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y

b

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APPLICATIONAPPLICATIONSS

Heat and mass transfer rates are Heat and mass transfer rates are enhanced by the oscillation of the enhanced by the oscillation of the surrounding fluid. Useful in combustion, surrounding fluid. Useful in combustion, drying and the passage of sound waves drying and the passage of sound waves through particulate systems.through particulate systems.

Particle-laden flows, Brownian motion, Particle-laden flows, Brownian motion, suspension rheometry, colloidal suspension rheometry, colloidal suspension, and particle motion in suspension, and particle motion in filters.filters.

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MODES OF MODES OF HEAT HEAT TRANSFERTRANSFER

RADIATIONRADIATION

CONDUCTIONCONDUCTION CONVECTION (forced & CONVECTION (forced &

free)free)

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GOVERNING EQUATIONS

Conservation of momentumConservation of momentum

wFpw)w(t

w 2

)w(w)2/w(w)w( 2

)w()w(w2

where

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First law of thermodynamicsFirst law of thermodynamics

Conservation of massConservation of mass

0w

T)Tw(tT 2

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VECTOR RELATIONS

2

1 2 3 1

2 3

1 1 2

1 3

2 2 3

1 2

3 3

1

h h h q

h hh q q

h hh q q

h hh q

( ) ( ) ( )

ds h dq dqij i jji

2 2

1

3

1

3

h

xq

yq

zqi

i i i

2 2 2 2 ( ) ( ) ( )

333

222

111

uqh

1u

qh

1u

qh

1

)hhF(q

)hhF(q

)hhF(qhhh

1F 213

3312

2321

1321

332211

321

332211

321

hFhFhFqqq

uhuhuh

hhh

1F

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w

1231

2132 q

whh,q

whh

Define:

332211

321

332211

321

hwhwhwqqq

uhuhuh

hhh1

w

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q

h

h h q q

h

h h qh h

1

2

1 3 1 2

1

2 3 21 2( ) ( )

h ht q q h q q h

qh

h h qh

qh

h h qh

hq

h Fq

h F

1 22 1 3 1 2 3

1

2

1 3 13

2

1

2 3 23

31

2 22

1 1

( ) ( )

then,

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and,

h h hT

t q

T

q q

T

q

q

h h

h

T

q q

h h

h

T

q

1 2 32 1 1 2

1

2 3

1 1 2

1 3

2 2

( ) ( )

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a

g

x r sin cos y r sin sin z r cos

SPHERICAL COORDINATES

SCALE FACTORSh h r h rr 1 , , sin

r

T

Ts

U U to cos( )

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ln( / )r a

/ U ao2

a Uo/

( ) / ( )T T T Ts

t U t ao /

U U U S to / cos( )

VARIABLES

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Re /2aUo

Gr g T T as ( )( ) /2 3 2

Pe = Re Pr

S a Uo /

Pr /

T

Ts

DIMENSIONLESS NUMBERS

kinematic viscosity

thermal diffusivity

volumetric expansion coef.

frequency of oscillation

surface temperature

far-field temperature

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e32

2

2

2 0

sin cot

ee

e

2

2

2

2

2 2

2

t

Gr2Re

2

sin( ) ( cot )

Recot

sin

sin cos

ee

Pe2

2

2

2

2

2

t

sincot

USING THE SPHERICAL COORDINATE SYSTEM, THE EQUATIONS REDUCE TO:

(1)

(2)

(3)

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0 1 0, and at

e S t and e S t

ore

S t

as

2 2 2

22

2

sin cos( ) , sin cos cos( )

, sin cos( )

0 0, and as

BOUNDARY CONDITIONS

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• Equations are two dimensional, time-dependent, nonlinear, coupled, and of infinite domain.

• No explicit boundary conditions for vorticity on the surface.

• Difficulties with finite- differences such as, indeterminate forms, etc..

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THE METHOD OF SOLUTION

e f t P dn n

zn

/ ( , ) ( )21

1

g t P zn nn

( , ) ( )1

1

z cos

h t P zn nn

( , ) ( )0

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HOW ?!

t

h

tPn

nn

0

ee

Pe2

2

2

2

2

2

t

sincot

eh

tP P dzi

i ji

2

1

1

0

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INTEGRALS NEEDED !!

