MIXED CONVECTION FLOW OF JEFFREY FLUID ALONG AN INCLINED STRETCHING CYLINDER WITH DOUBLE STRATIFICATION EFFECT

THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 849 MIXED CONVECTION FLOW OF JEFFREY FLUID ALONG AN 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 Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan b Nonlinear Analysis and Applied Mathematics Research Group, Fac ulty of Sci ence, King Abdulaziz Uni ver sity, Jeddah, Saudi Ara bia Orig i nal sci en tific pa per https://doi.org/10.98/tsci14110605h This pa per ad dresses dou ble strat i fied mixed con vec tion bound ary layer flow of Jeffrey fluid due to an im per me able in clined stretch ing cyl in der. Heat trans fer anal y sis is car ried out with heat gen er a tion/ab sorp tion. Vari able tem per a ture and con cen tra tion are as sumed at the sur face of cyl in der and am bi ent fluid. Non-lin ear par tial dif fer en tial equa tions are re duced into the non-lin ear or di nary dif fer en tial equa tions af ter us ing the suit able trans for ma tions. Con ver gent se ries so lu tions are com 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 Introduction 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 in pharmaceuticals, 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 the non-newtonian flu ids are di vided into three main types i. e., (1) rate type () 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 time only. 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 characteristics of flow over a permeable and impermeable stretching surfaces attained 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 tions in the ad vanced in dus trial and tech no log i cal pro cesses. Fur ther such flows with heat and mass trans fer are more sig nif i cant since the qual ity of fi nal prod uct greatly de pends upon the two fac tors * Corresponding author, e-mail: hfaroog@yahoo.com

850 THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 (1) cooling liquid () rate of stretching phenomenon. Applications of such phenomenon include chemical processing equipment, food-stuff processing, extrusion process, paper production, cooling 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 with stretch ing wall. Mukhopadhyay [7] ex am ined MHD flow in duced by a po rous stretch ing sheet with 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 cal pro 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 and force 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 vices cooled by fans, so lar power col lec tors, flows in the at mo sphere and ocean, etc. Bhattacharyya et al. [1] 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 with mixed 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 of micropolar 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 flow over 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 by com 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 in oceans 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 mal strat i fi ca tion. Dou ble strat i fied flow of nanofluid over a ver ti cal plate was stud ied by Ibrahim and Makinde [0]. Hayat et al. [1] ex am ined ra di a tive flow of Jeffrey fluid past a stretch ing sheet with dou ble strat i fi ca tion ef fects. Mukhopadhyay [] pre sented ther mally strat i fied MHD flow in duced by an ex po nen tially stretch ing sheet. In flu ence of dou ble strat i fi ca tion in MHD flow of micropolar fluid was ex am ined by Srinivasacharya and Upendar [3]. Hayat et al. [4] 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 of non-newtonian flu ids over the stretch ing sheet. It ap pears that the be hav ior of non-newtonian flu ids due to a stretch ing cyl in der is not in ves ti gated widely. There fore, the ob jec tive of pres ent anal y sis is to ex plore the char ac ter is tics of dou ble strat i fied mixed con vec tion flow of Jeffrey fluid 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

THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 851 se ries so lu tions are de vel oped by homotopy anal y sis method [5-31]. 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 for dif fer ent in volved pa ram e ters. Mathematical modeling We con sider the steady and in com press - ible mixed con vec tion flow of Jeffrey fluid past 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 face of cyl in der are as sumed higher than the am bi - ent fluid. Stretch ing ve loc ity is due to two forces on the cyl in der which are equal in mag ni tude but op po site in di rec tion when or i - gin is kept con stant. The con ser va tion laws af ter us ing the bound ary layer ap prox i ma - tions are given: Figure 1. Physical flow problem ( ru) ( rv) 0 (1) x r 3 u v u 3 u u u u u v x v u n u 1 u nl u r 3 r r x r r x r r 1 l1 r r r ( 1 l1 ) 1 r v u u u r x r [ gb ( T T ) gb ( C C )]cosa () with the bound ary con di tions T C u T v T k T T Q 1 0 T T x r rc r r r rc ( ) (3) u C x p v C C 1 C D r r r r u0x ax u( x, r) uw ( x), v( x, r) 0, T ( x, r) Tw ( x) T0 l l dx C( x, r) C w ( x) C0, at r R l bx ex u( x, r) 0, T ( x, r) T T0, C( x, r) C C0 (5) l l 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 = (mr) is ki ne matic vis cos ity, r the fluid den sity, m the dy namic vis cos - p (4)

