\unicode[STIX]{x1D716}^{2}), & \displaystyle\end{eqnarray}$$

(4.17b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\bar{p}}{\unicode[STIX]{x2202}\bar{x}}-\left[\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\bar{y}^{2}}+\unicode[STIX]{x1D716}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\bar{x}^{2}}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{y}}\right)\right]\bar{u}=O(\unicode[STIX]{x1D716}^{2}), & \displaystyle\end{eqnarray}$$

(4.17c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\bar{p}}{\unicode[STIX]{x2202}\bar{y}}-\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}^{2}\bar{v}}{\unicode[STIX]{x2202}\bar{y}^{2}}=O(\unicode[STIX]{x1D716}^{2}), & \displaystyle\end{eqnarray}$$

(4.18a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\hat{\bar{u}}}{\unicode[STIX]{x2202}\bar{x}}+\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}+\frac{1}{\unicode[STIX]{x1D70E}}\right)\hat{\bar{v}}=0, & \displaystyle\end{eqnarray}$$

(4.18b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\hat{\bar{p}}}{\unicode[STIX]{x2202}\bar{x}}-\unicode[STIX]{x1D716}\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}^{2}\hat{\bar{u}}}{\unicode[STIX]{x2202}\bar{x}^{2}}=O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D705}), & \displaystyle\end{eqnarray}$$

(4.18c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\hat{\bar{p}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\unicode[STIX]{x1D705}\left[\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\bar{x}^{2}}+\unicode[STIX]{x1D716}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}^{2}}+\frac{1}{\unicode[STIX]{x1D70E}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\frac{1}{\unicode[STIX]{x1D70E}^{2}}\right)\right]\hat{\bar{v}}=O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D705}), & \displaystyle\end{eqnarray}$$

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=0:\qquad \bar{u}=-1,\quad \bar{v}=0, & \displaystyle\end{eqnarray}$$

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=\bar{h}(\bar{x}):\qquad \bar{u}=0,\bar{v}=0, & \displaystyle\end{eqnarray}$$

(4.21) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=0:\qquad \frac{\unicode[STIX]{x2202}\hat{\bar{u}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}=0,\quad \hat{\bar{v}}=0, & \displaystyle\end{eqnarray}$$

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=1-\unicode[STIX]{x1D716}\bar{h}(\bar{x}):\qquad \hat{\bar{u}}=0,\quad \hat{\bar{v}}=0, & \displaystyle\end{eqnarray}$$

(4.23) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{at }\bar{y}=\bar{h}(\bar{x}),\unicode[STIX]{x1D70E}=1-\unicode[STIX]{x1D716}\bar{h}(\bar{x}):\nonumber\\ \displaystyle & & \displaystyle \quad -(\bar{p}-\hat{\bar{p}})+2\unicode[STIX]{x1D716}\left[\frac{\unicode[STIX]{x2202}\bar{v}}{\unicode[STIX]{x2202}\bar{y}}-\frac{\text{d}\bar{h}}{\text{d}\bar{x}}\frac{\unicode[STIX]{x2202}\bar{u}}{\unicode[STIX]{x2202}\bar{y}}-\unicode[STIX]{x1D705}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}+\frac{\text{d}\bar{h}}{\text{d}\bar{x}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{x}}\right)\hat{\bar{v}}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad \quad =\bar{\unicode[STIX]{x1D70F}}\left\{\left(1+\frac{\text{d}^{2}\bar{h}}{\text{d}\bar{x}^{2}}\right)+\unicode[STIX]{x1D716}\left[\bar{h}-\frac{1}{2}\left(\frac{\text{d}\bar{h}}{\text{d}\bar{x}}\right)^{2}\left(1+3\frac{\text{d}^{2}\bar{h}}{\text{d}\bar{x}^{2}}\right)\right]\right\}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad -\,\bar{\unicode[STIX]{x1D6FD}}\left\{\frac{\text{d}^{4}\bar{h}}{\text{d}\bar{x}^{4}}+\unicode[STIX]{x1D716}\left[\frac{1}{2}\left(1+\frac{\text{d}^{2}\bar{h}}{\text{d}\bar{x}^{2}}\right)-\frac{3}{2}\left(\frac{\text{d}^{2}\bar{h}}{\text{d}\bar{x}^{2}}\right)^{2}-2\frac{\text{d}\bar{h}}{\text{d}\bar{x}}\frac{\text{d}^{3}\bar{h}}{\text{d}\bar{x}^{3}}-\frac{5}{2}\left(\frac{\text{d}^{2}\bar{h}}{\text{d}\bar{x}^{2}}\right)^{3}\right.\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad -\left.\left.10\frac{\text{d}\bar{h}}{\text{d}\bar{x}}\frac{\text{d}^{2}\bar{h}}{\text{d}\bar{x}^{2}}\frac{\text{d}^{3}\bar{h}}{\text{d}\bar{x}^{3}}-\frac{5}{2}\left(\frac{\text{d}\bar{h}}{\text{d}\bar{x}}\right)^{2}\frac{\text{d}^{4}\bar{h}}{\text{d}\bar{x}^{4}}\right]\right\}+O(\unicode[STIX]{x1D716}^{2}\bar{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D716}^{2},\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D705}),\end{eqnarray}$$

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=\bar{h}(\bar{x}),\unicode[STIX]{x1D70E}=1-\unicode[STIX]{x1D716}\bar{h}(\bar{x}):\qquad \frac{\text{d}\bar{\unicode[STIX]{x1D70F}}}{\text{d}\bar{x}}-\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}\bar{u}}{\unicode[STIX]{x2202}\bar{y}}-\unicode[STIX]{x1D716}\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}\hat{\bar{v}}}{\unicode[STIX]{x2202}\bar{x}}=O(\unicode[STIX]{x1D716}^{2},\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D705}), & \displaystyle\end{eqnarray}$$

(4.25) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{x}=-\bar{\ell },\bar{x}^{\ast }=\bar{\ell }:\qquad \bar{h}=1, & \displaystyle\end{eqnarray}$$

(4.26) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{q}=-\int _{0}^{\bar{h}(\bar{x})}\bar{u}\,(1-\unicode[STIX]{x1D716}\bar{y})\,\text{d}\bar{y}=\text{const.}, & \displaystyle\end{eqnarray}$$

(4.27) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\bar{p}^{+}=2\unicode[STIX]{x1D716}\left.\int _{-\infty }^{\bar{\ell }}\frac{\unicode[STIX]{x2202}\bar{u}}{\unicode[STIX]{x2202}\bar{y}}\right|_{\bar{y}=0}\,\text{d}\bar{x}^{\ast }+2\unicode[STIX]{x1D716}\left.\int _{-\bar{\ell }}^{\infty }\frac{\unicode[STIX]{x2202}\bar{u}}{\unicode[STIX]{x2202}\bar{y}}\right|_{\bar{y}=0}\,\text{d}\bar{x}+O(\unicode[STIX]{x1D716}^{3/2})=\text{const}. & \displaystyle\end{eqnarray}$$

To leading order, the leakback flux $\bar{q}$ and extra pressure drop $\unicode[STIX]{x0394}\bar{p}^{+}$ are determined entirely from the inner solution. The error term omitted in (4.27) contains higher-order contributions from the inner region as well as the leading-order contribution from the outer region.

4.3 Matching conditions

The foregoing equations are valid in the outer and inner regions separately. In order to unify the solutions, matching conditions must be developed at the matching points $x=-1$ and $-\ell +1$ . Of particular importance are the matching conditions for the membrane shape:

(4.28a ) $$\begin{eqnarray}\lim _{x\rightarrow -1}1-r(x)=\lim _{\bar{x}\rightarrow \infty }\unicode[STIX]{x1D716}\bar{h}(\bar{x})\end{eqnarray}$$

at the front nose and

(4.28b ) $$\begin{eqnarray}\lim _{x\rightarrow -\ell +1}1-r(x)=\lim _{\bar{x}^{\ast }\rightarrow -\infty }\unicode[STIX]{x1D716}\bar{h}(\bar{x})\end{eqnarray}$$

at the rear tail. The limits are interpreted in the spirit of the matching principle of van Dyke (Reference van Dyke1964). With the inclusion of these matching conditions, the boundary-value problem is well posed.

5.1 Outer region

By expanding (4.6), (4.7) and (4.14) with respect to $\unicode[STIX]{x1D716}^{1/2}$ and setting $\unicode[STIX]{x1D716}=0$ , one immediately deduces that $\bar{p}_{0}$ , $\hat{\bar{p}}_{0}$ and $\bar{\unicode[STIX]{x1D70F}}_{0}$ are independent of spatial position. Thus, viscous stresses play no appreciable role in the outer region at this order. The leading-order problem reduces to the solution of the normal stress condition for the shape function $r_{0}(x)$ (cf. (4.13)),

(5.3) $$\begin{eqnarray}\bar{p}_{0}-\hat{\bar{p}}_{0}=-\bar{\unicode[STIX]{x1D70F}}_{0}\left(\frac{1}{r_{0}\sqrt{1+(\text{d}r_{0}/\text{d}x)^{2}}}-\frac{\text{d}^{2}r_{0}/\text{d}x^{2}}{\sqrt{1+(\text{d}r_{0}/\text{d}x)^{2}}^{3}}\right),\end{eqnarray}$$

with the boundary conditions (cf. (4.15)–(4.16) and (4.28)),

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }x=0:\qquad r_{0}=0,\quad \frac{\text{d}r_{0}}{\text{d}x}\rightarrow -\infty , & \displaystyle\end{eqnarray}$$

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }x=-\ell :\qquad r_{0}=0,\quad \frac{\text{d}r_{0}}{\text{d}x}\rightarrow \infty , & \displaystyle\end{eqnarray}$$

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }x=-1,-\ell +1:\qquad r_{0}=1. & \displaystyle\end{eqnarray}$$

Although the fluid pressures and membrane tension do not vary in the outer region, there exists the possibility of a mismatch between their ‘front’ and ‘rear’ values as a result of variations through the inner region. Thus, it is in general necessary to determine the outer solutions at the front nose and rear tail independently of each other and unify them through the inner solution.

The solution of (5.3)–(5.6) is

(5.7) $$\begin{eqnarray}r_{0}=\sqrt{1-\unicode[STIX]{x1D709}^{2}},\quad \bar{p}_{0}-\hat{\bar{p}}_{0}=-2\bar{\unicode[STIX]{x1D70F}}_{0},\end{eqnarray}$$

where

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D709}=\left\{\begin{array}{@{}ll@{}}x+1,\quad & \text{for }x\in [-1,0],\\ x+\ell -1,\quad & \text{for }x\in [-\ell ,-\ell +1],\\ 0,\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

Thus, the leading-order vesicle shape is a spherocylinder (a cylinder capped by two hemispheres). The membrane tension $\bar{\unicode[STIX]{x1D70F}}_{0}$ remains unknown and must be determined from matching to the inner solution.

