2.1 Region I: the lumen

In the lumen, we non-dimensionalise flow velocities by $\boldsymbol{u}_{l}^{\ast }=\boldsymbol{u}_{l}{\rm\Delta}PR^{2}/{\it\mu}_{l}L$ , where ${\rm\Delta}P$ is the average pressure drop between the inlet and the outlet of the microvessel, and lumen pressures and stresses by $P_{l}^{\ast }=P_{l}{\rm\Delta}PR/L$ , ${\bf\sigma}_{l}^{\ast }={\bf\sigma}_{l}{\rm\Delta}PR/L$ . Under these non-dimensionalisations, the flow in the lumen, $V_{l}$ , satisfies Stokes flow,

(2.1a,b ) $$\begin{eqnarray}\displaystyle {\rm\nabla}^{2}\boldsymbol{u}_{l}=\boldsymbol{{\rm\nabla}}P_{l},\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}_{l},=0. & & \displaystyle\end{eqnarray}$$

This flow can be expressed in the following boundary integral form (Pozrikidis Reference Pozrikidis2002):

(2.2) $$\begin{eqnarray}\displaystyle (u_{l}(\boldsymbol{x}_{0}))_{j} & = & \displaystyle -\frac{1}{4{\rm\pi}}\int _{\mathscr{S}_{1}}f_{i}(\boldsymbol{x})\,G_{ij}^{(1)}(\boldsymbol{x}-\boldsymbol{x}_{0})\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{4{\rm\pi}}\unicode[STIX]{x2A0D}_{\mathscr{S}_{1}}(u_{l}(\boldsymbol{x}))_{i}\,T_{ijk}^{(1)}(\boldsymbol{x}-\boldsymbol{x}_{0})n_{k}(\boldsymbol{x})\,\text{d}S(\boldsymbol{x}),\end{eqnarray}$$

(with repeated indices denoting summation). The symbol $\unicode[STIX]{x2A0D}$ denotes that the second integral is defined in the sense of a Cauchy principal integral, and $i,j,k=1,\ldots ,3$ . Here, $\boldsymbol{x}_{0}\in \mathscr{S}_{1}$ , where $\mathscr{S}_{1}$ represents the entire surface bounding region I. The unit normal vector in this equation, $\boldsymbol{n}$ , is taken to be the inward pointing normal. Moreover, $\boldsymbol{f}(\boldsymbol{x})={\bf\sigma}_{\boldsymbol{l}}\boldsymbol{\cdot }\boldsymbol{n}$ is the traction vector, and $G_{ij}^{(1)}$ and $T_{ijk}^{(1)}$ are the three-dimensional Stokeslet and stresslet for Stokes flow respectively, as given by Pozrikidis (Reference Pozrikidis2002).

The above formulation assumes that only plasma flows through the lumen, which may be intermittently the case in capillaries, where red blood cells flow through in single file. However, in larger microvessels such as postcapillary venules, multiple red blood cells may be present at any point along the vessel. A cell-depleted layer has been observed in such vessels, where the concentration of red blood cells is low (Kim et al. Reference Kim, Kong, Popel, Intaglietta and Johnson2007). This has led to the development of two-layer models. These two-layer models involve prescribing a spatially varying viscosity, which takes larger values in the core of the lumen where red blood cells are present, and a lower value in the cell-depleted layer close to the vessel wall. Such models have been shown to well approximate the experimentally observed distribution of red blood cells in the vessel (Pries et al. Reference Pries, Secomb, Gessner, Sperandio, Gross and Gaehtgens1994). We will present some results where such a two-layer viscosity profile has been used, in order to gauge the potential influence of red blood cells in the lumen upon the EGL’s impact. We use the model of Sharan & Popel (Reference Sharan and Popel2001), where the viscosity in the cell depletion layer is an emergent quantity, and is greater than that of plasma alone, due to the occasional intrusion of red blood cells into this depletion layer. Although Sharan & Popel (Reference Sharan and Popel2001) developed their two-layer model in the absence of an EGL, we shall see that in the low-permeability physiological regime, the lumen flow at the EGL interface satisfies the same conditions as on a solid surface (see 3.1), at leading order. As such, we take the thickness of the cell-depleted layer to be independent of the depth of the EGL.

2.2 Region II: the EGL

Region II consists of the EGL, which is a porous medium that allows vessel fluid to flow through it, but not freely. We model the poroelastohydrodynamics in the EGL using biphasic mixture theory (Drew Reference Drew1983; Ehlers & Bluhm Reference Ehlers and Bluhm2002; Kolev Reference Kolev2002), which consists of a fluid phase and a solid phase. We shall consider each phase in turn.

2.2.1 Fluid phase

Since elastic velocities can be assumed to be small in this setting, the fluid phase of the EGL can be modelled using Brinkman’s equations (Hariprasad & Secomb Reference Hariprasad and Secomb2012). Non-dimensionalising volume-averaged flow velocities in the EGL according to $\boldsymbol{u}_{f}^{\ast }=\boldsymbol{u}_{f}{\rm\Delta}PR^{2}/{\it\mu}_{f}L$ , and pressures and stresses by $P_{e}^{\ast }=P_{e}{\rm\Delta}PR/{\it\phi}_{f}L$ , ${\bf\sigma}_{f}^{\ast }={\bf\sigma}_{f}{\rm\Delta}PR/L$ (where ${\it\phi}_{f}$ is the fluid fraction in the EGL), the flow equations in the EGL consequently take the non-dimensional form

(2.3a,b ) $$\begin{eqnarray}\displaystyle {\rm\nabla}^{2}\boldsymbol{u}_{f}-{\it\lambda}^{2}\boldsymbol{u}_{f}=\boldsymbol{{\rm\nabla}}P_{e},\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}_{f}=0, & & \displaystyle\end{eqnarray}$$

where ${\it\lambda}^{2}=R^{2}/K_{P}$ , $K_{P}$ is the Darcy permeability within the EGL and ${\it\mu}_{f}$ is the dynamic viscosity of the fluid in the EGL. (Weinbaum et al. (Reference Weinbaum, Zhang, Yuefeng Han and Cowin2003) give an expression for the permeability in terms of the radius and spacing of the core proteins in the EGL’s ultrastructure.) We shall use ${\it\delta}$ to denote a characteristic EGL thickness.

As was the case for Stokes flow in the lumen, there is a boundary integral representation for Brinkman flow in the EGL (Pozrikidis Reference Pozrikidis1992),

(2.4) $$\begin{eqnarray}\displaystyle (u_{f}(\boldsymbol{x}_{0}))_{j} & = & \displaystyle -\frac{1}{4{\rm\pi}}\int _{\mathscr{S}_{2}}g_{i}(\boldsymbol{x})\,G_{ij}^{(2)}(\boldsymbol{x}-\boldsymbol{x}_{0})\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{4{\rm\pi}}\unicode[STIX]{x2A0D}_{\mathscr{S}_{2}}(u_{f}(\boldsymbol{x}))_{i}\,T_{ijk}^{(2)}(\boldsymbol{x}-\boldsymbol{x}_{0})n_{k}(\boldsymbol{x})\,\text{d}S(\boldsymbol{x}),\end{eqnarray}$$

where $G_{ij}^{(2)}$ and $T_{ij}^{(2)}$ are the Brinkman flow Stokeslet and stresslet respectively, as given by Pozrikidis (Reference Pozrikidis1992), and $\boldsymbol{g}(\boldsymbol{x})={\bf\sigma}_{\boldsymbol{f}}\boldsymbol{\cdot }\boldsymbol{n}$ . Here, $\boldsymbol{x}_{0}\in \mathscr{S}_{2}$ , where $\mathscr{S}_{2}$ represents the entire surface bounding region II.