P z P z dznk

mk( ) ( )( ) ( )

1

1

z P z P z dzn m2 1 1

1

1( ) ( )( ) ( )

P z P z P z dzn i j( ) ( )( ) ( ) ( )1 1

1

1

P z P z P z dzn i j( ) ( ) ( )( ) ( ) ( )1 1 2

1

1

and in general,

P z P z P z dzjm

jm

jm

1

1

2

2

3

3

1

1

( ) ( ) ( )

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P z P z P z dzj m j mj m j m

nj j n j j n

m m m m

n m mn m m

P z P z dz

jm

jm

jm

m m

n

jm

nm m

1

1

2

2

3

3

1 2

3

3 2 1

1

12 2 1 1

2 2 1 1

1 2 1 2

1 2 1 2

1 2

1 2 1

1

1 2 10 0 0

( ) ( ) ( )( )!( )!( )!( )!

( ) ( )

( )!( )!

( ) ( )

j j n j j1 2 1 2

where

After Mavromatis & Alassar:

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P z P z dzm m m m

j m j m

j m j m

Gk m m

k m m

Fm m k m m k m m

m mm

jm

jm

m

m m

k

k m m

1

1

2

2

2

2 1

2 1

1

1

2 1 2 1 2 1

1 1 2 2

1 1 2 2

2 1

2 1

3 22 1 2 1 2 1

2 1

1

21

2 2

3

2 2

1 1

1

2 2 21 1

3

( ) ( )( )

( ) ( )

( )!( )!

( )!( )!

( ( ) )( )!

( )!

, , ; ,

( )

2 1

21

m;

G kj j k j j k

m m m mm m

( ) ( )1 2 1

0 0 01 2 1 2 1 2

1 2 1 2j j k j j1 2 1 2

where

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Saalschutz’s Theorem

If c d a b n then

F a n b c dc a n c b n c c a b

c a c b c n c a b n

In thiscase

am m k

nk m m

bm m

c m m

dm m

1

1

1

2

2

21

1

3

2

3 2

1 2

1 2

1 2

1 2

1 2

,

, , ; , ;

:

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3-j SYMBOLS

Represent the probability amplitude that three angular momentum j1, j2, and j3 with projections m1, m2, and m3 are coupled to yield zero angular momentum. They are related to Clebsch-Gordan coefficients (C) by:

j j j

m m mCj m j

j j m j mj m1 2 3

1 2 3

2 12 1

1 3 3 1

3 1 1 2 2

3 3

( )

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Cabc b c

a b c a b

a c ca b b c

F a b c a c a b b c

a bc b

,

/

( )( )( )!

( )!( )!

( )!( )!( )( )!( )!( )!( )!

, , ; , ;

1

2 1

1 1 1

1 2

3 2

( )( )!( )!( )!

( )!

/

abca b c a b c a b c

a b c

1

1 2where

Clebsch-Gordan Coefficients

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2

22 5 21 2 1

fn f n n e gn

n n ( / ) ( ) /

eg

t

g gn n g

en

n hh

nn h

hS

n n nn

nn

nn

n

22

2

11

11

21

1

2 11

1

2 32

Re( )

( )( )

( )( )

Gr

2Re2

eh

t Pe

h hn n h Hn n n

n n2

2

2

21

( )

(1)

(2)

(3)

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S e fg

g gf

fn ijn

ij

j ijn

ji

ijiji

/ ( ) ( )2

1111

1

2

H en

ih

ff f

hn nj

ij

ii ij

ni

j

jiji

/ ( )2

0101

2 1

2 1

1

2

ijn n

j j

n n

n i j n i j

( )

( )

( )2 1

1

1 1 0 1 0 0 0

ijn n

j j j

n n i i

n i j n i j

( )

( )( )

( ) ( )2 1

1 2

1 1 1 1 2 0 0 0

2

ijn n

n i j

( )2 1

0 0 0

2

where,

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MODES BOUNDARY CONDITIONS

f tf

t and h tnn

n n( , ) ( , ) , ( , )0 0 0 0 0

f t e S t

f te S t as

n n

nn

( , ) cos( ) ,

( , )cos( )

