Transfer Masa Panas

1. CONSERVATION LAW
 1
X
A  2
x
A
    V
A
A
t
V
G
x
x 








 2
1
 
G
x
V
A
t










c
x
m
x
x ,
, 

 

x
m
x




 
 ,
x
c
x V


,

 
 
G
c
x
m
x
V
A
t











 ,
,
   
G
x
m
x
V
V
A
V
A
t













 ,      
G
x
m
x
x
V
AV
V
A
V
A
V
t

















 ,
   
G
x
x
V
AV
V
x
A
V
A
V
t


































  G
t

 






.
      G
m V
V
t

 












.
.
.
      G
V
V
t

 













.
.
.
3-D PROPERTY
CONSERVATION EQUATION
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)
Property
Accumulation
Rate
=
Inlet Property
Rate -
Outlet Property
Rate +
Property
Generation
Rate

2. HEAT TRANSPORT
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)
General heat transport phenomena is expressed as equation of change. Equation
of change concerning conduction and convection heat transport is called Energy
Equation basically as microscopic energy conservation equation.
      G
m V
V
t

 












.
.
.
=CpT, m
 T
k
q 



    
     G
T
V
CpT
CpT
CpT
V
t
CpT 
















.
.
. 




 
CpT



 G
G T

 

Cp
k

  = thermal diffusivity

3. HEAT TRANSPORT
       
v
v
p
q
v
U
t
U











:
.
.
ˆ
.
ˆ



     
v
v
T
p
T
q
T
v
t
T
v
C
V































:
.
.
.
ˆ
ˆ


      Dt
Dp
v
q
v
H
t
H










:
.
ˆ
.
ˆ



   
v
Dt
Dp
T
q
T
v
t
T
p
C
p
























:
ln
ln
.
.
ˆ 


OTHER FORMS OF ENERGY EQUATION
Expressed in 𝑈
:
Expressed in 𝐶𝑣 and T
Expressed in 𝐻
Expressed in 𝐶𝑝 and T
  V
V 


 
 : V
 Tabel $B.7
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)
V
T
k
Dt
DT
p
C 


 
 2
ˆ

4. HEAT TRANSPORT
2B
x
y
z
L
W
EXAMPLE -2
Determine temprature
distribution in a viscous liquid
flowing downward in steady
state and laminar between two
vertical paralell plate. The two
plates are maintained at
constant temprature of 𝑇0.
Assume constant  and k.
Solution:
First, we draw sketsc of flow
system,
Assumptions:
1.Steady state,
2.Laminar flow,
3.Vx=Vy=0, VZ is not function of y
4. Newtonian Fluid,
5.Constant , , k
6.
𝜕𝑇
𝜕𝑦
=
𝜕𝑇
𝜕𝑧
= 0
7. No slip at wall
8. End effect is neglected
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)

5. HEAT TRANSPORT
0












z
V
y
V
x
V
t
z
y
x 



0



z
Vz

0



z
Vz
z
z
z
z
z
z
z
y
x
x
z
g
z
V
y
V
x
V
z
P
z
V
V
y
V
V
x
V
V
t
V


 








































2
2
2
2
2
2
z
z
g
dx
V
d
z
P

 




 2
2
0
L
dx
V
d L
z 0
2
2




 z
g
P z




1
0
K
x
L
dx
dV L
z






0
0 


dx
dV
x z
x
L
dx
dV L
z

0




2
2
0
2
K
x
L
V L
z 





0


 z
V
B
x
2
0
2
2
B
L
K L






SOLUTION
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)
K1=0

6. HEAT TRANSPORT


















2
2
1
2 B
x
L
B
Vz
 















2
max
, 1
B
x
Vz
L
B
Vz

2
2
max
,



2
max
.
2
B
x
V
dx
dV
z
z







































2
2
2
2
2
2
z
T
y
T
x
T
k
z
T
V
y
T
V
x
T
V
t
T
C z
y
x
P





















































































2
2
2
2
2
2
2
y
V
z
V
x
V
z
V
x
V
y
V
z
V
y
V
x
V z
y
z
x
y
x
z
y
x


0
2
2
2








dx
dV
dx
T
d
k z
 0
4
. 4
2
2
max
,
2
2








B
x
V
dx
T
d
k z

3
3
4
2
max
,
3
4
K
x
kB
V
dx
dT z




4
3
4
4
2
max
,
3
K
x
K
x
kB
V
T z





ENERGY EQUATION
SOLUTION
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)
From table A-5