2.2.2 Solid phase

The solid phase of the EGL can be modelled as a linear elastic solid with extra forcing terms due to momentum transfer between the two phases (Damiano et al. Reference Damiano, Duling, Ley and Skalak1996). Non-dimensionalising the solid displacements in the EGL according to $\boldsymbol{u}_{s}^{\ast }=\boldsymbol{u}_{s}{\rm\Delta}PR^{2}/{\it\mu}_{s}L$ and the stresses by ${\bf\sigma}_{s}^{\ast }={\bf\sigma}_{s}{\rm\Delta}PR/L$ , Navier’s equation in the EGL takes the form

(2.5) $$\begin{eqnarray}\frac{1}{1-2{\it\nu}}\boldsymbol{{\rm\nabla}}(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}_{s})+{\rm\nabla}^{2}\boldsymbol{u}_{s}=\frac{{\it\phi}_{s}}{{\it\phi}_{f}}\boldsymbol{{\rm\nabla}}P_{e}-{\it\lambda}^{2}\boldsymbol{u}_{f},\end{eqnarray}$$

where ${\it\nu}$ is the Poisson’s ratio of the solid phase of the EGL. The pressure forcing on the right-hand side of (2.5) arises due to the presence of the fluid phase in the EGL, and the fluid velocity forcing arises due to momentum transfer between the two phases.

As before, there is a boundary integral representation for Navier’s equation, with

(2.6) $$\begin{eqnarray}\displaystyle (u_{s}(\boldsymbol{x}_{0}))_{j} & = & \displaystyle -\frac{1}{8{\rm\pi}(1-{\it\nu})}\int _{\mathscr{S}_{2}}h_{i}(\boldsymbol{x})\,G_{ij}^{(3)}(\boldsymbol{x}-\boldsymbol{x}_{0})\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{8{\rm\pi}(1-{\it\nu})}\unicode[STIX]{x2A0D}_{\mathscr{S}_{2}}(u_{s}(\boldsymbol{x}))_{i}\,T_{ijk}^{(3)}(\boldsymbol{x}-\boldsymbol{x}_{0})n_{k}(\boldsymbol{x})\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle -\,\frac{(1-2{\it\nu}){\it\phi}_{s}}{8{\rm\pi}(1-{\it\nu}){\it\phi}_{f}}\int _{\mathscr{S}_{2}}\frac{\partial P_{e}}{\partial \boldsymbol{n}}(\boldsymbol{x})\frac{(\boldsymbol{x}-\boldsymbol{x}_{0})_{j}}{r}\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle -\,\frac{(1-{\it\nu}){\it\phi}_{s}}{4{\rm\pi}(1-{\it\nu}){\it\phi}_{f}}\int _{\mathscr{S}_{2}}P_{e}(\boldsymbol{x})\left(\frac{{\it\delta}_{jk}}{r}+\frac{(\boldsymbol{x}-\boldsymbol{x}_{0})_{j}(\boldsymbol{x}-\boldsymbol{x}_{0})_{k}}{r^{3}}\right)n_{k}\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\lambda}^{2}}{8{\rm\pi}(1-{\it\nu})}\int _{\mathscr{S}_{2}}f_{i}(\boldsymbol{x})\,G_{ij}^{(4)}(\boldsymbol{x}-\boldsymbol{x}_{0})\,\text{d}S(\boldsymbol{x})\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\it\lambda}^{2}}{8{\rm\pi}(1-{\it\nu})}\int _{\mathscr{S}_{2}}(u_{f}(\boldsymbol{x}))_{i}\,T_{ijk}^{(4)}(\boldsymbol{x}-\boldsymbol{x}_{0})n_{k}(\boldsymbol{x})\,\text{d}S(\boldsymbol{x}),\end{eqnarray}$$

where $G_{ij}^{(3)}$ and $T_{ij}^{(3)}$ are the linear elasticity Green’s functions, as given by Pozrikidis (Reference Pozrikidis2002), $\boldsymbol{h}(\boldsymbol{x})={\bf\sigma}_{\boldsymbol{s}}\boldsymbol{\cdot }\boldsymbol{n}$ , and $G_{ij}^{(4)}$ and $T_{ij}^{(4)}$ are given by Sumets et al. (Reference Sumets, Cater, Long and Clarke2015). The forcing terms have been converted to boundary integrals using the approach developed by Sumets et al. (Reference Sumets, Cater, Long and Clarke2015).

2.3 Boundary conditions

The flow boundary conditions on the surface of the endothelium, which is assumed to be impermeable and stationary, are the usual no-slip and no-penetration conditions

(2.7) $$\begin{eqnarray}\boldsymbol{u}_{f}=\mathbf{0}\quad \text{on }S_{w}.\end{eqnarray}$$

Similarly, the solid phase is assumed to be attached to the endothelium, and so the solid phase is assumed to have no displacement on the surface of the endothelium,

(2.8) $$\begin{eqnarray}\boldsymbol{u}_{s}=\mathbf{0}\quad \text{on }S_{w}.\end{eqnarray}$$

The flow boundary conditions on the interface between the EGL and the lumen are as those given in Damiano et al. (Reference Damiano, Long, El-Khatib and Stace2004). Specifically, the homogenised velocity (fluid and solid velocity weighted by their respective volume fractions) is assumed to be continuous across the interface,

(2.9) $$\begin{eqnarray}{\it\phi}_{f}\boldsymbol{u}_{f}+{\it\phi}_{s}\dot{\boldsymbol{u}}_{s}=\boldsymbol{u}_{l}\quad \text{on }S_{i}.\end{eqnarray}$$

Under the assumption that we can neglect the elastic velocities (which are very small in this context), as we did in the momentum transfer terms, this leads to our second boundary condition for the fluid phase,

(2.10) $$\begin{eqnarray}{\it\phi}_{f}\boldsymbol{u}_{f}=\boldsymbol{u}_{l}\quad \text{on }S_{i}.\end{eqnarray}$$

The traction on the interface is shared between the two phases according to the volume fractions. For the fluid phase, this gives

(2.11) $$\begin{eqnarray}{\bf\sigma}_{f}\boldsymbol{\cdot }\boldsymbol{n}={\it\phi}_{f}{\bf\sigma}_{l}\boldsymbol{\cdot }\boldsymbol{n}\quad \text{on }S_{i},\end{eqnarray}$$

where ${\bf\sigma}_{l}=-P_{l}\unicode[STIX]{x1D644}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{l}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{l})^{\text{T}})$ and ${\bf\sigma}_{f}=-P_{e}\unicode[STIX]{x1D644}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{f}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{f})^{\text{T}})$ (T denotes a transpose) are the stress tensors for flow within the lumen and the EGL respectively, $\unicode[STIX]{x1D644}$ is the $3\times 3$ identity matrix and $\boldsymbol{n}$ is the inward unit normal to the surface $S_{i}$ for each region respectively. Similarly, for the solid phase this gives