/

/

3 21

3 21

3

2

g t and h t asn n( , ) , ( , ) 0 0

e g d S tnn n

( ) cos( )2

0

1

3

2

2

22 5 21 2 1

fn f n n e gn

n n ( / ) ( ) /

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NUMERICAL METHOD

eg

t

g gn n g

en

n hh

nn h

hS

n n nn

nn

nn

n

22

2

11

11

21

1

2 11

1

2 32

Re( )

( )( )

( )( )

Gr

2Re2

eh

t Pe

h hn n h Hn n n

n n2

2

2

21

( )

CRANK-NICOLSON F.D. SCHEME

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g

tq tn

n ( , )

q eg g

n n g Snn n

n n

22

2

21

Re

( ) *

1

12

tg t g t t

q t q t t

n n

n n

( , ) ( , )

( , ) ( , )

A t g t B t g t

C t g t D t t E tn n

n n n

( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , )

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SPECIALIZED STEP-BY-STEP METHOD

xf

n fnn

n

( / )1 2

Qf

n fnn

n

( / )1 2

fQ x

nnn n2 1 2( / )

2

22 5 21 2 1

fn f n n e g rn

n n n ( / ) ( ) ( )/

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x

n x rnn n ( / )1 2

Q

n Q rnn n ( / )1 2

at 0x Qx Q

n nn n

0

x h t x t e e r z t dzn nn n x

n

h

( , ) ( , ) ( , )( / ) ( / )

1 2 1 2

e n h( / )1 2 1

x h t x t r t nn n n( , ) ( , ) ( ) ( , ) / ( / ) 1 1 2

Q t e tn n( , ) cos( )/ 3 3 21

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x h t x th n h n n

r t

h n h n nr h t

h n h nr h t

n n n

n

n

( , ) ( , )( / ) ( / ) /

( , )

( )

( / ) ( / ) /( , )

( / ) ( / )( , )

01

1 2

1 3

2 1 2 1 20

2 1

1 2

2

1 2

1

1 2

1

1 2

1

2 1 22

2 3 2

2 3 2

2 3 2

x h t x th n h n n

r t

h n h nr h t

h n h n nr h t

n n n

n

n

( , ) ( , )( / ) ( / ) /

( , )

( )

( / )

( )

( / )( , )

( / ) ( / ) /( , )

21

1 2

3 1

2 1 2 1 2

2 1

1 2

2 1

1 2

1

1 2

3

2 1 2

1

1 22

22

2 3

2

2

2

2

2 3

2

2

2

2 3

2

2

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PHYSICAL PARAMETERS

Nh

Pun

nn

( , ) (cos )

2 20 0

N N du u( ) ( , )

1

0

N N du u( ) ( , ) sin

1

2 0

Nusselt Number

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Nh

u ( )

2 0

N N x dxu u( ) ( )/

/

1 2

1 2

N n F n nh

n

n

h

un

n

n

n

( ) ( ) ( / , , / ; , ; )

( / )

!

2 1 3 2 3 2 2 2 1

2 1 2

23 2

2

0

2

22

0

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3.00 3.25 3.50 3.75 4.00

-0.10

0.00

0.10

0.20

R e = 50

205

St

2

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CD

U aDo

2 2

C C CD DF DP

C dDF 4

0 2

0Re( , ) sin

CU

p dDPo

1

0 22

0

( , ) sin

Drag Coefficient

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( )Re Re

sin

p Gr

0

02

4

C gDF 16

301Re

( , )

C gg Gr

DP

8

30 0

2

311

2Re( , ) ( , )

Re

p p p dGr

d*

Re( )

Resin

4

2

pGr

Pg

gnn

n

Nn

n*( , )

Re( cos )

Re(cos ) ( ) ( , ) ( , )

2

1

14

1 0 0

Pressure Coefficient

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Re Present

Wong

et al.

Sayegh&

Gauvin

Dennis&

Walker Whitaker

Dennis&

Walker

1 2.263 - 2.232 2.260 2.402 2.22

10 3.326 3.282 3.323 3.358 3.349 3.18

20 4.046 3.971 4.022 4.065 3.950 3.86

30 4.584 - 4.560 - 4.421 4.36

60 5.855 5.889 - - 5.511 5.65

100 7.153 7.351 - - 6.625 7.11

Re Gr Present Wong et al.