7. HEAT TRANSPORT
4
3
4
4
2
max
,
3
K
x
K
x
kB
V
T z





0
0 


dx
dT
x
4
4
4
2
max
,
3
K
x
kB
V
T z




0
T
T
B
x 


B.C-1:
B.C.-2:
0
3 
K
k
V
T
K z
3
2
max
,
0
4




















4
2
max
,
0 1
3 B
x
k
V
T
T z

 



















4
2
4
0 1
12 B
x
k
L
B
T
T

SOLUTION
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)

8. HEAT TRANSPORT
Example 3
Inlet air at T1
Cooler
A system consist of two porous concentric spherical walls with radius R1 and R2.
Outer surfase of inner wall is maintained at temprature T1 and inner surface of
outer wall is maintained at tempratrure T2. Dry air at temprature T1 is blown
radially from inner wall to outer wall. Develop an expression for heat removal
rate needed from inner wall as a function of air mass flow rate. Assume steady
state laminar flow and low gas rate.
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)

9. HEAT TRANSPORT
Solution:
Assumption:
0
.
1 
 
 V
V )
(r
f
Vr 
2. T=T(r)
)
(
.
3 r



  0
1 2
2

r
V
r
dr
d
r
 
r
V
r 
2

4
r
w

constant
Continuity Equation:
  














 r
r
r V
r
dr
d
r
dr
d
dr
d
dr
dV
V 2
2
1


Equation of Motion
dr
d
dr
r
w
d
r
w
r
r 








2
2
4
4


dr
d
r
wr 



5
2
2
8 
  
 

 



2
2
5
2
2
8
R
r
r
R
r
r
dr
w
d


    












 4
4
2
2
2
2
1
1
32 r
R
w
r
R r


   



















4
2
4
2
2
2
2 1
32 r
R
R
w
R
r r


MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)
r
2
r
2
2
2
r
2
2
r
2
2
2
2
2
θ
r
r
θ
r
r
r
ρg
v
θ
sin
r
1
θ
v
sinθ
θ
sinθ
r
1
r
)
v
(r
r
1
μ
r
r
v
v
v
θ
sin
r
v
θ
v
r
v
r
v
v
t
v
ρ



































 
















p
𝜕
𝜕𝑟
1
𝑟2
𝜕
𝜕𝑟
𝑟2𝑣𝑟

10. HEAT TRANSPORT







dr
dT
r
dr
d
r
k
dr
dT
V
C r
p
2
2
1
ˆ
 






dr
dT
r
dr
d
C
w
k
dr
dT
p
r
2
ˆ
4
𝑢 = 𝑟2
𝑑𝑇
𝑑𝑟
= 𝐾1𝑒−
𝑅0
𝑟
2
0
1
0
2
0
0
2
1
2
R
R
R
R
R
R
r
R
e
e
e
e
T
T
T
T
/
/
/
/








 k
C
w
R P
r 
4
0 /
ˆ

1
2
1
4
R
r
r
q
R
Q


 
1
2
1
4
R
r
dr
dT
k
R

 
 
