Biological Systems: Multiscale Modeling Based on Mixture Theory - Multiscale Modeling in Biomechanics and Mechanobiology - page 271

Assuming the fluid phase ( F ) is comprised of the liquid ( L ) and the nutrient
phases ( N ) we obtain ( F

= +

L N )

S , L , N S , L , N S , L , N

T ʱ + ˁ ʱ + p ʱ − ˁ

S v S − ˁ F v F =

∇· b ˆ 0 . (11.80)

ʱ ʱ ʱ

F S , and

p S p N p F Since ˁ =− ˁ ˆ + ˆ + ˆ = 0, we obtain

S , L , N S , L , N

T ʱ + ˁ ʱ + ˁ

S

∇· b ( v F − v S ) = 0 . (11.81)

ʱ ʱ

The definition of the seepage velocity w FS provides the following equation

S , F S , F

T ʱ + ˁ ʱ + ˁ

S

∇· b ( w FS ) = 0 . (11.82)

ʱ ʱ

The strong form for the pressure equation can be written as follows

S 1
ˁ

F w FS

1

∇· ʷ + :

D S − ˁ

SR −

= .

I 0 (11.83)

NR

ˁ

The strong form of mass conservation equation for the solid phase is

D S

S S

(ʷ ) ˁ

S I

+ ʷ : D S =

SR .

(11.84)

Dt ˁ

Finally, the balance of mass for the nutrient phase can be described as

N w FS

D S

N N

(ʷ ) ˁ

N I

−

NR + ʷ

: D S +∇· ʷ = 0 . (11.85)

Dt ˁ

In the above, w FS is the seepage velocity, D S denotes the symmetric part of the
spatial velocity gradient, and

D S

()
Dt denotes the total derivative of quantities with
respect to the solid phase. The seepage velocity is obtained from

p F

1
S F

F

w FS = ʻ ∇ ʷ − ˆ , (11.86)

F is the vol-
ume fraction of the fluid. Equations 11.78 - 11.85 are required to be solved for the
bone remodeling problem with the mixture theory. The primary dependent variables
are

ʻ ʷ

Here, S F is the permeability tensor, denotes the pressure, and

, the solid displacements, interstitial pressure, and the solid and
nutrient volume fractions.

{ u S ,ʻ, n S , n N }