couette flow
DESCRIPTION
The presentation about Coutte FlowTRANSCRIPT
Group Members
Number Name Registration Number
1 Khawar Shahzad ME 113 009
2 Muhammad Ali ME 113 115
3 Mohsin Raza ME 113 080
4 Zohaib Ahmad ME 113 022
5 Aoun Abbas ME 113 074
6 Omer Ayub ME 113 089
7 Muhammad Ramzan ME 113 100
8 Khuram Yousaf ME 113 107
9 Zain Talib ME 113 108
10 Hassan Aamir ME 113 053
11 Adeel Anwar ME 113 039
2
Introduction
http://upload.wikimedia.org/wikipedia/commons/thumb/9/93/Laminar_shear.svg/800px-Laminar_shear.svg.png
3
Analytical Solution
The governing y-direction momentum equation is:
๐ ๐ ๐ฏ๐ ๐ญ
+๐๐ฎ ๐๐ฏ๐๐ฑ
+๐๐ฏ ๐ ๐ฏ๐ ๐ฒ
+๐๐ฐ ๐ ๐ฏ๐๐ณ
=โ ๐๐ฉ๐๐ฒ
+๐๐๐ฑ๐ฒ
๐๐ฑ+๐๐๐ฒ๐ฒ
๐ ๐ฒ+๐๐๐ณ๐ฒ
๐๐ณ+๐๐ ๐ฒ
Steady State Assumption No Body Forces
๐=๐๐ฉ๐๐ฒ
6
Analytical Solution
The governing x-direction momentum equation is:
๐ ๐๐ฎ๐๐ญ
+๐๐ฎ ๐๐ฎ๐๐ฑ
+๐ ๐ฏ ๐๐ฎ๐ ๐ฒ
+๐๐ฐ ๐๐ฎ๐๐ณ
=โ ๐๐๐ ๐
+๐๐๐๐
๐๐+๐๐ ๐๐
๐ ๐+๐๐๐๐
๐๐+๐ ๐ ๐
Steady State Assumption Y-Direction
Velocity (v) is Zero
No beginning or end of this flow in x-direction
Final Equation
๐๐ y ( ฮผ ๐u๐ y )=0
๐๐๐๐๐ฒ๐=๐
7
Using boundary condition at we have, we get.
Using boundary condition at we have, we get.
๐๐๐
=๐๐ซ
Analytical Solution
8
Analytical Solution โ Matlab Codeclcclear%Workspace and Command History Cleared
ue = 0.01 %Velocity of Upper Plate moving with 0.01m/s
D = 0.05 %Separation Distance between plate. 5cm%y is distance from lower plate to upper plate
y = linspace(0,D,500);for i=1:1:500 U(i)=(ue*y(i))/D;endplot (y,U)legend ('Couette Flow Velocity Profile','Fontsize',12')xlabel ('Width Between Plates (m)',' Fontsize ',12')ylabel ('Velocity Profile (m/s)',' Fontsize ',12')axis ([-0.01 0.06 -0.001 0.012])
-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
0
2
4
6
8
10
12x 10
-3
Width Between Plates (m)
Ve
locity P
rofile
(m
/s)
Couette Flow Velocity Profile
9
Numerical Implicit Method
Nature of Partial Differential Equation
Parabolic Nature Equation because
Conclusion: Time Marching Possible
10
The Numerical Formulation
๐ฎ ๐ฃ๐ง+๐=๐ฎ ๐ฃ
๐ง+๐ซ๐ญ
๐ (๐ซ๐ฒ )๐๐๐๐
๐ฎ ๐ฃ+๐๐ง+๐+
๐ซ๐ญ๐ (๐ซ ๐ฒ )๐๐๐๐
๐ฎ ๐ฃ+๐๐ง โ
๐ซ๐ญ๐ (๐ซ๐ฒ )๐๐๐๐
๐๐ฎ ๐ฃ๐ง+๐โ
๐ซ๐ญ๐ (๐ซ๐ฒ )๐๐๐๐
๐๐ฎ ๐ฃ๐ง+
๐ซ๐ญ๐ (๐ซ๐ฒ )๐๐๐๐
๐ฎ ๐ฃโ๐๐ง+๐+
๐ซ๐ญ๐ (๐ซ ๐ฒ )๐๐๐๐
๐ฎ ๐ฃโ๐๐ง
13
The Numerical Formulation
A B ๐ ๐ฃ
๐๐ฎ ๐ฃโ 1๐ง+1+๐๐ฎ ๐ฃ
๐ง+1+๐๐ฎ ๐ฃ+1๐ง+1=๐ ๐ฃ
14
The Numerical Formulation
๐๐ฎ ๐ฃโ 1๐ง+1+๐๐ฎ ๐ฃ
๐ง+ 1+๐ ๐ฎ ๐ฃ+1๐ง+1=๐ ๐ฃ
[ยฟ๐ยฟ๐ยฟ๐ยฟ๐ยฟ๐
ยฟ๐ยฟ๐ยฟ๐ยฟ๐ยฟ๐
ยฟ๐ยฟ๐ยฟ๐ยฟ๐ยฟ๐
ยฟ๐ยฟ๐ยฟ๐ยฟ๐ยฟ๐
ยฟ๐ยฟ๐ยฟ๐ยฟ๐ยฟ๐
] [ยฟ๐ฎ๐
๐ง+๐
ยฟ๐ฎ๐๐ง+๐
ยฟ๐ฎ๐๐ง+๐
ยฟ๐ฎ๐๐ง+๐
ยฟ๐ฎ๐๐ง+๐
]=[ยฟ๐ ๐โ๐ ๐ฎ๐
๐ง+๐
ยฟ๐๐
ยฟ๐๐
ยฟ๐๐
ยฟ๐๐
ยฟ๐ ๐โ๐ ๐ฎ๐๐ง+๐
]15