Merge branch 'StokesFlowDoc' into 'master'

[Web] Web documentation of the parallel-plate testcase.

See merge request !3658

web/content/docs/benchmarks/stokes-flow/parallel-plate-flow.md

++++
+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"
+
+[menu]
+  [menu.benchmarks]
+    parent = "Stokes Flow"
+
++++
+
+{{< data-link >}}
+
+## 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)
+$$
+\begin{equation}
+\nabla p - \mu \Delta \mathbf{v} = \mathbf{f},
+\end{equation}$$
+
+\begin{equation}
+\nabla \cdot \mathbf{v} = 0.
+\end{equation}
+
+## 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)
+$$
+\begin{equation}
+v \left(y\right) = \frac{1}{2\mu} \frac{P_{\mathrm{in}} - P_{\mathrm{out}}}{l} y \left( b - y\right).
+\end{equation}$$
+
+{{< 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.