Percentage

Difference

60 720 5.943 6.021 1.312

60 14400 7.331 7.183 2.019

100 2000 7.190 7.509 4.437

VALIDATION

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SOME RESULTSR e G r G r / R e 2

N uN N f o r c e du u/ ( )

5 0 . 0 0 . 0 2 . 2 3 9 4 1 . 0 0 0 05 1 2 . 5 0 . 5 2 . 5 5 8 6 1 . 1 4 2 65 2 5 1 . 0 2 . 6 9 5 7 1 . 2 0 3 85 1 2 5 5 . 0 3 . 1 1 8 6 1 . 3 9 2 65 2 5 0 1 0 . 0 3 . 3 7 4 1 1 . 5 0 6 7

2 0 0 . 0 0 . 0 2 . 5 2 9 6 1 . 0 0 0 02 0 1 0 0 0 . 2 5 3 . 1 6 0 2 1 . 2 4 9 32 0 2 0 0 0 . 5 3 . 3 8 6 2 1 . 3 3 8 62 0 4 0 0 1 . 0 3 . 6 5 8 7 1 . 4 4 6 42 0 2 0 0 0 5 . 0 4 . 5 3 8 0 1 . 7 9 4 02 0 4 0 0 0 1 0 . 0 5 . 0 8 5 8 2 . 0 1 0 5

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0 30 60 90 120 150 180

1

2

3

4

5

=0.125

0.0 , 1 .0

0 .25

0.375

0.5

0.625

0.75

0.875

fo r th e c a se o f R e = 5 0 , S = , G r = 0 d u rin g o n e co m p le te c y c le .

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19.0 19.5 20.0

2

3

4

5

6

R e = 200

100

50

2010

fo r th e case o f S = , G r = 0 a t d iffe ren t R ey n o ld s n u m b ers d u rin g cy c le n o . 2 0 .

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0 30 60 90 120 150 180

2

3

4

5

0.125

= 0.0 , 1 .0

0.25

0.375

0.5

0.625

0.75

0.875

fo r th e ca se o f R e = 5 , S = , G r = 2 5 0 d u rin g o n e co m p le te cy c le .

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9.00 9.25 9.50 9.75 10.00

2

3

4

5

6

G r = 4000

fo r th e case o f S = , R e = 2 0 a t d iffe ren t G rash o f n u m b ers d u rin g o n e co m p le te o sc illa tio n .

2000

400

200

100

0

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-10 -8 -6 -4 -2

r T em p e ra tu re d is trib u tio n a lo n g a t = 0 .0 fo r th e ca se o f R e = 2 0 .

G r = 0.0 100 400

0.0

0.2

0.4

0.6

0.8

1.0

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10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

r

T em p era tu re d is trib u tio n a lo n g a t = 0 .0 fo r th e ca se o f R e = 2 0 .

G r = 0.0 100

400

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0.23000.0000 0.2500

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0.2625 0.2725

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0.2750 0.3125

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0.5000 0.5550 0.5650

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0.6000 0.7250 0.7500

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FUTURE RESEARCH

•Heat transfer from a sphere in a spinning infinite fluid.

•Heat transfer from other geometries such as oblate and prolate spheroids.

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SPHEROIDS

x c cosh sin cos y c cosh sin sin z c sinh cos

h h c sinh cos2 2

h c cosh sin

c'

A x is o f sy m m e try

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(sinh cos ) (cosh sin

) (cosh sin

)

Re cosh(cosh )

sin(sin )

2 2

2 1 1

t

cosh sin (sinh cos ) cosh (cosh

)

sin (sin

)

2 2 1

10

Stream Function

Vorticity

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(sinh cos )cosh sin

tanh cot

2 2

2

2

2

2

1

2

t

Pe

1

22 2cosh sin ( )F t as

0 0, and as

0 1, at o

Boundary Conditions

Energy

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A NOTE ON THE PROBLEM OF SPINNING STREAM

ee2

2

2

2

2

2

t

sin Recot

e as2 2 sin

y t P zn nk

n

( , ) ( )1

332211

321

332211

321

hwhwhwqqq

uhuhuh

hhh1

w

wFpw)w(tw 2

w

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