MIXED CONVECTION FLOW OF JEFFREY FLUID ALONG AN …thermalscience.vinca.rs/pdfs/papers-2015/TSCI141106052H.pdf · AN INCLINED STRETCHING CYLINDER WITH DOUBLE STRATIFICATION EFFECT

MIXED CONVECTION FLOW OF JEFFREY FLUID ALONGAN INCLINED STRETCHING CYLINDER WITH

DOUBLE STRATIFICATION EFFECT

by

Tasawar HAYAT a,b, Sajid QAYYUM a, Mu ham mad FAROOQ a*,Ahmad ALSAEDI b, and Mu ham mad AYUB a

a De part ment of Math e mat ics, Quaid-I-Azam Uni ver sity, Islamabad, Pa ki stanb Non lin ear Anal y sis and Ap plied Math e mat ics Re search Group,

Fac ulty of Sci ence, King Abdulaziz Uni ver sity, Jeddah, Saudi Ara bia

Orig i nal sci en tific pa perhttps://doi.org/10.2298/TSCI141106052H

This pa per ad dresses dou ble strat i fied mixed con vec tion bound ary layer flow ofJeffrey fluid due to an im per me able in clined stretch ing cyl in der. Heat trans feranal y sis is car ried out with heat gen er a tion/ab sorp tion. Vari able tem per a ture andcon cen tra tion are as sumed at the sur face of cyl in der and am bi ent fluid. Non-lin earpar tial dif fer en tial equa tions are re duced into the non-lin ear or di nary dif fer en tialequa tions af ter us ing the suit able trans for ma tions. Con ver gent se ries so lu tions arecom puted. Effects of var i ous per ti nent pa ram e ters on the ve loc ity, tem per a ture, and con cen tra tion dis tri bu tions are an a lyzed graph i cally. Nu mer i cal val ues of skin fric -tion co ef fi cient, Nusselt, and Sherwood num bers are also com puted and dis cussed.

Key words: Jeffrey fluid model, stretch ing cyl in der, mixed con vec tion,heat gen er a tion/ab sorp tion, dou ble strat i fi ca tion

In tro duc tion

Anal y sis of non-newtonian flu ids is still a topic of great in ter est. Sci en tists have stim u -lated in this field of re search due to nu mer ous ap pli ca tions of non-newtonian flu ids inpharmaceuticals, phys i ol ogy, fi ber tech nol ogy, food prod ucts, coat ing of wires, crys tal growth,etc. Char ac ter is tics of non-newtonian flu ids can not be de scribed by a sin gle con sti tu tive re la -tion ship. Hence var i ous mod els of non-newtonain flu ids have been pro posed. Gen er ally thenon-newtonian flu ids are di vided into three main types i. e., (1) rate type (2) differential type,and (3) integral type. Rate type flu ids de scribe the be hav ior of re lax ation and re tar da tion times.Maxwell fluid is a sub class of rate type ma te rial which ex hib its the be hav ior of re lax ation timeonly. This model does not pres ent the be hav ior of re tar da tion time. Thus Jeffrey fluid model[1-5] is pro posed to fill this void. Jeffrey fluid model char ac ter izes the lin ear viscoelastic prop -er ties of flu ids which has wide spread ap pli ca tions in the poly mer in dus tries.

The char ac ter is tics of flow over a per me able and im per me able stretch ing sur faces at -tained lot of in ter est and in spi ra tion of re search ers and sci en tists due to its nu mer ous ap pli ca tionsin the ad vanced in dus trial and tech no log i cal pro cesses. Fur ther such flows with heat and masstrans fer are more sig nif i cant since the qual ity of fi nal prod uct greatly de pends upon the two fac tors

Hayat, T., et al.: Mixed Convection Flow of Jeffrey Fluid along an Inclined Stretching ...

THERMAL SCIENCE: Year 2017, Vol. 21, No. 2, pp. 849-862 849

* Corresponding author, e-mail: [email protected]

(1) cool ing liq uid (2) rate of stretch ing phe nom e non. Ap pli ca tions of such phe nom e non in cludechem i cal pro cess ing equip ment, food-stuff pro cess ing, ex tru sion pro cess, pa per pro duc tion, cool -ing of con tin u ous strips or fil a ments, de sign of heat exchangers, wire, and fi ber coat ing. Re search -ers have ex plored the flow be hav ior due to stretch ing phe nom e non in var i ous di rec tions.Freidoonimehr et al. [6] pre sented 3-D ro tat ing squeez ing nanofluid flow in a chan nel withstretch ing wall. Mukhopadhyay [7] ex am ined MHD flow in duced by a po rous stretch ing sheetwith slip ef fects and ther mal ra di a tion. Hayat et al. [8] stud ied MHD stag na tion point flow of sec -ond grade fluid past a stretch ing cyl in der. The 3-D MHD flow of viscoelastic fluid past a stretch -ing/shrink ing sur face with var i ous phys i cal ef fects was an a lyzed by Turkyilmazoglu [9].Mukhopadhyay [10] in ves ti gated chem i cally re ac tive bound ary layer flow past a stretch ing cyl in -der sat u rated with po rous me dium. Char ac ter is tics of heat and mass trans fer in MHD flow of vis -cous fluid over a stretch ing sur face were ex plored by Sheikholeslami et al. [11].