It happens that the $O(\unicode[STIX]{x1D716}^{1/2})$ correction to the shape function, $r_{1}(x)$ , may be determined without knowledge of the inner solution. The corrections $\bar{p}_{1}$ , $\hat{\bar{p}}_{1}$ and $\bar{\unicode[STIX]{x1D70F}}_{1}$ are spatially invariant, whence the $O(\unicode[STIX]{x1D716}^{1/2})$ term in (4.13) simplifies to

(5.9) $$\begin{eqnarray}\bar{p}_{1}-\hat{\bar{p}}_{1}=-2\bar{\unicode[STIX]{x1D70F}}_{1}+\bar{\unicode[STIX]{x1D70F}}_{0}\left(\frac{r_{1}}{r_{0}}-4r_{0}\sqrt{1-r_{0}^{2}}\frac{\text{d}r_{1}}{\text{d}x}+r_{0}^{3}\frac{\text{d}^{2}r_{1}}{\text{d}x^{2}}\right).\end{eqnarray}$$

With some rearrangement and introduction of the modified variables,

(5.10a,b ) $$\begin{eqnarray}f_{1}=r_{0}r_{1},\quad a_{1}=\frac{\bar{\unicode[STIX]{x1D70F}}_{1}}{\bar{\unicode[STIX]{x1D70F}}_{0}}-\frac{\bar{p}_{1}-\hat{\bar{p}}_{1}}{\bar{p}_{0}-\hat{\bar{p}}_{0}}=\text{const.},\end{eqnarray}$$

equation (5.9) may be recast as an inhomogeneous Legendre equation of order one,

(5.11) $$\begin{eqnarray}(1-\unicode[STIX]{x1D709}^{2})\frac{\text{d}^{2}f_{1}}{\text{d}\unicode[STIX]{x1D709}^{2}}-2\unicode[STIX]{x1D709}\frac{\text{d}f_{1}}{\text{d}\unicode[STIX]{x1D709}}+2f_{1}=2a_{1}.\end{eqnarray}$$

Here, the shifted coordinate $\unicode[STIX]{x1D709}$ is defined as before. The rear tail spans the region $\unicode[STIX]{x1D709}\in [-1,0]$ , while the front nose spans $\unicode[STIX]{x1D709}\in [0,1]$ . The boundary conditions for $f_{1}(\unicode[STIX]{x1D709})$ are

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D709}=0:\qquad f_{1}=0, & \displaystyle\end{eqnarray}$$

(5.13) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D709}=\pm 1:\qquad f_{1}=0. & \displaystyle\end{eqnarray}$$

The only solution of (5.11) which satisfies (5.12)–(5.13) is the trivial solution,

(5.14a,b ) $$\begin{eqnarray}f_{1}=0,\quad a_{1}=0.\end{eqnarray}$$

Therefore,

(5.15a,b ) $$\begin{eqnarray}r_{1}=0,\quad \bar{p}_{1}-\hat{\bar{p}}_{1}=-2\bar{\unicode[STIX]{x1D70F}}_{1},\end{eqnarray}$$

i.e. the $O(\unicode[STIX]{x1D716}^{1/2})$ solution admits no correction to the shape. The membrane tension $\bar{\unicode[STIX]{x1D70F}}_{1}$ remains unknown and must be determined from matching to the inner solution.

5.2 Inner region

In contrast to the ‘static’ outer region, viscous stresses play a significant role in the inner region. The governing equations and boundary conditions are

(5.16a-c ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x2202}\bar{v}_{0}}{\unicode[STIX]{x2202}\bar{y}}=0,\quad \frac{\unicode[STIX]{x2202}\bar{p}_{0}}{\unicode[STIX]{x2202}\bar{x}}-\frac{\unicode[STIX]{x2202}^{2}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{y}^{2}}=0,\quad \frac{\unicode[STIX]{x2202}\bar{p}_{0}}{\unicode[STIX]{x2202}\bar{y}}=0,\end{eqnarray}$$

(5.17a-c ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\hat{\bar{u}}_{0}}{\unicode[STIX]{x2202}\bar{x}}+\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}+\frac{1}{\unicode[STIX]{x1D70E}}\right)\hat{\bar{v}}_{0}=0,\quad \frac{\unicode[STIX]{x2202}\hat{\bar{p}}_{0}}{\unicode[STIX]{x2202}\bar{x}}=0,\quad \frac{\unicode[STIX]{x2202}\hat{\bar{p}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}^{2}\hat{\bar{v}}_{0}}{\unicode[STIX]{x2202}\bar{x}^{2}}=0,\end{eqnarray}$$

(5.18) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=0:\qquad \bar{u}_{0}=-1,\quad \bar{v}_{0}=0, & \displaystyle\end{eqnarray}$$

(5.19) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=\bar{h}_{0}(\bar{x}):\qquad \bar{u}_{0}=0,\quad \bar{v}_{0}=0, & \displaystyle\end{eqnarray}$$

(5.20) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=0:\qquad \frac{\unicode[STIX]{x2202}\hat{\bar{u}}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}=0,\quad \hat{\bar{v}}_{0}=0, & \displaystyle\end{eqnarray}$$

(5.21) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=1:\qquad \hat{\bar{u}}_{0}=0,\quad \hat{\bar{v}}_{0}=0, & \displaystyle\end{eqnarray}$$

(5.22) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=\bar{h}_{0}(\bar{x}):\qquad \bar{p}_{0}-\hat{\bar{p}}_{0}=-\bar{\unicode[STIX]{x1D70F}}_{0}\left(1+\frac{\text{d}^{2}\bar{h}_{0}}{\text{d}\bar{x}^{2}}\right)+\bar{\unicode[STIX]{x1D6FD}}\frac{\text{d}^{4}\bar{h}_{0}}{\text{d}\bar{x}^{4}},\quad \bar{\unicode[STIX]{x1D70F}}_{0}=\text{const.}, & \displaystyle\end{eqnarray}$$

(5.23) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{q}_{0}=-\int _{0}^{\bar{h}_{0}(\bar{x})}\bar{u}_{0}\,\text{d}\bar{y}=\text{const.}, & \displaystyle\end{eqnarray}$$

(5.24) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\bar{p}_{2}^{+}=2\left.\int _{-\infty }^{\bar{\ell }}\frac{\unicode[STIX]{x2202}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{y}}\right|_{\bar{y}=0}\,\text{d}\bar{x}^{\ast }+2\left.\int _{-\bar{\ell }}^{\infty }\frac{\unicode[STIX]{x2202}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{y}}\right|_{\bar{y}=0}\,\text{d}\bar{x}=\text{const}. & \displaystyle\end{eqnarray}$$

Equations (5.16)–(5.21) and (5.23)–(5.24) are the equations of lubrication theory. Following Lighthill (Reference Lighthill1968), they may be used to derive the so-called ‘Reynolds equation’ for the pressure gradient,

(5.25) $$\begin{eqnarray}\frac{\text{d}\bar{p}_{0}}{\text{d}\bar{x}}=\frac{\text{d}(\bar{p}_{0}-\hat{\bar{p}}_{0})}{\text{d}\bar{x}}=-\frac{6}{\bar{h}_{0}^{2}}\left(1-\frac{2\bar{q}_{0}}{\bar{h}_{0}}\right),\end{eqnarray}$$

as well as an expression for the extra pressure drop,

(5.26) $$\begin{eqnarray}\unicode[STIX]{x0394}\bar{p}_{2}^{+}=\int _{-\infty }^{\bar{\ell }}\frac{4}{\bar{h}_{0}}\left(2-\frac{3\bar{q}_{0}}{\bar{h}_{0}}\right)\,\text{d}\bar{x}^{\ast }+\int _{-\bar{\ell }}^{\infty }\frac{4}{\bar{h}_{0}}\left(2-\frac{3\bar{q}_{0}}{\bar{h}_{0}}\right)\,\text{d}\bar{x}.\end{eqnarray}$$

The gap height $\bar{h}_{0}(\bar{x})$ remains to be determined. Combining the Reynolds equation (5.25) with the stress condition (5.22) leads to a differential equation for $\bar{h}_{0}(\bar{x})$ that bears some resemblance to the one originally considered by Landau & Levich (Reference Landau and Levich1942) (with the addition of the bending term):

(5.27) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D6FD}}\frac{\text{d}^{5}\bar{h}_{0}}{\text{d}\bar{x}^{5}}-\bar{\unicode[STIX]{x1D70F}}_{0}\frac{\text{d}^{3}\bar{h}_{0}}{\text{d}\bar{x}^{3}}+\frac{6}{\bar{h}_{0}^{2}}\left(1-\frac{2\bar{q}_{0}}{\bar{h}_{0}}\right)=0.\end{eqnarray}$$

Dixit & Homsy (Reference Dixit and Homsy2013a ,Reference Dixit and Homsy b ) derived a similar expression to (5.27) for a stress-free interface in their study of particle-laden, dip-coating flow. The boundary conditions associated with equation (5.27) are as follows (cf. (4.25) and (4.28)):

(5.28) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{x}=-\bar{\ell },\bar{x}^{\ast }=\bar{\ell }:\qquad \bar{h}_{0}=1, & \displaystyle\end{eqnarray}$$

(5.29) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\bar{x}\rightarrow \infty :\qquad \bar{h}_{0}\rightarrow b_{0}+{\textstyle \frac{1}{2}}\bar{x}^{2}, & \displaystyle\end{eqnarray}$$

(5.30) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\bar{x}^{\ast }\rightarrow -\infty :\qquad \bar{h}_{0}\rightarrow b_{0}^{\ast }+{\textstyle \frac{1}{2}}\bar{x}^{\ast 2}, & \displaystyle\end{eqnarray}$$

where

(5.31a,b ) $$\begin{eqnarray}b_{0}=-r_{2}(-1),\quad b_{0}^{\ast }=-r_{2}(-\ell +1)\end{eqnarray}$$

are constants of integration. Since (5.27) is autonomous, it may be rewritten with $\bar{x}^{\ast }=\bar{x}-2\bar{\ell }$ substituted for $\bar{x}$ as the independent variable. Thus, matching with the ‘front’ and ‘rear’ outer solutions is facilitated by a simple shift of origin.