85 THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 ity, l the ra tio of re lax ation to re tar da tion times, l the re tar da tion time, g the ac cel er a tion due to grav ity, b T the ther mal ex pan sion co ef fi cient, b C the con cen tra tion ex pan sion co ef fi - cient, a the an gle of in cli na tion, c p the spe cific heat at con stant pres sure, k the ther mal con - duc tiv ity, Q 0 the heat gen er a tion/ab sorp tion co ef fi cient, u w (x) the lin ear stretch ing ve loc ity, u 0 the ref er ence ve loc ity, T w (x) and C w (x) are the vari able tem per a ture and con cen tra tion at the sur face of cyl in der, and T, T 0, and T are the fluid, ref er ence, and am bi ent tem per a tures, re spec - tively, D the mass diffusivity, C, C 0, and C 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: u r R 0 T T C C h, q( h), f( h) nl R Tw T0 Cw C0 u0x R u0n u0nx u f ( h), v f ( h), y ( h) Rf ( h) l r l l (6) Equa tion (1) is iden ti cally sat is fied while eqs. ()-(5) are re duced: ( 1 gh) f gf ( 1 l )( ff ( f ) ) gb( f f 3 ff ) 1 b( 1 gh)(( f ) ff iv ) ( 1 l ) l ( q Nf)cos a 0 1 T ( 1 gh) q gq Pr( f q f q Sf dq) 0 (8) ( 1 gh) f gf Sc( ff f f Pf ) 0 (9) f ( 0) 0, f ( 0) 1, q( 0) 1 S, f( 0) 1 P, f ( h) 0, q( h) 0, f( h) 0, as h (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, l the ra tio of re lax ation to re tar da tion times, l 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 tion pa ram e ter, P the solutal strat i fi ca tion pa ram e ter, l 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 Grashof num ber due to tem per a ture, Gr * the Grashof num ber due to con cen tra tion. These pa ram e ters are de fined: t w nl l u mc 0 p n lq0 b e g, b, Pr, Sc, d, S, P, u0r l k D cp ru0 a d Gr T T x Cw C T N Re, g ( ) ( l n b 3 Gr * T w 0 gb Gr,, Gr * C 0 )x 3 n Gr n Skin fric tion co ef fi cient, lo cal Nusselt, and Sherwood num bers are defineds: t w xq w xjw C f, Nu x Sh 1 k T T D C C u w w r 3 ( 0 ), (1) ( 0 ) w m u u u l w l v u, q 1 1 r r x r T k, j D C w (13) r r R r r R r R (7) (11)

THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 853 In dimensionless form these quan ti ties can be ex pressed: 1 1 Cf Re x [ f ( 0) b( f ( 0) f ( 0) g f ( 0) f ( 0) f ( 0) f ( 0)] 1 l1 Nu Sh q( 0), f( 0) Re Re x x (14) where Re x u x 0 / n l is the lo cal Reynolds num ber. Se ries so lu tions Homotopy anal y sis method was first pro posed by Liao in [3] at 199 which is used for the con struc tion of se ries so lu tion of highly non-lin ear prob lems. It is pre ferred over the other 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, q 0, f 0 ) and lin ear op er a tors (L f, L q L f ) for the mo men tum, en ergy, and con cen tra tion equa tions are ex pressed in the forms: f ( h) 1 exp( h), q ( h) ( 1 S )exp( h), and f ( h) ( 1 P)exp( h ) (15) 0 0 0 d 3 f df d q d f L f ( f ), L ( ) and L ( ) d 3 q q q f h dh dh f f (16) dh with L f [ A A exp( h) A exp( h )] (17) 1 3 0 in which A i (i = 1 7) are the ar bi trary con stants. Con ver gence anal y sis L q [ A exp( h) A exp( h)] (18) 4 5 0 L f [ A exp( h) A exp( h)] (19) 6 7 0 The se ries so lu tions by homotopy anal y sis method de pend upon the aux il iary pa ram e - ter,. This aux il iary pa ram e ter pro vides us great free dom to ad just and con trol the con ver gence re gion of the se ries so lu tions. There fore, we have plot ted the -curves at the 15 th or der of ap - prox i ma tions in the figs. ()-(4). It is seen that per mis si ble val ues of f, q, and f are 0.95 f, 0.95 q and 0.95 f. Results and discussion The main ob jec tive of this sec tion is to ex plore the im pacts of var i ous pa ram e ters on the 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 -

854 THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 Figure. The -curve for f Figure 3. The -curve for q Figure 4. The -curve for f in der while it in creases away from the sur face. Ve loc ity curves van ish as ymp tot i cally at some large val ues of h. It is also noted that bound ary layer 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 der with the fluid de creases which of fers less re sis - tance to the fluid mo tion. There fore, ve loc ity pro file in creases. In flu ence of l on ve loc ity dis tri bu tion is plot ted in fig. 6. It is shown that ve loc ity pro file de creases for larger val ues of l. Since l is the ra tio of re lax ation to re tar da - tion times so for larger val ues of l 1 the re lax - ation time in creases which pro duces more re sis tance to the fluid mo tion. Hence ve loc ity pro file de creases. Vari a tion of mixed con vec tion pa ram e ter l T 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 of Deborah 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 crease of 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 5. Effect of g on velocity profile Figure 6. Effect of l on velocity profile

THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 855 Figure 7. Effect of l T 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 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 is the ra tio of con cen tra tion to ther mal buoy ancy forces 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 ity pro file de creases with an in crease in a. Through in crease of a the grav ity af fect de creases which re sults in the re duc tion of ve loc ity pro file. Vari a - Figure 11. Effect of S on velocity profile 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. 1. Tem per a ture pro file de creases near the sur face of cyl in der and it in creases away from the sur face. Fig ure 13 pres ents the be hav ior of mixed con vec tion pa ram e ter l T on tem per a ture dis - tri bu tion. With the in crease of mixed con vec tion pa ram e ter l T 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.

856 THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 Figure 1. Effect of g on temperature profile Figure 13. Effect of l T 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 on temperature profile Both tem per a ture and ther mal bound ary layer thick ness de crease for higher val ues of Deborah num 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 ior of tem per a ture pro file. Due to the in crease in a the grav ity af fect de creases which re sults in the re duc tion of rate of heat trans fer. There fore, tem - per a ture pro file in creases. Fig ure 16 pro vides the anal y sis for the vari a tion of Prandtl num ber on the tem per a ture pro file. It is no ticed that a de - crease in the tem per a ture pro file and ther mal bound ary layer thick ness is ob served when Prandtl 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 the ther mal diffusivity de creases which re sults in the re duc tion of tem per a ture pro file. Flu ids with high Prandtl num ber cor re sponds to low ther mal diffusivity. Be hav ior of ther mal strat i fi ca tion pa ram e ter, S, on the tem per a ture dis tri bu tion is sketched in fig. 17. Higher val ues of ther mal strat i fi ca tion pa ram e ter re duce the tem per a ture and ther mal bound ary layer thick ness. This is due 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 -

THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 857 Figure 17. Effect of S on temperature profile Figure 18. Effect of d on temperature profile Figure 19. Effect of on concentration profile Figure 0. Effect of Schmidt number on concentration 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 and ther 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 it in creases away from the sur face. In flu ence of Schmidt num ber on con cen tra tion pro files is sketched in fig. 0. Con cen tra tion pro file de creases for larger val ues of Schmidt num ber. It is the 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. 1. 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 it is also noted that con cen tra tion bound ary layer thick ness de creases. Fig ures and 3 show the im 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 l T. 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 6 th or der of ap prox i ma tion is suf fi cient for mo men tum equa - tion and 8 th or der of ap prox i ma tion is suf fi cient for en ergy and con cen tra tion equa tions. Ta ble 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

858 THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 Figure 1. Effect of P on concentration profile Figure. Effect of a and b on skin friction Figure 3. Effect of l T and g on skin friction l 1, l T, and N. Neg a tive val ues of skin fric - tion phys i cally mean that cyl in der ex erts a drag force on the fluid par ti cles. Ta ble 3 shows the be hav ior of dif fer ent pa ram e ters 6 8 35 1.3979 1.3979 1.3979 1.1479 1.1478 1.1478 1.9 1.94 1.94 on Nusselt num ber. It is an a lyzed that Nusselt num ber in creases for higher val ues of g, l T, b, and Pr 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, l T, 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. Concluding remarks 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., l 1 = 1.1, l T = 0.1, b = 0.1, N = 0.1, a = /4. Pr = 1.5, S = 0.1, d = 0.1, Sc = 1.5, and P = 0.1 Or der of ap prox i ma tion f '(0) q'(0) f' (0) 1 1.666 1.0613 1.0950 5 1.3974 1.1497 1.193 10 1.3990 1.1496 1.66 15 1.3984 1.1484 1.79 0 1.3981 1.1480 1.86 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.

THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 859 Ta ble. Ef fects of var i ous pa ram e ters on the skin fric tion co ef fi cient when Pr = 1., Sc = 1., P = 0.1 g l 1 l T b N a S 0.5Re x 0.5 C f 0.1 1.1 0.1 0.1 0.1 /4 0.1 0.6931 0. 0.7304 0.5 0.7850 0. 1.1 0.1 0.1 0.1 /4 0.1 0.7304 1. 0.711 1.5 0.6643 0. 1.1 0.1 0.1 0.1 /4 0.1 0.7304 0. 0.6999 0.5 0.6159 0. 1.1 0.1 0.1 0.1 /4 0.1 0.7304 0. 0.7655 0.5 0.8643 0. 1.1 0.1 0.1 0.1 /4 0.1 0.7304 0. 0.773 0.5 0.7198 0. 1.1 0.1 0.1 0.1 0.0 0.1 0.7174 /4 0.7304 /3 0.7394 0. 1.1 0.1 0.1 0.1 /4 0.1 0.7304 0. 0.7345 0.5 0.7491 Table 3. Effect of various involved parameters on the lo cal Nusselt num ber when N = 0.1, Sc = 1., P = 0.1 g l 1 l T b a Pr S d q'(0) 0.0 1.1 0.1 0.1 /4 1. 0.1 0.1 0.973 0. 0.9954 0.3 1.096 0. 1.1 0.1 0.1 /4 1. 0.1 0.1 0.9954 1. 0.9873 1.5 0.9657 0. 1.1 0.1 0.1 /4 1. 0.1 0.1 0.9954 0. 1.0919 0.5 1.0649 0. 1.1 0.1 0.1 /4 1. 0.1 0.1 0.9954 0. 1.0106 0.5 1.0513 0. 1.1 0.1 0.1 0.0 1. 0.1 0.1 1.0064 /4 0.9954 /3 0.9857 0. 1.1 0.1 0.1 /4 0.8 0.1 0.1 0.7687 1.0 0.8857 1. 0.9954 0. 1.1 0.1 0.1 /4 1. 0.1 0.1 0.9954 0. 0.9640 0.5 0.8584 0. 1.1 0.1 0.1 /4 1. 0.1 0.1 0.9954 Skin friction coefficient, Nusselt, and Sherwood numbers increase via curvature 0. 0.8649 parameter. 0.3 0.6771 Higher Prandtl num ber re sults in the re 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 tion pa ram e ter in creases. Nomenclature a, b, d, e di men sional con stants, [m] C fluid con cen tra tion, [kgkg 1 ] C f skin fric tion, [ ] C w wall con cen tra tion C 0 ref er ence concentraion C am bi ent con cen tra tion, [kgkg 1 ] c p special heat at constant presure, [Jkg 1 C 1 ] D mass diffusivity, [m s 1 ] f dimensionless stream func tion, [ ] Gr tem per a ture Grashof numbers, [ ] Gr* concentration Grashof numbers, [ ] g grav i ta tional ac cel er a tion, [ms ] q aux il iary pa ram e ters for tem per a ture, [ ] f aux il iary pa ram e ters for con cen tra tion, [ ] f auxiliary parameter for momentum, [ ]

