Class 02 Handout

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Fluid Mechanics AS102

Class Note No: 02

Tuesday. July 31, 2007


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Review of Last Lecture: contact info

Luoyi Tao; Office: 104; Phone: 4003

Email: [email protected], [email protected]

Interactive & Open door policy


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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


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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


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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). TataMcGraw-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.


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Review of Last Lecture: why continuum treatment

What is a fluid (gas, liquid)?

shape determined by its confines? unable to resist shearing?


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Review of Last Lecture: why continuum treatment

why notdeal withindividualmolecules & atoms in this course

Interests Macro-scalemotions such as Air flow around an airplane (Water or Air) Pipe flows (of large size)

Atmospheric & Oceanic motion, etc.,where enoughnumber of molecules present detailsof micro-scale motionsnot essential& collectively

accountablethrough temperature, pressure...

integral, differential equations of motionadequate(continuum treatment)


8/26

Cartesian Tensor Calculus: review

What is a Cartesian tensor?

Cartesian? the rectangular coordinate systems (RCS)involved

O

x

y

z

i

j

k

O

x1

x2

x3

i1

i2

i3

Figure:Standard & Indicial


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Cartesian Tensor Calculus: reviewVector representation & operation rules:

Standard notation:

a= axi + ayj + azk, ax, ay, az R (1)

Indicial Notation:

a= a1i1+ a2i2+ a3i3=

3k=1

akik (2)

Summation Convention:

a= akik :=

3

k=1

akik=

3

j=1

ajij=ajij (3)

indexk repeated twice and twice only(!)

k taking onallthe values of its range

k dummy, replaceable by others


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Vector algebra:

addition:

a + b:= (ak+ bk) ik a + b= b + a

(a + b) + c= a + (b + c) (4)

subtraction:

a b:= (ak bk) ik b + a=a b (5)

multiplication by a (real) scalar:

ma:= (m ak) ik m(na) = (mn) a=n(ma)

(m+ n) a= ma + na

m(a + b) =ma + mb (6)


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Vector algebra:

dot product&cross productof vectors:

a

b

Figure:dot producta b

definition 1 of dot product:

a b:= |a| |b| cos , 0 (7)e. g.

ij ik=jk = ?, j, k (8)

jk the Kronecker delta


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Cartesian Tensor Calculus: reviewVector algebra:

givena = 101i1+ 3i2+239i3, b= 1i1+ 20i2+ 11i3

a b= ? or = ?...

definition 2 of dot product (in a r.c.s.):

a b:= akbk (summation convention) (9)

(independent of any specific rectangular coordinatesystems)

definition 1&definition 2are equivalent

a b= b a

a (b + c) =a b + a c

m(a b) = (ma) b=a (mb) = (a b) m (10)


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Vector algebra:

a

bv=a b

v

Figure:cross producta b

definition 1 of cross product:

a b:= |a| |b| sin v, 0 ,

|v| =1,

v thePlane(right-handed) (11)


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Vector algebra:

e. g.

ij ij=0, i1 i2=i3, i2 i3=i1, i3 i1=i2

or

ii ij=ijkik, i,j, k=1, 2, 3

ijk the alternating tensor;even, odd, not permutation= ?

C i T C l l i


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Vector algebra:

givena = 101i1+ 3i2+239i3, b= 1i1+ 20i2+ 11i3

a b= ? or = ?, v= ?...

definition 2 of cross product (in a r.c.s.):

a b:=ijk

aib

jik

(summation convention) (12)

definition 1&definition 2are equivalent

a b= b a

(a + b) c= a c + b c

m(a b) = (ma) b=a (mb) (13)

C i T C l l i


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Vector algebra:

Apply the rules to verify:

(a b) c=a (b c) =b (c a) =det [abc]

(a b)ici=

a (b c) = (a c) b (a b) c

ijk

ai(b c)

j=

C t i T C l l i


18/26


Vector calculus:

:=ik

xk=ikk=ik(),k

k := (),k:=

xk(14)

the gradient

grad:= = ikk= ik,k (15)

(from the scalar multiplication ?)


19/26

Cartesian Tensor Calc l s re ie


20/26


Vector calculus:

() =+,

() ,i= ,i+,i

(a) = () a + a,

(ai) ,i=,iai+ai,i

(a) = () a + a,

ijki

aj

=ijk

aj

,i=

(a b) =b ( a) a ( a) ,(a b)k,k=



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Vector calculus:

(a b) = (b ) a (a ) b + a ( b) b ( a)

ijki(a b)j=ijk

mnjambn

j,i=

(a

b

) = (a

)b

+ (b

)a

+a

(

b

) +b

(

a

)

2:= =jj= ,jj

= 0, ijki()j=ijk,ji=

( a) =0, ( a)k,k= ijkiaj,k=ijkaj,ik= ( a) = ( a) 2a



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Cartesian Tensor Calculus: reviewVector calculus:

Stokes theorem:

x1

x2

x3

Cdx

n

dSS

Figure:Stokes theorem (right-handed)

C

qdx=

S

n(q) dS,

C

qidxi=

S

niijkqk,jdS (18)



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Vector calculus:

the divergence theorem of Gauss:

LetSdenote the surface bounding a volume V, andnthe(outward) unit vector normal to the surface. Then

V

qdV =

S

nqdS,

V

qi,idV =

S

niqidS (19)


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Cartesian Tensor Calculus


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Cartesian Tensor Calculus

the gradientgradA:= A:=ikk

Aijiiij

= ikAij,kiiij=Aij,kikiiij (22)

the divergence

divA:= A:= ikkAijiiij= Akj,kij... (23) the divergence theorem

V

AdV =

S

nAdS,

V

Aki,kdV =

S

nkAkidS (24)

V

Aik,kdV =

S

nkAikdS? ... (25)

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