Equations (5.26)–(5.30) must be integrated numerically for specific values of the reduced length $\bar{\ell }$ and bending parameter $\bar{\unicode[STIX]{x1D6FD}}$ . The constants $\bar{q}_{0}$ , $\bar{\unicode[STIX]{x1D70F}}_{0}$ , $b_{0}$ , $b_{0}^{\ast }$ and $\unicode[STIX]{x0394}\bar{p}_{2}^{+}$ are determined simultaneously with $\bar{h}_{0}(\bar{x})$ during the integration procedure. Since the boundary-value problem is ill conditioned for many values of the parameters, the ‘multiple shooting method’ is employed for the numerical integration (Morrison, Riley & Zancanaro Reference Morrison, Riley and Zancanaro1962). A good description of the method is given in Stoer & Bulirsch (Reference Stoer and Bulirsch2002). In brief, the domain is first divided into two subdomains $\bar{x}\in [-\bar{\ell },\bar{X}]$ and $\bar{x}^{\ast }\in [-\bar{X}^{\ast },\bar{\ell }]$ . The boundary conditions are applied at $\bar{x}=\bar{X}$ , $\bar{x}=-\bar{\ell }$ , $\bar{x}^{\ast }=\bar{X}^{\ast }$ and $\bar{x}^{\ast }=\bar{\ell }$ . Care is taken to subtract off the error incurred by applying the far-field conditions (5.29)–(5.30) at finite boundaries $\bar{x}=\bar{X}$ and $\bar{x}=\bar{X}^{\ast }$ . Each subdomain is further divided into $O(10{-}100)$ segments separated by ‘shooting points’ at which continuity conditions are applied. Integration is carried out over each segment using a fourth-order Runge–Kutta scheme. Newton iteration with line search and backtracking is used to enforce the boundary and continuity conditions at the shooting points. The iteration is truncated when the norm of the subsequent Newton step or the norm of the function to be zeroed falls below $10^{-12}$ . In order to avoid numerical errors associated with the behaviour of the solution near the boundaries, integration always proceeds away from the boundary points towards a ‘fitting point’ located at the centre of the subdomain. Once a solution for particular values of $\bar{\ell }$ and $\bar{\unicode[STIX]{x1D6FD}}$ is known, additional solutions may be generated using the continuation method.

Representative profiles of $\bar{h}_{0}(\bar{x})$ are plotted in figure 3 for $\bar{\unicode[STIX]{x1D6FD}}=0$ and three different values of $\bar{\ell }$ . The constants $\bar{q}_{0}$ , $\bar{\unicode[STIX]{x1D70F}}_{0}$ , $b_{0}$ , $b_{0}^{\ast }$ and $\unicode[STIX]{x0394}\bar{p}_{2}^{+}$ are plotted against $\bar{\ell }$ for $\bar{\unicode[STIX]{x1D6FD}}=0$ in figure 4. Results for finite values of $\bar{\unicode[STIX]{x1D6FD}}$ are delayed until § 7.

Figure 3.

Gap height $\bar{h}$ plotted against the shifted axial coordinate $\bar{x}+\bar{\ell }$ for negligible bending stiffness ( $\bar{\unicode[STIX]{x1D6FD}}=0$ ). The left column is the leading-order solution $\bar{h}_{0}$ determined by integrating equations (5.26)–(5.30). The right column is the first correction $\bar{h}_{1}$ determined by integrating equations (5.48)–(5.52). The solid lines are the full numerical solutions. (a) Long, cylindrical vesicle ( $\bar{\ell }=10$ ), including the large- $\bar{\ell }$ approximation derived in § 6.1. (b) Intermediate length vesicle ( $\bar{\ell }=1$ ). (c) Short, spherical vesicle ( $\bar{\ell }=0.01$ ), including the small- $\bar{\ell }$ approximation derived in § 6.2 (cf. (6.18) and (6.26) truncated after the $O(1)$ term).

Figure 4.

(a) Leakback flux $\bar{q}$ , (b) membrane tension $\bar{\unicode[STIX]{x1D70F}}$ , (c) front intercept $b$ , (d) rear intercept $b^{\ast }$ and (e) extra pressure drop $\unicode[STIX]{x0394}\bar{p}^{+}$ plotted against the reduced length $\bar{\ell }$ for negligible bending stiffness ( $\bar{\unicode[STIX]{x1D6FD}}=0$ ). The left column is the leading-order solution; the right column is the first correction.

5.3 Higher-order corrections

Corrections to the leading-order solution may be derived by expanding the narrow-gap equations to higher order in $\unicode[STIX]{x1D716}^{1/2}$ . In the outer region, one derives an expression similar to (5.11) for the $O(\unicode[STIX]{x1D716})$ shape correction:

(5.32) $$\begin{eqnarray}(1-\unicode[STIX]{x1D709}^{2})\frac{\text{d}^{2}f_{2}}{\text{d}\unicode[STIX]{x1D709}^{2}}-2\unicode[STIX]{x1D709}\frac{\text{d}f_{2}}{\text{d}\unicode[STIX]{x1D709}}+2f_{2}=2a_{2},\end{eqnarray}$$

where

(5.33a,b ) $$\begin{eqnarray}f_{2}=r_{0}r_{2},\quad a_{2}=\frac{\bar{\unicode[STIX]{x1D70F}}_{2}}{\bar{\unicode[STIX]{x1D70F}}_{0}}-\frac{\bar{p}_{2}-\hat{\bar{p}}_{2}}{\bar{p}_{0}-\hat{\bar{p}}_{0}}=\text{const}.\end{eqnarray}$$

In deriving equation (5.32), equations (5.7) and (5.15a,b ) were used to eliminate $r_{0}$ and $r_{1}$ . The boundary conditions for $f_{2}(\unicode[STIX]{x1D709})$ are

(5.34) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\unicode[STIX]{x1D709}\rightarrow 0^{+}:\qquad f_{2}=-b_{0}, & \displaystyle\end{eqnarray}$$

(5.35) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\unicode[STIX]{x1D709}\rightarrow 0^{-}:\qquad f_{2}=-b_{0}^{\ast }, & \displaystyle\end{eqnarray}$$

(5.36) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D709}=\pm 1:\qquad f_{2}=0, & \displaystyle\end{eqnarray}$$

where a superscript ‘plus’ or ‘minus’ indicates that the boundary $\unicode[STIX]{x1D709}=0$ is approached in a limiting sense from positive or negative numbers, respectively. The constants $b_{0}$ and $b_{0}^{\ast }$ are known from the solution of the leading-order, inner problem. The only solution of (5.32) which satisfies (5.34)–(5.36) is

(5.37a ) $$\begin{eqnarray}\displaystyle & \displaystyle r_{2}=\frac{f_{2}}{\sqrt{1-\unicode[STIX]{x1D709}^{2}}}=\left\{\begin{array}{@{}ll@{}}\displaystyle -\frac{b_{0}(1-\unicode[STIX]{x1D709})}{\sqrt{1-\unicode[STIX]{x1D709}^{2}}},\quad & \text{for }\unicode[STIX]{x1D709}\in (0,1],\\ \displaystyle -\frac{b_{0}^{\ast }(1+\unicode[STIX]{x1D709})}{\sqrt{1-\unicode[STIX]{x1D709}^{2}}},\quad & \text{for }\unicode[STIX]{x1D709}\in [-1,0),\end{array}\right. & \displaystyle\end{eqnarray}$$

(5.37b ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{p}_{2}-\hat{\bar{p}}_{2}=\left\{\begin{array}{@{}ll@{}}-2\bar{\unicode[STIX]{x1D70F}}_{2}-2b_{0}\bar{\unicode[STIX]{x1D70F}}_{0},\quad & \text{for }\unicode[STIX]{x1D709}\in (0,1],\\ -2\bar{\unicode[STIX]{x1D70F}}_{2}-2b_{0}^{\ast }\bar{\unicode[STIX]{x1D70F}}_{0},\quad & \text{for }\unicode[STIX]{x1D709}\in [-1,0).\end{array}\right. & \displaystyle\end{eqnarray}$$

$\bar{\unicode[STIX]{x1D70F}}_{0}$

$\bar{\unicode[STIX]{x1D70F}}_{2}$

$\unicode[STIX]{x1D709}=0$

In the inner region, the $O(\unicode[STIX]{x1D716}^{1/2})$ correction fields are governed by the following equations and boundary conditions:

(5.38a-c ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\bar{u}_{1}}{\unicode[STIX]{x2202}\bar{x}}+\frac{\unicode[STIX]{x2202}\bar{v}_{1}}{\unicode[STIX]{x2202}\bar{y}}=0,\quad \frac{\unicode[STIX]{x2202}\bar{p}_{1}}{\unicode[STIX]{x2202}\bar{x}}-\frac{\unicode[STIX]{x2202}^{2}\bar{u}_{1}}{\unicode[STIX]{x2202}\bar{y}^{2}}=0,\quad \frac{\unicode[STIX]{x2202}\bar{p}_{1}}{\unicode[STIX]{x2202}\bar{y}}=0,\end{eqnarray}$$

(5.39a-c ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\hat{\bar{u}}_{1}}{\unicode[STIX]{x2202}\bar{x}}+\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}+\frac{1}{\unicode[STIX]{x1D70E}}\right)\hat{\bar{v}}_{1}=0,\quad \frac{\unicode[STIX]{x2202}\hat{\bar{p}}_{1}}{\unicode[STIX]{x2202}\bar{x}}=0,\quad \frac{\unicode[STIX]{x2202}\hat{\bar{p}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}-\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}^{2}\hat{\bar{v}}_{1}}{\unicode[STIX]{x2202}\bar{x}^{2}}=0,\end{eqnarray}$$

(5.40) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=0:\qquad \bar{u}_{1}=0,\quad \bar{v}_{1}=0, & \displaystyle\end{eqnarray}$$

(5.41) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=\bar{h}_{0}(\bar{x}):\qquad \bar{u}_{1}=-\bar{h}_{1}\frac{\unicode[STIX]{x2202}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{y}},\quad \bar{v}_{1}=-\bar{h}_{1}\frac{\unicode[STIX]{x2202}\bar{v}_{0}}{\unicode[STIX]{x2202}\bar{y}}, & \displaystyle\end{eqnarray}$$

(5.42) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=0:\qquad \frac{\unicode[STIX]{x2202}\hat{\bar{u}}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}=0,\quad \hat{\bar{v}}_{1}=0, & \displaystyle\end{eqnarray}$$

(5.43) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D70E}=1:\qquad \hat{\bar{u}}_{1}=0,\quad \hat{\bar{v}}_{1}=0, & \displaystyle\end{eqnarray}$$

(5.44) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{y}=\bar{h}_{0}(\bar{x}):\qquad \bar{p}_{1}-\hat{\bar{p}}_{1}=-\bar{\unicode[STIX]{x1D70F}}_{1}\left(1+\frac{\text{d}^{2}\bar{h}_{0}}{\text{d}\bar{x}^{2}}\right)-\bar{\unicode[STIX]{x1D70F}}_{0}\frac{\text{d}^{2}\bar{h}_{1}}{\text{d}\bar{x}^{2}}+\bar{\unicode[STIX]{x1D6FD}}\frac{\text{d}^{4}\bar{h}_{1}}{\text{d}\bar{x}^{4}},\quad \bar{\unicode[STIX]{x1D70F}}_{1}=\text{const.}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(5.45) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{q}_{1}=-\int _{0}^{\bar{h}_{0}(\bar{x})}\bar{u}_{1}\,\text{d}\bar{y}+\bar{h}_{1}\bar{u}_{0}|_{\bar{y}=\bar{h}_{0}(\bar{x})}=\text{const.}, & \displaystyle\end{eqnarray}$$