(2.12) $$\begin{eqnarray}\left({\bf\sigma}_{s}-\frac{{\it\phi}_{s}}{{\it\phi}_{f}}P_{e}\unicode[STIX]{x1D644}\right)\boldsymbol{\cdot }\boldsymbol{n}={\it\phi}_{s}{\bf\sigma}_{l}\boldsymbol{\cdot }\boldsymbol{n}\quad \text{on }S_{i},\end{eqnarray}$$

where ${\bf\sigma}_{s}=2{\it\nu}/(1-2{\it\nu})(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}_{s})\unicode[STIX]{x1D644}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{s}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{s})^{\text{T}})$ is the stress tensor of a linear elastic solid and ${\bf\sigma}_{s}-({\it\phi}_{s}/{\it\phi}_{f})P_{e}\unicode[STIX]{x1D644}$ is the total stress tensor of the solid phase of the EGL. Combination of (2.11) and (2.12) gives $({\bf\sigma}_{f}+{\bf\sigma}_{s}-({\it\phi}_{s}/{\it\phi}_{f})P_{e}\unicode[STIX]{x1D644})\boldsymbol{\cdot }\boldsymbol{n}={\bf\sigma}_{l}\boldsymbol{\cdot }\boldsymbol{n}$ , which shows that the total traction of the biphasic mixture balances the traction exerted on the interface by the fluid flow inside the lumen.

We are also required to specify flow conditions on the inlet, $\mathscr{S}_{0}$ , and outlet, $\mathscr{S}_{\infty }$ , surfaces,

(2.13a,b ) $$\begin{eqnarray}\boldsymbol{u}(\mathscr{S}_{0})=\boldsymbol{U}_{0},\quad \boldsymbol{u}(\mathscr{S}_{\infty })=\boldsymbol{U}_{1},\end{eqnarray}$$

where here $\boldsymbol{u}$ represents either $\boldsymbol{u}_{l}$ or $\boldsymbol{u}_{f}$ , depending upon whether $\mathscr{S}_{0}$ or $\mathscr{S}_{\infty }$ intersects region I or II respectively. See (4.1) in § 4.1 for the form of these inlet and outlet conditions.

Similarly, we specify displacement conditions on the portions of the inlet and outlet surfaces that intersect with the EGL, $\mathscr{S}_{0}^{\prime }$ and $\mathscr{S}_{\infty }^{\prime }$ respectively,

(2.14a,b ) $$\begin{eqnarray}\boldsymbol{u}_{s}(\mathscr{S}_{0}^{\prime })=\boldsymbol{V}_{0},\quad \boldsymbol{u}_{s}(\mathscr{S}_{\infty }^{\prime })=\boldsymbol{V}_{1}.\end{eqnarray}$$

We prescribe specific forms for these displacements in (4.2).

3.1 Lumen

In the lumen, we approximate the flow velocity and pressure by the following expansion (where we are required go to $O({\it\lambda}^{-2})$ in $\boldsymbol{u}_{l}$ in order to satisfy the continuity of velocity conditions):

(3.1a,b ) $$\begin{eqnarray}\boldsymbol{u}_{l}=\boldsymbol{u}_{l}^{(0)}+{\it\lambda}^{-1}\boldsymbol{u}_{l}^{(1)}+{\it\lambda}^{-2}\boldsymbol{u}_{l}^{(2)}+O({\it\lambda}^{-3}),\quad P_{l}=P_{l}^{(0)}+{\it\lambda}^{-1}P_{l}^{(1)}+O({\it\lambda}^{-2}),\end{eqnarray}$$

where ( $i=1,\ldots ,3$ )

(3.2) $$\begin{eqnarray}\mathbf{0}=-\boldsymbol{{\rm\nabla}}P^{(i)}+{\rm\nabla}^{2}\boldsymbol{u}^{(i)}.\end{eqnarray}$$

As we shall see from the scalings in the EGL, leading-order flow is required to satisfy homogeneous boundary conditions,

(3.3) $$\begin{eqnarray}\boldsymbol{u}^{(0)}(S_{i})=\mathbf{0}.\end{eqnarray}$$

Moreover, the first-order flow correction must satisfy no-penetration conditions at the interface,

(3.4) $$\begin{eqnarray}\boldsymbol{u}^{(1)}(S_{i})\boldsymbol{\cdot }\boldsymbol{n}=0.\end{eqnarray}$$

However, as will be shown in § 3.2.3, non-zero tangential components at the interface are driven by flows within a viscous layer adjacent to the lumen–EGL interface.

3.2 EGL: fluid phase

The flow in the EGL can be considered to consist of three distinct asymptotic regions: (i) a core flow, which is governed by pressure (i.e. a Darcy flow, where viscous effects are negligible); (ii) a thin viscous region adjacent to the endothelium, which ensures that the no-slip condition is satisfied; (iii) a thin viscous region adjacent to the lumen–EGL interface, where continuity of traction is enforced. We consider each of these regions in turn below.

3.2.1 EGL core

In the fluid phase of the EGL core, we rescale according to

(3.5a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{f}={\it\lambda}^{-2}\boldsymbol{u}_{f}^{(0)}+O({\it\lambda}^{-3}),\quad P=P_{e}^{(0)}+O({\it\lambda}^{-1}). & & \displaystyle\end{eqnarray}$$

The leading-order flow in the EGL is consequently governed by Darcy’s equations,

(3.6a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{f}^{(0)}=-\boldsymbol{{\rm\nabla}}P_{e}^{(0)},\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}_{f}^{(0)}=0. & & \displaystyle\end{eqnarray}$$

This flow is entirely driven by the kinematics, and can be reformulated in terms of pressure alone. Taking the divergence of the momentum equation yields

(3.7) $$\begin{eqnarray}\displaystyle {\rm\nabla}^{2}P_{e}^{(0)}=0. & & \displaystyle\end{eqnarray}$$

On the endothelial surface, we can satisfy the impermeability condition

(3.8) $$\begin{eqnarray}\displaystyle \boldsymbol{{\rm\nabla}}P_{e}^{(0)}(S_{w})\boldsymbol{\cdot }\boldsymbol{n}=0, & & \displaystyle\end{eqnarray}$$

but not the no-slip condition. Enforcement of this condition also necessitates the presence of another thin viscous region, this time adjacent to $S_{w}$ .