  
  1
1
4
2
1
1
0
1
2
0




R
R
R
R
T
T
k
R
Q
/
/
exp
  
2
1
1
2
1
0
1
4
R
R
T
T
k
R
Q
/




1
1
0
0






e
Q
Q
Q    
k
R
R
R
C
w
R
R
R
R P
r
1
2
1
1
1
0
4
1
1


/
ˆ
/ 



Energy Equation:
Substitution:
Heat transafer rate to inner wall is
Cooling requirement at inner wall,
If there is no air flow
MOLECULER AND CONVECTION HEAT TRANSPORT
PROCESS (APPLICATION OF EQUATION OF CHANGE)
𝑢
𝑟2
=
1
𝑅0
𝑑
𝑑𝑟
𝑢 →
𝑑𝑟
𝑟2
=
1
𝑅0
𝑑𝑢
𝑢
−
𝑅0
𝑟
+ ln 𝐾1 = ln 𝑢 → 𝑢 = 𝐾1𝑒−
𝑅0
𝑟
𝑑𝑇 = 𝐾1
𝑒−
𝑅0
𝑟
𝑟2 𝑑𝑟
𝑠 =
𝑅0
𝑟
→ 𝑑𝑠 = −
𝑅0
𝑟2 𝑑𝑟 𝑑𝑇 = 𝑇 = 𝐾1
𝑒−
𝑅0
𝑟
𝑟2 𝑑𝑟 = −
𝐾1
𝑅0
𝑒−𝑠
𝑑𝑠 =
𝐾1
𝑅0
𝑒−
𝑅0
𝑟 + 𝐾2
𝑢 = 𝑟2
𝑑𝑇
𝑑𝑟

11. EQUATION OF
CHANGE TABLE

12.      
0
z
ρv
y
ρv
x
ρv
t
ρ z
y
x












     
0
z
v
ρ
θ
v
ρ
r
1
r
v
r
ρ
r
1
t
ρ z
θ
r












     
0
v
ρ
rsinθ
1
θ
sinθ
v
ρ
rsinθ
1
r
v
r
ρ
r
1
t
ρ θ
r
2
2














Cartesian Coordinate System:
Cylindrical Coordinate System:
Spherical Coordinate System
Table A-1 Continuity Equation

13. -Table A2Equation of Motion (Equation of Momentum)
x
zx
yx
xx
x
z
x
y
x
z
x
ρg
z
τ
y
τ
x
τ
x
z
v
v
y
v
v
x
v
v
t
v
ρ 







































 p
y
zy
yy
xy
y
z
y
y
y
z
y
ρg
z
τ
y
τ
x
τ
y
z
v
v
y
v
v
x
v
v
t
v
ρ 







































 p
z
zz
yz
xz
z
z
z
y
z
z
z
ρg
z
τ
y
τ
x
τ
z
z
v
v
y
v
v
x
v
v
t
v
ρ 







































 p
Cartesian Coordinate System:

14.  
r
rz
θθ
rθ
rr
r
z
2
θ
r
θ
r
r
r
ρg
z
τ
r
τ
θ
τ
r
1
r
rτ
r
1
r
z
v
v
r
v
θ
v
r
v
r
v
v
t
v
ρ








































 p
 
θ
zθ
θθ
rθ
2
2
θ
z
θ
r
θ
θ
θ
r
θ
ρg
z
τ
θ
τ
r
1
r
τ
r
r
1
θ
r
1
z
v
v
r
v
v
θ
v
r
v
r
v
v
t
v
ρ







































 p
 
z
zz
θz
rz
z
z
z
θ
z
r
z
ρg
z
τ
θ
τ
r
1
x
rτ
r
1
z
z
v
v
θ
v
r
v
r
v
v
t
v
ρ 



































 p
Cylindrical Coordinate System
-Table A2Equation of Motion (Equation of Momentum)

15. Spherical Coordinate System
 
r
r
θθ
rθ
rr
2
2
2
2
θ
r
r
θ
r
r
r
ρg
τ
rsinθ
1
r
τ
τ
θ
sinθ
τ
rsinθ
1
r
τ
r
r
1
r
r
v
v
v
rsinθ
v
θ
v
r
v
r
v
v
t
v
ρ































 

















 p
 
θ
rθ
θ
θθ
rθ
2
2
2
θ
r
θ
θ
θ
θ
r
θ
ρg
τ
r
cotθ
r
τ
τ
rsinθ
1
θ
sinθ
τ
rsinθ
1
r
τ
r
r
1
θ
r
1
r
cotθ
v
r
v
v
v
rsinθ
v
θ
v
r
v
r
v
v
t
v
ρ



















































p
 














ρg
τ
r
cotθ
2
r
τ
τ
rsinθ
1
θ
τ
r
1
r
τ
r
r
1
rsinθ
1
cotθ
r
v
v
r
v
v
v
rsinθ
v
θ
v
r
v
r
v
v
t
v
ρ
θ
r
θ
r
2
2
θ
r
θ
φ
r
φ












































 p
-Table A2Equation of Motion (Equation of Momentum)

