Finite Element Modeling of Solutes in Hydrated Deformable Biological Tissues - Computer Models in Biomechanics - page 229

and solute molar fluxes,

w ,α

c α
d 0

˜

j α κ α d α ϕ w grad c α =˜ · − ˜ + = s,w. (17.16)

These expressions are relatively compact, yet they describe a broad set of phenom-
ena, including permeation (fluid flux driven by a pressure gradient), osmosis (fluid
flux driven by solute concentration gradients), electroosmosis (fluid flux driven by
an electric potential gradient), diffusion (solute flux driven by a concentration gra-
dient), electrophoresis (solute flux driven by an electric potential gradient), and
barophoresis (solute flux driven by a fluid pressure gradient).

17.3 Finite Element Formulation

The virtual work integral for a mixture of intrinsically incompressible constituents
combines the balance of momentum for the mixture, the balance of mass for the
mixture, and the balance of mass for each of the solutes. In addition, for charged
mixtures, the current condition of Eq. ( 17.6 ) may be enforced as a penalty constraint
on each solute mass balance equation:

δW =− δ v · div σ d v

b

p div v s w d v

− δ ˜ +

b

δc α 1
J s

z β div j β d v,

D s

Dt J s ϕ w

c α +

κ α div j α − +

b

α = s,w β = s,w

(17.17)

where δ v is the virtual velocity of the solid, δ p is the virtual effective fluid pres-
sure, and δc α is the virtual molar energy of solute α . Here, b represents the mixture
domain in the spatial frame and d v is an elemental volume in b . Applying the di-
vergence theorem, δW may be split into internal and external contributions to the
virtual work, δW = δW int − δW ext , where

w d v

D s J s
Dt

δ p

J s

˜

δW int = σ : δ D d v + · grad δ p ˜ −

b b

j α

c α d v

c α
J s

D s

Dt J s ϕ w

δ ˜

c α κ α + · grad δ ˜ − ˜ ˜

b

α = s,w

c α z β j β d v, + grad δ ˜ · (17.18)

b

α = s,w β = s,w