Brinkman Equations Theory
The dependent variables in the Brinkman equations are the Darcy velocity and the pressure. The flow in porous media is governed by a combination of the continuity equation and the momentum equation, which together form the Brinkman equations:
(4-34)
(4-35)
In these equations:
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μ (SI unit: kg/(m·s)) is the dynamic viscosity of the fluid
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u (SI unit: m/s) is the velocity vector
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ρ (SI unit: kg/m3) is the density of the fluid
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p (SI unit: Pa) is the pressure
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εp is the porosity
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κ (SI unit: m2) is the permeability of the porous medium, and
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Qm (SI unit: kg/(m3·s)) is a mass source or sink
Influence of gravity and other volume forces can be accounted for via the force term F (SI unit: kg/(m2·s2)).
When the Neglect inertial term (Stokes flow) checkbox is selected, the term (u · ∇)(u/εp) on the left-hand side of Equation 4-35 is disabled.
The mass source, Qm, accounts for mass deposit and mass creation within the domains. The mass exchange is assumed to occur at zero velocity.
The Forchheimer and Ergun drag options add a viscous force proportional to the square of the fluid velocity, F = −ρβ|u|u, on the right-hand side of Equation 4-35; see References for the Darcy’s Law Interface for details.
In case of a flow with variable density, Equation 4-34 and Equation 4-35 must be solved together with the equation of state that relates the density to the temperature and pressure (for instance the ideal gas law).
For incompressible flow, the density stays constant in any fluid particle, which can be expressed as
and the continuity equation (Equation 4-34) reduces to