[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" [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.
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