The pressure problem described in (3.7)–(3.8) can also be formulated in the well-known boundary integral form for a harmonic function:

(3.9) $$\begin{eqnarray}\displaystyle p(\boldsymbol{x}_{0})=\int _{\mathscr{S}_{2}}\boldsymbol{{\rm\nabla}}p(\boldsymbol{x})\boldsymbol{\cdot }\boldsymbol{n}(\boldsymbol{x})G(\boldsymbol{x},\boldsymbol{x}_{0})\,\text{d}S(\boldsymbol{x})-\unicode[STIX]{x2A0D}_{\mathscr{S}_{2}}\boldsymbol{{\rm\nabla}}G(\boldsymbol{x},\boldsymbol{x}_{0})\boldsymbol{\cdot }\boldsymbol{n}(\boldsymbol{x})p(\boldsymbol{x})\,\text{d}S(\boldsymbol{x}), & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{x}_{0}\in \mathscr{S}_{2}$ , $G(\boldsymbol{x},\boldsymbol{x}_{0})=1/2{\rm\pi}r$ , with $r=|\boldsymbol{x}-\boldsymbol{x}_{0}|$ .

3.2.2 EGL layer I: endothelial surface

In order to satisfy the no-slip condition, let us consider a local Cartesian coordinate system $(x_{1},x_{2},x_{3})$ , where $x_{3}$ lies in a direction normal to the surface. If we rescale according to

(3.10a-d ) $$\begin{eqnarray}\displaystyle x_{1}=X_{1},\quad x_{2}=X_{2},\quad x_{3}={\it\lambda}^{-1}X_{3},\quad \boldsymbol{u}_{f}=({\it\lambda}^{-2}U_{f},{\it\lambda}^{-2}V_{f},{\it\lambda}^{-3}W_{f}), & & \displaystyle\end{eqnarray}$$

the Brinkman equations (2.3), at leading order in ${\it\lambda}$ , take the form

(3.11a-d ) $$\begin{eqnarray}\displaystyle U_{f}=-\frac{\partial P_{e}}{\partial X_{1}}+\frac{\partial ^{2}U_{f}}{\partial X_{3}^{2}},\quad V_{f}=-\frac{\partial P_{e}}{\partial X_{2}}+\frac{\partial ^{2}V_{f}}{\partial X_{3}^{2}},\quad 0=\frac{\partial P_{e}}{\partial X_{3}},\quad \frac{\partial U_{f}}{\partial X_{1}}+\frac{\partial V_{f}}{\partial X_{2}}+\frac{\partial W_{f}}{\partial X_{3}}=0. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

These are subject to the following boundary and matching conditions:

(3.12a-c ) $$\begin{eqnarray}U_{f}(X_{3}=0)=0,\quad V_{f}(X_{3}=0)=0,\quad W_{f}(X_{3}=0)=0,\end{eqnarray}$$

(3.13a-c ) $$\begin{eqnarray}(U_{f}(X_{3}\rightarrow \infty ),\quad V_{f}(X_{3}\rightarrow \infty ))=\boldsymbol{v}_{s},\quad W_{f}(X_{3}\rightarrow \infty )=0,\end{eqnarray}$$

where $\boldsymbol{v}_{s}(x_{1},x_{2})=(u_{s},v_{s})=\boldsymbol{v}_{f}-(\boldsymbol{v}_{f}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{n}$ is the slip velocity induced in the Darcy region described by (3.7)–(3.8). The leading-order flow in this viscous region is consequently given by

(3.14a,b ) $$\begin{eqnarray}\displaystyle (U_{f}(X_{3}),V_{f}(X_{3}))=\boldsymbol{v}_{s}(1-\exp (-X_{3})),\quad W_{f}=(\boldsymbol{{\rm\nabla}}_{\Vert }\boldsymbol{\cdot }\boldsymbol{v}_{s})(1-X_{3}-\exp (-X_{3})), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{{\rm\nabla}}_{\Vert }=(\partial /\partial x_{1},\partial /\partial x_{2})$ . The leading-order tractions are then given by

(3.15) $$\begin{eqnarray}\displaystyle {\bf\sigma}\boldsymbol{\cdot }\boldsymbol{n}(S_{w})=({\it\lambda}^{-1}\partial U_{f}/\partial X_{3},{\it\lambda}^{-1}\partial V_{f}/\partial X_{3},-P_{e})=({\it\lambda}^{-1}u_{s}(S_{w}),{\it\lambda}^{-1}v_{s}(S_{w}),-P_{e}). & & \displaystyle\end{eqnarray}$$

3.2.3 EGL layer II: EGL interface

Here, we instead define ( $x_{1},x_{2},x_{3}$ ) to be a local coordinate system on the EGL interface, with $x_{3}$ in the direction of the local unit normal. In order to enforce continuity of the tangential components of traction, i.e. ${\it\phi}_{f}\boldsymbol{t}\boldsymbol{\cdot }{\bf\sigma}_{l}\boldsymbol{\cdot }\boldsymbol{n}(S_{i})=\boldsymbol{t}\boldsymbol{\cdot }{\bf\sigma}_{f}\boldsymbol{\cdot }\boldsymbol{n}(S_{i})$ , where $\boldsymbol{t}$ is a local unit tangent, we rescale according to

(3.16a-d ) $$\begin{eqnarray}\displaystyle x_{1}=X_{1},\quad x_{2}=X_{2},\quad x_{3}={\it\lambda}^{-1}X_{3},\quad P_{e}=P_{e}^{(0)}+({\it\lambda}^{-1}), & & \displaystyle\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle u_{f}={\it\lambda}^{-1}U_{f}^{(0)}+{\it\lambda}^{-2}U_{f}^{(1)}+O({\it\lambda}^{-3})+O({\it\lambda}^{-4}),\\ \displaystyle v_{f}={\it\lambda}^{-1}V_{f}^{(0)}+{\it\lambda}^{-2}V_{f}^{(1)}+O({\it\lambda}^{-3})+O({\it\lambda}^{-4}),\\ \displaystyle w_{f}={\it\lambda}^{-2}W_{f}^{(0)}+{\it\lambda}^{-3}W_{f}^{(1)}+O({\it\lambda}^{-4}).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The leading-order flow in this viscous layer is then governed by

(3.18a-d ) $$\begin{eqnarray}\displaystyle U_{f}^{(0)}=\frac{\partial ^{2}U_{f}^{(0)}}{\partial X_{3}^{2}},\quad V_{f}^{(0)}=\frac{\partial ^{2}V_{f}^{(0)}}{\partial X_{3}^{2}},\quad 0=\frac{\partial P_{e}^{(0)}}{\partial X_{3}},\quad \frac{\partial U_{f}^{(0)}}{\partial X_{1}}+\frac{\partial V_{f}^{(0)}}{\partial X_{2}}+\frac{\partial W_{f}^{(0)}}{\partial X_{3}}=0, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

subject to the boundary condition

(3.19a,b ) $$\begin{eqnarray}\displaystyle (U_{f}^{(0)}(X_{3}=0),V_{f}^{(0)}(X_{3}=0))=\frac{1}{{\it\phi}_{f}}(u_{l}^{(1)}(S_{i}),v_{l}^{(1)}(S_{i})),\quad W_{f}^{(0)}(X_{3}=0)=\frac{1}{{\it\phi}_{f}}w_{l}^{(2)}(S_{i}), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and matching conditions which recognise that the flow in the EGL is an order of magnitude smaller than in this viscous layer,