In re cent years the de po si tion of aero sol has a key role in the ad vanced tech no log i calpro cesses. Ex plic itly the de po si tion of con tam i nant par ti cle on the sur face of fi nal prod ucts has a piv otal role in the elec tronic in dus try. Mixed con vec tion (which is a com bi na tion of nat u ral andforce con vec tions) is one of the main fac tors which af fects the par ti cle de po si tion. Mixed con -vec tion flows ap pear in many nat u ral, in dus trial, and en gi neer ing pro cesses. Such flows oc cur in dry ing of po rous solid, nu clear re ac tors cooled dur ing emer gency shut down, elec tronic de vicescooled by fans, so lar power col lec tors, flows in the at mo sphere and ocean, etc. Bhattacharyya etal. [12] an a lyzed slip ef fect in mixed con vec tion flow past a ver ti cal plate. Hayat et al. [13] stud -ied melt ing heat trans fer char ac ter is tics in the stag na tion point flow of Maxwell fluid withmixed con vec tion. The MHD mixed con vec tion flow of viscoelastic fluid past a po rous stretch -ing sur face was ex am ined by Turkyilmazoglu [14]. Dou ble strat i fied mixed con vec tion flow ofmicropolar fluid with chem i cal re ac tion was ex plored by Rashad et al. [15]. Rashidi et al. [16]stud ied ra di a tive mixed con vec tion flow of viscoelastic fluid over a po rous wedge. Hayat et al.[17] pre sented 3-D ra di a tive mixed con vec tion flow of viscoelastic fluid in the pres ence of con -vec tive bound ary con di tion. Ellahi et al. [18] ex am ined mixed con vec tion bound ary layer flowover a ver ti cal slen der cyl in der. Singh and Makinde [19] ex plored slip ef fects in mixed con vec -tion flow of vis cous fluid past a mov ing plate with free stream.

Strat i fi ca tion is a phe nom e non which plays a key role in many nat u ral, en gi neer ing,and in dus trial pro cesses. It arises due to the vari a tions in tem per a ture and con cen tra tion or bycom bin ing the flu ids of dif fer ent den si ties. Such phe nom e non in cludes ther mal strat i fi ca tion inoceans and res er voirs, het er o ge neous mix tures in at mo sphere, ground wa ter res er voirs and en -ergy stor age. Con cen tra tion of the ox y gen level be comes low in the lower bot tom of the res er -voirs due to bi o log i cal pro cesses. This dif fi culty can be han dled with the im pli ca tion of ther malstrat i fi ca tion. Dou ble strat i fied flow of nanofluid over a ver ti cal plate was stud ied by Ibrahimand Makinde [20]. Hayat et al. [21] ex am ined ra di a tive flow of Jeffrey fluid past a stretch ingsheet with dou ble strat i fi ca tion ef fects. Mukhopadhyay [22] pre sented ther mally strat i fiedMHD flow in duced by an ex po nen tially stretch ing sheet. In flu ence of dou ble strat i fi ca tion inMHD flow of micropolar fluid was ex am ined by Srinivasacharya and Upendar [23]. Hayat et al.[24] stud ied the stag na tion point flow of an Oldroyd-B fluid with ther mally strat i fied me dium.

Lit er a ture sur vey in di cates that most of the re search ers ex am ined the flow be hav ior ofnon-newtonian flu ids over the stretch ing sheet. It ap pears that the be hav ior of non-newtonianflu ids due to a stretch ing cyl in der is not in ves ti gated widely. There fore, the ob jec tive of pres entanal y sis is to ex plore the char ac ter is tics of dou ble strat i fied mixed con vec tion flow of Jeffreyfluid past an in clined stretch ing cyl in der. Heat and mass trans fer is also con sid ered. Tem per a -ture and con cen tra tion at the sur face and away from cyl in der are as sumed vari able. Con ver gent


850 THERMAL SCIENCE: Year 2017, Vol. 21, No. 2, pp. 849-862

se ries so lu tions are de vel oped by homotopy anal y sis method [25-31]. Be hav iors of var i ous per -ti nent pa ram e ters on the ve loc ity, tem per a ture, and con cen tra tion dis tri bu tions are shown graph -i cally. Skin fric tion co ef fi cient, Nusselt, and Sherwood num bers are com puted nu mer i cally fordif fer ent in volved pa ram e ters.

