Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 282

multiresolution SPH formulation:
∂ω a C a
∂t

=4
b∈ Ω c ∪ Ω p

ω a ω b d a d b
ω a d a + ω b d b

C a − C b

x ab

x ab ·∇ a x ab ∂W ( x ab ,h ab )
∂x ab

V b

ω a r AB C a C a , − I = A, B (7.53)

a

=4
b∈ Ω c ∪ Ω p

∂ω a C a
∂t

ω a ω b d a d b
ω a d a + ω b d b

C a C b

x ab

−

x ab ·∇ a x ab ∂W ( x ab ,h ab )
∂x ab

V b

ω a k a
b∈ Ω s

+ ω a r AB C a C a ω a k a ( C a

− − C eq ) −

a

C eq ) W − 1
b

∆ b ( C a

− W ( x ab ,h r ) (7.54)

An extensive discussion of the multiresolution SPH method can be found
in Kitsionas and Whitworth (2002) and references therein.

7.5.10 Time Integration
Integration of the SPH equations (7.53), (7.54) can be carried out using various
explicit (Monaghan 2005) or fully implicit schemes (Chaniotis et al. 2003). To
improve the algorithm's eciency, adaptive particle time stepping can be used
(Kitsionas and Whitworth 2002).
In this study, we employed an explicit Euler time stepping integration
method,

C a ( t +∆ t )= C a ( t )+∆ t dC a ( t )
dt

(7.55)

where the time step, ∆ t , satisfies the following conditions:
∆ t< min ε 1

∆ x a
d a

ε 2
k AB
a

∆ x a
k

, ,ε 3 (7.56)

Our numerical experiments have shown that setting the constants ε 1 ,ε 2 ,
and ε 3 to ε 1 = ε 2 = ε 3 =0 . 25 provides a stable and accurate solution (see
also Monaghan [2005] where the first of these inequalities, ε 1 =0 . 25, was
postulated).

7.5.11 Numerical Example
We consider mixing-induced precipitation in the porous medium depicted in
Figure 7.9. A hybrid simulation of this process was conducted and validated
against numerical solution obtained with the single scale pore-scale model.

7.5.12 Pore-Scale SPH Simulations
First pore-scale simulations were conducted over the whole computational
domain, Ω T =[0 ,L ]

× [0 ,B ], in which diffusion of solutes A and B and the