(3.20) $$\begin{eqnarray}\displaystyle \boldsymbol{U}_{f}^{(0)}(X_{3}\rightarrow \infty )=\mathbf{0}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{U}_{f}^{(0)}=(U_{f}^{(0)},V_{f}^{(0)},W_{f}^{(0)})$ . These are solved by

(3.21a,b ) $$\begin{eqnarray}\displaystyle U_{f}^{(0)}=\frac{1}{{\it\phi}_{f}}u_{l}^{(1)}(S_{i})\exp (-X_{3}),\quad V_{f}^{(0)}=\frac{1}{{\it\phi}_{f}}v_{l}^{(1)}(S_{i})\exp (-X_{3}), & & \displaystyle\end{eqnarray}$$

(3.21c ) $$\begin{eqnarray}\displaystyle W_{f}^{(0)}=\frac{1}{{\it\phi}_{f}}\left(\boldsymbol{{\rm\nabla}}_{\Vert }\boldsymbol{\cdot }(u_{l}^{(1)},v_{l}^{(1)})\right)\exp (-X_{3}). & & \displaystyle\end{eqnarray}$$

Continuity of traction at $O(1)$ then gives at this order

(3.22a,b ) $$\begin{eqnarray}\displaystyle {\it\phi}_{f}({\it\sigma}_{l}^{(0)})_{13}=\frac{\partial U_{f}^{(0)}}{\partial X_{3}}(X_{3}=0)=-\frac{u_{l}^{(1)}}{{\it\phi}_{f}},\quad {\it\phi}_{f}({\it\sigma}_{l}^{(0)})_{23}=\frac{\partial V_{f}^{(0)}}{\partial X_{3}}(X_{3}=0)=-\frac{v_{l}^{(1)}}{{\it\phi}_{f}}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.22c ) $$\begin{eqnarray}\displaystyle {\it\phi}_{f}({\it\sigma}_{l}^{(0)})_{33}=-P_{e}^{(0)} & & \displaystyle\end{eqnarray}$$

for $\boldsymbol{x}$ on $S_{w}$ . As can be seen, enforcing continuity of tangential components of traction provides boundary conditions for the correction to the leading-order flow in the lumen. Continuity of the normal component of traction provides the required remaining boundary condition for pressure in the Darcy region. The ability of this viscous layer to ensure continuity of tangential velocities at $O({\it\lambda}^{-2})$ in light of the slip velocities induced in the Darcy region at $S_{i}$ requires us to consider the corrections to this leading-order flow. Moreover, continuity of normal velocity gives a permeability condition for the $O({\it\lambda}^{-2})$ lumen flow,

(3.23) $$\begin{eqnarray}\displaystyle w_{l}^{(2)}(S_{i})=\boldsymbol{{\rm\nabla}}_{\Vert }\boldsymbol{\cdot }(u_{l}^{(1)}(S_{i}),v_{l}^{(1)}(S_{i})). & & \displaystyle\end{eqnarray}$$

The flow corrections in this region are governed by

(3.24a-c ) $$\begin{eqnarray}\displaystyle U_{f}^{(1)}=-\frac{\partial P_{e}^{(0)}}{\partial \bar{x}_{1}}+\frac{\partial ^{2}U_{f}^{(1)}}{\partial X_{3}^{2}},\quad V_{f}^{(1)}=-\frac{\partial P_{e}^{(0)}}{\partial \bar{x}_{2}}+\frac{\partial ^{2}V_{f}^{(1)}}{\partial X_{3}^{2}},\quad \frac{\partial P_{e}^{(1)}}{\partial X_{3}}=\frac{\partial ^{2}W_{f}^{(0)}}{\partial X_{3}^{2}}-W_{f}^{(0)}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.24d ) $$\begin{eqnarray}\displaystyle \frac{\partial U_{f}^{(1)}}{\partial X_{1}}+\frac{\partial V_{f}^{(1)}}{\partial X_{2}}+\frac{\partial W_{f}^{(1)}}{\partial X_{3}}=0. & & \displaystyle\end{eqnarray}$$

The resulting flows allow us to bring to rest the slip flow generated on the interface within the Darcy region,

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle U_{f}^{(1)}=\left(u_{l}^{(2)}(S_{i})+\frac{\partial P_{e}^{(0)}}{\partial X_{1}}\right)\exp (-X_{3})-\frac{\partial P_{e}^{(0)}}{\partial X_{1}}, & \displaystyle\end{eqnarray}$$

(3.26) $$\begin{eqnarray}\displaystyle & \displaystyle V_{f}^{(1)}=\left(v_{l}^{(2)}(S_{i})+\frac{\partial P_{e}^{(0)}}{\partial X_{2}}\right)\exp (-X_{3})-\frac{\partial P_{e}^{(0)}}{\partial X_{2}}, & \displaystyle\end{eqnarray}$$

which can be seen to match to the flow in the EGL. We also note that continuity of tangential traction at $O({\it\lambda}^{-1})$ ,

(3.27) $$\begin{eqnarray}\displaystyle & \displaystyle ({\it\sigma}_{l}^{(1)}(S_{i}))_{13}=\frac{\partial U_{f}^{(1)}}{\partial X_{3}}(X_{3}=0)=-\!\left(u_{l}^{(2)}(S_{i})+\frac{\partial P_{e}^{(0)}}{\partial X_{1}}\right), & \displaystyle\end{eqnarray}$$

(3.28) $$\begin{eqnarray}\displaystyle & \displaystyle ({\it\sigma}_{l}^{(1)}(S_{i}))_{23}=\frac{\partial V_{f}^{(1)}}{\partial X_{3}}(X_{3}=0)=-\!\left(v_{l}^{(2)}(S_{i})+\frac{\partial P_{e}^{(0)}}{\partial X_{2}}\right), & \displaystyle\end{eqnarray}$$

together with (3.23), provides boundary conditions with which to solve the $O({\it\lambda}^{-2})$ flow in the lumen (although we do not need to do so here). As such, we have shown how the leading-order EGL flow consistently matches with the leading-order flow in the lumen, through flows within a thin viscous layer on the interface.

3.3 EGL: solid phase

Since the solid phase of the EGL is driven by flow in its fluid phase, there is an analogous three-layer asymptotic structure to the elastic displacements (see § 3.2).

