2.1 Governing equations

A schematic of the problem is shown in figure 1 with Cartesian axes $(x,y,z)=(x_{1},x_{2},x_{3})$ . A vesicle with volume $\unicode[STIX]{x1D6FA}_{0}$ and surface area $A_{0}$ enters a square duct and flows until it reaches some fully developed configuration (independent of end effects). The hydraulic radius of the duct is $R$ , defined as twice the ratio of the flow area to the wetted perimeter of the duct. In the creeping-flow regime, the fluid velocity $\boldsymbol{u}$ and stress $\unicode[STIX]{x1D64E}$ satisfy the Stokes equations (written below in Einstein notation),

(2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle \frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{j}}=0\quad \\ \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D61A}_{ij}}{\unicode[STIX]{x2202}x_{j}}=-\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u_{i}=0\quad \end{array}\right.\quad \text{in the suspending fluid}, & \displaystyle\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle \frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{j}}=0\quad \\ \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D61A}_{ij}}{\unicode[STIX]{x2202}x_{j}}=-\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{i}}+\hat{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D6FB}^{2}u_{i}=0\quad \end{array}\right.\quad \text{within the vesicle}, & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$\hat{\unicode[STIX]{x1D707}}$

$p$

$V$

(2.2) $$\begin{eqnarray}VA_{\times }=\int _{\mathscr{S}_{\times }}u_{1}\,\text{d}S\quad \text{as }x_{1}\rightarrow \pm \infty .\end{eqnarray}$$

Here, $\mathscr{S}_{\times }$ denotes a cross-sectional surface in the channel with surface area $A_{\times }$ . On the channel wall, the velocity $\boldsymbol{u}$ is subject to the no-slip condition,

(2.3) $$\begin{eqnarray}u_{i}=0\quad \text{on the channel wall}.\end{eqnarray}$$

The vesicle membrane is treated as an internally fluid, incompressible surface with unit normal $\boldsymbol{n}$ . Surface-area incompressibility implies that

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D617}_{ij}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}=0\quad \text{on the vesicle membrane},\end{eqnarray}$$

where $\unicode[STIX]{x1D64B}=\unicode[STIX]{x1D644}-\boldsymbol{n}\boldsymbol{n}$ is the surface projection tensor. The velocity $\boldsymbol{u}$ is continuous across the membrane surface, whereas the stress $\unicode[STIX]{x1D64E}$ undergoes a jump. Neglecting bending elasticity, the membrane boundary conditions read

(2.5) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x27E6}u_{i}\unicode[STIX]{x27E7}=0\quad \\ \displaystyle n_{i}\unicode[STIX]{x27E6}\,f_{i}\unicode[STIX]{x27E7}=-2H\unicode[STIX]{x1D70F}\quad \\ \displaystyle P_{ij}\unicode[STIX]{x27E6}\,f_{j}\unicode[STIX]{x27E7}=-P_{ij}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}x_{j}}\quad \end{array}\right.\quad \text{on the vesicle membrane},\end{eqnarray}$$

where $\unicode[STIX]{x27E6}\,\cdot \,\unicode[STIX]{x27E7}$ is the ‘jump’ operator, $H=-[(\unicode[STIX]{x1D64B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{n}]/2$ is the mean curvature, $\boldsymbol{f}=\unicode[STIX]{x1D64E}\boldsymbol{\cdot }\boldsymbol{n}$ is the fluid traction and $\unicode[STIX]{x1D70F}$ is the membrane tension (a Lagrange field that enforces surface-area incompressibility). The justification for neglecting bending elasticity in (2.5) is provided at the end of this section.

Figure 1.

Schematic of a vesicle driven by a pressure gradient through a square duct, drawn in a reference frame translating with the vesicle velocity $U$ .

Since the vesicle membrane is a free surface, the kinematic condition further requires that it translate with the local fluid velocity,

(2.6) $$\begin{eqnarray}\frac{\text{D}x_{si}}{\text{D}t}=\frac{\unicode[STIX]{x2202}x_{si}}{\unicode[STIX]{x2202}t}+u_{j}\frac{\unicode[STIX]{x2202}x_{si}}{\unicode[STIX]{x2202}x_{j}}=u_{i}\quad \text{on the vesicle membrane},\end{eqnarray}$$

where $\text{D}/\text{D}t$ is the material derivative and $\boldsymbol{x}_{s}$ denotes the position of a point on the membrane. The unit normal $\boldsymbol{n}$ can be related to $\boldsymbol{x}_{s}$ via standard relations from differential geometry (Kreyszig Reference Kreyszig1959).

Although the governing Stokes equations are steady, the boundary-value problem is time dependent through the kinematic condition (2.6). Thus, the system is evolved starting from some initial configuration for the vesicle shape. At the initial time point, the total vesicle volume $\unicode[STIX]{x1D6FA}_{0}$ and surface area $A_{0}$ are prescribed:

(2.7a,b ) $$\begin{eqnarray}\text{at }t=0:\int _{\mathscr{V}_{s}}\text{d}^{3}\boldsymbol{x}=\unicode[STIX]{x1D6FA}_{0},\quad \int _{\mathscr{S}_{s}}\,\text{d}S=A_{0},\end{eqnarray}$$

where $\mathscr{V}_{s}$ is the region occupied by the vesicle and $\mathscr{S}_{s}=\unicode[STIX]{x2202}\mathscr{V}_{s}$ . Thus, $\unicode[STIX]{x1D6FA}_{0}$ and $A_{0}$ take the role of parameters in the initial boundary-value problem. The conditions of surface-area and volume incompressibility require that $\unicode[STIX]{x1D6FA}_{0}$ and $A_{0}$ be time-invariant in the Lagrangian sense,

(2.8a,b ) $$\begin{eqnarray}\frac{\text{D}\unicode[STIX]{x1D6FA}_{0}}{\text{D}t}=0,\quad \frac{\text{D}A_{0}}{\text{D}t}=0.\end{eqnarray}$$

2.2 Vesicle velocity and extra pressure drop

The translational velocity of the vesicle is defined as the volume average of the internal velocity field,

(2.9) $$\begin{eqnarray}U_{i}=\frac{1}{\unicode[STIX]{x1D6FA}_{0}}\int _{\mathscr{V}_{s}}u_{i}\,\text{d}^{3}\boldsymbol{x}=\frac{1}{\unicode[STIX]{x1D6FA}_{0}}\int _{\mathscr{S}_{s}}x_{i}u_{j}n_{j}\,\text{d}S,\end{eqnarray}$$

which follows from the fact that the velocity is solenoidal. The rotational velocity of the vesicle is not of interest here. The axial component of $\boldsymbol{U}$ is denoted by $U$ , and $U/V$ denotes the relative velocity of the vesicle with respect to the mean flow.

The extra pressure drop $\unicode[STIX]{x0394}p^{+}$ is obtained by requiring the vesicle to be force-free. Applying a force balance to a control volume containing the suspending fluid, bounded externally by the duct walls and internally by the vesicle surface, results in the following condition,

(2.10) $$\begin{eqnarray}\unicode[STIX]{x0394}pA_{\times }\unicode[STIX]{x1D6FF}_{1i}=-\!\int _{\mathscr{S}_{w}}f_{i}\,\text{d}S,\end{eqnarray}$$

where $\mathscr{S}_{w}$ is the wall surface and $\unicode[STIX]{x0394}p$ is the pressure drop over a distance $L_{c}$ (the length of the channel). Taking the formal limit as $L_{c}\rightarrow \infty$ results in a diverging pressure drop. A bounded quantity may be obtained by first subtracting out the Poiseuille-flow contribution and taking the limit thereafter:

(2.11) $$\begin{eqnarray}\unicode[STIX]{x0394}p^{+}=\lim _{L_{c}\rightarrow \infty }(\unicode[STIX]{x0394}p-\unicode[STIX]{x0394}p^{\circ })=\lim _{L_{c}\rightarrow \infty }(\unicode[STIX]{x0394}p-L_{c}A_{\times }K_{hyd}V),\end{eqnarray}$$

which follows from (1.1)–(1.2). The dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ can be directly related to the apparent intrinsic viscosity of the vesicle–fluid system (Barakat Reference Barakat2018).

