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MIE508_20189_631575505914MIE508F_2018_FLUIDSOFBIOLOGICALSYSTEMS_E.pdf-M1E508 - Fluids of Biological Systems
Page 8
For
V,,
=
0, what type of flow does this problem become? Derive the velocity
profile u
=
u(y)
for
V
=
0.
(8 marks)
For
V
> 0, use the Navier-Stokes equations to
derive the appropriate ordinary
differential equation (ODE), and write the boundary conditions (BCs) needed
to solve u
=
u(y). (8 marks)
(Don't try to solve the ODE! Just show how you arrive at the ODE and the boundary
conditions.)
Note: For your interest, the solution to the ODE is:
u(y)
-
2
I
y
-
1
e
-
em1
Umax
-
Re
h
+
sinh Re
]
(ö]
Page 9
AID SHEET
Governing Equations
Navier-Stokes Equations, Cartesian coordinates:
Coordinates
F =
(x,y,z)
Velocity
=
(u,v,w)
(Note: for constant density and constant viscosity fluids)
Du Dv 3w
—+—+—
ax
ay
Dz
Iwo
Du
Du
DuI
p
[Du
at
x
ay
Dzj
Dv
Dv
Dy
1
p
[Dv
at
x
ay
Dzj
Dw
3w
3w1
p
[aw
x
Dy
Dzj
Dp
D
2
u
D
2
u
D
2
u
=
--+
ax
Dx
+—+--
+f
Dy
Dz
Dp
32v
(92V
D2v
=
—--
+i
+f
Dp
D
2
w
9
2
w
D
2
w
=
--
+i
Dz
Dx
—+----+--
+f
Dy
Dz
Navier-Stokes Equations, cylindrical coordinates:
Coordinates
i?
=
(r, 0, z)
Velocity
'7
=
(U
r
, u
9
, u)
(Note: for constant density and constant viscosity fluids)
1
D
1Du
0
Du
--(ru
r
)+—+
=
rDr
r 9 Dz
au,
au, u
9
Du
r
u
Du
r
=
at
Dr
r DO
r
Dz
Du
9
Du
9
u
9
Du
9
u
r
ue
N
O
Dr rDO r Dz
Du
Du u
0
Du
Du
=
at Dr rDO Dz
1DP
1V
_u
r
2Du
0
'
---
+1-i•
Ur
j+fr
pDr
r
rpDO
llDP
_uo 2Du
r
-----
+1'
U8
r2
r2
ao
1Dp
2
+ vV
u
2
+ f
p Dz
where
V
2
1D(D\ 1D
2
32
Reynold's Transport Theorem:
Drn
JVW
ap
Dt
DtS
Page 10
Stress Tensor
The shear stress tensor for a Newtonian fluid of constant density is defined as follows:
i'a
ni
=
(
+
\0Xi
3x
Trigonometric ]Identities
sin(A +
B)
=
sin
A
cos
B
+
cos
A
sin
B
cos (A +
B)
=
cos A cos B sin A sin B
sin
2
X + c0s
2
x
=
1
Calculus
Chain Rule:
f(g(x))
=
f(g(x))g'(x)
dx
Product Rule:
xx
=
f'xgx + f(xg'(x)
dx
Quotient Rule:
d (f(x))
f'(x)g(x)
-
f(x)g(x)
dx g(x)
-
(g(x))2
Porous Media Flows
Kozeny-Carman relationship for permeability:
E3
K
2n2S2
where e is the porosity, 'r is the tortuosity, and S is the surface area-to-volume ratio.
Flow rate through a single cylindrical straight pore:
-
Qpore
nr
4
/p
8p
L
Darcy's Law:
Q
KLp
AJ1L
10
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