(5.46) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\bar{p}_{3}^{+}=2\left.\int _{-\infty }^{\bar{\ell }}\frac{\unicode[STIX]{x2202}\bar{u}_{1}}{\unicode[STIX]{x2202}\bar{y}}\right|_{\bar{y}=0}\,\text{d}\bar{x}^{\ast }+2\left.\int _{-\bar{\ell }}^{\infty }\frac{\unicode[STIX]{x2202}\bar{u}_{1}}{\unicode[STIX]{x2202}\bar{y}}\right|_{\bar{y}=0}\,\text{d}\bar{x}+O(1)=\text{const.} & \displaystyle\end{eqnarray}$$

The terms multiplied by $\bar{h}_{1}$ on the right-hand side of (5.41) and (5.45) appear as a result of transferring the boundary from $\bar{y}=\bar{h}(\bar{x})$ to $\bar{y}=\bar{h}_{0}(\bar{x})$ (Bungay & Brenner Reference Bungay and Brenner1973). Following the same procedure as in § 5.2, the first correction to the pressure gradient and extra pressure drop are derived:

(5.47) $$\begin{eqnarray}\frac{\text{d}\bar{p}_{1}}{\text{d}\bar{x}}=\frac{\text{d}(\bar{p}_{1}-\hat{\bar{p}}_{1})}{\text{d}\bar{x}}=\frac{12}{\bar{h}_{0}^{2}}\left[\frac{\bar{h}_{1}}{\bar{h}_{0}}\left(1-\frac{3\bar{q}_{0}}{\bar{h}_{0}}\right)+\frac{\bar{q}_{1}}{\bar{h}_{0}}\right],\end{eqnarray}$$

(5.48) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\bar{p}_{3}^{+(inner)} & = & \displaystyle -\int _{-\infty }^{\bar{\ell }}\left[\frac{8\bar{h}_{1}}{\bar{h}_{0}^{2}}\left(1-\frac{3\bar{q}_{0}}{\bar{h}_{0}}\right)+\frac{12\bar{q}_{1}}{\bar{h}_{0}^{2}}\right]\,\text{d}\bar{x}^{\ast }\nonumber\\ \displaystyle & & \displaystyle -\,\int _{-\bar{\ell }}^{\infty }\left[\frac{8\bar{h}_{1}}{\bar{h}_{0}^{2}}\left(1-\frac{3\bar{q}_{0}}{\bar{h}_{0}}\right)+\frac{12\bar{q}_{1}}{\bar{h}_{0}^{2}}\right]\,\text{d}\bar{x},\end{eqnarray}$$

where the superscript ‘(inner)’ is ascribed to indicate that additional contributions from the outer region may contribute at this order.

It remains to calculate $\bar{h}_{1}(\bar{x})$ , the first correction to the gap height. Combining (5.47) with (5.44) yields an inhomogeneous differential equation for $\bar{h}_{1}(\bar{x})$ :

(5.49) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D6FD}}\frac{\text{d}^{5}\bar{h}_{1}}{\text{d}\bar{x}^{5}}-\bar{\unicode[STIX]{x1D70F}}_{0}\frac{\text{d}^{3}\bar{h}_{1}}{\text{d}\bar{x}^{3}}-\frac{12\bar{h}_{1}}{\bar{h}_{0}^{3}}\left(1-\frac{3\bar{q}_{0}}{\bar{h}_{0}}\right)=\bar{\unicode[STIX]{x1D70F}}_{1}\frac{\text{d}^{3}\bar{h}_{0}}{\text{d}\bar{x}^{3}}+\frac{12\bar{q}_{1}}{\bar{h}_{0}^{3}}.\end{eqnarray}$$

As before, the independent variable $\bar{x}$ may be exchanged for $\bar{x}^{\ast }$ by a simple shift of origin. The boundary conditions associated with equation (5.49) are

(5.50) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{x}=-\bar{\ell },\bar{x}^{\ast }=\bar{\ell }:\quad \bar{h}_{1}=0, & \displaystyle\end{eqnarray}$$

(5.51) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\bar{x}\rightarrow \infty :\qquad \bar{h}_{1}\rightarrow b_{1}-b_{0}\bar{x}, & \displaystyle\end{eqnarray}$$

(5.52) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\bar{x}^{\ast }\rightarrow -\infty :\qquad \bar{h}_{1}\rightarrow b_{1}^{\ast }+b_{0}^{\ast }\bar{x}^{\ast }, & \displaystyle\end{eqnarray}$$

where

(5.53a,b ) $$\begin{eqnarray}b_{1}=-r_{3}(-1),\quad b_{1}^{\ast }=-r_{3}(-\ell +1)\end{eqnarray}$$

are constants of integration.

Equations (5.48)–(5.52) must be integrated numerically for specific values of the reduced length $\bar{\ell }$ and bending parameter $\bar{\unicode[STIX]{x1D6FD}}$ . As with all perturbation theories, the correction fields are forced by source terms that depend on the leading-order fields. Thus, any numerical solution requires accurate tabulation of $\bar{h}_{0}$ at discrete points in $\bar{x}$ . Above $\bar{\ell }\simeq 20$ , the system becomes numerically stiff and roundoff errors accumulate. Numerical results for $\bar{\unicode[STIX]{x1D6FD}}=0$ are shown in figures 3 and 4 alongside the leading-order solutions. Results for finite values of $\bar{\unicode[STIX]{x1D6FD}}$ are delayed until § 7.

At the next order in $\unicode[STIX]{x1D716}$ , the outer correction to the shape $r_{3}$ forces a condition on the slope of $\bar{h}_{2}$ in the far field. The key result from the $O(\unicode[STIX]{x1D716})$ inner problem is a non-constant correction to the membrane tension $\bar{\unicode[STIX]{x1D70F}}_{2}$ that increases monotonically from the rear tail to the front nose. The pressure jump $\bar{p}_{2}-\hat{\bar{p}}_{2}$ , in turn, decreases along the axial coordinate, resulting in a pressure drop of $O(\unicode[STIX]{x1D716})$ . Calculation of $r_{3}$ requires consideration of the outer flow field and is beyond the scope of the present work. Thus, further corrections are not pursued here.

6.1 Long, cylindrical vesicles $(\bar{\ell }\gg 1)$

Asymptotic solutions of (5.26)–(5.30) and (5.48)–(5.52) may be determined when the reduced length $\bar{\ell }$ is either very large or very small, the former being a regular limit and the latter being a singular one. The asymptotic solution ‘for large $\bar{\ell }$ ’ is the subject of this section. Solutions ‘for small $\bar{\ell }$ ’ are discussed in § 6.2. In this and the subsequent section, the bending terms in the shape equations are neglected by setting $\bar{\unicode[STIX]{x1D6FD}}=0$ . This limit might raise concerns, since $\bar{\unicode[STIX]{x1D6FD}}$ is multiplying the highest derivative in (5.27) and (5.49). However, it turns out that the limit $\bar{\unicode[STIX]{x1D6FD}}\rightarrow 0$ is a regular one, so setting $\bar{\unicode[STIX]{x1D6FD}}=0$ does not preclude satisfaction of the boundary conditions. Results for finite values of $\bar{\unicode[STIX]{x1D6FD}}$ are discussed in § 7.

Figure 5.

Separation of the front and rear parts of the inner region for a long, cylindrical vesicle ( $\bar{\ell }\gg 1$ ). The dashed curves are the linearized solutions near $\bar{h}=1$ .

As $\bar{\ell }\rightarrow \infty$ , the midsection of the vesicle adopts a cylindrical shape and the supporting film becomes asymptotically uniform with thickness $\bar{h}=1$ and flux $\bar{q}=1/2+O(\unicode[STIX]{x1D716})$ . A numerical solution for $\bar{h}_{0}$ and $\bar{h}_{1}$ may be derived using the procedure originally outlined by Bretherton (Reference Bretherton1961). In brief, the domain is separated into a ‘front’ and ‘rear’ part separated by a long, middle section where $\bar{h}=1$ , as shown in figure 5. Linearizing (5.27) (with $\bar{\unicode[STIX]{x1D6FD}}=0$ ) about the fixed point $(\bar{h}_{0},\bar{q}_{0})=(1,1/2)$ yields

(6.1) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D70F}}_{0}\frac{\text{d}^{3}\bar{h}_{0}}{\text{d}\bar{x}^{3}}-6(\bar{h}_{0}-1)=0.\end{eqnarray}$$

Starting from the front end of the vesicle, the dominant trajectory approaching the uniform film is an exponential profile,

(6.2) $$\begin{eqnarray}\text{as }\bar{x}\rightarrow -\infty :\qquad \bar{h}_{0}-1\rightarrow c_{0}\text{e}^{\unicode[STIX]{x1D6FE}_{0}\bar{x}},\end{eqnarray}$$

where $c_{0}$ is a constant of integration and $\unicode[STIX]{x1D6FE}_{0}=(6/\bar{\unicode[STIX]{x1D70F}}_{0})^{1/3}.$ From the rear end, the trajectory towards the uniform film oscillates in space,

(6.3) $$\begin{eqnarray}\text{as }\bar{x}^{\ast }\rightarrow \infty :\quad \bar{h}_{0}-1\rightarrow \text{e}^{-\unicode[STIX]{x1D6FE}_{0}\bar{x}^{\ast }/2}\left[c_{0}^{\ast }\cos \left(\sqrt{3}\unicode[STIX]{x1D6FE}_{0}\bar{x}^{\ast }/2\right)+d_{0}^{\ast }\sin \left(\sqrt{3}\unicode[STIX]{x1D6FE}_{0}\bar{x}^{\ast }/2\right)\right],\end{eqnarray}$$

where $c_{0}^{\ast }$ and $d_{0}^{\ast }$ are constants of integration. Equations (6.2)–(6.3) are meant to supplant (5.28) in the limit $(\bar{\unicode[STIX]{x1D6FD}},\bar{\ell })\rightarrow (0,\infty )$ . The front part of the shape is determined first and forced to match to (6.2), which then specifies a unique rear profile that matches to (6.3).