2.3 Scalings and dimensionless groups

To recast the problem in dimensionless form, all variables with units of distance are scaled by the channel hydraulic radius $R$ , fluid velocity $\boldsymbol{u}$ by the mean velocity $V$ , fluid pressure $p$ by $\unicode[STIX]{x1D707}V/R$ and membrane tension $\unicode[STIX]{x1D70F}$ by $\unicode[STIX]{x1D707}V$ . The relevant dimensionless groups are

(2.12) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D705}=\frac{\hat{\unicode[STIX]{x1D707}}}{\unicode[STIX]{x1D707}}\quad \text{(the viscosity ratio)},\\ \displaystyle \unicode[STIX]{x1D706}=\frac{R_{0}}{R}\quad \text{(the radius ratio)},\\ \displaystyle \unicode[STIX]{x1D710}=\frac{\unicode[STIX]{x1D6FA}_{0}}{{\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}R_{0}^{3}}\quad \text{(the reduced volume)}.\end{array}\right\}\end{eqnarray}$$

Here, $R_{0}=\sqrt{A_{0}/(4\unicode[STIX]{x03C0})}$ is the effective vesicle radius based on surface area. The physical significance of these dimensionless groups is easily interpreted. The viscosity ratio $\unicode[STIX]{x1D705}$ is a measure of the relative dissipation between the interior and exterior fluids. The reduced volume $\unicode[STIX]{x1D710}$ varies between 0 and 1 and gauges the degree of sphericity of the vesicle shape (a reduced volume of unity corresponds to a perfectly spherical vesicle). Since the vesicle volume $\unicode[STIX]{x1D6FA}_{0}$ and surface area $A_{0}$ are assumed to be fixed quantities, $\unicode[STIX]{x1D710}$ is a geometric parameter that is unaffected by flow and membrane deformation. The radius ratio $\unicode[STIX]{x1D706}$ , which was introduced previously, relates the characteristic dimension of the vesicle to the characteristic dimension of the channel cross-section. Thus, $\unicode[STIX]{x1D706}$ is a measure for flow confinement: as $\unicode[STIX]{x1D706}$ increases, the vesicle’s motion is more significantly hindered by vesicle–wall interactions.

Under typical experimental conditions, using water as the typical working fluid (with some combination of dissolved electrolytes), $\unicode[STIX]{x1D707}\approx \hat{\unicode[STIX]{x1D707}}\approx 1~\text{mN}~\text{s}~\text{m}^{-2}$ and one may reasonably assume that $\unicode[STIX]{x1D705}\approx 1$ . The square microchannels considered in this work have dimensions $R\simeq 12~\unicode[STIX]{x03BC}\text{m}$ . Giant, unilamellar vesicles span a distribution of length scales ranging from $1{-}100~\unicode[STIX]{x03BC}\text{m}$ . Thus, the radius ratio $\unicode[STIX]{x1D706}$ can take on a range of values. When prepared using electroformation, these vesicles are typically spherical or nearly spherical, meaning that their reduced volume $\unicode[STIX]{x1D710}$ is of $O(1)$ . In this work, reduced volumes in the range $0.85\leqslant \unicode[STIX]{x1D710}\leqslant 1$ are considered.

It is apparent that no dimensionless group relating the intrinsic vesicular mechanics to the strength of flow has been introduced, and this choice requires some clarification. Namely, it is well known that vesicle membranes have intrinsic mechanical properties that can compete with hydrodynamic stresses created by fluid flow. Among them, the bending stiffness $E_{B}$ of the membrane is perhaps the most significant, as it directly influences the shape through the normal component of the local stress balance on the membrane. The effect of bending elasticity on vesicle motion through circular tubes was previously investigated by two of the authors, who determined that this effect is suppressed when $\unicode[STIX]{x1D706}$ or $\unicode[STIX]{x1D710}$ are large (Barakat & Shaqfeh Reference Barakat and Shaqfeh2018a ,Reference Barakat and Shaqfeh b ). To gauge the strength of bending forces relative to viscous forces, one has to compare $E_{B}$ to the characteristic viscous energy scale $\unicode[STIX]{x1D707}VR^{2}$ . A typical bending modulus of giant, unilamellar vesicles is $E_{B}\simeq 10^{-13}~\unicode[STIX]{x03BC}\text{N}~\text{m}$ (Evans & Needham Reference Evans and Needham1987). Typical mean fluid velocities in the microfluidic devices presented in § 5 are $V=1{-}5~\text{cm}~\text{s}^{-1}$ , which gives a viscous energy scale $\unicode[STIX]{x1D707}VR^{2}\simeq 10^{-9}~\unicode[STIX]{x03BC}\text{N}~\text{m}$ . (One could make the argument that the vesicle radius $R_{0}$ is the proper length scale to use here. However, the channel hydraulic radius $R$ typically bounds the membrane radius of curvature, making it the more suitable choice of length scale.) Thus, the dimensionless group $E_{B}/(\unicode[STIX]{x1D707}VR^{2})=O(10^{-4})$ is small enough to justify the neglect of membrane bending elasticity. Exceptional circumstances can arise when the local (deviatoric) curvature of the vesicle membrane becomes large. However, these situations never arise for vesicles that are ‘sphere-like’ ( $\unicode[STIX]{x1D710}=O(1)$ ), which are the subjects of the present study.

By neglecting bending elasticity, the only intrinsic characteristic of the vesicle membrane is that it is an incompressible surface. Strictly speaking, the vesicle membrane has a measurable dilatational modulus $E_{D}$ , akin to that of red blood cells, and is typically approximately $200~\text{mN}~\text{m}^{-1}$ . The analogous viscous scale is $\unicode[STIX]{x1D707}V\simeq 10^{-2}~\text{mN}~\text{m}^{-1}$ , giving a ratio $E_{D}/(\unicode[STIX]{x1D707}V)=O(10^{4})$ . This estimate suggests that the membrane cannot stretch significantly in the flow regime of interest (Marmottant & Hilgenfeldt Reference Marmottant and Hilgenfeldt2003; Vitkova, Mader & Podgorski Reference Vitkova, Mader and Podgorski2004). It is possible that large extensional stresses can be created locally, e.g. when the separation between the vesicle membrane and the duct wall becomes small. Under such circumstances, local dilatation can change the membrane surface area. However, this change is bounded by the maximally allowed areal strain (approximately 1 %–2 %) before the membrane ruptures (Evans & Needham Reference Evans and Needham1986). Thus, it is expected that the vesicle membrane conserves surface area locally to within a reasonable degree of accuracy.

4.1 Formulation

Cylindrical coordinates $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719},x)$ are conveniently adopted here, with $\unicode[STIX]{x1D70C}=\sqrt{y^{2}+z^{2}}$ being the radial distance, $\unicode[STIX]{x1D719}=\arctan (z/y)$ the azimuthal angle and $x$ the axial distance. The theory will be generally formulated for a polygonal duct with $N$ sides at steady state, assuming that the system has $N$ -fold symmetry (i.e. the geometry possesses reflectional symmetry at the planes $\unicode[STIX]{x1D719}=j\unicode[STIX]{x03C0}/N$ , $j=1,2,\ldots ,2N$ ). Unlike in § 3, where the vesicle was given a Lagrangian representation, in the forthcoming development the surface position vector $\boldsymbol{x}_{s}$ will be given the Eulerian representation,

(4.2) $$\begin{eqnarray}x_{si}(\unicode[STIX]{x1D719},x)=x\unicode[STIX]{x1D6FF}_{1i}+\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x)\cos \unicode[STIX]{x1D719}\unicode[STIX]{x1D6FF}_{2i}+\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x)\sin \unicode[STIX]{x1D719}\unicode[STIX]{x1D6FF}_{3i},\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x)$ defines the location of the vesicle membrane in space. The basic assumption of lubrication theory is to neglect components and gradients of velocity in the $\unicode[STIX]{x1D719}$ and $x$ directions. This assumption is rooted in assuming that spatial variations in the velocity field are most rapid in the transverse ( $\unicode[STIX]{x1D70C}$ ) direction as compared to the lateral ( $\unicode[STIX]{x1D719}$ , $x$ ) directions. In this approximation, the relevant velocity components $u_{\unicode[STIX]{x1D719}}$ and $u_{x}$ are governed by the simplified equations of motion,