Math e mat i cal mod el ing

We con sider the steady and in com press -ible mixed con vec tion flow of Jeffrey fluidpast an in clined stretch ing cyl in der fig. 1.Flow anal y sis is car ried out with dou ble strat -i fi ca tion and heat gen er a tion-ab sorp tion.Tem per a ture and con cen tra tion at the sur faceof cyl in der are as sumed higher than the am bi -ent fluid. Stretch ing ve loc ity is due to twoforces on the cyl in der which are equal inmag ni tude but op po site in di rec tion when or i -gin is kept con stant. The con ser va tion lawsaf ter us ing the bound ary layer ap prox i ma -tions are given:

¶

¶

¶

¶

( ) ( )ru

x

rv

r+ = 0 (1)

uu

xv

u

r

u

r r

u

r

vu

¶

¶

¶

¶

¶

¶

¶

¶

¶

+ =+

+æ

èç

ö

ø÷ +

+

n

l

nl

l1

1

11

2

2

2

1

3

( )

¶

¶

¶

¶

¶

¶

¶ ¶

¶

¶

¶

¶ ¶¶

¶

¶r

v

r

u

ru

u

x r

u

r

u

x r

rv

u

ru

3

2

2

3

2

2

2

2

1

+ + + +

+ +2 u

x r

T T C CT C

¶ ¶

æ

èç

ö

ø÷

é

ë

êêêê

ù

û

úúúú

+ - + -[ ( ) ( )]cosg gb b a4 4

+

(2)

uT

xv

T

r

k

c

T

r r

T

r

Q

cT T

p p

¶

¶

¶

¶

¶

¶

¶

¶

2

2+ = +

æ

èç

ö

ø÷ + -

r r

1 0 ( )4 (3)

uC

xv

C

rD

C

r r

C

r

¶

¶

¶

¶

¶

¶

¶

¶

2

2+ = +

æ

èç

ö

ø÷

1(4)

with the bound ary con di tions

u x r u xu x

lv x r T x r T x T

ax

l

C x

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

( ,

= = = = = +w w0

00

r C x Cdx

lr R

u x r T x r T Tbx

lC x

) ( ) ,

( , ) , ( , ) , (

= = + =

® ® = +

w at0

00 4 , )r C Cex

l® = +4 0 (5)

In the pre vi ous ex pres sions u and v are the ve loc ity com po nents in the x and r di rec -tions re spec tively, n = (m/r) is ki ne matic vis cos ity, r – the fluid den sity, m – the dy namic vis cos -



Figure 1. Physical flow problem

ity, l1 – the ra tio of re lax ation to re tar da tion times, l2 – the re tar da tion time, g – the ac cel er a tion due to grav ity, bT – the ther mal ex pan sion co ef fi cient, bC – the con cen tra tion ex pan sion co ef fi -cient, a – the an gle of in cli na tion, cp – the spe cific heat at con stant pres sure, k – the ther mal con -duc tiv ity, Q0 – the heat gen er a tion/ab sorp tion co ef fi cient, uw(x) – the lin ear stretch ing ve loc ity,u0 – the ref er ence ve loc ity, Tw(x) and Cw(x) are the vari able tem per a ture and con cen tra tion at the sur face of cyl in der, and T, T0, and T4 are the fluid, ref er ence, and am bi ent tem per a tures, re spec -tively, D – the mass diffusivity, C, C0, and C4 are the fluid, ref er ence, and am bi ent con cen tra -tions, re spec tively, l – the char ac ter is tics length, a, b, d, and e are the di men sional con stants. Us -ing the trans for ma tions:

hn

q h f h=-æ

èç

ö

ø÷ =

-

-=

-

-

u

l

r R

R

T T

T T

C C

C C

u

w w

02 2

0 02, ( ) , ( )4 4

= ¢ = - =u x

lf v

R

r

u

lf

u x

lRf0 0 0

2

( ), ( ), ( ) ( )hn

h y hn

h

(6)

Equa tion (1) is iden ti cally sat is fied while eqs. (2)-(5) are re duced:

( ) ( )( ( ) ) (1 2 2 1 312- ¢¢¢ + ¢¢ + + ¢¢ - ¢ + ¢ ¢¢ - ¢¢gh g l gbf f ff f f f f ¢ +

+ + ¢¢ - + + + =

f

f ff Niv

)

( )(( ) ) ( ) ( )cosb gh l l q f a1 2 1 021 T

(7)

( ) Pr( )1 2 2 0+ ¢¢ + ¢ + ¢ - ¢ - ¢ + =gh q gq q q dqf f Sf (8)