A similar analysis may be applied to $\bar{h}_{1}(\bar{x})$ . Linearizing equation (5.49) (with $\bar{\unicode[STIX]{x1D6FD}}=0$ ) near the fixed point $(\bar{h}_{1},\bar{q}_{1})=(0,0)$ yields

(6.4) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D70F}}_{0}\frac{\text{d}^{3}\bar{h}_{1}}{\text{d}\bar{x}^{3}}-6\bar{h}_{1}=-\frac{6\bar{\unicode[STIX]{x1D70F}}_{1}}{\bar{\unicode[STIX]{x1D70F}}_{0}}(\bar{h}_{0}-1).\end{eqnarray}$$

Equation (6.4) admits the linearized trajectories,

(6.5) $$\begin{eqnarray}\text{as }\bar{x}\rightarrow -\infty :\qquad \bar{h}_{1}\rightarrow c_{1}\text{e}^{\unicode[STIX]{x1D6FE}_{0}\bar{x}}-{\textstyle \frac{1}{3}}c_{0}(\bar{\unicode[STIX]{x1D70F}}_{1}/\bar{\unicode[STIX]{x1D70F}}_{0})\unicode[STIX]{x1D6FE}_{0}\bar{x}\text{e}^{\unicode[STIX]{x1D6FE}_{0}\bar{x}},\end{eqnarray}$$

starting from the front end and

(6.6) $$\begin{eqnarray}\text{as }\bar{x}^{\ast }\rightarrow \infty :\qquad \bar{h}_{1}\rightarrow \text{e}^{-\unicode[STIX]{x1D6FE}_{0}\bar{x}^{\ast }/2}\left[c_{1}^{\ast }\cos \left(\sqrt{3}\unicode[STIX]{x1D6FE}_{0}\bar{x}^{\ast }/2\right)+d_{1}^{\ast }\sin \left(\sqrt{3}\unicode[STIX]{x1D6FE}_{0}\bar{x}^{\ast }/2\right)\right],\end{eqnarray}$$

from the rear end. Here, $c_{1}$ , $c_{1}^{\ast }$ and $d_{1}^{\ast }$ are constants of integration. Equations (6.5)–(6.6) supplant (5.50) in the limit $(\bar{\unicode[STIX]{x1D6FD}},\bar{\ell })\rightarrow (0,\infty )$ . The second term on the right-hand side of (6.5) is the particular solution forced by the $\bar{\unicode[STIX]{x1D70F}}_{1}(\text{d}^{3}\bar{h}_{0}/\text{d}\bar{x}^{3})$ term in (5.49). Carrying out the numerical integration at the front end reveals that $\bar{\unicode[STIX]{x1D70F}}_{1}=0$ , which is why no such term appears in (6.6).

Table 1.

Numerical results for long, cylindrical vesicles ( $\bar{\ell }\gg 1$ ) with negligible bending stiffness ( $\bar{\unicode[STIX]{x1D6FD}}=0$ ).

Numerical solutions for $\bar{h}_{0}$ and $\bar{h}_{1}$ in the limit $\bar{\ell }\rightarrow \infty$ are plotted in figure 3(a) atop the numerical solutions for finite $\bar{\ell }$ . Numerical results for the constants $\bar{q}$ , $\bar{\unicode[STIX]{x1D70F}}$ , $b$ , $b^{\ast }$ , $c$ , $c^{\ast }$ , $d^{\ast }$ and $\unicode[STIX]{x0394}\bar{p}^{+}$ are tabulated in table 1 and plotted in figure 4 for large $\bar{\ell }$ . Of particular interest is the extra pressure drop,

(6.7) $$\begin{eqnarray}\unicode[STIX]{x0394}\bar{p}^{+}=2(\ell -2)\unicode[STIX]{x1D716}^{1/2}+20.3442\unicode[STIX]{x1D716}+4.1086\unicode[STIX]{x1D716}^{3/2}+O(\unicode[STIX]{x1D716}^{3/2}),\end{eqnarray}$$

where $(\ell /2-1)/\unicode[STIX]{x1D716}^{1/2}$ has been exchanged for $\bar{\ell }$ since $\bar{\ell }\gg 1$ . The first term on the right-hand side of (6.7) is dominant for small $\unicode[STIX]{x1D716}$ and arises from the dissipation in the uniform-film region. The order of this term (in physical units) is $O[\unicode[STIX]{x1D707}UL/(Rh^{\circ })]$ , which is in agreement with the prediction made by Halpern & Secomb (Reference Halpern and Secomb1989) and invalidates the fallacious order assignment $O(\unicode[STIX]{x1D707}UL^{3}/R^{4})$ given by Bruinsma (Reference Bruinsma1996). The second and third terms in (6.7) are corrections due to dissipation through the ‘transitions regions’, in which $\bar{h}_{0}$ deviates from unity in order to match with the outer solution. These corrections are independent of the vesicle length. Finally, the $O(\unicode[STIX]{x1D716}^{3/2})$ error term is meant to account for additional ‘outer contributions’, which were not computed here.

An expression similar to (6.7) was derived by Cantat (Reference Cantat2013) for the extra pressure drop across ‘no-slip bubble’ whose surface is loaded with a high concentration of insoluble surfactant. She computed a numerical value of 19.9 for the second term (when expressed in the present nomenclature). The claim here is that the discrepancy between her result and (6.7) is due to the difference in the computed values of $b_{0}$ and $b_{0}^{\ast }$ . Cantat reported $b_{0}=2.88$ and $b_{0}^{\ast }=-0.82$ , which may be compared to the values $b_{0}=2.8996$ and $b_{0}^{\ast }=-0.8453$ reported in table 1 and even Bretherton’s original calculations of $b_{0}=2.79$ and $b_{0}^{\ast }=-0.73$ . The latter values have been cited as inaccurate by Wong, Radke & Morris (Reference Wong, Radke and Morris1995), among others. The constants $b_{0}$ and $b_{0}^{\ast }$ are determined through application of the far-field conditions (5.29)–(5.30) at finite domain boundaries $\bar{X}$ and $-\bar{X}^{\ast }$ . Since $\bar{h}_{0}$ grows like $\bar{x}^{2}$ in the far field, the error in $\bar{h}_{0}$ incurred by truncating the domain decays like $1/\bar{X}$ (cf. (5.27)), which is a slow, algebraic decay. In computing the coefficients in table 1, it was found necessary to subtract off the error in successive powers of $1/\bar{X}$ until the computed values of $b_{0}$ and $b_{0}^{\ast }$ became independent of domain size. Thus, the reported values in table 1 were determined to an accuracy of four (or more) decimal places.

To conclude this section, it is worth quoting Bretherton’s result for the extra pressure drop across a long bubble in a tube, which when expressed in the present notation is given by

(6.8) $$\begin{eqnarray}\unicode[STIX]{x0394}\bar{p}^{+}=11.586\unicode[STIX]{x1D716}+O(\unicode[STIX]{x1D716}^{3/2}).\end{eqnarray}$$

(Equation (6.8) (in terms of outer variables) is typically expressed as $\unicode[STIX]{x0394}p^{+}=10.02{\mathcal{C}}^{-1/3}+O(1)$ , where ${\mathcal{C}}=\unicode[STIX]{x1D707}U/T$ is the capillary number and $T$ is the bubble surface tension. The capillary number has been eliminated using the well-known result for the clearance, $\unicode[STIX]{x1D716}=1.337{\mathcal{C}}^{2/3}+O({\mathcal{C}})$ .) Two important differences between (6.8) and (6.7) are worth mentioning. Firstly, it is apparent that equation (6.8) does not depend on the bubble length. This result is somewhat intuitive: since the bubble surface is shear-free, the long, cylindrical portion of the bubble does not drag fluid and so, in principle, would not contribute to the extra pressure drop. Later studies, however, have indicated a dependence on bubble length when $\unicode[STIX]{x1D716}$ is not vanishingly small (Schwartz, Princen & Kiss Reference Schwartz, Princen and Kiss1986). The second important difference is the numerical prefactor multiplying $\unicode[STIX]{x1D716}$ in (6.8), which is nearly half that of (6.7). Again, the difference can be traced to the fact that a bubble surface is a ‘shear-free surface’, whereas the vesicle membrane is treated here as a ‘no-slip surface’. The latter admits higher dissipation in the transition regions between the uniform-film and end-cap regions.

6.2 Short, spherical vesicles $(\bar{\ell }\ll \,1)$

As $\bar{\ell }\rightarrow 0$ , the front nose and rear tail interact through a very short distance. The vesicle is forced into a nearly spherical shape and, consequently, the ability of the surface curvature to change in response to large lubrication pressures is dampened. In response, the membrane tension diverges to balance normal stresses on the membrane. Since the membrane is immobile, the flow field, leakback flux $\bar{q}$ and extra pressure drop $\unicode[STIX]{x0394}\bar{p}^{+}$ are adequately described by the theory for closely fitting, rigid spheres that are freely suspended in a fluid-filled tube (Hochmuth & Sutera Reference Hochmuth and Sutera1970; Bungay & Brenner Reference Bungay and Brenner1973).

The shape deformation $\bar{h}$ and resulting membrane tension $\bar{\unicode[STIX]{x1D70F}}$ for small but finite values of $\bar{\ell }$ are not immediately obvious. In fact, their solution in the limit of vanishing $\bar{\ell }$ requires a singular perturbation theory in terms of a second small parameter $\bar{\ell }^{1/2}\ll 1$ . In the limit $\bar{\ell }\rightarrow 0$ , it is convenient to shift the origin to the midplane of the vesicle,

(6.9) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D709}}=\bar{x}+\bar{\ell }=\bar{x}^{\ast }-\bar{\ell },\end{eqnarray}$$

which represents the ‘outer coordinate’ for small $\bar{\ell }$ . The associated ‘inner coordinate’ is given by

(6.10) $$\begin{eqnarray}\tilde{\bar{\unicode[STIX]{x1D709}}}=\bar{\ell }^{1/2}\bar{\unicode[STIX]{x1D709}},\end{eqnarray}$$

where an ‘overtilde’ is used to denote a rescaling with respect to $\bar{\ell }^{1/2}$ . A schematic of the new coordinate system is shown in figure 6. The ‘inner region’ for small $\bar{\ell }$ is located in the far field ( $\bar{\unicode[STIX]{x1D709}}\rightarrow \pm \infty$ ), where $\bar{h}$ is very large. The width of this region is of $O(\bar{\ell }^{-1/2})$ . The inner solution must be matched to an outer solution near $\tilde{\bar{\unicode[STIX]{x1D709}}}=0$ , where $\bar{h}=O(1)$ .

Figure 6.

Coordinate system for a short, spherical vesicle ( $\bar{\ell }\ll 1$ ). The ‘outer region’ is located inside the dashed box. The ‘inner region’ is located outside the dashed box within an $O(\bar{\ell }^{-1/2})$ boundary layer.

