Biomedical Engineering Reference
In-Depth Information
represent each of the three major types of second-order partial differential
equations—the elliptic, the parabolic, and the hyperbolic. These three types of
differential equations are characteristic of three different types of physical
situations. Elliptic partial differential equations typically occur in equilibrium or
steady-state situations where time does not enter the problem. Laplace's equation,
2
f
2
f
2
f
Þ¼
@
x
2
þ
@
y
2
þ
@
2
f
r
ð
x
;
y
;
z
z
2
¼
0
;
(6.1)
@
@
@
2
f
(
x
,
y
,
z
)
and Poisson's equation,
g
(
x
,
y
,
z
), are typical elliptic partial differen-
tial equations. Parabolic partial differential equations,
r
¼
Þ¼
@
f
2
f
r
ð
x
;
y
;
z
t
;
(6.2)
@
typically occur in diffusion problems, such as thermal diffusion in a heat
conducting material or fluid pressure diffusion in a rigid porous medium as will
be seen in the following section. Hyperbolic partial differential equations,
2
f
Þ¼
@
2
f
r
ð
x
;
y
;
z
t
2
;
(6.3)
@
often characterize dynamic situations with propagating waves, and are called the
wave equations
.A
boundary value problem
is the problem of finding a solution to a
differential equation or to a set of differential equations subject to certain specified
boundary and/or initial conditions. The theories developed in this chapter all lead to
boundary value problems. Thus, in this chapter, conservation principles, constitu-
tive equations, and some kinematics relations from the previous chapters are
employed to formulate continuum theories that lead to physically motivated and
properly formulated boundary value problems.
6.2 The Theory of Fluid Flow Through Rigid Porous Media
The theory of fluid flow through rigid porous media reduces to a typical diffusion
problem; the diffusion of the pore fluid pressure through the porous medium. The
solid component of the continuum is assumed to be porous, rigid, and stationary,
thus the strain and rate of deformation are both zero. The pores in the solid cannot
be closed,—most of them must be open and connected; so it is possible for the pore
fluid to flow about the medium. The assumption of an immobile rigid porous
continuum is not necessary as the constitutive equation for the porous medium
may be combined with the equations of elasticity to form a theory for poroelastic
r
f
denotes the density of the fluid in the pores of the
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