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## Problem definition

## Mathematical description

## Results

## References

This benchmark deals with fluid flow through an open parallel-plate channel. The figure below gives a pictorial view of the considered scenario.

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 |

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}

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) $$ \begin{equation} v \left(y\right) = \frac{1}{2\mu} \frac{P_{\mathrm{in}} - P_{\mathrm{out}}}{l} y \left( b - y\right). \end{equation}$$

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.

This article was written by Renchao Lu, Dmitri Naumov. If you are missing something or you find an error please let us know.
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