It is instructive at this point to derive the perturbative solution to (5.27)–(5.30) for the gap height $\bar{h}_{0}(\bar{\unicode[STIX]{x1D709}})$ in the limit $\bar{\ell }\rightarrow 0$ , setting $\bar{\unicode[STIX]{x1D6FD}}=0$ . Rescaling the dependent variables gives

(6.11) $$\begin{eqnarray}(\tilde{\bar{h}}_{0},\tilde{\bar{q}}_{0},\tilde{\bar{\unicode[STIX]{x1D70F}}}_{0},\tilde{b}_{0},\tilde{b}_{0}^{\ast })=(\bar{\ell }\bar{h}_{0},\bar{q}_{0},\bar{\ell }^{1/2}\bar{\unicode[STIX]{x1D70F}}_{0},b_{0},b_{0}^{\ast }).\end{eqnarray}$$

These scalings are necessary in order to satisfy all of the boundary conditions. The outer equations, which govern the unstretched gap height $\bar{h}_{0}(\bar{\unicode[STIX]{x1D709}})$ , are given by

(6.12) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\bar{\unicode[STIX]{x1D70F}}}_{0}\frac{\text{d}^{3}\bar{h}_{0}}{\text{d}\bar{\unicode[STIX]{x1D709}}^{3}}=\frac{6\bar{\ell }^{1/2}}{\bar{h}_{0}^{2}}\left(1-\frac{2\bar{q}_{0}}{\bar{h}_{0}}\right), & \displaystyle\end{eqnarray}$$

(6.13) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{\unicode[STIX]{x1D709}}=0:\qquad \bar{h}_{0}=1. & \displaystyle\end{eqnarray}$$

The corresponding inner equations for the stretched gap height $\tilde{\bar{h}}_{0}(\tilde{\bar{\unicode[STIX]{x1D709}}})$ are

(6.14) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\bar{\unicode[STIX]{x1D70F}}}_{0}\frac{\text{d}^{3}\tilde{\bar{h}}_{0}}{\text{d}\tilde{\bar{\unicode[STIX]{x1D709}}}^{3}}=\frac{6\bar{\ell }^{2}}{\tilde{\bar{h}}_{0}^{2}}-\frac{12\bar{\ell }^{3}\tilde{\bar{q}}_{0}}{\tilde{\bar{h}}_{0}^{3}}, & \displaystyle\end{eqnarray}$$

(6.15) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\tilde{\bar{\unicode[STIX]{x1D709}}}\rightarrow \infty :\qquad \tilde{\bar{h}}_{0}\rightarrow {\textstyle \frac{1}{2}}\tilde{\bar{\unicode[STIX]{x1D709}}}^{2}+\bar{\ell }\tilde{b}_{0}-\bar{\ell }^{3/2}\tilde{\bar{\unicode[STIX]{x1D709}}}+{\textstyle \frac{1}{2}}\bar{\ell }^{3}, & \displaystyle\end{eqnarray}$$

(6.16) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\tilde{\bar{\unicode[STIX]{x1D709}}}\rightarrow -\infty :\qquad \tilde{\bar{h}}_{0}\rightarrow {\textstyle \frac{1}{2}}\tilde{\bar{\unicode[STIX]{x1D709}}}^{2}+\bar{\ell }\tilde{b}_{0}^{\ast }+\bar{\ell }^{3/2}\tilde{\bar{\unicode[STIX]{x1D709}}}+{\textstyle \frac{1}{2}}\bar{\ell }^{3}. & \displaystyle\end{eqnarray}$$

The outer and inner solutions are then matched according to the matching conditions,

(6.17a ) $$\begin{eqnarray}\displaystyle & \displaystyle \lim _{\tilde{\bar{\unicode[STIX]{x1D709}}}\rightarrow 0^{+}}\tilde{\bar{h}}_{0}=\lim _{\bar{\unicode[STIX]{x1D709}}\rightarrow \infty }\bar{\ell }\bar{h}_{0}, & \displaystyle\end{eqnarray}$$

(6.17b ) $$\begin{eqnarray}\displaystyle & \displaystyle \lim _{\tilde{\bar{\unicode[STIX]{x1D709}}}\rightarrow 0^{-}}\tilde{\bar{h}}_{0}=\lim _{\bar{\unicode[STIX]{x1D709}}^{\ast }\rightarrow -\infty }\bar{\ell }\bar{h}_{0}. & \displaystyle\end{eqnarray}$$

$\bar{\ell }^{1/2}$

$\bar{\unicode[STIX]{x1D6FD}}$

$\bar{h}_{0}(\bar{\unicode[STIX]{x1D709}})$

(6.18) $$\begin{eqnarray}\displaystyle \bar{h}_{0} & = & \displaystyle 1+{\textstyle \frac{1}{2}}\bar{\unicode[STIX]{x1D709}}^{2}+\left(\frac{\sqrt{2}\bar{\ell }}{\unicode[STIX]{x03C0}}\right)^{1/2}4\arctan \left(\frac{\bar{\unicode[STIX]{x1D709}}}{\sqrt{2}}\right)+\left(\frac{\sqrt{2}\bar{\ell }}{\unicode[STIX]{x03C0}}\right)\left[-\frac{3\unicode[STIX]{x03C0}^{2}\bar{\unicode[STIX]{x1D709}}^{2}}{2}+\frac{11}{3}\left(\frac{\bar{\unicode[STIX]{x1D709}}^{2}}{1+\frac{1}{2}\bar{\unicode[STIX]{x1D709}}^{2}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left.\left(22-\frac{8}{1+\frac{1}{2}\bar{\unicode[STIX]{x1D709}}^{2}}\right)\frac{\bar{\unicode[STIX]{x1D709}}}{\sqrt{2}}\arctan \left(\frac{\bar{\unicode[STIX]{x1D709}}}{\sqrt{2}}\right)+6\bar{\unicode[STIX]{x1D709}}^{2}\arctan ^{2}\left(\frac{\bar{\unicode[STIX]{x1D709}}}{\sqrt{2}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle +\,O\left[\left(\frac{\sqrt{2}\bar{\ell }}{\unicode[STIX]{x03C0}}\right)^{3/2}\right].\end{eqnarray}$$

The algebra leading to (6.18) (as well as (6.26), below) is available from the authors upon request.

A similar procedure may be applied to the solution of (5.49)–(5.52) for $\bar{h}_{1}(\bar{\unicode[STIX]{x1D709}})$ after setting $\bar{\unicode[STIX]{x1D6FD}}=0$ . The rescaled dependent variables are

(6.19) $$\begin{eqnarray}(\tilde{\bar{h}}_{1},\tilde{\bar{q}}_{1},\tilde{\bar{\unicode[STIX]{x1D70F}}}_{1},\tilde{b}_{1},\tilde{b}_{1}^{\ast })=(\bar{\ell }^{1/2}\bar{h}_{1},\bar{q}_{1},\bar{\ell }^{3/2}\bar{\unicode[STIX]{x1D70F}}_{1},\bar{\ell }^{1/2}b_{1},\bar{\ell }^{1/2}b_{1}^{\ast }).\end{eqnarray}$$

As it turns out, it is necessary to stretch $\bar{h}_{1}$ to $\tilde{\bar{h}}_{1}$ in both regions of the domain in order to formulate a consistent expansion. Using the above definitions, it is straightforward to derive the outer equations,

(6.20) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\bar{\unicode[STIX]{x1D70F}}}_{0}\frac{\text{d}^{3}\tilde{\bar{h}}_{1}}{\text{d}\bar{\unicode[STIX]{x1D709}}^{3}}=-\frac{6\tilde{\bar{\unicode[STIX]{x1D70F}}}_{1}}{\bar{h}_{0}^{2}\tilde{\bar{\unicode[STIX]{x1D70F}}}_{0}}\left(1-\frac{2\bar{q}_{0}}{\bar{h}_{0}}\right)-\frac{12\bar{\ell }^{1/2}\tilde{\bar{h}}_{1}}{\bar{h}_{0}^{3}}\left(1-\frac{3\bar{q}_{0}}{\bar{h}_{0}}\right)-\frac{12\bar{\ell }\bar{q}_{1}}{\bar{h}_{0}^{3}}, & \displaystyle\end{eqnarray}$$

(6.21) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\bar{\unicode[STIX]{x1D709}}=0:\qquad \tilde{\bar{h}}_{1}=0, & \displaystyle\end{eqnarray}$$

the inner equations,

(6.22) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\bar{\unicode[STIX]{x1D70F}}}_{0}\frac{\text{d}^{3}\tilde{\bar{h}}_{1}}{\text{d}\tilde{\bar{\unicode[STIX]{x1D709}}}^{3}}=-\frac{6\bar{\ell }^{1/2}\tilde{\bar{\unicode[STIX]{x1D70F}}}_{1}}{\tilde{\bar{h}}_{0}^{2}\tilde{\bar{\unicode[STIX]{x1D70F}}}_{0}}+\frac{12\bar{\ell }^{3/2}\tilde{\bar{q}}_{0}\tilde{\bar{\unicode[STIX]{x1D70F}}}_{1}}{\tilde{\bar{h}}_{0}^{3}\tilde{\bar{\unicode[STIX]{x1D70F}}}_{0}}-\frac{12\bar{\ell }^{2}\tilde{\bar{h}}_{1}}{\tilde{\bar{h}}_{0}^{3}}-\frac{12\bar{\ell }^{5/2}\tilde{\bar{q}}_{1}}{\tilde{\bar{h}}_{0}^{3}}+\frac{36\bar{\ell }^{3}\tilde{\bar{h}}_{1}\tilde{\bar{q}}_{0}}{\tilde{\bar{h}}_{0}^{4}}, & \displaystyle\end{eqnarray}$$

(6.23) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\tilde{\bar{\unicode[STIX]{x1D709}}}\rightarrow \infty :\qquad \tilde{\bar{h}}_{1}\rightarrow \tilde{b}_{1}-\tilde{b}_{0}\tilde{\bar{\unicode[STIX]{x1D709}}}+\bar{\ell }^{3/2}\tilde{b}_{0}, & \displaystyle\end{eqnarray}$$

(6.24) $$\begin{eqnarray}\displaystyle & \displaystyle \text{as }\tilde{\bar{\unicode[STIX]{x1D709}}}\rightarrow -\infty :\qquad \tilde{\bar{h}}_{1}\rightarrow \tilde{b}_{1}^{\ast }+\tilde{b}_{0}^{\ast }\tilde{\bar{\unicode[STIX]{x1D709}}}+\bar{\ell }^{3/2}\tilde{b}_{0}^{\ast }, & \displaystyle\end{eqnarray}$$

and the matching conditions,

(6.25a ) $$\begin{eqnarray}\displaystyle \lim _{\tilde{\bar{\unicode[STIX]{x1D709}}}\rightarrow 0^{+}}\tilde{\bar{h}}_{1}=\lim _{\bar{\unicode[STIX]{x1D709}}\rightarrow \infty }\tilde{\bar{h}}_{1}, & & \displaystyle\end{eqnarray}$$

(6.25b ) $$\begin{eqnarray}\displaystyle \lim _{\tilde{\bar{\unicode[STIX]{x1D709}}}\rightarrow 0^{-}}\tilde{\bar{h}}_{1}=\lim _{\bar{\unicode[STIX]{x1D709}}^{\ast }\rightarrow -\infty }\tilde{\bar{h}}_{1}. & & \displaystyle\end{eqnarray}$$

$\bar{h}_{0}$

$\bar{h}_{1}(\bar{\unicode[STIX]{x1D709}})$

(6.26) $$\begin{eqnarray}\displaystyle \bar{h}_{1} & = & \displaystyle \left(\frac{\sqrt{2}}{\unicode[STIX]{x03C0}\bar{\ell }}\right)^{1/2}\left\{2\arctan \left(\frac{\bar{\unicode[STIX]{x1D709}}}{\sqrt{2}}\right)+\left(\frac{\sqrt{2}\bar{\ell }}{\unicode[STIX]{x03C0}}\right)^{1/2}\left[-\frac{3\unicode[STIX]{x03C0}^{2}\bar{\unicode[STIX]{x1D709}}^{2}}{2}+\frac{11}{3}\left(\frac{\bar{\unicode[STIX]{x1D709}}^{2}}{1+\frac{1}{2}\bar{\unicode[STIX]{x1D709}}^{2}}\right)\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left.\left.\left(22-\frac{8}{1+\frac{1}{2}\bar{\unicode[STIX]{x1D709}}^{2}}\right)\frac{\bar{\unicode[STIX]{x1D709}}}{\sqrt{2}}\arctan \left(\frac{\bar{\unicode[STIX]{x1D709}}}{\sqrt{2}}\right)+6\bar{\unicode[STIX]{x1D709}}^{2}\arctan ^{2}\left(\frac{\bar{\unicode[STIX]{x1D709}}}{\sqrt{2}}\right)\right]+O\left(\frac{\sqrt{2}\bar{\ell }}{\unicode[STIX]{x03C0}}\right)\!\!\right\}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Equations (6.18) and (6.26), truncated after the $O(1)$ term, are plotted in figure 3(c) atop the full numerical solutions for finite $\bar{\ell }$ . Analytical results for the constants $\bar{q}$ , $\bar{\unicode[STIX]{x1D70F}}$ , $b$ , $b^{\ast }$ and $\unicode[STIX]{x0394}\bar{p}^{+}$ are tabulated in table 2 and plotted in figure 4 for small $\bar{\ell }$ . The corrections $\bar{q}_{1}$ and $\unicode[STIX]{x0394}\bar{p}_{3}^{+(inner)}$ were not computed.