860 THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 Ta ble 4. Ef fect of var i ous in volved pa ram e ters on the Sherwood num ber when l 1 = 1.1, Pr = 1., S = 0.1, d g l 1 b N a Sc P f'(0) 0.1 0.1 0.1 0.1 p/4 1. 0.1 1.0385 0. 1.0750 0.5 1.1795 0. 0.1 0.1 0.1 p/4 1. 0.1 1.0750 0. 1.0948 0.5 1.1308 g l 1 b N a Sc P f'(0) 0. 0.1 0.1 0.1 p/4 1. 0.1 1.0750 0. 1.0866 0.5 1.1180 0. 0.1 0.1 0.1 p/4 1. 0.1 1.0750 0. 1.0768 0.5 1.0798 0. 0.1 0.1 0.1 0.0 1. 0.1 1.0810 p/4 1.0750 p/3 1.0707 0. 0.1 0.1 0.1 p/4 0.8 0.1 0.8348 1.0 0.9600 1. 1.0750 0. 0.1 0.1 0.1 p/4 1. 0.1 1.0750 0. 1.0383 0.5 0.933 j w mass flux, [kgs 1 m ] k ther mal con duc tiv ity, [Wm 1 k 1 ] L f lin ear op er a tor for mo men tum L q lin ear op er a tors for en ergy...and con cen tra tion l char ac ter is tics length, [m] N con cen tra tion to ther mal buoy ancy ra tio Nu x Nusselt num ber, [ ] Pr Prandtl num ber, [ ] Q 0 heat gen er a tion/ab sorp tion co ef fi cient, [J] q w sur face heat flux, [Wm ] P solutal strat i fied pa ram e ters, [ ] r, x space co -or di nates, [m] Re x 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] T w wall tem per a ture T 0 ref er ence tem per a ture T am bi ent tem per a ture, [K] u, v ve loc ity com po nents, [ms 1 ] u 0 ref er ence ve loc ity, [ms 1 ] u w stretch ing sur face ve loc ity, [ms 1 ] x stream funtion, [ ] Greeks sym bols a an gle of in cli na tion b Deborah num ber in terms of re tar da tion...time, [ ] b C solutal expansion coefficient, [ ]

THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 861 b T 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, [ ] l T mixed con vec tion pa ram e ter, [ ] l ra tio of re lax ation to re tar da tion times, [ ] l re tar da tion time, [s] m dy namic vis cos it, [kgm 1 s 1 ] r den sity, [kgm 3 ] t w sur face shear stress, [Nm ] n ki ne matic vis cos ity, [m s 1 ] y stream function, [ ] c dimensionless vari able, [ ] Ref er ences [1] Hussain, T., et al., Ra di a tive Hydromagnetic Flow of Jeffrey Nanofluid by an Ex po nen tially Stretch ing Sheet, Plos one., 9 (014), 8, pp. e103719 [] Turkyilmazoglu, M., Pop, I., Ex act An a lyt i cal So lu tions for the Flow and Heat Trans fer Near the Stag na - tion Point on a Stretch ing / Shrink ing Sheet in a Jeffrey Fluid, Int. J. Heat Mass Trans fer., 57 (013), 1, pp. 8-88 [3] Ellahi, R., et al., Blood Flow of Jeffrey Fluid in a Catherized Ta pered Ar tery with the Sus pen sion of Nanoparticles, Phys. Let ter A., 378 (014), 40, pp. 973-980 [4] Hayat, T., et al., Three Di men sional Stretched Flow of Jeffrey Fluid with Vari able Ther mal Con duc tiv ity and Ther mal Ra di a tion, Appl. Math. Mech. Engl. Ed., 34 (013), 7, pp. 83-83 [5] Hayat, T., et al., Ther mal Ra di a tion Ef fects in Squeez ing Flow of a Jeffrey Fluid, Eur. Phys. J. Plus., 18 (013), 8, p. 85 [6] Freidoonimehr, N., et al., An a lyt i cal Mod el ling of Three-Di men sional Squeez ing Nanofluid Flow in a Ro - tat ing Chan nel on a Lower Stretch ing Po rous Wall, Math. Prob. Eng., 014 (014), ID6978 [7] Mukhopadhyay, S., Slip Ef fects on MHD Bound ary Layer Flow over an Ex po nen tially Stretch ing Sheet with Suc tion/blow ing and Ther mal Radiation, Ain Shams Eng. J., 4 (013), 3, pp. 485-491 [8] Hayat, T., et al., MHD Stag na tion Point Flow of Sec ond Grade Fluid over a Stretch ing Cyl in der with Heat and Mass Trans fer, Int. J. Non lin ear Sci. Num ber. Simul., 15 (014), 6, pp. 365-376 [9] Turkyilmazoglu, M., Three Di men sional MHD Flow and Heat Trans fer over a Stretch ing/ Shrink ing Sur - face in a Viscoelastic Fluid with Var i ous Phys i cal Ef fects, Int. J. Heat Mass Trans fer, 78 (014), Nov., pp. 150-155 [10] Mukhopadhyay, S., Chem i cally Re ac tive Sol ute Trans fer in Bound ary Layer Flow along a Stretch ing Cyl - in der in Po rous Me dium, Afr. Mat., 5 (014), 1, pp. 1-10 [11] Sheikholeslami, M., et al., In ves ti ga tion of Heat and Mass Trans fer of Ro tat ing MHD Vis cous Flow be - tween a Stretch ing Sheet and a Po rous Sur face, Eng. Comput., 30 (013), 3, pp. 357-378 [1] Bhattacharyya, K., et al., Sim i lar ity So lu tion of Mixed Con vec tion Bound ary Layer Slip Flow over a Ver - ti cal Plate, Ain Shams Eng. J., 4 (013),, pp. 99-305 [13] Hayat, T., et al., Melt ing Heat Trans fer in the Stag na tion-point Flow Maxwell Fluid with Dou ble-dif fu - sive Con vec tion, In ter na tional Jour nal of Nu mer i cal Meth ods for Heat and Fluid Flow, 4 (014), 3, pp. 760-774 [14] Turkyilmazoglu, M., The An a lyt i cal So lu tion of Mixed Con vec tion Heat Trans fer and Fluid Flow of a MHD Viscoelastic Fluid over a Per me able Stretch ing Sur face, Int. J. Mech. Sci., 77 (013), Dec., pp. 63-68 [15] Rashad, A. M., et al., Mixed Con vec tion Flow of a Micropolar Fluid over a con tin u ously Mov ing Ver ti cal Sur face Im mersed in a Ther mally and Solutally Strat i fied Me dium with Chem i cal Re ac tion, J. Tai wan Inst. Chem. Eng., 45 (014), 5, pp. 163-169 [16] Rashidi, M. M., et al., Mixed Con vec tion Heat Trans fer for MHD Viscoelastic Fluid Flow over a Po rous Wedge with Ther mal Ra di a tion, Adv. Mech. Eng., 014 (014), ID735939 [17] Hayat, T., et al., Three-Di men sional Mixed Con vec tion Flow of Viscoelastic Fluid with Ther mal Ra di a - tion and Con vec tive Con di tions, Plos one., 9 (014), 3, pp. e90038 [18] Ellahi, R., et al., A Study on the Mixed Con vec tion Bound ary Layer Flow and Heat Trans fer over a Ver ti - cal Slen der Cyl in der, Ther mal Sci ence, 18 (014), 4. pp. 147-158 [19] Singh, G., Makinde, O. D., Mixed Con vec tion Slip Flow with Tem per a ture Jump along a Mov ing Plate in Pres ence of Free Stream, Ther mal Sci ence, 19 (015), 1, pp. 119-18