16. x
2
x
2
x
2
2
x
2
x
z
x
y
x
z
x
ρg
z
v
y
v
x
v
μ
x
z
v
v
y
v
v
x
v
v
t
v
ρ 







































 p
y
2
y
2
2
y
2
2
y
2
y
z
y
y
y
z
y
ρg
z
v
y
v
x
v
μ
y
z
v
v
y
v
v
x
v
v
t
v
ρ 







































 p
z
2
z
2
2
z
2
2
z
2
z
z
z
y
z
z
z
ρg
z
v
y
v
x
v
μ
z
z
v
v
y
v
v
x
v
v
t
v
ρ 







































 p
-Table A2Equation of Motion (Equation of Momentum)
(Incompressible Newtonian Fluid)
Cartesian Coordinate System:

17. Table A-3 Equation of Motion (Equation of Momentum)
(Incompresiible Newtonian Fluid)
Cylindrical Coordinate:
  r
2
r
2
θ
2
2
r
2
2
r
r
z
2
θ
r
θ
r
r
r
ρg
z
v
θ
v
r
2
θ
v
r
1
rv
r
r
1
r
μ
r
z
v
v
r
v
θ
v
r
v
r
v
v
t
v
ρ


















































 p
  θ
2
θ
2
r
2
2
θ
2
2
θ
θ
z
θ
r
θ
θ
θ
r
θ
ρg
z
v
θ
v
r
2
θ
v
r
1
rv
r
r
1
r
μ
θ
r
1
z
v
v
r
v
v
θ
v
r
v
r
v
v
t
v
ρ
















































 p
x
2
z
2
2
z
2
2
z
z
z
z
θ
z
r
z
ρg
z
v
θ
v
r
1
r
v
r
r
r
1
μ
z
z
v
v
θ
v
r
v
r
v
v
t
v
ρ 











































 p

18. Table A-3 Equation of Motion ( Equation of Momentum)
(Incompressible Newtonian Fluid)
Spherical Coordinate
r
2
r
2
2
2
r
2
2
r
2
2
2
2
2
θ
r
r
θ
r
r
r
ρg
v
θ
sin
r
1
θ
v
sinθ
θ
sinθ
r
1
r
)
v
(r
r
1
μ
r
r
v
v
v
θ
sin
r
v
θ
v
r
v
r
v
v
t
v
ρ



































 
















p
θ
2
2
r
2
2
θ
2
2
2
θ
2
θ
2
2
2
θ
r
θ
θ
θ
θ
r
θ
ρg
v
θ
sin
r
cosθ
2
θ
v
r
2
v
θ
sin
r
1
θ
sinθ
v
sinθ
1
θ
r
1
r
v
r
r
r
1
μ
θ
r
1
r
cotθ
v
r
v
v
v
rsinθ
v
θ
v
r
v
r
v
v
t
v
ρ






































































 p
















ρg
v
θ
sin
r
cosθ
2
v
sinθ
r
2
v
θ
sin
r
1
θ
sinθ
v
sinθ
1
θ
r
1
r
v
r
r
r
1
μ
rsinθ
1
cotθ
r
v
v
r
v
v
v
rsinθ
v
θ
v
r
v
r
v
v
t
v
ρ
θ
2
2
r
2
2
2
2
2
2
2
2
θ
r
θ
r




































