3.3.1 EGL core

In the core EGL, we expand the solid displacements according to

(3.29a,b ) $$\begin{eqnarray}\boldsymbol{u}_{s}=\boldsymbol{u}_{s}^{(0)}+{\it\lambda}^{-1}\boldsymbol{u}_{s}^{(1)}+O({\it\lambda}^{-2}),\quad P_{e}=P_{e}^{(0)}+{\it\lambda}^{-1}P_{e}^{(1)}+O({\it\lambda}^{-2}).\end{eqnarray}$$

Combined with the rescaling of fluid quantities within the EGL (3.5), the leading-order elasticity equations then take the form

(3.30) $$\begin{eqnarray}\frac{1}{1-2{\it\nu}}\boldsymbol{{\rm\nabla}}(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}_{s}^{(0)})+{\rm\nabla}^{2}\boldsymbol{u}_{s}^{(0)}=\frac{{\it\phi}_{s}}{{\it\phi}_{f}}\boldsymbol{{\rm\nabla}}P_{e}^{(0)}-\boldsymbol{u}_{f}^{(0)}.\end{eqnarray}$$

Upon substitution of the leading-order flow in the EGL core (3.6), we obtain

(3.31) $$\begin{eqnarray}\frac{1}{1-2{\it\nu}}\boldsymbol{{\rm\nabla}}(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}_{s}^{(0)})+{\rm\nabla}^{2}\boldsymbol{u}_{s}^{(0)}=\frac{1}{{\it\phi}_{f}}\boldsymbol{{\rm\nabla}}P_{e}^{(0)}.\end{eqnarray}$$

The boundary conditions on the elastic displacements are determined by the viscous flows in the asymptotic layers of the EGL’s fluid phase. As we shall see in the next sections, the displacement boundary condition remains the same on the endothelium, giving

(3.32) $$\begin{eqnarray}\boldsymbol{u}_{s}^{(0)}(S_{i})=\mathbf{0}.\end{eqnarray}$$

However, on the interface, the traction condition changes due to the presence of the forcing terms in the equation. At leading order, this traction boundary condition is

(3.33) $$\begin{eqnarray}\left({\bf\sigma}_{s}^{(0)}-\frac{1}{{\it\phi}_{f}}P_{e}^{(0)}\unicode[STIX]{x1D644}\right)\boldsymbol{\cdot }\boldsymbol{n}={\bf\sigma}_{l}^{(0)}\boldsymbol{\cdot }\boldsymbol{n}\quad \text{on }S_{i}.\end{eqnarray}$$

3.3.2 EGL layer I: endothelial surface

We use the same local Cartesian coordinate system as before (3.10). Within this layer, no $O(1)$ displacements are driven by fluid forcing. Therefore, the $O(1)$ elastic displacements in the EGL core satisfy a zero displacement condition on the endothelium, as given in (3.32).

However, in order to match the leading-order traction coming from the core and understand how this traction is transmitted to the endothelium, we must still consider the $O({\it\lambda}^{-1})$ displacements. As such, in this layer we rescale the solid displacements according to

(3.34) $$\begin{eqnarray}\boldsymbol{u}_{s}=({\it\lambda}^{-1}U_{s},{\it\lambda}^{-1}V_{s},{\it\lambda}^{-1}W_{s}).\end{eqnarray}$$

The governing equations (2.5) at $O({\it\lambda})$ take the form

(3.35a-c ) $$\begin{eqnarray}\displaystyle \frac{\partial ^{2}U_{s}^{(0)}}{\partial X_{3}^{2}}=0,\quad \frac{\partial ^{2}V_{s}^{(0)}}{\partial X_{3}^{2}}=0,\quad \frac{\partial ^{2}W_{s}^{(0)}}{\partial X_{3}^{2}}=0, & & \displaystyle\end{eqnarray}$$

subject to the following boundary and matching conditions:

(3.36a-c ) $$\begin{eqnarray}\displaystyle U_{s}(X_{3}=0)=0,\quad V_{s}(X_{3}=0)=0,\quad W_{s}(X_{3}=0)=0, & & \displaystyle\end{eqnarray}$$

(3.36d ) $$\begin{eqnarray}\displaystyle \left(\frac{\partial U_{s}}{\partial X_{3}}(X_{3}\rightarrow \infty ),\frac{\partial V_{s}}{\partial X_{3}}(X_{3}\rightarrow \infty ),\frac{\partial W_{s}}{\partial X_{3}}(X_{3}\rightarrow \infty )\right)=(\boldsymbol{n}\boldsymbol{\cdot }{\bf\sigma}_{s})(S_{w}). & & \displaystyle\end{eqnarray}$$

This is solved by

(3.37a-c ) $$\begin{eqnarray}\displaystyle U_{s}^{(0)}=({\bf\sigma}_{s})_{13}(S_{w})X_{3},\quad V_{s}^{(0)}=({\bf\sigma}_{s})_{23}(S_{w})X_{3},\quad W_{s}^{(0)}=\frac{(3-4{\it\nu})}{(1-2{\it\nu})}({\bf\sigma}_{s})_{33}(S_{w})X_{3}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

This shows that the leading-order traction is simply transmitted through this layer unchanged.

3.3.3 EGL layer II: EGL interface

Again, we use the same coordinate and pressure rescalings as used for the fluid phase in this layer (3.16). We approximate the solid displacement as in the core using the expansion

(3.38) $$\begin{eqnarray}\boldsymbol{u}_{s}=(U_{s}^{(0)},V_{s}^{(0)},W_{s}^{(0)})+({\it\lambda}^{-1}U_{s}^{(1)},{\it\lambda}^{-1}V_{s}^{(1)},{\it\lambda}^{-1}W_{s}^{(1)})+O({\it\lambda}^{-2}).\end{eqnarray}$$

The governing equations (2.5) at $O({\it\lambda}^{2})$ take the form

(3.39a-c ) $$\begin{eqnarray}\displaystyle \frac{\partial ^{2}U_{s}^{(0)}}{\partial X_{3}^{2}}=0,\quad \frac{\partial ^{2}V_{s}^{(0)}}{\partial X_{3}^{2}}=0,\quad \frac{\partial ^{2}W_{s}^{(0)}}{\partial X_{3}^{2}}=0. & & \displaystyle\end{eqnarray}$$

These equations are solved by displacements that are linear in $X_{3}$ . However, the linear term in these equations would give rise to $O({\it\lambda})$ tractions, which cannot be matched at the interface by the $O(1)$ lumen tractions. This means that these linear terms must be zero. Hence, the $O(1)$ elastic displacements in this layer are constant and do not contribute to the traction.

As such, in order to understand how the traction is transmitted through this layer, we must consider the first-order corrections. The governing equations (2.5) at $O({\it\lambda})$ take the form

(3.40a-c ) $$\begin{eqnarray}\displaystyle \frac{\partial ^{2}U_{s}^{(1)}}{\partial X_{3}^{2}}=-U_{f}^{(0)},\quad \frac{\partial ^{2}V_{s}^{(1)}}{\partial X_{3}^{2}}=-V_{f}^{(0)},\quad \frac{\partial ^{2}W_{s}^{(1)}}{\partial X_{3}^{2}}=0, & & \displaystyle\end{eqnarray}$$

where the fluid velocities $U_{f}^{(0)}$ and $V_{f}^{(0)}$ are given by (3.21). These are solved by

(3.41) $$\begin{eqnarray}\displaystyle & \displaystyle U_{s}^{(1)}=({\bf\sigma}_{l})_{13}(S_{i})X_{3}+B_{1}-\frac{1}{{\it\phi}_{f}}u_{l}^{(1)}(S_{i})\exp (-X_{3}), & \displaystyle\end{eqnarray}$$

(3.42) $$\begin{eqnarray}\displaystyle & \displaystyle V_{s}^{(1)}=({\bf\sigma}_{l})_{23}(S_{i})X_{3}+B_{2}-\frac{1}{{\it\phi}_{f}}v_{l}^{(1)}(S_{i})\exp (-X_{3}), & \displaystyle\end{eqnarray}$$

