Commit 48f80909 authored by's avatar Committed by Dmitry Yu. Naumov
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[Web] Web documentation of the parallel-plate testcase.

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author = "Renchao Lu, Dmitri Naumov"
weight = 142
project = "StokesFlow/ParallelPlate.prj"
date = "2021-06-09T14:41:09+01:00"
title = "Fluid flow through an open parallel-plate channel"
parent = "Stokes Flow"
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## Problem definition
This benchmark deals with fluid flow through an open parallel-plate channel. The figure below gives a pictorial view of the considered scenario.
{{< img src="../Fig1_SchematicDiagram.png" title="Schematic diagram of the parallel-plate flow channel in two-dimensional space.">}}
The model parameters used in the simulation are summarized in the table below.
| Parameter | Unit | Value |
| ----------------------------------------------------|:-----------| --------:|
| Hydraulic pressure at the inlet $P_{\mathrm{in}}$ | Pa | 200039.8 |
| Hydraulic pressure at the outlet $P_{\mathrm{out}}$ | Pa | 200000 |
| Fluid dynamic viscosity $\mu$ | Pa$\cdot$s | 5e-3 |
## Mathematical description
The fluid motion in the parallel-plate channel can be described by the Stokes equation. To close the system of equations, the continuity equation for incompressible and steady-state flow is applied. The governing equations of incompressible flow in the entire domain are given as (Yuan et al., 2016)
\nabla p - \mu \Delta \mathbf{v} = \mathbf{f},
\nabla \cdot \mathbf{v} = 0.
## Results
Figure 2(a) shows the hydraulic pressure profile through the parallel-plate flow channel, wherein the pressure drop is linearly distributed. Figure 2(b) gives the transverse velocity component profile over the cross-section of the plane flow channel which shows a parabolic shape. The transverse velocity component reaches a maximum value of 0.004975 m/s at the center which conforms to the value obtained from the analytical solution of the transverse velocity component. The analytical solution of the velocity is given as (Sarkar et al., 2004)
v \left(y\right) = \frac{1}{2\mu} \frac{P_{\mathrm{in}} - P_{\mathrm{out}}}{l} y \left( b - y\right).
{{< img src="../Fig2_SimulationResults.png" title="Simulation results: (a) Hydrualic pressure profile through the parallel-plate flow channel; (b) Transverse velocity component profile over the cross-section of the plane flow channel.">}}
## References
Sarkar, S., Toksoz, M. N., & Burns, D. R. (2004). Fluid flow modeling in fractures. Massachusetts Institute of Technology. Earth Resources Laboratory.
Yuan, T., Ning, Y., & Qin, G. (2016). Numerical modeling and simulation of coupled processes of mineral dissolution and fluid flow in fractured carbonate formations. Transport in Porous Media, 114(3), 747-775.
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