3.5 Balance Laws
119
including the effects of fluid viscosity as well as any viscoelasticity or other
structural damping in the spring. (Friction between the mass and box also could be
included.) Lumped parameter models can be useful in studies of global behavior. For
example, a Windkessel model consisting of compliance and resistance components
often is used to relate blood pressure and flow in the vascular system (Fung 1990).
However, detailed analysis of the stress distribution in the blood or the wall of
a blood vessel requires a continuum analysis. This is an important issue because
abnormally high stresses can damage blood cells or the vessel wall. Moreover, blood
vessels grow and remodel in response to stress (Taber 1995; Humphrey 2002).
Energy Balance in 1D
To specialize Eq. (3.163) for a 1D continuum, consider the differential element
shown in Fig. 3.21. In the current configuration, the length and cross-sectional area
are dx and dA, respectively. In addition to the stress σ (x, t) and body force b(x, t),
the schematic is modified to include the velocity v(x, t) and two sources of heat,
q(x, t) and r(x, t). Here, q represents the rate at which heat flows across the surfaces
at the ends of the element, and r is the heat production rate from internal sources.
The heat flux q is defined per unit deformed surface area (like stress), while r is
defined per unit deformed volume (like body force). In general, the stress, velocity,
and heat flux differ slightly between the ends of the element (Fig. 3.21).
The kinetic energy for the element is
K =
1
2 ρv
2 dA dx,
(3.168)
where ρ dAdx is the mass and v is the velocity of the center of mass. For dx → 0,
v is defined as the average velocity
1
2 [v + v + (∂v/∂x)dx] → v. The internal energy
is given by
U = ρu dAdx,
(3.169)
in which u is the internal energy per unit mass. In this section, note that the symbol
u does not represent displacement.
Fig. 3.21 Differential
element for energy balance in
1D
σ +
∂σ
∂x
dx
b
dx
σ
dA
v
v +
∂v
∂x
dx
q +
∂q
∂x
dx
q
r
119
including the effects of fluid viscosity as well as any viscoelasticity or other
structural damping in the spring. (Friction between the mass and box also could be
included.) Lumped parameter models can be useful in studies of global behavior. For
example, a Windkessel model consisting of compliance and resistance components
often is used to relate blood pressure and flow in the vascular system (Fung 1990).
However, detailed analysis of the stress distribution in the blood or the wall of
a blood vessel requires a continuum analysis. This is an important issue because
abnormally high stresses can damage blood cells or the vessel wall. Moreover, blood
vessels grow and remodel in response to stress (Taber 1995; Humphrey 2002).
Energy Balance in 1D
To specialize Eq. (3.163) for a 1D continuum, consider the differential element
shown in Fig. 3.21. In the current configuration, the length and cross-sectional area
are dx and dA, respectively. In addition to the stress σ (x, t) and body force b(x, t),
the schematic is modified to include the velocity v(x, t) and two sources of heat,
q(x, t) and r(x, t). Here, q represents the rate at which heat flows across the surfaces
at the ends of the element, and r is the heat production rate from internal sources.
The heat flux q is defined per unit deformed surface area (like stress), while r is
defined per unit deformed volume (like body force). In general, the stress, velocity,
and heat flux differ slightly between the ends of the element (Fig. 3.21).
The kinetic energy for the element is
K =
1
2 ρv
2 dA dx,
(3.168)
where ρ dAdx is the mass and v is the velocity of the center of mass. For dx → 0,
v is defined as the average velocity
1
2 [v + v + (∂v/∂x)dx] → v. The internal energy
is given by
U = ρu dAdx,
(3.169)
in which u is the internal energy per unit mass. In this section, note that the symbol
u does not represent displacement.
Fig. 3.21 Differential
element for energy balance in
1D
σ +
∂σ
∂x
dx
b
dx
σ
dA
v
v +
∂v
∂x
dx
q +
∂q
∂x
dx
q
r