(4.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}=0, & \displaystyle\end{eqnarray}$$

(4.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D707}\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}\left(\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D719}})\right), & \displaystyle\end{eqnarray}$$

(4.3c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}=\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}\left(\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}\right), & \displaystyle\end{eqnarray}$$

(4.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{x}=u_{\unicode[STIX]{x1D719}}=0\quad \text{at }\unicode[STIX]{x1D70C}=R(\unicode[STIX]{x1D719}), & \displaystyle\end{eqnarray}$$

(4.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}\left(\frac{u_{\unicode[STIX]{x1D719}}}{\unicode[STIX]{x1D70C}}\right)=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{\unicode[STIX]{x1D719}}}\quad \text{at }\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x), & \displaystyle\end{eqnarray}$$

(4.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{x}}\quad \text{at }\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x). & \displaystyle\end{eqnarray}$$

$\text{d}s_{\unicode[STIX]{x1D719}}=\sqrt{\unicode[STIX]{x1D70C}_{s}^{2}+(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D719})^{2}}\,\text{d}\unicode[STIX]{x1D719}$

$\text{d}s_{x}=\sqrt{1+(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x2202}x)^{2}}\,\text{d}x$

$R_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D719})$

(4.5a,b ) $$\begin{eqnarray}R_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D719})=R\sec \unicode[STIX]{x1D719},\quad 0\leqslant \unicode[STIX]{x1D719}\leqslant \frac{\unicode[STIX]{x03C0}}{N},\end{eqnarray}$$

where $N=4$ for a square duct. Due to symmetry, one need only consider the section of the domain bounded by $\unicode[STIX]{x1D719}=0$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/N$ . It is expected that these equations supply a good approximation of the local flow conditions provided that $|\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x2202}x|$ and $|\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D719}|$ are not very large.

The pressure $p$ must be determined by conserving volume in the suspending fluid. (Consideration of the fluid inside the vesicle is immaterial, since the shear rates in the suspending fluid are presumably much larger than those in the interior fluid.) Depth-averaging the equation of continuity and applying the kinematic condition for a steady vesicle shape ( $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{u}=0$ on the membrane) yields a local mass balance condition,

(4.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(\frac{q_{\unicode[STIX]{x1D719}}}{R}\right)+\frac{\unicode[STIX]{x2202}q_{x}}{\unicode[STIX]{x2202}x}=0,\end{eqnarray}$$

where

(4.7a,b ) $$\begin{eqnarray}q_{\unicode[STIX]{x1D719}}=\int _{\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x)}^{R_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D719})}Ru_{\unicode[STIX]{x1D719}}\,\text{d}\unicode[STIX]{x1D70C},\quad q_{x}=\int _{\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x)}^{R_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D719})}\unicode[STIX]{x1D70C}(u_{x}-U)\,\text{d}\unicode[STIX]{x1D70C}\end{eqnarray}$$

are the leakback fluxes (here, the factor of $R$ is introduced arbitrarily so that $q_{\unicode[STIX]{x1D719}}$ and $q_{x}$ have the same dimensions). Multiplying (4.6) by $\text{d}\unicode[STIX]{x1D719}$ , integrating from $\unicode[STIX]{x1D719}=0$ to $2\unicode[STIX]{x03C0}$ and applying periodicity conditions yields a macroscopic mass balance,

(4.8) $$\begin{eqnarray}\int _{0}^{2\unicode[STIX]{x03C0}}q_{x}\,\text{d}\unicode[STIX]{x1D719}=\int _{0}^{\unicode[STIX]{x03C0}/N}2Nq_{x}\,\text{d}\unicode[STIX]{x1D719}=Q-A_{\times }U=A_{\times }(V-U),\end{eqnarray}$$

where

(4.9) $$\begin{eqnarray}A_{\times }=\int _{0}^{\unicode[STIX]{x03C0}/N}NR_{\unicode[STIX]{x1D719}}^{2}\,\text{d}\unicode[STIX]{x1D719}\end{eqnarray}$$

is the cross-sectional surface area. Equation (4.8) holds for all $-L\leqslant x\leqslant 0$ , where $L$ is the length of the vesicle (a free boundary). Obviously, the velocities $U$ and $V$ cannot be specified independently. Rather, by specifying one, the other must be determined as part of the solution to the boundary-value problem. Here, it is most convenient to specify $U$ a priori and determine $V$ a posteriori.

The membrane tension $\unicode[STIX]{x1D70F}$ must also be determined, by conserving surface area locally on the membrane. This requirement is enforced by (2.4), which when written in cylindrical coordinates (with $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{u}=0$ on the membrane) becomes

(4.10) $$\begin{eqnarray}\frac{1}{G_{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(\frac{G_{s}\bar{u}_{s\unicode[STIX]{x1D719}}}{\unicode[STIX]{x1D70C}_{s}}\right)+\frac{1}{G_{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}(G_{s}\bar{u}_{sx})=0,\end{eqnarray}$$

where $G_{s}=\sqrt{\unicode[STIX]{x1D70C}_{s}^{2}[1+(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x2202}x)^{2}]+(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x2202}\unicode[STIX]{x1D719})^{2}}$ is the surface metric and

(4.11a,b ) $$\begin{eqnarray}\bar{u}_{s\unicode[STIX]{x1D719}}=u_{\unicode[STIX]{x1D719}}|_{\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x)},\quad \bar{u}_{sx}=(u_{x}-U)|_{\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D719},x)}\end{eqnarray}$$

are the membrane slip velocities. Multiplying (4.10) by $G_{s}\,\text{d}\unicode[STIX]{x1D719}$ , integrating from $\unicode[STIX]{x1D719}=0$ to $2\unicode[STIX]{x03C0}$ and applying periodicity conditions gives

(4.12) $$\begin{eqnarray}\int _{0}^{2\unicode[STIX]{x03C0}}G_{s}\bar{u}_{sx}\,\text{d}\unicode[STIX]{x1D719}=\int _{0}^{\unicode[STIX]{x03C0}/N}2NG_{s}\bar{u}_{sx}\,\text{d}\unicode[STIX]{x1D719}=0.\end{eqnarray}$$

The right-hand side of (4.12) has been set equal to zero by the requirement that the vesicle remain stationary in the moving reference frame. This condition holds for all $-L\leqslant x\leqslant 0$ .

The equations (4.10)–(4.12) are exact at steady state, but require as input the local flow field $\boldsymbol{u}$ (cf. 4.7 and 4.11). The lubrication approximation allows one to analytically compute the flow field by considering only the dominant shear stresses in the $x\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D719}\unicode[STIX]{x1D70C}$ planes. Thus, expressions for the components of the leakback flux vector $\boldsymbol{q}$ and membrane slip velocity vector $\bar{\boldsymbol{u}}_{s}$ can be obtained directly by integrating (4.3)–(4.4) with respect to $\unicode[STIX]{x1D70C}$ :

(4.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{q_{\unicode[STIX]{x1D719}}}{R}=\frac{m_{\unicode[STIX]{x1D719}}^{(qp)}}{\unicode[STIX]{x1D70C}_{s}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}+m_{\unicode[STIX]{x1D719}}^{(q\unicode[STIX]{x1D70F})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{\unicode[STIX]{x1D719}}},\quad q_{x}=m_{x}^{(qp)}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}+m_{x}^{(q\unicode[STIX]{x1D70F})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{x}}-\frac{1}{2}U(R_{\unicode[STIX]{x1D719}}^{2}-\unicode[STIX]{x1D70C}_{s}^{2}), & \displaystyle\end{eqnarray}$$

(4.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\bar{u}_{s\unicode[STIX]{x1D719}}}{\unicode[STIX]{x1D70C}_{s}}=\frac{m_{\unicode[STIX]{x1D719}}^{(up)}}{\unicode[STIX]{x1D70C}_{s}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}+m_{\unicode[STIX]{x1D719}}^{(u\unicode[STIX]{x1D70F})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{\unicode[STIX]{x1D719}}},\quad \bar{u}_{sx}=m_{x}^{(up)}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}+m_{x}^{(u\unicode[STIX]{x1D70F})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{x}}-U, & \displaystyle\end{eqnarray}$$