 p

19. Table A-4 Components of stress tensor for Newtonian Fluid
Koordinat Silinder Koordinat Bola
 










 v
.
κ
x
v
2
μ
τ x
xx
 










 v
.
κ
y
v
2
μ
τ
y
yy
 










 v
z
vz
xx .
2 
















x
v
y
v
μ
τ
τ
y
x
yx
xy














y
v
z
v
μ
τ
τ z
y
zy
yz














z
v
x
v
μ
τ
τ x
z
xz
zx
 










 v
.
κ
r
v
2
μ
τ r
rr
 

















 v
.
κ
r
v
θ
v
r
1
2
μ
τ r
θ
θθ
 










 v
.
κ
z
v
2
μ
τ z
zz
 














θ
v
r
1
r
/r
v
r
μ
τ
τ r
θ
θr
rθ














θ
v
r
1
z
v
μ
τ
τ z
θ
zθ
θz














z
v
r
v
μ
τ
τ r
z
rz
zr
 










 v
.
κ
r
v
2
μ
τ r
rr
 

















 v
.
κ
r
v
θ
v
r
1
2
μ
τ r
θ
θθ
 




















 v
.
κ
r
cotθ
v
r
v
v
rsinθ
1
2
μ
τ θ
r

















θ
v
r
1
r
/r)
(v
r
μ
τ
τ r
θ
θr
rθ


























θ
θ
θ
v
rsinθ
1
sinθ
v
θ
r
sinθ
μ
τ
τ














r
/r)
(v
r
φ
v
rsinθ
1
μ
τ
τ
φ
r
r
r 

3
2
κ 
Cartesian Coordinate System

20. Cylindrical Coordinate
Spherical Coordinate
 
v
.

z
v
y
v
x
v z
y
x








 
z
v
θ
v
r
1
rv
r
r
1 z
θ
r








   









 v
rsinθ
1
sinθ
v
θ
rsinθ
1
v
r
r
r
1
θ
r
2
2
Cartesian Coordinate System:

21. Tabel A-5
Fungsi untuk fluida Newton
Coordinate System
Cartesian
Sylindrical
Spherical
v

 2
2
z
x
2
y
z
2
x
y
2
z
2
y
2
x
v
.
3
2
x
v
z
v
z
v
y
v
y
v
x
v
z
v
y
v
x
v
2










































































 2
2
z
r
2
θ
z
2
r
θ
2
z
2
r
θ
2
r
v
.
3
2
r
v
z
v
z
v
θ
v
r
1
θ
v
r
1
r
v
r
r
z
v
r
v
θ
v
r
1
r
v
2














































































 2
2
r
2
θ
2
r
θ
2
θ
r
2
r
θ
2
r
v
.
3
2
r
v
r
r
v
rsi
n θ
1
v
rsi
n θ
1
si
n θ
v
θ
r
si
n θ
θ
v
r
1
r
v
r
r
r
cotθ
v
r
v
v
rsi
n θ
1
r
v
θ
v
r
1
r
v
2










































































































22. Table A-6
Component of Energy Flux
Cartesian Cylindrical Spherical
z
T
k
q
y
T
k
q
x
T
k
q
z
y
x












z
T
k
q
θ
T
r
1
k
q
r
T
k
q
z
θ
r

























T
rsinθ
1
k
q
θ
T
r
1
k
q
r
T
k
q
φ
θ
r

23. Tabel A-7
Energy Equation in energy and momentum flux
 










































































































y
v
z
v
τ
x
v
z
v
τ
x
v
y
v
τ
z
v
τ
y
v
τ
x
v
τ
v
.
T
p
T
z
q
y
q
x
q
z
T
v
y
T
v
x
T
v
t
T
ρC
z
y
yz
z
x
xz
y
x
xy
z
zz
y
yy
x
xx
ρ
z
y
x
z
y
x
v
   

















































































































z
v
θ
v
r
1
τ
z
v
r
v
τ
θ
v
r
1
r
v
r
r
τ
z
v
τ
v
θ
v
r
1
τ
r
v
τ
v
.
T
p
T
z
q
θ
q
r
1
rq
r
r
1
z
T
v
θ
T
r
v
r
T
v
t
T
ρC
θ
z
θz
r
z
rz
r
θ
rθ
z
zz
r
θ
θθ
r
rr
ρ
z
θ
r
z
θ
r
v
 
 





















































































































































v
r
cotθ
v
rsinθ
1
θ
v
r
1
τ
r
v
v
rsinθ
1
r
v
τ
r
v
θ
v
r
1
r
v
τ
r
cotθ
v
r
v
v
rsinθ
1
τ
v
θ
v
r
1
τ
r
v
τ
v
.
T
p
T
q
rsinθ
1
θ
sinθ
q
rsinθ
1
q
r
r
r
1
T
rsinθ
v
θ
T
r
v
r
T
v
t
T
ρC
θ
θ
r
r
θ
r
θ
rθ
θ
r
r
θ
θθ
r
rr
ρ
θ
r
2
2
θ
r
v
Cartesian
Cylindrical
Spherical