( ) ( )1 2 2 0+ ¢¢ + ¢ + ¢ - ¢ - ¢ =gh f gf f fSc f f Pf (9)

f f S P f( ) , ( ) , ( ) , ( ) , ( ) , ( ) ,0 0 0 1 0 1 0 1 0 0= / ¢ = = - = - ¢ ® ®q f h q h f( ) ,h h® ®0 as 4 (10)

where g is the cur va ture pa ram e ter, b – the Deborah num ber in terms of re tar da tion time, l1 – thera tio of re lax ation to re tar da tion times, l2 – the re tar da tion time, Pr – the Prandtl num ber, Sc –the Schmidt num ber, d – the heat gen er a tion/ab sorp tion pa ram e ter, S – the ther mal strat i fi ca tionpa ram e ter, P – the solutal strat i fi ca tion pa ram e ter, lT – the ther mal buoy ancy (or mixed con vec -tion) pa ram e ter, and N – the ra tio of con cen tra tion to ther mal buoy ancy forces, Gr – the Grashofnum ber due to tem per a ture, Gr* – the Grashof num ber due to con cen tra tion. These pa ram e tersare de fined:

gn

bl m

k

nd

r= = = = = = =

l

u R

u

l

c

D

lQ

c uS

b

aP

ep

p02

2 0 0

0

, , Pr , , , ,Scd

T T xN

C CT

w

,

Re,

( ), ,

(*l n

b

n

b= =

-= =

-GrGr

g Gr

GrGr

gT w*

C

2

03

2

03

2

)x

n

(11)

Skin fric tion co ef fi cient, lo cal Nusselt, and Sherwood num bers are defineds:

C

u

xq

k T T

xj

D C Cw

fw

xw

w

w

w

Nu Sh= =-

-=

-

-

t

r1

23 0 0

,( )

,( )

(12)

tm

llw w=

++ +

æ

èç

ö

ø÷

é

ëê

ù

ûú =

=1 1

2 2

¶

¶

¶

¶

¶

¶ ¶

2 2u

rv

u

ru

u

x rq

r R

, -æ

èç

ö

ø÷ = -

æ

èç

ö

ø÷

= =

kT

rj D

C

rr R r R

¶

¶

¶

¶, w (13)



In dimensionless form these quan ti ties can be ex pressed:

1

2

1

10 0 0 0 0

1

C f f f f f ff xRe [ ( ) ( ( ) ( ) ( ) ( )=+

¢¢ + - ¢¢¢ - ¢¢ + ¢l

b g ( ) ( )]

Re( ), ( )

0 0

0 0

¢¢

= - ¢ = - ¢

f

Nu Sh

Rex x

q f(14)

where Re /x = u x l02 n is the lo cal Reynolds num ber.

Se ries so lu tions

Homotopy anal y sis method was first pro posed by Liao in [23] at 1992 which is usedfor the con struc tion of se ries so lu tion of highly non-lin ear prob lems. It is pre ferred over theother meth ods due to the fol low ing ad van tages.– It does not de pend upon the small or large pa ram e ters.– It en sures the con ver gence of se ries so lu tions.– It pro vides us great choice to se lect the base func tion and lin ear op er a tor.

To pro ceed with such method, it is es sen tial to de fine the ini tial guess and lin ear op er a -tor. So ini tial guesses ( , , )f 0 0 0q f and lin ear op er a tors (Lf, Lq, Lf) for the mo men tum, en ergy,and con cen tra tion equa tions are ex pressed in the forms:

f S P0 0 01 1 1( ) exp( ), ( ) ( )exp( ), ( ) ( )eh h q h h f h= - - = - - = -and xp( )-h (15)

L L Lf ff f

( ) , ( ) ( )= - = - = -d

d

d

d

d

dand

d

d

3

3

2

2

2

2h hq

q

hq f

f

hfq f (16)

with

L f [ exp( ) exp( )]A A A1 2 3 0+ - + =h h (17)

Lq h h[ exp( ) exp( )]A A4 5 0- + = (18)

Lf h h[ exp( ) exp( )]A A6 7 0- + = (19)

in which Ai (i = 1 – 7) are the ar bi trary con stants.

Con ver gence anal y sis

The se ries so lu tions by homotopy anal y sis method de pend upon the aux il iary pa ram e -ter, h. This aux il iary pa ram e ter pro vides us great free dom to ad just and con trol the con ver gencere gion of the se ries so lu tions. There fore, we have plot ted the h-curves at the 15th or der of ap -prox i ma tions in the figs. (2)-(4). It is seen that per mis si ble val ues of hf, hq, and hf are –0.95 £ hf £ -0.3, –0.95 £ hq £ -0.25, and –0.95 £ hf £ -0.35.