(4.14a ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D719}}^{(qp)}=-\frac{R_{\unicode[STIX]{x1D719}}^{3}}{8\unicode[STIX]{x1D707}}\left[\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}-\frac{\unicode[STIX]{x1D70C}_{s}^{5}}{R_{\unicode[STIX]{x1D719}}^{5}}+\frac{4\unicode[STIX]{x1D70C}_{s}^{3}}{R_{\unicode[STIX]{x1D719}}^{3}}\log \left(\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}\right)\right], & \displaystyle\end{eqnarray}$$

(4.14b ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D719}}^{(q\unicode[STIX]{x1D70F})}=-\frac{R_{\unicode[STIX]{x1D719}}^{2}}{4\unicode[STIX]{x1D707}}\left[\frac{\unicode[STIX]{x1D70C}_{s}^{2}}{R_{\unicode[STIX]{x1D719}}^{2}}-\frac{\unicode[STIX]{x1D70C}_{s}^{4}}{R_{\unicode[STIX]{x1D719}}^{4}}+\frac{2\unicode[STIX]{x1D70C}_{s}^{2}}{R_{\unicode[STIX]{x1D719}}^{2}}\log \left(\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}\right)\right], & \displaystyle\end{eqnarray}$$

(4.14c ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{x}^{(qp)}=-\frac{R_{\unicode[STIX]{x1D719}}^{4}}{16\unicode[STIX]{x1D707}}\left[\left(1-\frac{3\unicode[STIX]{x1D70C}_{s}^{2}}{R_{\unicode[STIX]{x1D719}}^{2}}\right)\left(1-\frac{\unicode[STIX]{x1D70C}_{s}^{2}}{R_{\unicode[STIX]{x1D719}}^{2}}\right)-\frac{4\unicode[STIX]{x1D70C}_{s}^{4}}{R_{\unicode[STIX]{x1D719}}^{4}}\log \left(\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}\right)\right], & \displaystyle\end{eqnarray}$$

(4.14d ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{x}^{(q\unicode[STIX]{x1D70F})}=\frac{R_{\unicode[STIX]{x1D719}}^{3}}{4\unicode[STIX]{x1D707}}\left[\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}-\frac{\unicode[STIX]{x1D70C}_{s}^{3}}{R_{\unicode[STIX]{x1D719}}^{3}}+\frac{2\unicode[STIX]{x1D70C}_{s}^{3}}{R_{\unicode[STIX]{x1D719}}^{3}}\log \left(\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}\right)\right], & \displaystyle\end{eqnarray}$$

(4.14e ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D719}}^{(up)}=\frac{R_{\unicode[STIX]{x1D719}}}{4\unicode[STIX]{x1D707}}\left[\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}-\frac{\unicode[STIX]{x1D70C}_{s}^{3}}{R_{\unicode[STIX]{x1D719}}^{3}}+\frac{2\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}\log \left(\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}\right)\right], & \displaystyle\end{eqnarray}$$

(4.14f ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D719}}^{(u\unicode[STIX]{x1D70F})}=\frac{1}{2\unicode[STIX]{x1D707}}\left(1-\frac{\unicode[STIX]{x1D70C}_{s}^{2}}{R_{\unicode[STIX]{x1D719}}^{2}}\right), & \displaystyle\end{eqnarray}$$

(4.14g ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{x}^{(up)}=-\frac{R_{\unicode[STIX]{x1D719}}^{2}}{4\unicode[STIX]{x1D707}}\left[1-\frac{\unicode[STIX]{x1D70C}_{s}^{2}}{R_{\unicode[STIX]{x1D719}}^{2}}+\frac{2\unicode[STIX]{x1D70C}_{s}^{2}}{R_{\unicode[STIX]{x1D719}}^{2}}\log \left(\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}\right)\right], & \displaystyle\end{eqnarray}$$

(4.14h ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{x}^{(u\unicode[STIX]{x1D70F})}=-\frac{\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x1D707}}\log \left(\frac{\unicode[STIX]{x1D70C}_{s}}{R_{\unicode[STIX]{x1D719}}}\right) & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x1D70C}_{s}$

$R_{\unicode[STIX]{x1D719}}$

$p$

$\unicode[STIX]{x1D70F}$

(4.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(\frac{m_{\unicode[STIX]{x1D719}}^{(qp)}}{\unicode[STIX]{x1D70C}_{s}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left(m_{x}^{(qp)}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}\right)\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(m_{\unicode[STIX]{x1D719}}^{(q\unicode[STIX]{x1D70F})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{\unicode[STIX]{x1D719}}}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left(m_{x}^{(q\unicode[STIX]{x1D70F})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{x}}\right)=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left[\frac{1}{2}U(R_{\unicode[STIX]{x1D719}}^{2}-\unicode[STIX]{x1D70C}_{s}^{2})\right],\end{eqnarray}$$

(4.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(\frac{G_{s}m_{\unicode[STIX]{x1D719}}^{(up)}}{\unicode[STIX]{x1D70C}_{s}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left(G_{s}m_{x}^{(up)}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}\right)\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(G_{s}m_{\unicode[STIX]{x1D719}}^{(u\unicode[STIX]{x1D70F})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{\unicode[STIX]{x1D719}}}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left(G_{s}m_{x}^{(u\unicode[STIX]{x1D70F})}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}s_{x}}\right)=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}(G_{s}U).\end{eqnarray}$$

Although these equations are linear in $p$ and $\unicode[STIX]{x1D70F}$ , the boundary-value problem is nonlinear because the membrane radius $\unicode[STIX]{x1D70C}_{s}$ must also be determined as part of the solution. The governing PDE for $\unicode[STIX]{x1D70C}_{s}$ is the normal stress balance condition on the membrane:

(4.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{1}{G_{s}^{3}}\left\{\unicode[STIX]{x1D70C}_{s}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}x^{2}}\left[\unicode[STIX]{x1D70C}_{s}^{2}+\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\right)^{2}\right]+\unicode[STIX]{x1D70C}_{s}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}^{2}}\left[1+\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}x}\right)^{2}\right]-2\unicode[STIX]{x1D70C}_{s}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}x\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\right)^{2}\right\}-\frac{1}{G_{s}}=\frac{p}{\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

The left-hand side of (4.17) is twice the mean curvature $2H$ expressed in cylindrical coordinates. The reference pressure is chosen to be the interior vesicle pressure (assumed spatially homogeneous), so that the pressure $p$ the right-hand side of (4.17) (approximately) reflects the jump in the normal component of the membrane surface traction.

Although, strictly speaking, the lubrication equations (4.15)–(4.17) are only valid in regions where the bounding surfaces are nearly parallel, two of the authors have previously found success in extending these equations outside the lubrication region under axisymmetric conditions (Barakat & Shaqfeh Reference Barakat and Shaqfeh2018a ). There is substantial literature precedent for this solution methodology in application to red blood cells (Secomb et al. Reference Secomb, Skalak, Oozkaya and Gross1986) and bubbles (Ratulowski & Chang Reference Ratulowski and Chang1989) in axisymmetric geometries. The basic justification for applying (4.15)–(4.17) outside the regions of lubrication hinges on the assumption that errors made in approximating the flow field outside the lubrication zone are of little consequence to the shape deformation and force moment integrals. This assumption breaks down when the region of closest separation between the vesicle membrane and the duct wall degenerates to points, as was discussed at the beginning of this section. Thus, equations (4.15)–(4.17) are applied over the whole domain $-L\leqslant x\leqslant 0$ and $0\leqslant \unicode[STIX]{x1D719}\leqslant \unicode[STIX]{x03C0}/N$ .

