Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 155

with boundary conditions

n u C ¼ e 3 R ð p I þ p C Þ n ðr x þ e r y Þ p I ¼ e 2 W ð p I p C Þ

and

ð 15 Þ

on the boundary :

We use the perturbation expansions given by

u C ¼ u 0 ð x ; y Þþ eu 1 ð x ; y Þþ e 2 u 2 ð x ; y Þþ; ð 16 Þ

p C ¼ p 0 ð x ; y Þþ ep 1 ð x ; y Þþ e 2 p 2 ð x ; y Þþ;

ð 17 Þ

p I ¼ p I 0 ð x ; y Þþ ep I 1 ð x ; y Þþ e 2 p I 2 ð x ; y Þþ: ð 18 Þ

Substituting Eqs. ( 16 )-( 18 ) into Eqs. ( 13 )-( 15 ) we find the first order, i.e.,
O ð e 0 Þ , capillary flow equations to be

r x p 0 ¼ 0 ; r x u 0 ¼ 0 ; ð 19 Þ

with n u 0 ¼ 0 on the boundary. This equation essentially says that p 0 ¼ p 0 ð y Þ ,
i.e., at the leading order the pressure depends only on the macroscopic space scale
y and has no rapid local variations.
The first order interstitial flow equation is

r x p I 0 ¼ 0 ;

ð 20 Þ

with n r x p I 0 ¼ 0 on the boundary and thus similarly to the capillary flow problem
p I 0 ¼ p I 0 ð y Þ , i.e., the interstitial pressure is also only varying on the macroscale.
The O ð e Þ capillary flow equation together with O ð 1 Þ continuity equation gives
us two equations for u 0 and p 1 , i.e.,

r x p 1 r y p 0 þr x u 0 ¼ 0 ; r x u 0 ¼ 0 ;

ð 21 Þ

with n u 0 ¼ 0 on the boundary. The equations suggest that we can look for
separable solutions, i.e.,

u 0 ð x ; y Þ¼ n j ð x Þ o p 0
oy j

p 1 ¼ p j ð x Þ o p 0
oy j

; ; ð 22 Þ

where n j ð x Þ and p j ð x Þ are called local corrector functions that only depend on the
microscale variable x. The local corrector functions are defined/determined by the
specific local solution that depends on the specific microstructure, i.e.,

n j ¼ 0 ; r x p j ¼r x n j þ e j ; r x ð 23 Þ

with n j ¼ 0 on the internal microstructure surface, and all variables periodic on the
external (on the unit square/rectangle) surfaces; e j is the unit vector in the j
coordinate direction. For any specific geometric situation one would in general