24. Tabel A-8
V




































μ
z
T
y
T
x
T
k
z
T
v
y
T
v
x
T
v
t
T
ρC 2
2
2
2
2
2
z
y
x
p
V










































μ
z
T
θ
T
r
1
r
T
r
r
r
1
k
z
T
v
θ
T
r
v
r
T
v
t
T
ρC 2
2
2
2
2
z
θ
r
p
V




















































μ
T
θ
sin
r
1
θ
T
sinθ
θ
sinθ
r
1
r
T
r
r
r
1
k
T
rsinθ
v
θ
T
r
v
r
T
v
t
T
ρ 2
2
2
2
2
2
2
θ
r
p



Energy Equation For Incompressible Newtonian Fluid with constant properties
Spherical
Cylindrical
Cartesian
𝜌𝐶𝑝
V




















































μ
T
θ
sin
r
1
θ
T
sinθ
θ
sinθ
r
1
r
T
r
r
r
1
k
T
rsinθ
v
θ
T
r
v
r
T
v
t
T
ρ 2
2
2
2
2
2
2
θ
r
p




25. Tabel A-9
Components of molecular Diffusion Flux
Spherical
z
C
D
J
y
C
D
J
x
C
D
J
A
AB
Az
A
AB
Ay
A
AB
Ax












z
C
D
J
θ
C
r
1
D
J
r
C
D
J
A
AB
Az
A
AB
Aθ
A
AB
Ar

























A
AB
Az
A
AB
Aθ
A
AB
Ar
C
rsinθ
1
D
J
θ
C
r
1
D
J
r
C
D
J
Cartesian Cylindrical

26. Tabel A-10
Continuity Equation of Component A
A
Az
Ay
Ax
A
R
z
N
y
N
x
N
t
C




















  A
Az
Aθ
Ar
A
R
z
N
θ
N
r
1
rN
r
r
1
t
C


















    A
A
Aθ
Ar
2
2
A
R
N
rsinθ
1
sinθ
N
θ
rsinθ
1
N
r
r
r
1
t
C






















Cartesian
Cylindrical
Spherical

27. TabLE A-11
Continuity equation of component A for constant properties
A
2
A
2
2
A
2
2
A
2
AB
A
z
A
y
A
x
A
R
z
C
y
C
x
C
D
z
C
v
y
C
v
x
C
v
t
C





































A
2
A
2
2
A
2
2
A
AB
A
z
A
θ
A
r
A
R
z
C
θ
C
r
1
r
C
r
r
r
1
D
z
C
v
θ
C
r
1
v
r
C
v
t
C











































A
2
A
2
2
2
A
2
A
2
2
AB
A
A
θ
A
r
A
R
C
θ
sin
r
1
θ
C
sinθ
θ
sinθ
r
1
r
C
r
r
r
1
D
C
rsin θ
1
v
θ
C
r
1
v
r
C
v
t
C
























































Cartesian
Cylindrical
Spherical

28. P(x,y,z)
y
z
x
y
x
z
CARTESIAN COORDINATE SYSTEM

29. z
 
z
r
P ,
,

r
CYLINDRICAL COORDINATE SYSTEM

30.  

,
,
r
P


SPHERICAL COORDINATE SYSTEM

31. NON NEWTONIAN FLUID
 




















:
2
1
0
0


   2
0
:
2
1


 
0

   2
0
:
2
1


 
0
0 

 




y
Vx
yx 2
0
2

 
yx
0



y
Vx 2
0
2

 
yx
  












1
:
2
1
n
m

 























 

2
0
0
:
2
1
1







BINGHAM PLASTIC
POWER LAW
Model Reiner-Philippof

32. NON NEWTONIAN FLUID
 




















:
2
1
0
0


   2
0
:
2
1


 
0

   2
0
:
2
1


 
0
0 

 




y
Vx
yx 2
0
2

 
yx
0



y
Vx 2
0
2

 
yx
  












1
:
2
1
n
m

 























 

2
0
0
:
2
1
1







BINGHAM PLASTIC
POWER LAW
Model Reiner-Philippof