Biomedical Engineering Reference
In-Depth Information
@r
@
p
_
p
þ rð
p
Þðr
v
Þ¼
0
:
(6.38)
This system of equations will become more complicated if thermal effects are
considered. They may also be simplified in different ways. An easy simplification is
to assume that the viscosity of the fluid is small and can be neglected (that is to say
the fluid is assumed to be inviscid), then
l
and
m
are set equal to zero, (
6.37
) becomes
r _
v
¼r
p
þ r
d
;
(6.39)
and (
6.38
) is unchanged. The system of equations (
6.37
) and (
6.38
) may also be
simplified using the assumption of incompressibility of the fluid. An incompressible
material is one which is not permitted to have changes in its volume, tr
D
¼r
v
¼
0. If the volume cannot change, the density
r
of the fluid cannot change. It follows
that the barotropic relationship
(
p
) is not appropriate and the pressure is no
longer determined by the density. The pressure field p in an incompressible material
is a Lagrange multiplier (see Example 6.4.1) that serves the function of maintaining
the incompressibility constraint,
r ¼ r
r
v
¼
0. Because the volume of the fluid cannot
change, p does no work on the fluid; it is a function of
x
and
t
,
p
(
x
,
t
), to be
determined by the solution of the system of differential equations and boundary/
initial conditions. The reduced Navier-Stokes equation for viscous fluid flow and
incompressibility constraint now becomes a system of four equations
2
v
r v
¼r
p
þ mr
þ r
d
;r
v
¼
0
(6.40)
for the four unknown fields, the three components of
v
(
x
,
t
) and
p
(
x
,
t
). The typical
boundary condition applied in viscous fluid theory is the “no slip” condition. This
condition requires that a viscous fluid at a solid surface must stick to the surface and
have no velocity,
v
(
x
*,
t
)
¼
0 for
x
*
2
∂
O
s
, where
∂
O
s
stands for the solid
boundary of the fluid domain.
Example 6.4.1 Pressure as a Lagrange Multiplier in Incompressible Fluids
The constraint of incompressibility is imposed using a Lagrange multiplier. In order
to describe what this means and how it is accomplished, Lagrange's method of
calculating extrema in problems in which there is a constraint summarized briefly.
Lagrange's method is for the solution of a type of problem in which one must find the
extremal values of a function
f
(
x
,
y
,
z
) subject to the constraint
g
(
x
,
y
,
z
)
¼
c
. The
solution to the problem is obtained by forming the function
q
(
x, y, z
)
¼
f
(
x, y, z
)
þ
L
is a constant, called the Lagrange multiplier, whose value is to be
determined. Treating
x
,
y
, and
z
as independent variables four independent
conditions
g
(
x, y, z
) where
L
q
/
x
¼
0,
q
/
y
¼
0,
q
/
z
¼
0, and
g
(
x, y, z
)
¼
c
are available to
∂
∂
∂
∂
∂
∂
find the four unknowns,
x, y, z
, and
. As an example of the application of
Lagrange's method, consider the problem of finding the maximum or minimum
distance from the point (
a, b, c
) to a point on the surface of the unit sphere,
x
2
L
y
2
þ
z
2
þ
¼
1. The function that is to be extremized is the square of the distance between
Search WWH ::
Custom Search