Couette Flow Analytical & Numerical Solution

RASIKH TARIQrasikhtariq@gmail.com

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

Introduction

Laminar

Viscous

Incompressible Fluid

4

Introduction

5

Analytical Solution

The governing y-direction momentum equation is:

𝛒 𝛛 𝐯𝛛 𝐭

+𝛒𝐮 𝛛𝐯𝛛𝐱

+𝛒𝐯 𝛛 𝐯𝛛 𝐲

+𝛒𝐰 𝛛 𝐯𝛛𝐳

=− 𝛛𝐩𝛛𝐲

+𝛛𝛕𝐱𝐲

𝛛𝐱+𝛛𝛕𝐲𝐲

𝛛 𝐲+𝛛𝛕𝐳𝐲

𝛛𝐳+𝛒𝐟 𝐲

Steady State Assumption No Body Forces

𝟎=𝛛𝐩𝛛𝐲

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Analytical Solution

The governing x-direction momentum equation is:

𝛒 𝛛𝐮𝛛𝐭

+𝛒𝐮 𝛛𝐮𝛛𝐱

+𝛒 𝐯 𝛛𝐮𝛛 𝐲

+𝛒𝐰 𝛛𝐮𝛛𝐳

=− 𝝏𝒑𝝏 𝒙

+𝝏𝝉𝒙𝒙

𝝏𝒙+𝝏𝝉 𝒚𝒙

𝝏 𝒚+𝝏𝝉𝒛𝒙

𝝏𝒛+𝝆 𝒇 𝒙

Steady State Assumption Y-Direction

Velocity (v) is Zero

No beginning or end of this flow in x-direction

Final Equation

𝜕𝜕 y ( μ 𝜕u𝜕 y )=0

𝛛𝟐𝒖𝛛𝐲𝟐=𝟎

7

Using boundary condition at we have, we get.

Using boundary condition at we have, we get.

𝒖𝒖𝒆

=𝒚𝑫

Analytical Solution

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Analytical Solution – 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

11

The Numerical Formulation

12

The Numerical Formulation

𝐮 𝐣𝐧+𝟏=𝐮 𝐣

𝐧+𝚫𝐭

𝟐 (𝚫𝐲 )𝟐𝐑𝐞𝐃

𝐮 𝐣+𝟏𝐧+𝟏+

𝚫𝐭𝟐 (𝚫 𝐲 )𝟐𝐑𝐞𝐃

𝐮 𝐣+𝟏𝐧 −

𝚫𝐭𝟐 (𝚫𝐲 )𝟐𝐑𝐞𝐃

𝟐𝐮 𝐣𝐧+𝟏−

𝚫𝐭𝟐 (𝚫𝐲 )𝟐𝐑𝐞𝐃

𝟐𝐮 𝐣𝐧+

𝚫𝐭𝟐 (𝚫𝐲 )𝟐𝐑𝐞𝐃

𝐮 𝐣−𝟏𝐧+𝟏+

𝚫𝐭𝟐 (𝚫 𝐲 )𝟐𝐑𝐞𝐃

𝐮 𝐣−𝟏𝐧

13

The Numerical Formulation

A B 𝐊 𝐣

𝐀𝐮 𝐣− 1𝐧+1+𝐁𝐮 𝐣

𝐧+1+𝐀𝐮 𝐣+1𝐧+1=𝐊 𝐣

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The Numerical Formulation

𝐀𝐮 𝐣− 1𝐧+1+𝐁𝐮 𝐣

𝐧+ 1+𝐀 𝐮 𝐣+1𝐧+1=𝐊 𝐣

[¿𝐁¿𝐀¿𝟎¿𝟎¿𝟎

¿𝐀¿𝐁¿𝐀¿𝟎¿𝟎

¿𝟎¿𝐀¿𝐁¿𝐀¿𝟎

¿𝟎¿𝟎¿𝐀¿𝐁¿𝐀

¿𝟎¿𝟎¿𝟎¿𝐀¿𝐁

] [¿𝐮𝟐

𝐧+𝟏

¿𝐮𝟑𝐧+𝟏

¿𝐮𝟒𝐧+𝟏

¿𝐮𝟓𝐧+𝟏

¿𝐮𝟔𝐧+𝟏

]=[¿𝐊 𝟐−𝐀 𝐮𝟏

𝐧+𝟏

¿𝐊𝟑

¿𝐊𝟒

¿𝐊𝟓

¿𝐊𝟔

¿𝐊 𝟔−𝐀 𝐮𝟕𝐧+𝟏

]15

Thank you!

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