Page 326 - Advances in Biomechanics and Tissue Regeneration

324 16. ON THE SIMULATION OF ORGAN-ON-CHIP CELL PROCESSES

And Eq. (16.41) becomes

N r N
X ∂C X r
^
ϕ i  C , i ¼ 1,…,n (16.44)
ðiÞ
r
s ∂t ϕ ¼ sr i
r¼1 r¼1
where we have defined
^
 sr ¼ ϕ ϕ (16.45)
s r
P N r P N r
f ϕ becomes
r
i
Finally, Eq. (16.30) with ϕ ¼ ϕ s , C i ¼ r¼1 C ϕ and f i ¼ r¼1 i r
r
r
C ¼ f , i ¼ 1,…,n (16.46)
i i
P N r
r
i
and Eq. (16.31) with ϕ ¼ ϕ s and C i ¼ r¼1 C ϕ :
N Z ! Z Z
X r ∂C i
κ i ϕ ϕ dΓ C + ϕ s dΓ ¼ ϕ g i dΓ, i ¼ 1,…,n (16.47)
s
s r
i
r¼1 Γ R i Γ R i ∂n Γ R i
16.5.1.2.2 CHEMICAL SPECIES
In the same way as above, plugging now Eq. (16.36) into Eq. (16.33) we obtain
!
N
Z r Z Z Z
X ∂S i ∂S i
ϕ ϕ dΩ —ϕ B i dΩ ¼ ϕb i dΩ ϕ dΓ, i ¼ 1,…,m (16.48)
r
Ω r¼1 ∂t Ω Ω Γ 0 ∂n
R
i
Using again the linear properties of — operator, Eq. (16.25) writes
N
X r
ðS K Þ—ϕ , i ¼ 1,…,m (16.49)
0 0
i
q ¼ i D,i r
r¼1
So, as B i ¼ S i v + q it is obtained
0
i
!
N N
X r X r
S ϕ ðS K 0 Þ—ϕ , i ¼ 1,…,m (16.50)
B i ¼ i r v i D,i r
r¼1 r¼1
So, —ϕ B i writes
N
X r
S —ϕ ϕ v —ϕ ðK 0 —ϕ Þ, i ¼ 1,…,m (16.51)
—ϕ B i ¼ i r D,i r
r¼1
r
r
Besides, as S ¼ S ðtÞ, ϕ ∂S r i writes
i i ∂t
N r N
X ∂S X r
_
ϕ i ϕ ¼ ϕ S ϕ , i ¼ 1,…,m (16.52)
r
r
∂t i
r¼1 r¼1
Choosing as before ϕ ¼ ϕ s , s ¼ 1, …, N (Galerkin method) and reorganizing terms in Eq. (16.51) we may write
N
X
^
 0ðiÞ r
s S , i ¼ 1,…,m (16.53)
—ϕ B i ¼ sr i
r¼1
where we have defined
^
 0ðiÞ ¼ð—ϕ ðϕ vÞ —ϕ K 0 D,i —ϕ Þ, i ¼ 1,…,m (16.54)
s
r
r
s
sr
And Eq. (16.52) becomes
N r N
X ∂S X r
^
ϕ i  S , i ¼ 1,…,m (16.55)
ðiÞ
r
s ∂t ϕ ¼ sr i
r¼1 r¼1
II. MECHANOBIOLOGY AND TISSUE REGENERATION