Flow in a wavy-walled channel lined with a poroelastic layer

Abstract

Motivated by physiological flows in capillaries, venules and the pleural space, the pressure-driven flow of a Newtonian fluid in a two-dimensional wavy-walled channel is investigated theoretically. The sinusoidal wavy shape is due to the configuration of underlying cells, their nuclei and intercellular junctions or clefts. The walls are lined with a thin poroelastic layer that models the glycocalyx coating of the cell surface. The upper and lower wavy walls are offset axially by the phase angle $\Phi $, where $\Phi\,{ =}\, 0$ ($\upi$) yields an antisymmetric (symmetric) channel. Biphasic theory is employed for the poroelastic layer and the flow is solved by a lubrication approximation using a small parameter, $\delta\,{\ll}\,1$, where $\delta$ is the channel width/wavelength ratio. The velocity fields in the core and layer are determined as perturbation expansions in $\delta^2$ and finite-Reynolds-number effects occur at $O(\delta^2)$ assuming $\delta^2\hbox{\it Re}\,{\ll}\,1$. When the hydraulic resistivity, $\alpha$, the ratio of the channel width to the Darcy permeability, is sufficiently large and $\Phi$ is near enough to $\upi$, the flow develops a trapped recirculation eddy within the glycocalyx layer near the widest part of the channel. This can be of significance to transport through the cellular boundary, since that location corresponds to intercellular clefts through which important fluid and solute exchange occurs. Increasing $|\Phi\,{-}\,\upi |$ diminishes the recirculation region. Increasing the Reynolds number moves the recirculation slightly upstream. Both layer velocity and wall shear stresses decrease as $\alpha$ increases and support the appearance of flow recirculation. Further, the wavy geometry allows a portion of the flow to enter and exit the layer, which provides a mechanism for convective transport between these two regions that otherwise have only diffusive interactions. The relevant Péclet number is $\hbox{\it Pe}\,{=}\,V^*_nb/D$ where $D$ is molecular diffusivity and $V^*_n$ is the normal velocity to the glycocalyx layer. For large molecules, $\hbox{\it Pe}\,{=}\,O(10^2)$ or higher, so the convective transport is important. The solid displacement, dictated by the layer flow field, increases as $\alpha$ increases.