Re sults and dis cus sion

The main ob jec tive of this sec tion is to ex plore the im pacts of var i ous pa ram e ters onthe ve loc ity, tem per a ture, and con cen tra tion dis tri bu tions. Fig ure 5 shows the ef fect of cur va -ture pa ram e ter, g, on the ve loc ity pro file. Ve loc ity dis tri bu tion de creases near the sur face of cyl -



in der while it in creases away from the sur face.Ve loc ity curves van ish as ymp tot i cally at somelarge val ues of h. It is also noted that bound arylayer thick ness in creases. In fact for higher val -ues of cur va ture pa ram e ter the ra dius of cyl in der de creases. So con tact sur face area of cyl in derwith the fluid de creases which of fers less re sis -tance to the fluid mo tion. There fore, ve loc itypro file in creases. In flu ence of l1 on ve loc itydis tri bu tion is plot ted in fig. 6. It is shown thatve loc ity pro file de creases for larger val ues of l1. Since l1 is the ra tio of re lax ation to re tar da -tion times so for larger val ues of l1 the re lax -

ation time in creases which pro duces more re sis tance to the fluid mo tion. Hence ve loc ity pro filede creases. Vari a tion of mixed con vec tion pa ram e ter lT on ve loc ity pro file is sketched in fig. 7.It is an a lyzed that ve loc ity pro file is higher for larger val ues of mixed con vec tion pa ram e ter. It is due to the fact that larger val ues of mixed con vec tion pa ram e ter cor re sponds to the higher ther -mal buoy ancy force which is re spon si ble in the en hance ment of ve loc ity pro file. Ef fect ofDeborah num ber, b, (in terms of re tar da tion time) on ve loc ity dis tri bu tion is dis played in fig. 8.Ve loc ity and mo men tum bound ary thick ness are higher for larger val ues of b. With the in creaseof b the re tar da tion time in creases (or elas tic ity of the ma te rial in creases) which is re spon si ble in the en hance ment of ve loc ity pro file. Fig ure 9 shows the in flu ence of ra tio of buoy ancy forces, N,



Figure 2. The h-curve for f Figure 3. The h-curve for q

Figure 4. The h-curve for f

Figure 5. Effect of g on velocity profile Figure 6. Effect of l1 on velocity profile

on the ve loc ity dis tri bu tion. It is noted that ve loc -ity pro file and mo men tum bound ary layer thick -ness are higher for larger val ues of N. Since N isthe ra tio of con cen tra tion to ther mal buoy ancyforces so with the in crease of N the con cen tra tion buoy ancy force in creases which re sults in the en -hance ment of ve loc ity pro file. Fig ure 10 dis plays the ef fect of an gle of in cli na tion, a, on the ve loc -ity dis tri bu tion. It is ob served that the ve loc itypro file de creases with an in crease in a. Throughin crease of a the grav ity af fect de creases whichre sults in the re duc tion of ve loc ity pro file. Vari a -tion of ther mal strat i fi ca tion pa ram e ter S on ve -loc ity dis tri bu tion is sketched in fig. 11. It is an a lyzed that the ve loc ity and mo men tum bound ary layer thick ness de crease with an in crease in ther mal strat i fi ca tion pa ram e ter, S. In fact con vec -tive po ten tial be tween the sur face of cyl in der and am bi ent fluid de creases. Hence ve loc ity pro -file de creases. In flu ence of cur va ture pa ram e ter g on the tem per a ture pro file is dis played in fig.12. Tem per a ture pro file de creases near the sur face of cyl in der and it in creases away from thesur face. Fig ure 13 pres ents the be hav ior of mixed con vec tion pa ram e ter lT on tem per a ture dis -tri bu tion. With the in crease of mixed con vec tion pa ram e ter lT the ther mal buoy ancy force in -creases which is re spon si ble for high rate of heat trans fer. There fore, tem per a ture pro file de -creases. Fig ure 14 shows the char ac ter is tics of Deborah num ber, b, on the tem per a ture pro file.



Figure 7. Effect of lT on velocity profile Figure 8. Effect of b on velocity profile

Figure 9. Effect of N on velocity profile Figure 10. Effect of a on velocity profile

Figure 11. Effect of S on velocity profile

Both tem per a ture and ther mal bound ary layerthick ness de crease for higher val ues of Deborahnum ber, b. Vari a tion of an gle of in cli na tion, a,on tem per a ture dis tri bu tion is ex pressed in fig.15. In crease in a shows the in creas ing be hav iorof tem per a ture pro file. Due to the in crease in athe grav ity af fect de creases which re sults in there duc tion of rate of heat trans fer. There fore, tem -per a ture pro file in creases. Fig ure 16 pro vides theanal y sis for the vari a tion of Prandtl num ber onthe tem per a ture pro file. It is no ticed that a de -crease in the tem per a ture pro file and ther malbound ary layer thick ness is ob served whenPrandtl num ber in creases. Prandtl num ber is the

