Biomedical Engineering Reference
In-Depth Information
¥
§
¦
u
U
µ
·
¶
u
T
u
u
U
Du
Dt
1
S
yx
xx
zx
F
(5.34)
x
u
u
x
y
z
However, pressure term is included in
σ
xx
. Substituting relational
expression between stress tensor (5.25), (5.26), (5.27) and
deformation rate (5.28), (5.29), (5.30) into the equation above, we
obtain the following. These are Navier-Stokes equations.
¥
2
2
2
µ
Du
1
u
p
1
N
u2
N
u u u
u
u
u
F
-
·
(5.35)
¦
¶
x
2
2
2
Dt
S
u
x
3
S
u
x
S
§
u u u
x
y
z
¥
2
2
2
µ
Dv
1
u
p
1
N
u2
N
u u u
v
v
v
F
-
·
(5.36)
¦
¶
y
2
2
2
Dt
S
u
y
3
S
u
y
S
§
u u u
x
y
z
¥
2
2
2
µ
w
1
u
p
1
N
u2u
u u
N
w
ww
F
-
·
(5.37)
¦
¶
z
2
2
2
Dt
S
u
z
3
S
u
z
S
§
u u u
x
y
z
Or else,
N
N
D
v
1
1
2
2
F
-
grad p
grad
v
(5.38)
S
S
S
Dt
3
Here,
D
Dt
u
u
u
u
u
u
u
u
u
u
u
v
w
v
grad
(5.39)
t
x
y
zt
¥
§
¦
2
2
2
µ
·
¶
u
u
u
u
u
u
2
(5.40)
2
2
2
x
y
z
¥
§
¦
µ
·
¶
u
u a
u
u a
u
u
grad
(5.41)
xyz
2
u
u
u
u
x
u
u
v
y
w
z
(5.42)
u
v
=
μ
/
ρ
is kinematic viscosity. In the case of incompressible luid,
Navier-Stokes equations are rewritten as follows:
¥
2
2
2
µ
Du
1
u
p
N
u u u
u
u
u
F
-
·
(5.43)
¦
¶
x
2
2
2
Dt
S
u
x
S
§
u u u
x
y
z
¥
2
2
2
µ
Dv
1
u
p
N
u u u
v
v
v
F
-
·
(5.44)
¦
¶
y
2
2
2
Dt
S
u
y
S
§
u u u
x
y
z
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