Class 03 Handout

7/25/2019 Class 03 Handout

1/19

Fluid Mechanics AS102

Class Note No: 03

Wednesday, August 1, 2007


2/19

Review of Last Lecture: contact info

Luoyi Tao; Office: 104; Phone: 4003

Email: [email protected], [email protected]

Interactive & Open door policy


3/19

Review of Last Lecture: grading policy...

Homework assignments & Tutorials, 20%

Two quizes, 20%each

Final exam, 40%

Plus the extra points

Homework assignments collected; Randomly select

problems to grade


4/19

Review of Last Lecture: grading policy...

Tutorial time:

Friday 1:00pm - 1:50pm

Special Attention:

key concepts,to be identified, will be repeatedly tested inthe quizes and the final;

problems in assignments & tutorials will appear in thequizes and the final, with slight modifications

Attendancy:I & you follow the rules


5/19

Review of Last Lecture: reference books

Yuan, S. W., Foundations of Fluid Mechanics.

Granger R. A.,Fluid Mechanics. (Dover edition?) Subramanya, K.,1000 Solved Problems in Fluid

Mechanics (includes hydraulic machines). Tata

McGraw-Hill Publishing Company Limited.

Giles R. V., Evett J. B. , and Liu C., Schaums Outline of

Theory and Problems of Fluid Mechanics and Hydraulics.

Third Edition. Tata McGraw-Hill Publishing Company

Limited.

Mase, G. E.,Schaums ouline of Continuum Mechanics.

Tata McGraw-Hill Publishing Company Limited. Kay, D. C.,Schaums ouline of Tensor Calculus. Tata

McGraw-Hill Publishing Company Limited.

any fluid mechanics books in the library and Wiley-India

Edition in the bookstore


6/19

Review of Last Lecture: calculus

vector algebra & calculus refreshment:

# in rectangular coordinate systems

# indicial notations

{x, y, z} {x1, x2, x3} , {i, j, k} {i1, i2, i3}

# summation rules/conventionsa= axix+ ayiy+ aziz a= a1i1+ a2i2+ a3i3 =akik

# addition, subtraction, multiplication, dot & cross products


7/19


vector algebra & calculus:

# the Kronecker delta

ij :=

1 ifi=j0 ifi

=j

# the alternating tensor

ijk :=

0 if two or more have the same value1 ifijk=123, 231, 312

1 others


8/19


vector algebra & calculus refreshment:# the gradients

grad := = ikk= ik,k

# the divergence

divq:= q= ikiqk=kqk=qk,k

# the curl / rotation

curlq:= rotq:= q= ijkiqjik=ijkqj,iik


9/19


vector algebra & calculus refreshment:

# Stokes theorem:

C

q

dx=

S

n

(

q) dS,

C

qidxi= S

niijkqk,jdS

# the divergence theorem of Gauss:

V

qdV =

S

n

qdS,

V

qi,idV = S

niqidS


10/19


2nd order Cartesian tensors sketch:

A=Aklikil=A

kli

ki

l, A

kl=QkmQlnAmn

# the gradientgrad A:= A:= ikk

Aijiiij= ikAij,kiiij=Aij,kikiiij

# the divergencedivA:= A:= ikk

Aijiiij

= Aij,kkiij=Akj,kij

# the divergence theoremV

AdV =

S

nAdS,

V

Aki,kdV =

S

nkAkidS

V

Aik,kdV =

S

nkAikdS...


11/19

Curvilinear Coordinate Systems & Tensor Analysis

Purpose:

outline the scheme for writing equations of motion in curvilinearc.s.

Example:

in RCS:

= := 2

xixi=

2

x2 +

2

y2 +

2

z2

in cylindrical c.s.:

= 2

r2 +1

r

r + 1

r22

2 +2

z2 ?

Will learn the techniques


12/19


example: cylindrical c.s.

x1 =rcos, x2 =rsin, x3 =z,r=

(x1)2 + (x2)2, =tan

1 x2x1

, z=x3

O

x1

x2

x3

r

P

r

z

i1

i2

i3

gr

ggz r u1 =u1(x1, x2)

u2 =u2(x1, x2)z u3 =u3(x3)

Figure:cylindrical coordinates


13/19

Curvilinear Coordinate Systems & Tensor AnalysisIn general, (short-hand writing & 1-1)

ui =uixj , xi=xiuj

,r= xiii=xi

uj

ii=: r

uj

(1)

O

x1

x2

x3

r

Pu1

u2

u3

i1

i2

i3

g1g2

g3

Figure:curvilinear coordinates


14/19


convention 1:

will use {xk}, xkto denote the rectangular coordinates convention 2:

will use

uk

,

uk

to denote the curvilinear coordinates

# WARNING:pay attention to the positions of the indices in c.c.s.

Ak, Aj

Ajk, Amn, Akl

you will know the reasons


15/19


fix=0, z=0 & varyr in cylindrical c.s a space curver= x1i1 in r.c.s. x the tangent to the curve is i1

fixu2

, u

3

& varyu

1

a space curver= ruj=xiuj ii in r.c.s. x the tangent to the curve (of varyingu1):

g1 = r

u1 = (xiii)

u1 =

u1xi

uj

ii, u2, u3 fixed (2)

C ili C di S T A l i


16/19


tangent to the curve ofuk:

gk := r

uk =

xiuk

ii, k=1, 2, 3 (3)

called the covariant / natural base vector for uk, k=1, 2, 3important to the tensor analysis

ii= uk

xigk (4)

gkgl= xiuk

xiul

(5)

C ili C di S & T A l i


17/19


taker=

10 in cylindrical c.s.

the surface of (x1)2 + (x2)2 =10 in r.c.s. normal to the surface is

1

10(x1i1+ x2i2)

takeu1 =constant the surface of u1

xj

= constant in r.c.s.

normal to the surface isg1 =im

u1

xm

C ili C di t S t & T A l i


18/19


normal to the surface of uk = const:

gk := uk = uk

xiii (6)

contravariant / dural base vector for uk, k=1, 2, 3

important to the tensor analysis

gkgl =lk =?kl (7)

gkgl = uk

xi

ul

xi(8)

C ili C di t S t & T A l i


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metric tensor:

ds2 :=dxdx=gkldukdul,gkl :=gkgl= xiuk

xiul

=glk (9)

gkl =gkgl =glk = ukxi

ulxi

,

gklglm=km... (10)

gk =gkl gl, gk=gklgl ... (11)

Specific examples: next class

Class 03 Handout

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