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Lecture 1

COMPUTATIONAL

MECHANICS MKM410

mohsen.sharifpur@up.ac.za

Dr M. Sharifpur

Department ofMechanical and Aeronautical Engineering

University ofPretoria

February 24th, 2011

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Computers and Humans

"Computers are incrediblyfast,accurate, and stupid;

humans are incredibly slow,

inaccurate and brilliant;together they are powerful

beyond imagination."

1921 Nobel Prize in Physics for photoelectric effect. In 1999 Einstein

was named Time magazine's "Person of the Century", and a poll of

prominent physicists named him the greatest physicist of all time

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Wrong Info

Wrong Initial & BC

Wrong Answer

The important thing is:

The problem The answer

Did you solve your own problem?

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First order linear differential equation

Solution;

Example;

Simple

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Sometimes we have the experimental data, and we wantto find the generalized solution.

Sometimes we have the Mathematical model but we do

not have the exact analytical solution.

t

Tce

z

Tk

zy

Tk

yx

Tk

x

gen

Example; General Heat Conduction Equation

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t

Tce

z

Tk

zy

Tk

yx

Tk

xgen

In your Text book this equation (General Heat Conduction Equation)

is represented by equation 2.22(page 24) as

zxandyxxxeq gen 321 ,,

Therefore:

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t

Tcez

Tkzy

Tkyx

Tkx

gen

and Initial condition

Initial and Boundary conditions

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Boundary and Initial Conditions

t

Tc

k

e

y

Tb

x

Ta

x

T gen

2

21) BC=1, I=1

3) BC=2, I=0

4) BC=2, I=15) BC=3, I=0

6) BC=3, I=1

7) BC=4, I=08) BC=4, I=1

0

gene

z

Td

y

Tc

x

Tk

x

02

2

gene

z

Td

y

Tc

x

T

x

k

x

Tk

(Mathematically we call them Conditions)

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9

Boundary and Initial Conditions

t

T

k

e

y

T

x

T gen

12

2

2

2 B=4, I=1

Mechanically

h, T

Insulated

a

b

h, T

1

23

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Boundary and Initial Conditions

2 2 2

2 2 2

1geneT T T T

x y z k t

t

T

k

eT

z

T

y

T

x

T gen

12

2

2

2

2

21

x

T

t

T

k

e

z

T

y

T

y

T gen

02

2

k

e

y

T

x

T gen BC=3, I=0

BC=4, I=1

BC=6, I=1

BC=5, I=1

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Unannounced Test 1Allowance Time 2 minutes

01

2

2

t

T

k

eT

z

Td

y

Tc

y

Tb

x

Ta

gen

For solving following partialdifferential equation analytically,

How many Initial and Boundary

Conditions do we need?

Specify how many at all , and how

many in each direction? and why?

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01

2

2

t

T

k

eTz

Tdy

Tcy

Tbx

Ta

gen

Initial Condition: 1

Initial andBoundary Conditions:

Boundary Conditions:

x- direction: 1

y- direction: 2

z- direction: 1

BC = 4 at all

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and Initial condition

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For Heat transfer and Fluid flow

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General Heat Conduction Equation:

t

Tce

z

Tk

zy

Tk

yx

Tk

x

gen

In rectangularcoordinates

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Special Cases

2 2 2

2 2 2

1geneT T T T

x y z k t

2 2 2

2 2 2 0

geneT T T

x y z k

2 2 2

2 2 2

1T T T T

x y z t

2 2 2

2 2 20

T T T

x y z

Two-dimensional

Three-dimensional

1) Steady-state with heat generation

2) Transient, no heat generation:

3) Steady-state, no heat generation:

Constant thermal conductivity:

t

Tce

z

Tk

zy

Tk

yx

Tk

xgen

General:

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Special Cases

For Plain Wall and constant thermal conductivity:

tTce

zTk

zyTk

yxTk

xgen

General:

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In Cylindrical

Coordinates

2

1 1gen

T T T T T rk k k e c

r r r r z z t

General Heat Conduction Equation:

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Usually for homogenous material, with

symmetry boundary condition

2

1 1gen

T T T T T rk k k e c

r r r r z z t

Special Cases in CylindricalCoordinates

2

1 1gen

T T T T T rk k k e c

r r r r z z t

+for long cylinder (L>>D)

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In Spherical

Coordinates

2

2 2 2 2

1 1 1sin

sin singen

T T T Tkr k k e c

r r r r r t

General Heat Conduction Equation:

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2

2 2 2 2

1 1 1sin

sin singen

T T T Tkr k k e c

r r r r r t

Special Case in SphericalCoordinates

Usually for homogenous material, withsymmetry boundary condition

2

2 2 2 2

1 1 1sin

sin singen

T T T Tkr k k e c

r r r r r t

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Boundary Conditions

Specified Temperature Boundary Condition

Specified Heat Flux Boundary Condition Convection Boundary Condition

Radiation Boundary Condition

Interface Boundary Conditions

Generalized Boundary Conditions

t

Tce

z

Tk

zy

Tk

yx

Tk

xgen

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Specified Temperature Boundary Condition

For one-dimensional heat transfer

through a plane wall of thickness

L, for example, the specified

temperature boundary conditionscan be expressed as

T(0, t) = T1T(L, t) = T2

t

Tce

z

Tk

zy

Tk

yx

Tk

xgen

t

Tc

x

Tk

2

2

BC

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Specified Heat Flux Boundary Condition

dTq k

dx

Heat flux in

the positive

x-direction

The sign of the specified heat flux is determined byinspection:positiveif the heat flux is in the positive

directionof the coordinate axis, and negativeif it is in

the oppositedirection.

The heat flux in the positivex-direction anywhere in the medium,

including the boundaries, can be

expressed by Fouriers lawof heat

conduction as

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Two Special Cases

Insulatedboundary Thermalsymmetry

(0, ) (0, )0 or 0

T t T t k

x x

0),0( tQx

,20

LT t

x