Biomedical Engineering Reference
In-Depth Information
Re ½ o t u C þð u C u C ¼r p C þr 2 u C ; r u C ¼ 0 ;
ð 7 Þ
where Re ¼ qdU = l is the Reynolds number based on the microscopic length scale.
Clearly, the interstitial fluid flow equation ( 4 ) remains unchanged by the pressure
scaling. The dimensional boundary condition ( 2 ) becomes the following dimen-
sionless boundary condition
n u C ¼ R ð p I p C Þ
on the boundary ;
ð 8 Þ
where R ¼ al = d.
The microscale Reynolds number Re ¼ qdU = l is based on the microscopic
(unit) length scale d. However, we expect the dominant fluid pressure variation
within lymphatic capillaries to occur over the macroscale, and therefore it makes
sense to rescale pressures with 1 = e, where e ¼ d = L is the ratio of the microscale
(i.e. the unit cell scale) d to macroscale (i.e. tissue scale) L. After such rescaling
the dimensionless Navier-Stokes equations become
eRe ½ o t u C þð u C u C ¼r p C þ e r 2 u C ; r u C ¼ 0 ;
ð 9 Þ
with the boundary condition on the capillary surface oX C
given by
n u C ¼ R ð p I p C Þ
on the boundary ;
ð 10 Þ
R ¼ R = e ¼ alL = d 2
where
is
the
dimensionless
modified
lymphatic
capillary
permeability.
The dimensionless interstitial equations ( 4 ) and associated boundary condition
( 5 ) are
r 2 p I ¼ 0 ;
ð 11 Þ
n r p I ¼ W ð p I p C Þ
on the boundary ;
ð 12 Þ
where W ¼ ald = k.
3.1.3 Parameter Values
There is a high level of uncertainty about some of the parameters, such as lym-
phatic wall permeability a, but the other parameters such as interstitial perme-
ability and geometry are reasonably well known. We present estimates for all the
parameters required for the model in Table 1 .
Based on the values in Table 1 the values of the dimensionless parameters are
as follows: e 10 2 , Re 10 4 to 10 3 , R 2 10 6 and W 0 : 5 ð 1to10 3 Þ .
Thus, since e and Re are always small we can always neglect the inertial terms in
the Navier-Stokes equations and use Stokes equations. Evidently also, the other
parameters are comparable to e, i.e., we can rewrite them as R ¼ Re 3 and W ¼ We b
Search WWH ::




Custom Search