(3.43) $$\begin{eqnarray}\displaystyle & \displaystyle W_{s}^{(1)}=B_{3}, & \displaystyle\end{eqnarray}$$

where the $B_{i}$   $(i=1,2,3)$ are given by matching the first-order correction to the core elastic displacements. These equations give the traction boundary condition for the core elastic region as

(3.44) $$\begin{eqnarray}\left({\bf\sigma}_{s}-\frac{1}{{\it\phi}_{f}}P_{e}\unicode[STIX]{x1D644}\right)\boldsymbol{\cdot }\boldsymbol{n}={\bf\sigma}_{l}\boldsymbol{\cdot }\boldsymbol{n}\quad \text{on }S_{i}.\end{eqnarray}$$

3.4 Lumen (two-layer model)

Whereas in capillaries, red blood cells pass through intermittently and must be treated individually (which is beyond the scope of what we do here), in larger postcapillary venules, the persistent presence of multiple red blood cells may have an influence on the bulk properties of the fluid which can be captured using a two-layer viscosity model (Pries et al. Reference Pries, Secomb, Gessner, Sperandio, Gross and Gaehtgens1994; Sharan & Popel Reference Sharan and Popel2001). Therefore, in addition to the model given in § 2.1, which considers the flow of plasma alone through the lumen, we also consider an extended model that seeks to capture these effects.

In this two-layer model, the blood flow is still modelled as a Stokes flow. However, within a microvessel, the red blood cells tend to move away from the vessel wall, leading to a red-blood-cell-rich core layer at the centre of the vessel and a cell-free layer surrounding this core. Due to the presence of red blood cells, this core layer has a higher viscosity than blood plasma, which we will call ${\it\mu}_{c}$ . The cell-free layer viscosity, ${\it\mu}_{o}$ , is lower than this core layer viscosity, although it is still modelled to be higher than the plasma viscosity to account for the periodic invasion of red blood cells from the core into this outer layer (Sharan & Popel Reference Sharan and Popel2001). These viscosities, along with the cell-free layer thickness, can be chosen to fit experimental data (Sharan & Popel Reference Sharan and Popel2001).

One of the advantages of the asymptotic formulation is that this can be carried out in a computationally tractable manner, since the flow in the lumen decouples from the flow within the EGL at leading order.

Non-dimensionalising as in § 2.1, with the fluid velocity of each layer non-dimensionalised on its respective viscosity, we find that the governing equations in non-dimensional form are

(3.45a,b ) $$\begin{eqnarray}\displaystyle {\rm\nabla}^{2}\boldsymbol{u}_{c}=\boldsymbol{{\rm\nabla}}P_{c},\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }u_{c}=0\quad \text{in }V_{c}, & & \displaystyle\end{eqnarray}$$

(3.46a,b ) $$\begin{eqnarray}\displaystyle {\rm\nabla}^{2}\boldsymbol{u}_{o}=\boldsymbol{{\rm\nabla}}P_{o},\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }u_{o}=0\quad \text{in }V_{o}, & & \displaystyle\end{eqnarray}$$

where the $c$ and $o$ subscripts denote quantities in the core and cell-free layers respectively. The non-dimensional boundary conditions at the interface, $S_{c}$ , between these two layers are

(3.47) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}_{c}=\frac{{\it\mu}_{o}}{{\it\mu}_{c}}\boldsymbol{u}_{o}\quad \text{on }S_{c}, & \displaystyle\end{eqnarray}$$

(3.48) $$\begin{eqnarray}\displaystyle & \displaystyle {\bf\sigma}_{c}\boldsymbol{\cdot }\boldsymbol{n}={\bf\sigma}_{o}\boldsymbol{\cdot }\boldsymbol{n}\quad \text{on }S_{c}, & \displaystyle\end{eqnarray}$$

which give continuity of velocity and traction respectively. It should be noted that the viscosity ratio in (3.47) is due to the difference in non-dimensionalisations between the two velocities. The outer boundary conditions between the lumen and the EGL remain the same as given in § 3.1. Analytic solutions of this model for a circular tube are given in (4.5).

4.1 Microvessel geometry

Our vessel geometry is informed by confocal microscopy images of a postcapillary venule, taken from mouse cremaster muscle. An image of the data can be seen in figure 2(a). Images were supplied by Dr Bodkin and Professor Nourshargh, Centre for Microvascular Research, William Harvey Research Institute, Barts and The London Medical School, Queen Mary University of London. The experimental method used to image the vessel is described in detail in Woodfin et al. (Reference Woodfin, Voisin, Beyrau, Colom, Caille, Diapouli, Nash, Chavakis, Albelda and Rainger2011) and Colom et al. (Reference Colom, Bodkin, Beyrau, Woodfin, Ody, Rourke, Chavakis, Brohi, Imhof and Nourshargh2015). The experimental method used to collect the data is summarised as follows. A relatively straight postcapillary venule from mouse cremaster muscle was exteriorised, and a Z-stack of images was collected by confocal microscopy using a single-beam Leica TCS-SP5 confocal laser-scanning microscope. The cell–cell junctions had been previously stained by intrascrotal injection of Alexa Fluor 555-labelled monoclonal antibody (mAb) to Platelet and Endothelial Cell Adhesion Molecule 1 (PECAM-1), which allows the cell–cell junctions of the endothelial cells to be visualised.

Figure 2.

(a) AMIRA visualisation of cell–cell junctions in a postcapillary venule taken from mouse cremaster muscle, obtained by Bodkin and Nourshargh using confocal microscopy. (b) The cell–cell junction data (red crosses), as viewed down the longitudinal axis. The microvessel shape can be seen to be well approximated by a fitted elliptical cylinder (black line, with its corresponding major axis shown as a blue line). (c,d) Plots showing the triangularisation used for the microvessel. The colour rendering denotes cell height and the green dashed line denotes the area of the cell that is raised due to cell nuclei. These show an EGL mesh of 250 000 (c) and 1 000 000 (d) elements shown on the same section of vessel. (e) An example synthesised endothelium, with colour renderings indicating heights (dominated by the cell nuclei – see also figure 9 in appendix B).

From these data, we first extracted the cell–cell junctions as line segments in three-dimensional space, using the data visualisation package AMIRA. It was found that the endothelial surface was largely elliptical in shape (excluding a small section at one end of the microvessel), with a major axis of $36.2~{\rm\mu}\text{m}$ and a minor axis of $22.8~{\rm\mu}\text{m}$ (see figure 2 b, where the fit was carried out using the fitellipse.m routine in Matlab (Brown Reference Brown2007)). Hence, we hereafter assume this elliptical representation of the microvessel basal surface in our numerical computations. In order to minimise entry and exit length effects (which are expected to scale with the vessel radius at these low Reynolds numbers), we also added vessel extensions to smoothly transition the geometry between the perfectly circular inlet and outlet, and the non-uniform endothelial surface. This also allowed us to prescribe the following analytic expressions for the inlet and outlet flow boundary conditions: $\boldsymbol{U}_{0}=\boldsymbol{U}_{1}=(0,0,U)$ , where (Damiano et al. Reference Damiano, Duling, Ley and Skalak1996)

