diff --git a/web/content/docs/benchmarks/stokes-flow/Fig1_SchematicDiagram.png b/web/content/docs/benchmarks/stokes-flow/Fig1_SchematicDiagram.png new file mode 100644 index 0000000000000000000000000000000000000000..a04a13182c837ff7ae3c4eac1196fa1888879c5f Binary files /dev/null and b/web/content/docs/benchmarks/stokes-flow/Fig1_SchematicDiagram.png differ diff --git a/web/content/docs/benchmarks/stokes-flow/Fig2_SimulationResults.png b/web/content/docs/benchmarks/stokes-flow/Fig2_SimulationResults.png new file mode 100644 index 0000000000000000000000000000000000000000..885016f195f33459f517730d412bde0d54a14d22 Binary files /dev/null and b/web/content/docs/benchmarks/stokes-flow/Fig2_SimulationResults.png differ diff --git a/web/content/docs/benchmarks/stokes-flow/parallel-plate-flow.md b/web/content/docs/benchmarks/stokes-flow/parallel-plate-flow.md new file mode 100644 index 0000000000000000000000000000000000000000..eb5f011022044f9f7ebc2316097a28944535519e --- /dev/null +++ b/web/content/docs/benchmarks/stokes-flow/parallel-plate-flow.md @@ -0,0 +1,56 @@ ++++ +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.