Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 48

S f ¼ð p b þ p Þ I þ 2l f e ð v Þ; ð 9 Þ

where S f is the stress tensor in the fluid, v is the fluid velocity, p b is the
hydraulic pressure, p is the Donnan pressure and l f is the fluid dynamic
viscosity. Note that the osmotic Donnan swelling pressure p ¼
2RTn b ð cosh u 1 Þ due to the double layer electrostatics repulsion [ 79 ] is here
introduced.
Moreover, the canalicular space is partially occupied by pericellular fibers
generating a sub-microscopic friction effect and thus slowing down the fluid
movement [ 85 , 166 ]. To take into account this effect occurring in the fluid
volume, a viscous force F b ¼ð l f = k f Þ v depending on the fluid velocity v is
introduced. Here, k f represents the isotropic pericellular fibers permeability.
Note that for few fibers, the pore space is very permeable (i.e. k f is high) and
F b vanishes. In parallel, noting E f the electric field in the fluid, the electric
body force F e ¼ q f E f has to be taken into account. Thus the equilibrium
equation for the fluid reads:

r S f þ F b þ F e ¼ 0 : ð 10 Þ

In addition, assuming the electrolyte to be incompressible, the mass conser-
vation equation for the fluid reads:

r v ¼ 0 : ð 11 Þ

• Ionic transport Regarding the ionic transport, the total cationic and anionic flux
vectors are the sum of the convection flux vector, the diffusion flux vector and
the electric current vector. Thus, the Nernst-Planck convection-diffusion-elec-
tromigration equations that govern the ionic transport are [ 76 ]:

;

o n

ot þrð n v Þ¼r D ðr n n r / f Þ

ð 12 Þ

where t is the time, D are the water-ions diffusion tensors for cations and
anions, respectively. Using the Boltzmann distributions of the ionic species of
Eq. ( 6 ) and the electric potential decomposition of Eq. ( 7 ), this equation
becomes:

o ð n b exp ð u Þ
ot

þrð n b exp ð u Þ v Þ¼r D ð exp ð u Þðr n b n b r W b ÞÞ ½ ;

ð 13 Þ

In this equation, the terms in the divergence corresponds to the electro-diffusive
ionic flux vectors J ¼ D ð exp ð u Þðr n b n b r W b ÞÞ: