Application of Porous Media Theories in Marine Biological Modeling - Porous Media: Applications in Biological Systems and Biotechnology - page 370

The Navier-Stokes equations can be derived from the Chapman-Enskog
procedure (Chopard and Droz 1998), which leads to

∂ρ
∂t +

∇· ( ρ u ) = 0 (10.6)

∂ ( ρ u )
∂t

+ ∇· ( ρ uu )= −∇ p + ν ∇· [ ρ ( ∇ u + u ∇ )] (10.7)

where p = c s ρ is the pressure, and the effective viscosity is defined as
ν = 3 τ

2 δ t

1

− (10.8)

10.2.2 LBM for Incompressible Flows in Porous Media
Flow in porous media is usually modeled by some semiempirical models
because of the complex structure of a porous medium based on the volume
averaging at the scale of representative elementary volume (REV). Several
widely used models have been introduced in the literature, such as the Darcy,
the Brinkman-extended Darcy, and the Forchheimer-extended Darcy models.
A recent achievement in modeling flow in porous media is the so-called gener-
alized model, in which all fluid forces and the solid drag force are considered
in the momentum equation given by:

∇· ( u ) = 0 (10.9)

uu
φ

=

∂ ( u )
∂t

1
ρ ∇

2 u + F

+ ∇· − ( φp )+ ν e ∇ (10.10)

In the above equation, ν e is the effective viscosity and F represents the total
body force given by the following:

φν
K u

φF φ
√ K |

F = − − u | u + φ G (10.11)

in which the three terms on the right side represent Darcy, Forchheimer, and
gravity force, respectively. The geometric function F φ and permeability K can
be expressed as follows:

1 . 75
150 φ 3 F φ = (10.12)

φ 3 d p
150(1

K = (10.13)

φ ) 2 −

where d p is the solid particle diameter.
In the LBM notation, the momentum equation for the fluid flow in a porous
medium can be expressed as:

f i ( x, t )

( x, t ) + δ t F i

1
τ

f ( eq )
i

f i ( x + e i δ t ,t + δ t ) − f i ( x ,t )= − − (10.14)