(4.1) $$\begin{eqnarray}\displaystyle U(r)=\left\{\begin{array}{@{}ll@{}}\displaystyle \!{\it\lambda}^{2}({\it\phi}_{f})^{2}\!\left[\!\left(\!\frac{{\it\alpha}}{2{\it\lambda}}+\frac{K_{1}({\it\alpha}/{\it\lambda})}{K_{0}(1/{\it\lambda})}\!\right)\frac{J_{0}({\it\alpha})}{J_{1}({\it\alpha})}+\frac{K_{0}({\it\alpha}/{\it\lambda})}{K_{0}(1/{\it\lambda})}-1\!\right]+\frac{1}{4}(r^{2}-{\it\alpha}^{2}),\quad & 0\leqslant r\leqslant {\it\alpha},\\ \displaystyle \!{\it\lambda}^{2}({\it\phi}_{f})\!\left[\!\left(\!\frac{{\it\alpha}}{2{\it\lambda}}+\frac{K_{1}({\it\alpha}/{\it\lambda})}{K_{0}(1/{\it\lambda})}\!\right)\frac{J_{0}(r)}{J_{1}({\it\alpha})}+\frac{K_{0}({\it\alpha}/{\it\lambda})}{K_{0}(1/{\it\lambda})}-1\!\right],\quad & {\it\alpha}\leqslant r\leqslant 1,\end{array}\right. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with $J_{0}(x)=I_{0}(x/{\it\lambda})-{\it\epsilon}K_{0}(x/{\it\lambda})$ and $J_{1}(x)=I_{1}(x/{\it\lambda})+{\it\epsilon}K_{1}(x/{\it\lambda})$ . Here, $K_{0}$ and $K_{1}$ are modified Bessel functions, ${\it\epsilon}=I_{0}(1/{\it\lambda})/K_{0}(1/{\it\lambda})$ and ${\it\alpha}=1-{\it\delta}_{0}$ , where ${\it\delta}_{0}$ is the EGL thickness at the inlet and outlet.

In addition, we also specify the inlet and outlet displacement conditions $\boldsymbol{V}_{0}=\boldsymbol{V}_{1}=(0,0,V)$ , where

(4.2) $$\begin{eqnarray}\displaystyle V(r)=\frac{1}{4}(r^{2}-1)+\frac{1}{{\it\phi}_{f}}U(r),\quad {\it\alpha}\leqslant r\leqslant 1. & & \displaystyle\end{eqnarray}$$

For the asymptotic model, we prescribe analytic solutions for the inlet and outlet as before. In the lumen, we specify Poiseuille conditions,

(4.3) $$\begin{eqnarray}\displaystyle U(r)=\frac{1}{4}(r^{2}-{\it\alpha}^{2}),\quad 0\leqslant r\leqslant {\it\alpha}, & & \displaystyle\end{eqnarray}$$

while the boundary condition for flow in the EGL is a constant pressure condition. The inlet and outlet displacement conditions are given by

(4.4) $$\begin{eqnarray}\displaystyle V(r)=\frac{1}{4}(r^{2}-1)+{\it\alpha}^{2}\ln (r),\quad {\it\alpha}\leqslant r\leqslant 1. & & \displaystyle\end{eqnarray}$$

For the two-layer Stokes model, the inlet and outlet boundary conditions are given by

(4.5) $$\begin{eqnarray}\displaystyle U(r)=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{4}(r^{2}-{\it\beta}^{2})+\frac{{\it\mu}_{c}}{4{\it\mu}_{o}}({\it\beta}^{2}-{\it\alpha}^{2}),\quad & 0\leqslant r\leqslant {\it\beta},\\ \displaystyle \frac{1}{4}(r^{2}-{\it\alpha}),\quad & {\it\beta}\leqslant r\leqslant {\it\alpha},\end{array}\right. & & \displaystyle\end{eqnarray}$$

where ${\it\beta}=1-r_{o}$ , $r_{o}$ is the cell-free layer thickness.

Experimental constraints meant that confocal microscopy data were only available for the lower half of the vessel, and cell membrane surfaces were not readily obtainable. As such, we synthesised cells in a manner that observed the spatial statistics seen in the biological data, as well as the cell profiles reported in the earlier literature. (It should be noted that we performed a statistical analysis of the resulting cell patterns, in order to confirm that they reproduced the distributions seen in the biological data.) More details can be found in appendix B. In figure 2(e) we include an example synthesised endothelium geometry.

We consider two possible scenarios for the EGL distribution in vivo. Model A considers an EGL that has redistributed to the relatively flat regions between cell nuclei, producing a poroelastic layer that varies in thickness. Here, the EGL is thinnest at the cell peaks, where it has depth $t_{min}$ , and thickest between the cells, where its depth is ${\it\alpha}_{0}t_{min}$ , where ${\it\alpha}_{0}$ is some amplitude factor that alters the shape of the redistributed EGL (see figure 1 a for an illustration). It should be noted that a typical value of ${\it\alpha}_{0}$ is not currently well known, due to the experimental challenges associated with visualising the EGL in vivo, and so in what follows we have performed simulations using different values of ${\it\alpha}_{0}$ . For the same reasons, we have chosen a cubic interpolation for the EGL profile between cell peaks and cell–cell junctions. However, we have checked that the trends reported are not overly sensitive to this choice by performing additional simulations using linear interpolants. By contrast, in model B, the EGL follows the topology of the endothelial surface, and has constant thickness $t_{min}$ throughout the vessel (see figure 1 b). For comparison, we have also considered a bare vessel, which does not contain an EGL at all.

Finally, we consider some scenarios where the flow in the lumen consists of a higher-viscosity inner region and a lower-viscosity outer region, as discussed in § 3.4. Such two-layer models have been shown to well approximate the influence of red blood cells outside of a cell-depleted region in the lumen. We choose to implement the profiles proposed by Sharan & Popel (Reference Sharan and Popel2001), where the lower viscosity in the cell-depleted layer remains higher than that of plasma alone, to take account of occasional red blood cell intrusion into this layer. We choose a cell-depleted layer that follows the topology of the EGL interface, with its thickness being that predicted by earlier studies (Pries et al. Reference Pries, Secomb, Gessner, Sperandio, Gross and Gaehtgens1994; Sharan & Popel Reference Sharan and Popel2001). Using a discharge haematocrit of 45 % and a vessel radius of $29.47~{\rm\mu}\text{m}$ , we find a cell-free layer thickness of $r_{o}=4.3~{\rm\mu}\text{m}$ , a core viscosity of ${\it\mu}_{c}=3.71{\it\mu}_{pl}$ and a cell-free layer viscosity of ${\it\mu}_{o}=1.45{\it\mu}_{pl}$ (where ${\it\mu}_{pl}$ indicates the plasma viscosity).

4.2 Boundary element scheme

The governing boundary integral equations were solved numerically by first discretising the vessel surfaces into triangular elements (as can be seen in figure 2) and then assuming a constant basis function across these triangular elements. The required integrals were then computed using numerical quadratures, and the resulting linear system was solved iteratively and in parallel over multiple cores using GMRES with a Jacobi preconditioner. This process is described in more detail in appendix A along with a description of how these solvers were verified, and indicative values of mesh sizes and solution times.