Equations (4.15)–(4.17) are nonlinear, elliptic PDEs for the fields $p$ , $\unicode[STIX]{x1D70F}$ , and $\unicode[STIX]{x1D70C}_{s}$ . In applying these equations to the whole domain, boundary conditions are required at the planes $x=0$ , $x=-L$ , $\unicode[STIX]{x1D719}=0$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/N$ . Simple geometric considerations yield the following conditions:

(4.18a ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D719}=0:\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}=0,\quad \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}=0,\quad \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}=0, & \displaystyle\end{eqnarray}$$

(4.18b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }\unicode[STIX]{x1D719}=\frac{\unicode[STIX]{x03C0}}{N}:\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}=0,\quad \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}=0,\quad \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}=0, & \displaystyle\end{eqnarray}$$

(4.18c ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }x=0:\unicode[STIX]{x1D70C}_{s}=0,\quad \frac{\unicode[STIX]{x1D70C}_{s}}{G_{s}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}x}=-1,\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(\frac{p}{\unicode[STIX]{x1D70F}}\right)=0, & \displaystyle\end{eqnarray}$$

(4.18d ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{at }x=-L:\unicode[STIX]{x1D70C}_{s}=0,\quad \frac{\unicode[STIX]{x1D70C}_{s}}{G_{s}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}}{\unicode[STIX]{x2202}x}=+1,\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\left(\frac{p}{\unicode[STIX]{x1D70F}}\right)=0. & \displaystyle\end{eqnarray}$$

$x$

$x=-L/2$

$\unicode[STIX]{x1D6FA}_{0}$

$A_{0}$

$\unicode[STIX]{x1D70C}_{s}$

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FA}_{0}=\int _{0}^{\unicode[STIX]{x03C0}/N}\int _{-L}^{0}N\unicode[STIX]{x1D70C}_{s}^{2}\,\text{d}x\,\text{d}\unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle A_{0}=\int _{0}^{\unicode[STIX]{x03C0}/N}\int _{-L}^{0}2NG_{s}\,\text{d}x\,\text{d}\unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

which must be satisfied simultaneously with (4.8), (4.12) and (4.15)–(4.18). The boundary-value problem is now closed. Once a solution for the fields $p$ , $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D70C}_{s}$ is obtained, the extra pressure drop $\unicode[STIX]{x0394}p^{+}$ is approximated by the difference in total pressure at the planes tangent to the front and rear ends of the vesicle:

(4.21) $$\begin{eqnarray}\unicode[STIX]{x0394}p^{+}\simeq \frac{1}{A_{\times }}\int _{0}^{\unicode[STIX]{x03C0}/N}NR_{\unicode[STIX]{x1D719}}^{2}(p|_{x=-L}-p|_{x=0})\,\text{d}\unicode[STIX]{x1D719}-LK_{hyd}A_{\times }V.\end{eqnarray}$$

The second term on the right-hand side is the Poiseuille pressure drop across a vesicle length $L$ . For a circular tube of radius $R$ , this contribution is equal to $8\unicode[STIX]{x1D707}VL/R^{2}$ . For a square duct ( $N=4$ ), it is $7.12\unicode[STIX]{x1D707}VL/R^{2}$ .

It is noteworthy that, although the derivation above was performed for a conduit whose cross-section is a polygon with $N$ sides, in principle this derivation could be applied to any cross-section. The applicability of the lubrication solution, however, is inherently dependent on the validity of neglecting additional viscous stresses in (4.3)–(4.4). The limitations of this assumption are discussed above.

As is common for problems in cylindrical coordinates, polar singularities occur at $x=0$ and $-L$ , resulting in a divergence of the membrane slope $|\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x2202}x|\sim |x|^{-1/2}$ near these points. Following Wong et al. (Reference Wong, Morris and Radke1992), a coordinate transformation removes the polar singularity:

(4.22a,b ) $$\begin{eqnarray}\displaystyle x=-\frac{L}{2}(1-\cos \unicode[STIX]{x1D709}),\quad \unicode[STIX]{x1D709}=\arccos \left(1+\frac{2x}{L}\right), & & \displaystyle\end{eqnarray}$$

where $0\leqslant \unicode[STIX]{x1D709}\leqslant \unicode[STIX]{x03C0}$ . The coordinate transformation (4.22) has the additional benefit of clustering grid points near $x=0$ and $x=-L$ , which in turn enables resolution of sharp gradients near the poles. Conversion between $x$ - and $\unicode[STIX]{x1D709}$ -operations (i.e. derivatives and integrals) is easily achieved via the chain rule. In the transformed domain, the polar singularity is removed by application of l’Hôpital’s rule (Wong et al. Reference Wong, Morris and Radke1992). However, two additional concerns remain. Firstly, removing the polar singularity does nothing to alleviate the additional stress singularity that results from the failure of the lubrication approximation at the poles. Although this latter kind of singularity cannot be removed, it is of little consequence to the solution of the boundary-value problem. The stresses computed via lubrication theory diverge within a very small region near the front nose and rear tail of the vesicle, and hence their overall contribution to the shape deformation and force moment integrals is negligible. Secondly, the one-to-one mapping between $x$ and $\unicode[STIX]{x1D709}$ excludes the possibility of multivalued solutions to the boundary-value problem. Such solutions can occur, for instance, when the vesicle develops concave curvature in the rear end (the so-called ‘parachute shape’). This restriction is somewhat mild, however, as the vesicles considered in the present work have reduced volumes in the range $0.85\leqslant \unicode[STIX]{x1D710}\leqslant 1$ . Such vesicles are approximately spherical and do not typically develop concave curvature under flow.

Figure 3.

Relative velocity $U/V$ plotted against the radius ratio $\unicode[STIX]{x1D706}$ for $\unicode[STIX]{x1D710}=0.99$ and (a) $N=4$ (square), (b) $N=6$ (hexagon), (c) $N=12$ (dodecagon) and (d) $N\rightarrow \infty$ (circle). The symbols are the BEM simulations, while the solid lines are the predictions from lubrication theory.

Equations (4.8), (4.12) and (4.15)–(4.20) were solved using the finite difference method. The poles $\unicode[STIX]{x1D709}=0$ and $\unicode[STIX]{x03C0}$ were excluded from the domain in order to avoid the stress singularity at these points. The membrane shape at the poles was approximated by a spherical section with constant curvature. The 2-D domain was discretized into a uniform grid and second-order accurate spatial difference operators were used to approximate derivatives and integrals with respect to $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D709}$ . Applying the difference analogy to the governing equations (4.8), (4.12) and (4.15)–(4.20) results in a system of nonlinear algebraic equations, which was subsequently solved iteratively using a Newton–Raphson procedure. For each Newton step, the Jacobian matrix was pre-computed analytically and subsequently discretized using finite differences. Iteration proceeded until the merit function to be zeroed fell below $10^{-6}$ or the Newton step in the descent direction fell below $10^{-6}$ . Starting from some initial guess (e.g. solutions of the analogous axisymmetric problem considered by Barakat & Shaqfeh (Reference Barakat and Shaqfeh2018a )), convergence was typically achieved in under 10 iterations. A family of solutions parametrized by $\unicode[STIX]{x1D710}$ and $\unicode[STIX]{x1D706}$ was obtained using the continuation method. Typically, higher spatial resolution was necessary in order to achieve convergence for large values of $\unicode[STIX]{x1D706}$ . For further details on the numerical method, the reader is referred to Barakat (Reference Barakat2018).

Figure 4.

Dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ plotted against the radius ratio $\unicode[STIX]{x1D706}$ for $\unicode[STIX]{x1D710}=0.99$ and (a) $N=4$ (square), (b) $N=6$ (hexagon), (c) $N=12$ (dodecagon) and (d) $N\rightarrow \infty$ (circle). The symbols are the BEM simulations, while the solid lines are the predictions from lubrication theory.

4.2 Error analysis

The lubrication approximation is not an exact solution of Stokes’ equations of motion. Viscous stresses induced by velocity gradients neglected in the approximation can contribute to the momentum flux in the $x$ and $\unicode[STIX]{x1D719}$ directions. These stresses are not expected to contribute significantly when the flow is axisymmetric. For the non-axisymmetric flow in square ducts, these stresses are expected to play a more significant role. There are large portions of the domain (e.g. the corners and the ends of the vesicle) where the gap separating the membrane and the duct wall is not narrow, nor is it slowly varying in the axial or circumferential directions. Consequently, while the local flow field in the narrow-gap region is expected to be well approximated by the lubrication theory, the macroscopic mass and momentum balances may incur substantial errors due to neglected viscous stresses, particularly for motion tangential to the wall (less so for normal motion, which is not considered here) (O’Neill & Stewartson Reference O’Neill and Stewartson1967). These errors are expected to decrease if the flow becomes axisymmetric, the narrow gap thins, or the apparent-contact area $A^{\ast }$ increases.

