# Fluid flow through an open parallel-plate channel

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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.

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

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.