^{1}and George E. Kapellos

^{1,a)}

### Abstract

An analytical solution is presented for the problem of fully developed plane Couette-Poiseuille flow past a homogeneous, permeable poroelastic layer. Main novel feature of this work is that the compressibility, which is related to the action of the free-fluid pressure on the poroelastic layer, is taken into account. Therefore, the solid stress problem is two-dimensional, although the fluid flow problem remains one-dimensional in the limit of infinitesimal strain. The pressure-related compressibility affects strongly the distribution of the von Mises stress in the poroelastic layer by shifting the local maximum towards the high-pressure region of the channel. Furthermore, the established analytical solution is used to investigate some aspects of the mechanotransducing role of the endothelial surface layer. A compressible surface layer might serve as a “bumper’’ by reducing the magnitude of the overall shearing force (viscous and elastic) acting on endothelial cells, as compared to the magnitude of the fluid shear force that would be exerted in its absence.

The authors are grateful to Professor George Dassios of Patras University for providing very useful feedback on a first draft of this work.

I. INTRODUCTION

II. STATEMENT OF THE PROBLEM

III. ANALYTICAL SOLUTION

A. Analytical solution of the fluid-flow problem

B. Analytical solution of the solid-stress problem

IV. RESULTS AND DISCUSSION

A. Distribution of the fluid velocity in the channel

B. Distribution of the von Mises stress in the poroelastic layer

C. Implications for the mechanotransducing role of the endothelial glycocalyx

V. CONCLUSIONS

### Key Topics

- Poiseuille flow
- 14.0
- Boundary value problems
- 12.0
- Elasticity
- 12.0
- Tensor methods
- 9.0
- Stress strain relations
- 8.0

## Figures

Geometry for plane Couette-Poiseuille flow past a poroelastic layer (illustrated in terms of dimensionless quantities).

Geometry for plane Couette-Poiseuille flow past a poroelastic layer (illustrated in terms of dimensionless quantities).

Effect of the flow regime parameter f cp and the Darcy number DaH on the distribution of the fluid velocity in the channel. Other parameters: δπ = 0.4, φυ = 0.7, G p = −1. The fluid velocity is v x = ⟨v υ⟩ x for 0 ⩽ y < δπ and v x = v υ, x for δπ ⩽ y ⩽ 1.

Effect of the flow regime parameter f cp and the Darcy number DaH on the distribution of the fluid velocity in the channel. Other parameters: δπ = 0.4, φυ = 0.7, G p = −1. The fluid velocity is v x = ⟨v υ⟩ x for 0 ⩽ y < δπ and v x = v υ, x for δπ ⩽ y ⩽ 1.

Contours of the von Mises stress in the poroelastic layer for different values of the compressibility parameter: (a) α p = 0, (b) α p = φσ/2, and (c) α p = φσ. The values of the other parameters are: δπ = 0.4, f cp = 0, DaH = 10−3, ν s = 0.3, φυ = 0.7, L = 5, G p = −1.

Contours of the von Mises stress in the poroelastic layer for different values of the compressibility parameter: (a) α p = 0, (b) α p = φσ/2, and (c) α p = φσ. The values of the other parameters are: δπ = 0.4, f cp = 0, DaH = 10−3, ν s = 0.3, φυ = 0.7, L = 5, G p = −1.

Contours of the von Mises stress in the poroelastic layer for different values of Poisson's ratio: (a) ν s = 0.2 and (b) ν s = 0.4. The values of the other parameters are: δπ = 0.4, f cp = 0, DaH = 10−3, α p = 0, φυ = 0.7, L = 5, G p = −1.

Contours of the von Mises stress in the poroelastic layer for different values of Poisson's ratio: (a) ν s = 0.2 and (b) ν s = 0.4. The values of the other parameters are: δπ = 0.4, f cp = 0, DaH = 10−3, α p = 0, φυ = 0.7, L = 5, G p = −1.

Contours of the von Mises stress in the poroelastic layer for different values of the flow regime parameter: (a) f cp = (1 − δπ)2, (b) f cp = (1 − δπ)2/2, (c) f cp = 0, (d) f cp = −(1 − δπ)2/4, (e) f cp = −(1 − δπ)2/2, and (f) f cp = −2(1 − δπ)2. The values of the other parameters are: δπ = 0.4, DaH = 10−3, α p = 0, ν s = 0.3, φυ = 0.7, L = 5, G p = −1.

Contours of the von Mises stress in the poroelastic layer for different values of the flow regime parameter: (a) f cp = (1 − δπ)2, (b) f cp = (1 − δπ)2/2, (c) f cp = 0, (d) f cp = −(1 − δπ)2/4, (e) f cp = −(1 − δπ)2/2, and (f) f cp = −2(1 − δπ)2. The values of the other parameters are: δπ = 0.4, DaH = 10−3, α p = 0, ν s = 0.3, φυ = 0.7, L = 5, G p = −1.

Contours of the von Mises stress in the poroelastic layer for different values of the Darcy number: (a) DaH = 10−2 and (b) DaH = 10−4. The values of the other parameters are: δπ = 0.4, f cp = 0, α p = 0, ν s = 0.3, φυ = 0.7, L = 5, G p = −1.

Contours of the von Mises stress in the poroelastic layer for different values of the Darcy number: (a) DaH = 10−2 and (b) DaH = 10−4. The values of the other parameters are: δπ = 0.4, f cp = 0, α p = 0, ν s = 0.3, φυ = 0.7, L = 5, G p = −1.

Effect of the Darcy number and the porosity of the endothelial surface layer on: (a) the fluid shear stress, (b) the elastic shear stress for α p = φσ, and (c) the elastic shear stress for α p = 0, which are exerted on the glycocalyx-endothelial interface. The thickness of the layer was set to δπ = 0.1.

Effect of the Darcy number and the porosity of the endothelial surface layer on: (a) the fluid shear stress, (b) the elastic shear stress for α p = φσ, and (c) the elastic shear stress for α p = 0, which are exerted on the glycocalyx-endothelial interface. The thickness of the layer was set to δπ = 0.1.

Effect of the thickness of the endothelial surface layer on the stress ratio for different values of the solid volume fraction and the compressibility parameter. Here, is the overall shear stress exerted by the glycocalyx at the cell surface and is the shear stress exerted by the free-fluid at the cell surface in the absence of a glycocalyx.

Effect of the thickness of the endothelial surface layer on the stress ratio for different values of the solid volume fraction and the compressibility parameter. Here, is the overall shear stress exerted by the glycocalyx at the cell surface and is the shear stress exerted by the free-fluid at the cell surface in the absence of a glycocalyx.

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