The macroscopic mass and momentum balances lead directly to the integral figures of merit: the relative velocity $U/V$ and dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ . In order to estimate the errors made in the lubrication approximation, these quantities were computed at different values of $N$ (polygonal ducts with varying number of edges in a cross-section) and compared to the analogous BEM simulations. Results for the relative velocity $U/V$ and dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ are shown in figures 3 and 4, respectively, for $N=4$ (square cross-section), 6 (hexagonal), 12 (dodecagonal) and $\infty$ (circular). From comparing the two methods, it is clear that the lubrication theory underpredicts both the relative velocity and extra pressure drop. The error is most significant for the square cross-section, and decreases as $N$ increases until the system approaches the cylindrically symmetric system. For the circular-tube case, the error in the relative velocity nearly vanishes, whereas there remains a very small error in the extra pressure drop. This error is likely due to the approximation (4.21), which neglects wall stresses upstream and downstream of the ends of the vesicle.

From figures 3 and 4, one may reasonably conclude that the errors in the lubrication approximation arise from the neglect of velocity gradients in the azimuthal direction (e.g.  $\unicode[STIX]{x2202}u_{x}/\unicode[STIX]{x2202}\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x2202}\unicode[STIX]{x1D719}$ ) in the equations of motion and stress conditions. These terms are not necessarily small and should play a significant role when the duct is not axisymmetric. However, one expects that these terms become less important as the vesicle becomes more confined. As the radius ratio $\unicode[STIX]{x1D706}$ increases, the vesicle fills the entire cross-section and the gap separating the membrane from the duct wall becomes very small in certain regions. Velocity gradients with respect to the radial distance (e.g.  $\unicode[STIX]{x2202}u_{x}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}$ ) are dominant in this case, and lubrication theory is expected to predict the singular contribution to integral force moments.

Unfortunately, it is not possible to compare the lubrication theory to the BEM simulations at very high confinement, since the BEM calculations become prohibitively expensive in this regime. Instead, the lubrication theory is compared to experimental measurements at similarly high confinement (delayed until § 6). Details of the experimental methods are given in the next section.

6.1 Vesicle flow through a contraction

To begin, the BEM simulations for straight channels with square cross-section are compared to the contraction–expansion simulations. The goal here is to correlate the instantaneous vesicle confinement, relative velocity $U/V$ , and dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ , as well as quantify the time to reach steady state.

Figure 7.

(a) Time sequence of a vesicle (reduced volume $\unicode[STIX]{x1D710}=0.99$ ) passing through a square constriction channel, as computed via BEM simulation. The left and right columns correspond to $\unicode[STIX]{x1D706}=0.6$ and $0.8$ , respectively. (b) Relative velocity $U/V$ and dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ plotted against the dimensionless time $tV/R$ for $\unicode[STIX]{x1D710}=0.99$ and $\unicode[STIX]{x1D706}=0.6,0.8$ . The dashed, horizontal lines correspond to the steady-state results for an infinitely long, square conduit.

Figure 7 shows BEM simulation results for a nearly spherical vesicle (reduced volume $\unicode[STIX]{x1D710}=0.99$ ) entering a constriction, at two different radius ratios ( $\unicode[STIX]{x1D706}=0.6$ and 0.8). The mesh geometry (shown in figure 2 c) has the same dimensions as the ‘test channel’ of the microfluidic manometer (figures 5 and 6). The straight-channel portion of the geometry is sufficiently long such that the expansion flow downstream does not affect the vesicle motion. In the experiment shown in figure 6, the radius ratio $\unicode[STIX]{x1D706}$ is higher than that reported in figure 7. High-confinement simulations require an exorbitant number of mesh elements and are prohibitively costly with the current BEM implementation. As such, one should not directly compare figure 7 to figure 6. Nevertheless, the simulations can give insight into the transient effects at the entrance of the manometer, at lower confinement.

The vesicle shape (figure 7 a) is tracked together with the relative velocity $U/V$ and dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ (figure 7 b). As the vesicle approaches the entrance of the narrow channel, it accelerates due to the converging flow field; the extra pressure drop rises simultaneously. At $tV/R=0$ , the vesicle enters the square duct and the pressure drop fluctuates; these fluctuations are dampened as confinement increases. Steady state is achieved after the vesicle travels several channel radii (roughly $tV/R\simeq 3$ ), which is consistent with the experiment shown in figure 6 (physically, this distance is approximately $40~\unicode[STIX]{x03BC}\text{m}$ for a channel hydraulic radius of $12~\unicode[STIX]{x03BC}\text{m}$ ). Increasing the flow confinement ( $\unicode[STIX]{x1D706}$ ) results in a slight reduction in the relative velocity $U/V$ and a dramatic increase in the dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ . Similar trends are observed for lower reduced volumes (not shown here).

Quéguiner & Barthès-Biesel (Reference Quéguiner and Barthès-Biesel1997) performed BEM simulations of a capsule (a model for a red blood cell) entering an axisymmetric contraction–expansion geometry. These authors modelled the capsule membrane as either a neo-Hookean solid or as an incompressible sheet; the latter is equivalent to the membrane model used in the present study. They tracked the pressure drop as a function of distance travelled by the capsule and showed that a steady state was achieved after several channel radii, as is shown here. To the best of the authors’ knowledge, the present simulations are the first to model a vesicle flowing through rectangular contraction–expansion geometry, which is fully three-dimensional and hence cannot be investigated using the axisymmetric boundary integral equations.

6.2 Steady vesicle shapes, surface tractions and surface velocities

Figures 8–11 show the theoretical predictions for the vesicle shape in square conduits at two different reduced volumes ( $\unicode[STIX]{x1D710}=0.99$ and $0.90$ ) and a range of radius ratios $\unicode[STIX]{x1D706}$ . The images are ordered (top-to-bottom) with increasing radius ratio $\unicode[STIX]{x1D706}$ ; thus, vesicle confinement increases down the page. The colours indicate the normal component of the membrane surface traction jump (i.e. the ‘total pressure’), $f_{sn}=\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{f}_{\!s}$ . Larger values of $f_{sn}$ (towards positively valued numbers) indicate higher pressures in the suspending fluid phase. The results shown in figures 8 and 10 were obtained using BEM simulations; those shown in figures 9 and 11 were obtained using lubrication theory. In order to compare results from the BEM and lubrication theory, one may juxtapose figures 8(d) and 9(a) (for $\unicode[STIX]{x1D710}=0.99$ , $\unicode[STIX]{x1D706}=0.9$ ), as well as figures 10(e) and 11(a) (for $\unicode[STIX]{x1D710}=0.90$ , $\unicode[STIX]{x1D706}=1.1$ ). Clearly, the two methods produce qualitatively similar shapes and pressure distributions. Quantitative differences in the computed pressures are also observed.

Figure 8.

Vesicles ( $\unicode[STIX]{x1D710}=0.99$ ) in square ducts as determined by 3-D BEM simulation, viewed from $yz$ (left) and $xy$ (right) planes. The colours indicate the normal component of the membrane surface traction jump $f_{sn}=\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{f}_{s}$ . Shown are results for $\unicode[STIX]{x1D706}=$ (a) 0.6, (b) 0.7, (c) 0.8 and (d) 0.9.

Figure 9.

Vesicles ( $\unicode[STIX]{x1D710}=0.99$ ) in square ducts as determined by lubrication theory, viewed from $yz$ (left) and $xy$ (right) planes. The colours have the same meaning as in figure 8. Shown are results for $\unicode[STIX]{x1D706}=$ (a) 0.9, (b) 1.0, (c) 1.1 and (d) 1.12.

Figure 10.

Vesicles ( $\unicode[STIX]{x1D710}=0.90$ ) in square ducts as determined by 3-D BEM simulation, viewed from $yz$ (left) and $xy$ (right) planes. The colours indicate the normal component of the membrane surface traction jump $f_{sn}=\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{f}_{s}$ . Shown are results for $\unicode[STIX]{x1D706}=$ (a) 0.7, (b) 0.8, (c) 0.9, (d) 1.0 and (e) 1.1.

Figure 11.