ra tio of mo men tum diffusivity to ther mal diffusivity. So with the in crease of Prandtl num ber thether mal diffusivity de creases which re sults in the re duc tion of tem per a ture pro file. Flu ids withhigh Prandtl num ber cor re sponds to low ther mal diffusivity. Be hav ior of ther mal strat i fi ca tionpa ram e ter, S, on the tem per a ture dis tri bu tion is sketched in fig. 17. Higher val ues of ther malstrat i fi ca tion pa ram e ter re duce the tem per a ture and ther mal bound ary layer thick ness. This isdue to the fact that the tem per a ture dif fer ence grad u ally de creases be tween the sur face of cyl in -der and am bi ent fluid which causes a re duc tion in the tem per a ture pro file. Ef fect of heat gen er a -



Figure 12. Effect of g on temperature profile Figure 13. Effect of lT on temperature profile

Figure 14. Effect of b on temperature profile Figure 15. Effect of a on temperature profile

Figure 16. Effect of Prandtl number ontemperature profile

tion/ab sorp tion pa ram e ter, d, on tem per ate pro file is shown in fig. 18. Both tem per a ture andther mal bound ary layer thick ness in crease when gen er a tion/ab sorp tion pa ram e ter, d, is in -creased. Here more heat is pro duced dur ing the heat gen er a tion pro cess which in creases the tem -per a ture pro file. Fig ure 19 pres ents the ef fect of cur va ture pa ram e ter, g, on con cen tra tion dis tri -bu tion. It is ob served that con cen tra tion pro file de creases near the sur face of cyl in der and itin creases away from the sur face. In flu ence of Schmidt num ber on con cen tra tion pro files issketched in fig. 20. Con cen tra tion pro file de creases for larger val ues of Schmidt num ber. It isthe ra tio of mo men tum diffusivity to mass diffusivity. Higher val ues of Schmidt num ber cor re -sponds to lower mass diffusivity which re sults in the re duc tion of con cen tra tion pro file. Char ac -ter is tic of solutal strat i fi ca tion pa ram e ter, P, on con cen tra tion pro file is dis played in fig. 21.Con cen tra tion pro file de creases when solutal strat i fi ca tion pa ram e ter, P, is in creased. Fur ther itis also noted that con cen tra tion bound ary layer thick ness de creases. Fig ures 22 and 23 show theim pacts of var i ous pa ram e ters on skin fric tion co ef fi cient. It is an a lyzed that skin fric tion co ef fi -cient is higher for larger val ues of a, b, and g while it de creases with an in crease in lT.

Ta ble 1 shows the con ver gence of se ries so lu tions for mo men tum, en ergy, and con cen -tra tion equa tions. It is noted that 26th or der of ap prox i ma tion is suf fi cient for mo men tum equa -tion and 28th or der of ap prox i ma tion is suf fi cient for en ergy and con cen tra tion equa tions. Ta ble2 pres ents the ef fects of var i ous pa ram e ters on skin fric tion co ef fi cient. It is noted that skin fric -tion co ef fi cient in creases for larger g, b, a, and S while it de creases with in creas ing the val ues of



Figure 17. Effect of S on temperature profile Figure 18. Effect of d on temperature profile

Figure 19. Effect of g on concentration profileFigure 20. Effect of Schmidt number onconcentration profile

l1, lT , and N. Neg a tive val ues of skin fric -tion phys i cally mean that cyl in der ex erts adrag force on the fluid par ti cles. Ta ble 3shows the be hav ior of dif fer ent pa ram e terson Nusselt num ber. It is an a lyzed that Nusselt num ber in creases for higher val ues of g, lT, b, andPr while it de creases with an in crease in l, a, S, and d. Ta ble 4 is con structed to ex am ine the be -hav ior of var i ous pa ram e ters on Sherwood num ber. It is ob served that Sherwood num ber in -creases with the in crease in g, lT, b, N, and Sc while it de creases when a and P are in creased. It is also noted that neg a tive val ues of Nusselt and Sherwood num bers rep re sent heat and mass trans -fer from cyl in der sur face to the fluid i. e., nor mal to the sur face.

Con clud ing re marks

Here we in ves ti gated the dou ble strat i fied mixed con vec tion flow of Jeffrey fluid in -duced by an im per me able in clined stretch ing cyl in der. Heat trans fer char ac ter is tics are ex plored with heat gen er a tion/ab sorp tion. The key points are sum ma rized as fol lows:· Ve loc ity, tem per a ture, and con cen tra tion pro files in crease for larger cur va ture pa ram e ter g

away from the cyl in der.· Ther mal and solutal strat i fi ca tion pa ram e ters re duce the tem per a ture and con cen tra tion,

re spec tively.· Higher val ues of Deborah num ber re sult in the en hance ment of ve loc ity dis tri bu tion.· Tem per a ture pro file en hances with the in crease of heat gen er a tion/ab sorp tion pa ram e ter.



