Mass Transport and Heat Transfer in the Microcirculation - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 228

where L actual is the minimum actual distance between two points that are connected via
pores and L is the straight-line distance between the two points.
Fluid flow through porous media is most typically quantified via Darcy's law, which
relates the pressure gradient (as the driving force for fluid flow) to the flow rate of fluid
through the porous media. Darcy's law is an idealized case that is not applicable for non-
Newtonian fluids and fluids when there is a large momentum transfer between the fluid
and the solid particles. Under these conditions, Darcy's law states that the fluid velocity
ðvÞ

will be defined by

- r -

K ð 7 : 24 Þ

52

where K is the hydraulic conductivity and p is the driving pressure. If the porous media is
non-isotropic, then the hydraulic conductivity is not constant and must be included within
the gradient operator. Also, if blood flow and lymph flow is accounted for in Darcy's law,
we get

- blood 2 - lymph 52 rð

K -

Þ ð : Þ

7 25

Gravitational forces can be included within Equation 7.25 if necessary. The hydraulic
conductivity is related to the viscosity of the fluid that is moving through the porous
media. The hydraulic conductivity can be defined as

3

cε

K ð 7 : 26 Þ

5

V 2

SA

μ

where c is the viscosity, SA is the interface surface

area of the pores, and V is the total volume of the porous media. The flow through porous
media can also be described, in a more general case, by the Brinkman equation:

is a shape factor, ε is the porosity, μ

1
κ

- -

2 r -

2

μr 0 ð 7 : 27 Þ

2 5

7.6 MICROCIRCULATORY HEAT TRANSFER

Heat transfer is an important principle in biological systems. As discussed in an earlier
chapter, one of the most important functions of the cardiovascular system is to maintain
the temperature of the body. Also, air entering your lungs must be warmed (or cooled) to
body temperature. This is accomplished through all blood vessels, but we will focus on
the thinnest walled vessels to make our analysis easier. There are three basic mechanisms
for heat transfer to occur. They are conduction, which is the transfer of energy between
adjacent non-moving particles; convection, which is the transfer of energy between a solid
surface and flowing fluid; and radiation, which is the transfer of energy through electro-
magnetic waves (e.g., photons). Each of these heat-transfer mechanisms can be sub-
divided further into special cases of heat transfer. We will focus here on internal forced
convection because this is the most applicable heat-transfer modality within the