Vesicles ( $\unicode[STIX]{x1D710}=0.90$ ) in square ducts as determined by lubrication theory, viewed from $yz$ (left) and $xy$ (right) planes. The colours have the same meaning as in figure 10. Shown are results for $\unicode[STIX]{x1D706}=$ (a) 1.1, (b) 1.2, (c) 1.3, (d) 1.4 and (e) 1.43.

Figures 8(a), 10(a) show that the pressure distribution in the vicinity of the vesicle surface is nearly axisymmetric at low flow confinement. Were the vesicle to shrink to a point, local analysis of the flow field in a square duct yields an axisymmetric flow field at the channel centreline. Finite-sized vesicles sample a non-axisymmetric flow field. Figure 10(a,b) shows a fourfold symmetric ‘parachute shape’ for $\unicode[STIX]{x1D710}=0.90$ , with four lobes appearing at the rear end. This is not observed in circular tubes, wherein a vesicle of this reduced volume would develop an axisymmetric parachute shape.

As the confinement is increased, the pressure distribution becomes increasingly asymmetric with ‘hot spots’ (regions of high pressure) localized at four discrete areas where the gap widths are narrow. These hot spots become less localized at lower reduced volume (comparing $\unicode[STIX]{x1D710}=0.99$ – $0.90$ ), indicating that there is a smaller variation in the local distance between the vesicle membrane and the duct wall. Moreover, the magnitude of the pressure is significantly lessened as the reduced volume is decreased (at fixed radius ratio). This result is expected, since lower reduced-volume vesicles can form more streamlined bodies in pressure-driven flow. The non-axisymmetric pressure distribution confirms that the velocity field will exhibit a strong dependence on the azimuthal angle $\unicode[STIX]{x1D719}$ . This dependence is neglected in the lubrication approximation, and so quantitative discrepancies between the lubrication theory and the BEM simulations are expected at low confinement.

At very high confinement, the lubrication solutions indicate that the vesicle conforms to the duct boundary and makes ‘apparent contact’, from a macroscopic point of view (figures 9 d, 11 e). The apparent-contact area $A^{\ast }$ widens as the reduced volume $\unicode[STIX]{x1D710}$ is decreased. Microscopically, a thin layer of suspending fluid separates the membrane from the wall. The colour contours indicate very large pressures in this region. The distribution of pressure is also interesting: the highest pressures are located downstream and distributed uniformly over a wide area (the ‘orange ellipse’ in figures 9 d, 11 e), while a thin ‘suction’ layer (the ‘blue croissant’) develops upstream.

Figure 12.

Normalized membrane slip velocity vectors $\bar{\boldsymbol{u}}_{s}/\Vert \bar{\boldsymbol{u}}_{s}\Vert$ , as computed by lubrication theory. (a) $\unicode[STIX]{x1D710}=0.99$ , $\unicode[STIX]{x1D706}=0.9$ (left) and 1.12 (right). (b) $\unicode[STIX]{x1D710}=0.90$ , $\unicode[STIX]{x1D706}=1.1$ (left) and 1.43 (right). The corresponding surface traction distributions are shown in figures 9(a,d) and 11(a,e).

The pressure distributions described above allude to a backflow (opposite in direction to the mean flow) in the region of apparent contact. In figure 12, the membrane slip velocity $\bar{\boldsymbol{u}}_{s}=\boldsymbol{u}_{s}-\boldsymbol{U}$ is plotted for the parameter sets shown in figures 9(a,d) and 11(a,e) (computed via lubrication theory). The vectors are normalized so that only the direction of flow is indicated. The velocity vectors indeed indicate a backflow in the apparent-contact region. In the ‘corner regions’, the membrane flows in the direction of the mean flow; the local speed (not shown) is also high in the corner regions, relative to the contact regions. Thus, the wall ‘drags’ fluid backward, with forward flow supplied in the corner regions, and the vesicle ‘tank-treads’ through the duct. This motion is markedly unlike that of rigid particles, and is permitted by the spatial asymmetry of the duct cross-section.

6.3 Critical geometry

The lubrication theory can be used to estimate the vesicle geometry at maximum confinement (minimal clearance between the membrane and the wall). The term ‘estimate’ is used here, because it is numerically impossible to set the gap distance to zero in the theoretical calculations. However, one can incrementally ‘march up’ in $\unicode[STIX]{x1D706}$ (at fixed $\unicode[STIX]{x1D710}$ ) using the continuation method, decreasing the gap distance until some acceptable threshold. Here, the minimum gap distance is chosen to be $0.001R$ , at which point the continuation procedure is truncated and the value of $\unicode[STIX]{x1D706}$ is denoted as the ‘apparent critical radius ratio’ $\unicode[STIX]{x1D706}^{\ast }$ . To reiterate, the exact value of $\unicode[STIX]{x1D706}^{\ast }$ is theoretically achieved by continuing until the film thickness drops to zero. Examples of the near-critical geometry are shown in figures 9(d) and 11(e) for $\unicode[STIX]{x1D710}=0.99$ and $0.90$ , respectively.

Figure 13.

Estimates of (a) the critical radius ratio $\unicode[STIX]{x1D706}^{\ast }$ , (b) the dimensionless contact area $A^{\ast }/R^{2}$ and (c) the aspect ratio $L^{\ast }/(2R)$ plotted against the reduced volume $\unicode[STIX]{x1D710}$ , as determined via lubrication theory. Results for both square and circular cross-sections are shown.

From the numerical solutions at near-critical confinement, one can extract three important geometric parameters: the critical radius ratio $\unicode[STIX]{x1D706}^{\ast }$ , the lubrication contact area $A^{\ast }$ and the critical vesicle length $L^{\ast }$ . The apparent-contact area $A^{\ast }$ can be estimated by computing the total area of the region in which the normal stress jump $f_{sn}\approx -p$ exceeds a predetermined threshold (say, $100~\unicode[STIX]{x03BC}V/R$ ); this region is denoted as the ‘contact region’, since the vesicle makes apparent contact with the wall from a macroscopic viewpoint. Figure 13 shows plots of the estimates for $\unicode[STIX]{x1D706}^{\ast }$ , $A^{\ast }$ , and $L^{\ast }$ against the reduced volume $\unicode[STIX]{x1D710}$ , with the analogous circular-tube results shown for comparison.

At fixed reduced volume, the critical radius ratio $\unicode[STIX]{x1D706}^{\ast }$ is higher in the square-duct geometry than in the circular-tube geometry (figure 13 a). Physically, this implies that vesicles will have an easier time squeezing through a square duct than a circular tube, due to the larger available cross-sectional area (relative to the square of the hydraulic radius $R^{2}$ ). The contact area $A^{\ast }$ in the square-duct geometry is much smaller than in the circular tube (figure 13 b), due to a larger fraction of ‘free’ surface in the corner regions. As $\unicode[STIX]{x1D710}$ approaches unity, the contact area $A^{\ast }$ approaches zero in the square-duct geometry, indicating that the surfaces of contact degenerate to points. As was mentioned near the beginning of § 4, the applicability of lubrication theory when $A^{\ast }=0$ is dubious in the evaluation of integral force moments. For $\unicode[STIX]{x1D710}=0.99$ , the apparent-contact area $A^{\ast }$ is small, but finite.

The critical vesicle length $L^{\ast }$ does not differ significantly between the square and circular cross-sections (figure 13 c), indicating that the vesicle shape is very nearly a spherocylinder at maximal confinement. For vesicles in circular tubes, there are exact equations relating the reduced volume $\unicode[STIX]{x1D710}$ and critical radius ratio $\unicode[STIX]{x1D706}^{\ast }$ to the critical aspect ratio $L^{\ast }/(2R)$ :

(6.1) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D710}=\frac{3}{2}\left(\frac{L^{\ast }}{2R}\right)^{-1/2}-\frac{1}{2}\left(\frac{L^{\ast }}{2R}\right)^{-3/2}\quad \\ \displaystyle \unicode[STIX]{x1D706}^{\ast }=\left(\frac{L^{\ast }}{2R}\right)^{1/2}\quad \end{array}\right.\quad \text{(circular tubes)}.\end{eqnarray}$$

For the square-duct geometry, the numerical data are well fitted by the following functions relating $\unicode[STIX]{x1D710}$ and $\unicode[STIX]{x1D706}^{\ast }$ to $L^{\ast }/(2R)$ over the range $0.85\leqslant \unicode[STIX]{x1D710}\leqslant 1$ :

(6.2) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D710}=1.502\left(\frac{L^{\ast }}{2R}\right)^{-0.498}-0.502\left(\frac{L^{\ast }}{2R}\right)^{-1.518}\quad \\ \displaystyle \unicode[STIX]{x1D706}^{\ast }=1.010\left(\frac{L^{\ast }}{2R}\right)^{0.545}\quad \end{array}\right.\quad \text{(square ducts)}.\end{eqnarray}$$

The maximum relative difference in the reduced volume $\unicode[STIX]{x1D710}$ between (6.1) and (6.2) (within the range of interpolation) is only 1 %; the maximum relative difference in $\unicode[STIX]{x1D706}^{\ast }$ is 5 %.