Figure 21. Effect of P on concentration profile Figure 22. Effect of a and b on skin friction

Figure 23. Effect of lT and g on skin friction

Ta ble 1. Con ver gence of the se ries so lu tions for dif -fer ent or der of ap prox i ma tions when g = 0.2, l1 = 1.1, lT = 0.1, b = 0.1, N = 0.1, a = p/4. Pr = 1.5,S = 0.1, d = 0.1, Sc = 1.5, and P = 0.1

Or der ofap prox i ma tion

f '(0) q'(0) f' (0)

1 –1.2666 –1.0613 –1.0950

5 –1.3974 –1.1497 –1.2193

10 –1.3990 –1.1496 –1.2266

15 –1.3984 –1.1484 –1.2279

20 –1.3981 –1.1480 –1.2286

26 –1.3979 –1.1479 –1.2292

28 –1.3979 –1.1478 –1.2294

35 –1.3979 –1.1478 –1.2294

· Skin friction coefficient, Nusselt, andSherwood numbers increase via curvatureparameter.

· Higher Prandtl num ber re sults in there duc tion of tem per a ture pro file while Nusselt num ber in creases.

· Ve loc ity pro file in creases while tem per a ture pro file de creases when mixed con vec tionpa ram e ter – in creases.



Ta ble 2. Ef fects of var i ous pa ram e ters on the skinfric tion co ef fi cient when Pr = 1.2, Sc = 1.2, P = 0.1

g l1 lT b N a S –0.5Rex0.5Cf

0.1 1.1 0.1 0.1 0.1 p/4 0.1 0.6931

0.2 0.7304

0.5 0.7850

0.2 1.1 0.1 0.1 0.1 p/4 0.1 0.7304

1.2 0.7121

1.5 0.6643

0.2 1.1 0.1 0.1 0.1 p/4 0.1 0.7304

0.2 0.6999

0.5 0.6159

0.2 1.1 0.1 0.1 0.1 p/4 0.1 0.7304

0.2 0.7655

0.5 0.8643

0.2 1.1 0.1 0.1 0.1 p/4 0.1 0.7304

0.2 0.7273

0.5 0.7198

0.2 1.1 0.1 0.1 0.1 0.0 0.1 0.7174

p/4 0.7304

p/3 0.7394

0.2 1.1 0.1 0.1 0.1 p/4 0.1 0.7304

0.2 0.7345

0.5 0.7491

Ta ble 3. Ef fect of var i ous in volved pa ram e ters onthe lo cal Nusselt num ber when N = 0.1, Sc = 1.2,P = 0.1

g l1 lT b a Pr S d –q'(0)

0.0 1.1 0.1 0.1 p/4 1.2 0.1 0.1 0.9273

0.2 0.9954

0.3 1.0296

0.2 1.1 0.1 0.1 p/4 1.2 0.1 0.1 0.9954

1.2 0.9873

1.5 0.9657

0.2 1.1 0.1 0.1 p/4 1.2 0.1 0.1 0.9954

0.2 1.0919

0.5 1.0649

0.2 1.1 0.1 0.1 p/4 1.2 0.1 0.1 0.9954

0.2 1.0106

0.5 1.0513

0.2 1.1 0.1 0.1 0.0 1.2 0.1 0.1 1.0064

p/4 0.9954

p/3 0.9857

0.2 1.1 0.1 0.1 p/4 0.8 0.1 0.1 0.7687

1.0 0.8857

1.2 0.9954

0.2 1.1 0.1 0.1 p/4 1.2 0.1 0.1 0.9954

0.2 0.9640

0.5 0.8584

0.2 1.1 0.1 0.1 p/4 1.2 0.1 0.1 0.9954

0.2 0.8649

0.3 0.6771

No men cla ture

a, b, d, e – di men sional con stants, [m]C – fluid con cen tra tion, [kgkg–1]Cf – skin fric tion, [–]Cw – wall con cen tra tionC0 – ref er ence concentraionC4 – am bi ent con cen tra tion, [kgkg–1]cp – special heat at constant presure,

[Jkg–1°C–1]D – mass diffusivity, [m2s–1]

f – dimensionless stream func tion, [–]Gr – tem per a ture Grashof num bers, [–]Gr* – con cen tra tion Grashof num bers, [–]g – grav i ta tional ac cel er a tion, [ms–2]hq – aux il iary pa ram e ters for

tem per a ture, [–]hf – aux il iary pa ram e ters for

con cen tra tion, [–]hf – aux il iary pa ram e ter for mo men tum, [–]