Table 2.

Analytical results for short, spherical vesicles ( $\bar{\ell }\ll 1$ ) with negligible bending stiffness ( $\bar{\unicode[STIX]{x1D6FD}}=0$ ).

The extra pressure drop, to leading order, is given by

(6.27) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\bar{p}^{+} & = & \displaystyle 4\sqrt{2}\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}+O(\unicode[STIX]{x1D716}^{3/2})\nonumber\\ \displaystyle & \simeq & \displaystyle 17.7715\unicode[STIX]{x1D716}+O(\unicode[STIX]{x1D716}^{3/2}),\end{eqnarray}$$

which is simply the result for a rigid sphere that is freely suspended in a Poiseuille flow (Hochmuth & Sutera Reference Hochmuth and Sutera1970; Bungay & Brenner Reference Bungay and Brenner1973). It is noteworthy that the $O(\unicode[STIX]{x1D716})$ term in (6.27) is smaller than the like-ordered term in (6.7). This reduction in drag arises because the outer solution is forced to match to an inner solution with finite curvature rather than to a uniform film.

9.1 Review of assumptions

The narrow-gap theory presented in this paper was developed after making several assumptions other than the obvious restriction $\unicode[STIX]{x1D716}\ll 1$ . These include

1. (i) inertialessness,

2. (ii) weightlessness,

3. (iii) axisymmetry,

4. (iv) steadiness.

It is instructive at this point to justify these assumptions. In doing so, it will become necessary to introduce some new dimensional parameters that were previously omitted. In the forthcoming section, tube radii in the range $R=1{-}100~\unicode[STIX]{x03BC}\text{m}$ and vesicle speeds in the range $U=1~\unicode[STIX]{x03BC}\text{m}~\text{s}^{-1}$ to $1~\text{cm}~\text{s}^{-1}$ will be assumed, as these are typical working parameters in microfluidic experiments (Squires & Quake Reference Squires and Quake2005).

(i) Neglect of inertial forces in the equations of motion (2.4) is justified so long as the Reynolds number ${\mathcal{R}}=\unicode[STIX]{x1D70C}RU/\unicode[STIX]{x1D707}$ is sufficiently small, where $\unicode[STIX]{x1D70C}$ is the exterior fluid density (the interior fluid density being $\hat{\unicode[STIX]{x1D70C}}$ ). In the context of the narrow-gap assumption, one requires

(9.1) $$\begin{eqnarray}\bar{{\mathcal{R}}}=\unicode[STIX]{x1D716}{\mathcal{R}}\ll 1\end{eqnarray}$$

for inertial forces to be neglected in the exterior fluid. A similar condition may be written for the interior fluid. Using water as the typical working fluid (viscosity $\unicode[STIX]{x1D707}=1~\text{cP}$ and density $\unicode[STIX]{x1D70C}=1~\text{g}~\text{cm}^{-3}$ ), the Reynolds number is estimated to be in the range ${\mathcal{R}}=10^{-6}-1$ . Thus, equation (9.1) is always satisfied so long as $\unicode[STIX]{x1D716}\ll 1$ .

(ii) The fluid pressures $p$ and $\hat{p}$ were made ‘dynamic’ by subtracting the hydrostatic contribution $\unicode[STIX]{x1D70C}\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{x}$ and $\hat{\unicode[STIX]{x1D70C}}\boldsymbol{g}\boldsymbol{\cdot }\boldsymbol{x}$ , respectively, where $\boldsymbol{g}$ is the acceleration due to gravity. Defining the ‘gravity number’ as ${\mathcal{G}}=|\unicode[STIX]{x1D70C}-\hat{\unicode[STIX]{x1D70C}}|gR^{2}/(\unicode[STIX]{x1D707}U)$ (Leal Reference Leal2007) with $g\equiv |\boldsymbol{g}|=9.8~\text{m}~\text{s}^{-2}$ , gravitational terms may be neglected in the normal stress condition (2.9a ) so long as

(9.2) $$\begin{eqnarray}\bar{{\mathcal{G}}}=\unicode[STIX]{x1D716}^{3/2}{\mathcal{G}}\ll 1.\end{eqnarray}$$

Equation (9.2) is typically satisfied if the exterior and interior fluids have nearly equal density (a neutrally buoyant vesicle). The factor of $\unicode[STIX]{x1D716}^{3/2}$ in (9.2) diminishes the effect of non-neutral buoyancy. Assuming the largest density difference to be $|\unicode[STIX]{x1D70C}-\hat{\unicode[STIX]{x1D70C}}|=0.1~\text{g}~\text{cm}^{-3}$ , the gravity number is estimated to be in the range ${\mathcal{G}}=10^{-4}{-}10^{4}$ , with the largest gravity numbers corresponding to small velocities and large tube radii. It is possible for gravitational effects to play a role under these latter conditions.

(iii) Only axisymmetric vesicle shapes are reported in this work, raising the question as to whether non-axisymmetric solutions exist. Since the azimuthal coordinate $\unicode[STIX]{x1D719}$ is not stretched in the small- $\unicode[STIX]{x1D716}$ expansion,

(9.3) $$\begin{eqnarray}\frac{|\unicode[STIX]{x2202}^{2}h/\unicode[STIX]{x2202}\unicode[STIX]{x1D719}^{2}|}{|\unicode[STIX]{x2202}^{2}h/\unicode[STIX]{x2202}x^{2}|}=\unicode[STIX]{x1D716}\left(\frac{|\unicode[STIX]{x2202}^{2}\bar{h}/\unicode[STIX]{x2202}\unicode[STIX]{x1D719}^{2}|}{|\unicode[STIX]{x2202}^{2}\bar{h}/\unicode[STIX]{x2202}\bar{x}^{2}|}\right)=O(\unicode[STIX]{x1D716}),\end{eqnarray}$$

which implies that the change in the azimuthal curvature due to variations of $h$ along $\unicode[STIX]{x1D719}$ is small relative to the meridional curvature. Thus, non-axisymmetric solutions do not enter into the leading-order theory if $\unicode[STIX]{x1D716}\ll 1$ .

(iv) The steady-state form of the kinematic condition, $\text{d}q/\text{d}x=0$ , was used in the development of the narrow-gap theory. The unsteady form is

(9.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}q}{\unicode[STIX]{x2202}x}=0,\end{eqnarray}$$

where $t$ is the time variable (scaled by $R/U$ ). Rescaling $\bar{t}=t/\unicode[STIX]{x1D716}$ and eliminating $q$ from the expression, one obtains at leading order,

(9.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\bar{h}_{0}}{\unicode[STIX]{x2202}\bar{t}}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{x}}\left[\frac{\bar{h}_{0}}{2}+\frac{\bar{h}_{0}^{3}}{12}\left(\bar{\unicode[STIX]{x1D6FD}}\frac{\unicode[STIX]{x2202}^{5}\bar{h}_{0}}{\unicode[STIX]{x2202}\bar{x}^{5}}-\bar{\unicode[STIX]{x1D70F}}_{0}\frac{\unicode[STIX]{x2202}^{3}\bar{h}_{0}}{\unicode[STIX]{x2202}\bar{x}^{3}}\right)\right]=0.\end{eqnarray}$$

Equation (9.5) is the unsteady analogue of (5.27). The linear stability of $\bar{h}_{0}(\bar{x},\bar{t})$ near the fixed point $\bar{h}_{0}=1$ is analysed by introducing a Fourier perturbation $\bar{h}_{0}(\bar{x},\bar{t})=1+\unicode[STIX]{x1D6FF}\text{e}^{\unicode[STIX]{x1D714}\bar{t}+\text{i}k\bar{x}}$ , where $\unicode[STIX]{x1D6FF}$ is the amplitude, $\unicode[STIX]{x1D714}$ is the frequency and $k$ is the wavenumber. After neglecting $O(\unicode[STIX]{x1D6FF}^{2})$ terms, the dispersion relationship between $\unicode[STIX]{x1D714}$ and $k$ is found to be

(9.6) $$\begin{eqnarray}\unicode[STIX]{x1D714}=-{\textstyle \frac{1}{12}}(\bar{\unicode[STIX]{x1D6FD}}k^{6}+\bar{\unicode[STIX]{x1D70F}}_{0}k^{4})+{\textstyle \frac{1}{2}}\text{i}k.\end{eqnarray}$$

Equation (9.6) indicates that the Fourier perturbations are travelling waves with phase speed $k/2$ . The waves decay in time so long as $\bar{\unicode[STIX]{x1D70F}}_{0}+\bar{\unicode[STIX]{x1D6FD}}k^{2}>0$ . Thus, the membrane is stable to small-amplitude perturbations when under tension ( $\bar{\unicode[STIX]{x1D70F}}_{0}>0$ ). It is not possible, under the auspices of the theory, for the vesicle to be put under compression ( $\bar{\unicode[STIX]{x1D70F}}_{0}<0$ ), as can occur when the vesicle is freely suspended in a quiescent fluid. Since it was previously stipulated that $\bar{\unicode[STIX]{x1D6FD}}=O(\unicode[STIX]{x1D716}^{-1/2})$ , the viscous stress term in (5.27) remains dominant for any choice of the parameters. As $\bar{\unicode[STIX]{x1D6FD}}$ increases, the membrane tension $\bar{\unicode[STIX]{x1D70F}}_{0}$ decreases in magnitude (but remains positively valued), resulting in a balance between curvature elasticity and viscous stress in (5.27).

