📂Fluid Mechanics

Derivation of Darcy-Brinkman-Forchheimer equation

Theorem ¹

We aim to describe the motion of a fluid in a porous medium. $$ \mathbf{u} = \mathbf{u} \left( t ; \mathbf{x} \right) = \left( u_{1} \left( t ; \mathbf{x} \right) , u_{2} \left( t ; \mathbf{x} \right) , u_{3} \left( t ; \mathbf{x} \right) \right) $$ In particular, consider the velocity field in three-dimensional space at time $t$ and spatial coordinates $\mathbf{x} = \left( x_{1} , x_{2} , x_{3} \right)$ represented by the velocity vector as above (see $\mathbf{x} = \left( x_{1} , x_{2} , x_{3} \right)$). Similarly, $p : \mathbb{R}^{3} \to \mathbb{R}$ denotes the pressure applied at each coordinate (see $p = p \left( \mathbf{x} \right)$). If $\mathbf{u}$ is the velocity of an incompressible Newtonian fluid, it satisfies the following governing equations. $$ \rho \left[ {\frac{ \partial \mathbf{u} }{ \partial t }} + \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} \right] = - \nabla p + \mu \nabla^{2} \mathbf{u} - {\frac{ \mu }{ k }} \mathbf{u} - \beta \rho |u| u + \rho \mathbf{g} $$ Here $\rho$ is the density, $\nabla \cdot$ is the divergence, $\mu$ is the viscosity coefficient, $k$ is the permeability, $\beta$ is an empirical constant, and $\mathbf{g}$ is the gravitational acceleration.

Explanation

Navier–Stokes equations: $$ {\frac{ \partial \mathbf{u} }{ \partial t }} + \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} = - \nabla w + \nu \nabla^{2} \mathbf{u} + \mathbf{g} $$

The Darcy–Brinkman–Forchheimer equation is analogous to the Navier–Stokes equations but is the governing equation when the porous medium is taken into account.

Derivation

This derivation is by no means rigorous; since empirical laws are applied when the Forchheimer term is added, it is sufficient to accept it only to get an intuitive sense.

Euler equation: $$ {\frac{ \partial \mathbf{u} }{ \partial t }} + \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} = - {\frac{ 1 }{ \rho }} \nabla p + \mathbf{g} $$

Start from the simplest Euler equation. Various terms will be added to the right-hand side; here we examine the pressure $p '$ not of the fluid itself but of the porous medium. $$ \rho \left[ {\frac{ \partial \mathbf{u} }{ \partial t }} + \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} \right] = - \nabla p + \nabla p ' + \rho \mathbf{g} $$ Of course, $\nabla p$ and $\nabla p '$ have the same dimensions, so adding them on the right-hand side is natural from a dimensional analysis viewpoint. From now on we focus only on how $\nabla p '$ appears.

Darcy term

Darcy’s law: $$ Q = \frac{k A}{\mu L} \Delta p $$

For a length $L$ we have $\Delta p ' = p ' \left( \mathbf{x} \right) - p ' \left( \mathbf{x} + L \right)$. We know $L$ is not a vector, so expressions like $\mathbf{x} + L$ do not strictly make sense, but as mentioned earlier we do not require rigor here, so let that pass. Expressed for $p '$ this yields: $$ {\frac{ p ' \left( \mathbf{x} + L \right) - p ' \left( \mathbf{x} \right) }{ L }} = - {\frac{ \mu }{ k }} {\frac{ Q }{ A }} $$

Continuity equation for flow rate: $$ Q = A_{1} U_{1} = A_{2} U_{2} $$

On the left-hand side take $L \to \infty$, and on the right-hand side, since the velocity $\mathbf{u}$ can be expressed as the ratio of flow rate to cross-sectional area, it can be written as: $$ \nabla p ' = - {\frac{ \mu }{ k }} \mathbf{u} $$

Brinkman term ²

Fundamentally, this is the same as adding the viscous term from the Navier–Stokes equations. $\nabla p '$ can be represented by adding the Brinkman term $\mu \nabla^{2} \mathbf{u}$ as follows. $$ \nabla p ' = - {\frac{ \mu }{ k }} \mathbf{u} + \mu \nabla^{2} \mathbf{u} $$

Forchheimer term

This is an empirical law to account for inertial effects. An additional term proportional to the square of the velocity, the Forchheimer term $- \beta \rho |u| u$, is added to the right-hand side. $$ \nabla p ' = - {\frac{ \mu }{ k }} \mathbf{u} + \mu \nabla^{2} \mathbf{u} - \beta \rho |u| u $$

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