Ta ble 4. Ef fect of var i ous in volved pa ram e ters on the Sherwood num ber when l1 = 1.1, Pr = 1.2, S = 0.1, d = 0.1

g l1 b N a Sc P –f'(0)

0.1 0.1 0.1 0.1 p/4 1.2 0.1 1.0385

0.2 1.0750

0.5 1.1795

0.2 0.1 0.1 0.1 p/4 1.2 0.1 1.0750

0.2 1.0948

0.5 1.1308

g l1 b N a Sc P –f'(0)

0.2 0.1 0.1 0.1 p/4 1.2 0.1 1.0750

0.2 1.0866

0.5 1.1180

0.2 0.1 0.1 0.1 p/4 1.2 0.1 1.0750

0.2 1.0768

0.5 1.0798

0.2 0.1 0.1 0.1 0.0 1.2 0.1 1.0810

p/4 1.0750

p/3 1.0707

0.2 0.1 0.1 0.1 p/4 0.8 0.1 0.8348

1.0 0.9600

1.2 1.0750

0.2 0.1 0.1 0.1 p/4 1.2 0.1 1.0750

0.2 1.0383

0.5 0.9233

jw – mass flux, [kgs–1m–2]k – ther mal con duc tiv ity, [Wm–1k–1]Lf – lin ear op er a tor for mo men tumLq – lin ear op er a tors for en ergy

....and con cen tra tionl – char ac ter is tics length, [m]N – con cen tra tion to ther mal buoy ancy ra tioNux – Nusselt num ber, [–]Pr – Prandtl num ber, [–]Q0 – heat gen er a tion/ab sorp tion

co ef fi cient, [J]qw – sur face heat flux, [Wm–2]P – solutal strat i fied pa ram e ters, [–]r, x – space co -or di nates, [m]Rex – lo cal Reynold num ber, [–]S – ther mal strat i fied pa ram e ters, [–]Sc – Schmidt num ber, [–]

Sh – Sherwood num ber, [–]T – fluid tem per a ture, [K]Tw – wall tem per a tureT0 – ref er ence tem per a ture T4 – am bi ent tem per a ture, [K]u, v – ve loc ity com po nents, [ms–1]u0 – ref er ence ve loc ity, [ms–1]uw – stretch ing sur face ve loc ity, [ms–1]x – stream funtion, [–]

Greeks sym bols

a – an gle of in cli na tionb – Deborah num ber in terms of re tar da tion

...time, [–]bC – solutal ex pan sion co ef fi cient, [–]

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bT – ther mal ex pan sion co ef fi cient, [–] g – cur va ture pa ram e ter, [–]d – heat genearation/ab sorp tion|

pa ram e ter, [–]h – dimensionless co -or di nate, [–]q – dimensionless tem per a ture, [–]f – dimensionless con cen tra tion, [–]lT – mixed con vec tion pa ram e ter, [–]l1 – ra tio of re lax ation to re tar da tion

times, [–]

l2 – re tar da tion time, [s]m – dy namic vis cos it, [kgm–1s–1]r – den sity, [kgm–3]tw – sur face shear stress, [Nm–2]n – ki ne matic vis cos ity, [m2s–1]y – stream function, [–]c – dimensionless vari able, [–]

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[22] Mukhopadhyay, S., MHD Bound ary Layer Flow and Heat Trans fer over an Ex po nen tially Stretch ingSheet Em bed ded in a Ther mally Strat i fied Me dium, Alex. Eng. J., 52 (2103), 3, pp. 259-265

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[27] Abbasbandy, S., Jalili, M., De ter mi na tion of Op ti mal Con ver gence-Con trol Pa ram e ter Value in HomotopyAnal y sis Method, Numer. Algor., 64 (2013), 4, pp. 593-605

[28] Hayat, T., et al., Melt ing Heat Trans fer in the Stag na tion Point Flow of Powell-Eyring Fluid, J. Thermophys. Heat Trans., 27 (2013), 4, pp. 761-766

[29] Turkyilmazoglu, M., So lu tion of the Thomas-Fermi Equa tion with a Con ver gent Ap proach, Commun.Non lin ear Sci. Numer. Simulat., 17 (2012), 11, pp. 4097-4103

[30] Hayat, T., et al., MHD Un steady Squeez ing Flow over a Po rous Stretch ing Plate, Eur. Phys. J. Plus., 128(2013), ID157

[31] Hayat, T., et al., MHD Flow of Nanofluid over Per me able Stretch ing Sheet with Con vec tive Bound aryCon di tions, Ther mal Sci ence, 20 (2016), 6, pp. 1835-1845

Paper submitted: November 6, 2014 Paper revised: March 11, 2015 Paper accepted: March 12, 2015

© 2017 So ci ety of Ther mal En gi neers of Ser biaPublished by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia.

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