The crucial criteria for shape stability were $\unicode[STIX]{x1D716}\ll 1$ and $\bar{\unicode[STIX]{x1D6FD}}=O(\unicode[STIX]{x1D716}^{-1/2})$ . For larger values of $\unicode[STIX]{x1D716}$ , it is possible for the membrane tension to balance (even locally) the curvature-elastic term in (5.27). When this balance occurs, the membrane tension can become negatively valued and locally flat sections of membrane become linearly unstable to large-wavenumber perturbations. Schwartz et al. (Reference Schwartz, Princen and Kiss1986) also showed that variations in the azimuthal curvature can become non-negligible and destabilizing when $\unicode[STIX]{x1D716}$ is not small.

9.2 Comparison to direct numerical simulations

Trozzo et al. (Reference Trozzo, Boedec, Leonetti and Jaeger2015) performed axisymmetric boundary element simulations of vesicles flowing in capillary tubes. A comparison of the present theory to their results for the radius of curvature $r=1-h$ and membrane traction $\boldsymbol{f}=(\unicode[STIX]{x1D64E}-\hat{\unicode[STIX]{x1D64E}})\boldsymbol{\cdot }\boldsymbol{n}$ is shown in figure 11 for two sets of the parameters $\unicode[STIX]{x1D710}$ , $\unicode[STIX]{x1D6EC}$ and $\unicode[STIX]{x1D6FD}$ (also reported are the associated parameters $\unicode[STIX]{x1D716}$ , $\bar{\ell }$ and $\bar{\unicode[STIX]{x1D6FD}}$ used in the perturbation theory).

Figure 11.

Comparison of the present theoretical predictions for vesicle shape $r$ and traction components $f_{x}$ and $f_{\unicode[STIX]{x1D70E}}$ to the boundary element simulations of Trozzo et al. (Reference Trozzo, Boedec, Leonetti and Jaeger2015) (viscosity ratio $\unicode[STIX]{x1D705}=1$ ). Results are shown for (a) $\unicode[STIX]{x1D710}=0.84$ , $\unicode[STIX]{x1D6EC}=1.2$ , $\unicode[STIX]{x1D6FD}=0.01$ ( $\unicode[STIX]{x1D716}=0.22$ , $\bar{\ell }=1.6$ , $\bar{\unicode[STIX]{x1D6FD}}=0.006$ ) and (b) $\unicode[STIX]{x1D710}=0.61$ , $\unicode[STIX]{x1D6EC}=2.1$ , $\unicode[STIX]{x1D6FD}=0.03$ ( $\unicode[STIX]{x1D716}=0.057$ , $\bar{\ell }=16$ , $\bar{\unicode[STIX]{x1D6FD}}=0.008$ ).

The agreement between the theory and the simulations improves with decreasing $\unicode[STIX]{x1D716}$ , as expected. The theoretical prediction for the radial component of the membrane traction $f_{\unicode[STIX]{x1D70E}}$ exhibits the largest discrepancy with the simulation result. Whereas the simulation reports a linearly increasing profile over the majority of the thin-film region, the perturbative result is mostly flat. This discrepancy is due to the neglect of higher-order perturbative corrections that scale linearly with $\bar{\ell }$ . The radial traction $f_{\unicode[STIX]{x1D70E}}$ is approximately given by the radial projection of the pressure jump vector $-(p-\hat{p})|_{\unicode[STIX]{x1D70E}=r(x)}\boldsymbol{n}$ , which is expanded up to first order in $\unicode[STIX]{x1D716}^{1/2}$ in this work:

(9.7) $$\begin{eqnarray}(p-\hat{p})|_{\unicode[STIX]{x1D70E}=r(x)}=\unicode[STIX]{x1D716}^{-3/2}[\bar{p}_{0}-\hat{\bar{p}}_{0}+\unicode[STIX]{x1D716}^{1/2}(\bar{p}_{1}-\hat{\bar{p}}_{1})+O(\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}\bar{\ell })].\end{eqnarray}$$

Since $\unicode[STIX]{x0394}\bar{p}_{2}^{+}\sim 4\bar{\ell }$ , according to (6.7), the coefficient $\bar{p}_{2}-\hat{\bar{p}}_{2}$ of the next term in the asymptotic series must be a function that decreases by an amount $4\bar{\ell }$ through the thin-film region. This inner correction forces an outer correction to the rear pressure that is $O(\unicode[STIX]{x1D716}\bar{\ell })$ .

By contrast, both the vesicle radius $r$ and the axial component of the membrane traction $f_{x}$ are well approximated by the perturbation theory within the front nose and thin-film regions. Significant errors in these functions are only observed at the rear tail – again, due to the neglect of $O(\unicode[STIX]{x1D716}\bar{\ell })$ terms in the perturbative solution. In the thin-film region, the dominant contribution to $f_{x}$ is the membrane shear stress $(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}\unicode[STIX]{x1D70E})|_{\unicode[STIX]{x1D70E}=r(x)}$ :

(9.8) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}\right|_{\unicode[STIX]{x1D70E}=r(x)}=-\unicode[STIX]{x1D716}^{-1}\left(\left.\frac{\unicode[STIX]{x2202}\bar{u}_{0}}{\unicode[STIX]{x2202}\bar{y}}\right|_{\bar{y}=\bar{h}_{0}(\bar{x})}+\unicode[STIX]{x1D716}^{1/2}\left.\frac{\unicode[STIX]{x2202}\bar{u}_{1}}{\unicode[STIX]{x2202}\bar{y}}\right|_{\bar{y}=\bar{h}_{0}(\bar{x})}+O(\unicode[STIX]{x1D716},\unicode[STIX]{x1D716}\bar{\ell })\right).\end{eqnarray}$$

The agreement with the perturbation theory is therefore expected, since the small- $\unicode[STIX]{x1D716}$ assumption resolves parallel flow (and, therefore, shear stresses) within the narrow gap. The fact that shear stresses are more well resolved by the perturbation theory than normal stresses is important for calculating the dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}U/V$ , which is given by the resultant of shear stresses on the tube wall.

9.3 Comparison to experiments and previous theories

In figure 12, predictions for the speed ratio $U/V$ and dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}U/V$ are compared to available experimental measurements for vesicles and red blood cells in capillary flow. Both the composite expansions (solid lines) as well as the asymptotic interpolants (8.8)–(8.9) (dashed lines) are plotted.

Figure 12.

(a) Velocity ratio $U/V$ and (b) dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}U/V$ plotted against the radius ratio $\unicode[STIX]{x1D6EC}$ for $\unicode[STIX]{x1D6FD}=0$ and several reduced volumes $\unicode[STIX]{x1D710}$ . Also shown are the experimental measurements of Vitkova et al. (Reference Vitkova, Mader and Podgorski2004), Ahmmed et al. (Reference Ahmmed, Suteria, Garbin and Vanapalli2017) (for vesicles), Albrecht et al. (Reference Albrecht, Gaehtgens, Pries and Heuser1979) and Pries, Neuhaus & Gaehtgens (Reference Pries, Neuhaus and Gaehtgens1992) (for red blood cells) as well as previous theories due to Secomb et al. (Reference Secomb, Skalak, Oozkaya and Gross1986) and Halpern & Secomb (Reference Halpern and Secomb1989) (for red blood cells).

Measurements of $U/V$ for vesicles in circular capillaries ( $\unicode[STIX]{x1D710}=0.924{-}0.999$ ) and square channels ( $\unicode[STIX]{x1D710}=0.80{-}1$ ) were collected by Vitkova et al. (Reference Vitkova, Mader and Podgorski2004) and Ahmmed et al. (Reference Ahmmed, Suteria, Garbin and Vanapalli2017), respectively. Unfortunately, experimental measurements of $\unicode[STIX]{x0394}p^{+}U/V$ for vesicles in channel flow are currently unavailable. The experimental data compiled from Albrecht et al. (Reference Albrecht, Gaehtgens, Pries and Heuser1979) and Pries et al. (Reference Pries, Neuhaus and Gaehtgens1992) are for suspensions of red blood cells flowing in capillary tubes at various volume concentrations (or ‘haematocrits’). Other reported measurements (e.g. those of Tomaiuolo et al. (Reference Tomaiuolo, Simeone, Martinelli, Rotoli and Guido2009)) were rejected on the basis that the red blood cells were not highly confined. Also shown are the theories of Secomb et al. (Reference Secomb, Skalak, Oozkaya and Gross1986) and Halpern & Secomb (Reference Halpern and Secomb1989) for red blood cells with vanishing shear and bending elasticity. The predictions of Bruinsma (Reference Bruinsma1996), which are not of the correct scale, are omitted for the sake of clarity.

It is clear that $U/V$ and $\unicode[STIX]{x0394}p^{+}U/V$ are highly sensitive to $\unicode[STIX]{x1D710}$ and $\unicode[STIX]{x1D6EC}$ . Figure 12(a) shows that a 5 % change in the reduced volume $\unicode[STIX]{x1D710}$ can lead to as much as a 10 % change in the predicted speed ratio $U/V$ . As illustrated by the vesicle shapes in figure 10, decreasing $\unicode[STIX]{x1D710}$ tends to ‘streamline’ the body of the vesicle, which enhances its mobility in the tube. Variation in the measured values of $\unicode[STIX]{x1D710}$ and $\unicode[STIX]{x1D6EC}$ can explain the scatter in the experimental measurements of Vitkova et al. (Reference Vitkova, Mader and Podgorski2004) and Ahmmed et al. (Reference Ahmmed, Suteria, Garbin and Vanapalli2017). It is important to keep in mind that the measurements of Ahmmed et al. (Reference Ahmmed, Suteria, Garbin and Vanapalli2017) were conducted in square conduits rather than circular capillaries, which would enhance the vesicle mobility for a given hydraulic channel radius.

The present theory, which is valid for all reduced volumes, complements and extends previous results for red blood cells, which are typically restricted to a narrow window of reduced volumes centred about $\unicode[STIX]{x1D710}\simeq 0.61$ (Canham & Burton Reference Canham and Burton1968). Since shear and bending elasticity play only a secondary role relative to the isotropic membrane tension under high confinement, one expects the results for red blood cells and vesicles to be comparable. The present theory converges to the results of Halpern & Secomb (Reference Halpern and Secomb1989) as $\unicode[STIX]{x1D6EC}\rightarrow \unicode[STIX]{x1D6EC}^{\ast }=2.3047$ for $\unicode[STIX]{x1D710}=0.61$ . Relatively good agreement with the data reported by Albrecht et al. (Reference Albrecht, Gaehtgens, Pries and Heuser1979) and Pries et al. (Reference Pries, Neuhaus and Gaehtgens1992) is observed, although the present theory overpredicts the data of Pries et al. (Reference Pries, Neuhaus and Gaehtgens1992) for smaller values of $\unicode[STIX]{x1D6EC}$ due to the neglect of higher-order perturbative corrections. In this regime, the parallel-flow theory of Secomb et al. (Reference Secomb, Skalak, Oozkaya and Gross1986), in which the transverse component of the velocity is neglected in lieu of invoking the narrow-gap assumption, more accurately captures the experimental data.