86 THERMAL SCIENCE: Year 017, Vol. 1, No., pp. 849-86 [0] Ibrahim, W., Makinde, O. D., The Ef fect of Dou ble Strat i fi ca tion on Bound ary Layer Flow and Heat Trans fer of Nanofluid over a Ver ti cal Plate, Computers and Fluids., 86 (013), Nov., pp. 433-441 [1] Hayat, T., et al., Ther mal and Con cen tration Stratifications Effects in Radiative Flow of Jeffrey Fluid over a Stretch ing Sheet, Plos one., 9 (014), Oct., pp. e107858 [] Mukhopadhyay, S., MHD Bound ary Layer Flow and Heat Trans fer over an Ex po nen tially Stretch ing Sheet Em bed ded in a Ther mally Strat i fied Me dium, Alex. Eng. J., 5 (103), 3, pp. 59-65 [3] Srinivasacharya, D., Upendar, M., Ef fect of Dou ble Strat i fi ca tion on MHD Free Con vec tion in a Micropolar Fluid, J. Egyp. Math. Soc., 1 (013), 3, pp. 370-378 [4] Hayat, T., et al., Ther mally Strat i fied Stag na tion Point Flow of an Oldroyd-B Fluid, Int. J. Non lin ear Sci. Numer. Simulat., 15 (014), 1, pp. 77-86 [5] Liao, S. J., Homotopy Anal y sis Method in Non-Lin ear Dif fer en tial Equa tions, Springer and Higher Ed u - ca tion Press, Hei del berg, Ger many, 01 [6] Rashidi, M. M., et al., In ves ti ga tion of En tropy Gen er a tion in MHD and Slip Flow over a Ro tat ing Po rous Disk with Vari able Prop er ties, Int. J. Heat Mass Trans., 70 (014), Mar., pp. 89-917 [7] Abbasbandy, S., Jalili, M., Determination of Optimal Convergence-Control Parameter Value in Homotopy Analysis Method, Numer. Algor., 64 (013), 4, pp. 593-605 [8] Hayat, T., et al., Melt ing Heat Trans fer in the Stag na tion Point Flow of Powell-Eyring Fluid, J. Thermophys. Heat Trans., 7 (013), 4, pp. 761-766 [9] 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 (01), 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., 18 (013), ID157 [31] Hayat, T., et al., MHD Flow of Nanofluid over Per me able Stretch ing Sheet with Con vec tive Bound ary Con di tions, Ther mal Sci ence, 0 (016), 6, pp. 1835-1845 Paper submitted: November 6, 014 Paper revised: March 11, 015 Paper accepted: March 1, 015 017 So ci ety of Ther mal En gi neers of Ser bia Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This is an open ac cess ar ti cle dis trib uted un der the CC BY-NC-ND 4.0 terms and con di tions