In equations (6.1)–(6.2), $\unicode[STIX]{x1D710}$ and $\unicode[STIX]{x1D706}^{\ast }$ are expressed in terms of $L^{\ast }$ as though $L^{\ast }$ were the independent variable (in reality, $\unicode[STIX]{x1D710}$ is the control variable since the vesicle volume and surface area are presumed to be unchanged by the deformation). The reason for this choice is motivated by the experimental measurements, in that it is relatively easy to measure the vesicle length $L$ using optical microscopy. By contrast, the vesicle reduced volume $\unicode[STIX]{x1D710}$ is less accurately measured. This quantity requires measurement of both the vesicle volume and surface area, which are difficult to obtain with high accuracy from 2-D micrographs. A large set of micrographs is required in order to improve the statistical certainty of a reduced-volume measurement (Dahl et al. Reference Dahl, Narsimhan, Gouveia, Kumar, Shaqfeh and Muller2016). However, in the presently considered flow geometry, only a single image of the vesicle can be taken within a given time interval.

Figure 14.

Aspect ratio $L/(2R)$ plotted against the radius ratio $\unicode[STIX]{x1D706}$ . In the experimental measurements, the aspect ratio $L/(2R)$ is measured and the reduced volume $\unicode[STIX]{x1D710}$ is estimated based on the correlation (6.2) from the lubrication theory; the diamond markers are coloured accordingly.

In order to estimate the reduced volume from the experiments, it is necessary to make an assumption about the vesicle shape. It is assumed here that the vesicle conforms to the duct walls and is given by the ‘critical geometry’ predicted via lubrication theory. The analogous assumption for vesicle flow in circular tubes is that the vesicle is a spherocylinder, which is the wall-conformal shape for that geometry. Thus, in the experiments it is assumed that

(6.3) $$\begin{eqnarray}L\approx L^{\ast }.\end{eqnarray}$$

By measuring the aspect ratio $L/(2R)$ in the experiments, one obtains the reduced volume $\unicode[STIX]{x1D710}$ from the correlation (6.2) (the analogous correlation for circular tubes is (6.1)). The results from this procedure are shown in figure 14, where the aspect ratio $L/(2R)$ (determined by BEM, lubrication theory and experiments) is plotted against the radius ratio $\unicode[STIX]{x1D706}$ . The solid lines indicate the lubrication solution above the critical limit (maximum confinement) for a circular tube of equivalent hydraulic radius, while the dashed lines indicate solutions below this limit. The experimental data are coloured by reduced volume using the correlation (6.2) for the assumed critical geometry. The general trend shown is that as the vesicle lengthens, the reduced volume decreases.

There are obvious uncertainties with the assumption made above. Essentially, the reduced volume has been approximated by assuming that the vesicle is critically confined, even though the radius ratio is varied in the experiments. The main justification for this assumption is that only those experiments which give a measurable extra pressure drop are selected; the extra pressure drop must be sufficiently large in order to produce an observable deflection in the comparator (see figure 6). Since the measured pressure drops are very large, it is reasonable to assume that the vesicle fills a large portion of the cross-section. The slight geometric variations between the actual shape and the shape assumed here are, therefore, probably small. Thus, one expects that the above approximation will give a good sense of the distribution of reduced volumes in the vesicle populations studied experimentally. This will be important when quantifying the relative velocity $U/V$ and dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ (presented below), which depend strongly on the reduced volume.

6.4 Vesicle velocity and extra pressure drop

Figure 15 shows the relative velocity $U/V$ plotted against the radius ratio $\unicode[STIX]{x1D706}$ as determined by 3-D BEM simulations, lubrication theory and experiments. The solid lines indicate the lubrication solution above the critical limit for a circular tube of equivalent hydraulic radius, while the dashed lines indicate solutions below this limit. Also shown are the experimental measurements of Ahmmed et al. (Reference Ahmmed, Suteria, Garbin and Vanapalli2018), which were conducted at smaller radius ratios $\unicode[STIX]{x1D706}$ . Their measurements were performed using a different microfluidic geometry than that reported in the present study. In their experiments, vesicles were flowed through a linear, straight channel with a wide cross-section ( $\text{height}=25.0\pm 0.4~\unicode[STIX]{x03BC}\text{m}$ , $\text{width}=25.3\pm 0.4~\unicode[STIX]{x03BC}\text{m}$ , $\text{length}=150~\unicode[STIX]{x03BC}\text{m}$ ) and only the velocity ratio $U/V$ was measured (the dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ could not be measured due to the absence of a reference channel). In the presently reported experiments, a test channel with a smaller cross-section ( $\text{height}=12.6\pm 0.6~\unicode[STIX]{x03BC}\text{m}$ , $\text{width}=12.8\pm 0.7~\unicode[STIX]{x03BC}\text{m}$ , $\text{length}=330~\unicode[STIX]{x03BC}\text{m}$ ) was used.

Figure 15.

Relative velocity $U/V$ plotted against the radius ratio $\unicode[STIX]{x1D706}$ .

Accounting for the distribution of reduced volumes in the experiments, the agreement between the lubrication theory and the experiments in figure 15 is remarkable. Significantly, the theory correctly predicts subunit relative velocities ( $U/V<1$ ) at high confinement. That is to say, the vesicle travels a speed slower than the mean channel speed in this regime. At lower confinement ( $\unicode[STIX]{x1D706}$ ), the BEM simulations agree qualitatively with previous experimental measurements reported by Ahmmed et al. (Reference Ahmmed, Suteria, Garbin and Vanapalli2018). The apparent discrepancy between the BEM and the lubrication theory at intermediate $\unicode[STIX]{x1D706}$ is attributed to the neglect of velocity gradients along the azimuth in the lubrication theory, as discussed previously. These errors were quantified in § 4.2 as a function of duct symmetry. Unfortunately, it is not feasible at present to perform BEM simulations at much higher values of $\unicode[STIX]{x1D706}$ to within a reasonable degree of accuracy.

Figure 16.

Dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ plotted against the radius ratio $\unicode[STIX]{x1D706}$ .

Figure 16 shows the dimensionless extra pressure drop $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ plotted against $\unicode[STIX]{x1D706}$ , again comparing the simulations, theory and experiments. The discrepancy between the BEM simulations and the lubrication theory at low values of $\unicode[STIX]{x1D706}$ has already been discussed in detail. As expected, the distribution of the extra-pressure-drop measurements can be attributed to variations in the reduced volume $\unicode[STIX]{x1D710}$ and the radius ratio $\unicode[STIX]{x1D706}$ . All of the experimental measurements fall within the predictions of the lubrication theory, whereas the BEM predictions lie in a regime outside the range accessible by experiment. At low radius ratios $\unicode[STIX]{x1D706}$ , the vesicle is too small to occlude the channel cross-section and therefore admits little resistance to flow. Consequently, the experimentally measured extra-pressure-drop signal ( $\unicode[STIX]{x0394}p^{+}R/(\unicode[STIX]{x1D707}V)$ in figure 6 b) is too weak to be distinguished from the noise when $\unicode[STIX]{x1D706}$ is small. As $\unicode[STIX]{x1D706}$ increases to the point where the vesicle occupies a significant fraction of the channel cross-section, the resistance to flow increases substantially and the experimental signal-to-noise ratio improves. The lubrication calculations indicate that this precipitous rise in resistance is principally due to dissipation in the thin-film region where the membrane nearly makes contact with the channel walls.