Biomedical Engineering Reference
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into the primary lymphatics and this relationship is usually a function of the
pressure difference. Thus, as a lymphatic capillary surface oX boundary condition
we take
n u C ¼ a ð p I p C Þ
on the boundary ;
ð 2 Þ
where p I is the interstitial fluid pressure, n is the unit inward pointing normal to the
capillary surface and a is the lymphatic capillary permeability, i.e., we include the
primary lymphatic valve function via this effective capillary lumen permeability
parameter. We also supplement this with a no-slip boundary condition on the
lumen surface.
To describe the fluid flow in the interstitial space we use Darcy's law, i.e. we
will assume that the fluid flux in the interstitium is related to the intersititial
pressure gradient, i.e.,
u I ¼ k
l r p I ;
ð 3 Þ
where k is the hydraulic permeability (cm 2 ) and l is the intersititial fluid viscosity.
We will consider the viscosity of the interstitial fluid and lymphatic fluid to be the
same. Combining ( 3 ) with the fluid conservation equation r u I ¼ 0 we get
r 2 p I ¼ 0 ;
ð 4 Þ
when k = l is constant.
Clearly, on the lymphatic capillary lumen boundary the fluid flux out of the
interstitium should be the same as fluid flux into the interstitium, i.e. on oX C
we
have (looking from the interstitial side)
¼ a ð p I p C Þ:
n u C ¼ n u I ¼ n k
l r p I
ð 5 Þ
3.1.2 Dimensionless Equations
Our next task in the homogenisation procedure is to non-dimensionalise the Na-
vier-Stokes and Darcy equations ( 1 ) and ( 4 ) with suitable scales. We begin by
defining the standard Navier-Stokes scalings, i.e.,
p C P ;
p I P ;
u C U ;
x d ;
t d = U
and
ð 6 Þ
where d is the length scale of the typical single lymphatic structure (i.e. the so
called unit cell scale), U is the typical fluid flow velocity in the lymphatic cap-
illaries, and the classical pressure scale P is given by P ¼ lU = d. Thus, the
dimensionless Navier-Stokes equations are
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