MIXED CONVECION FLOW OF JEFFREY FLUID ALONG AN INCLINED SRECHING CYLINDER WIH DOUBLE SRAIFICAION EFFEC by asawar HAYA a,b,sajid QAYYUM a, Muhammad FAROOQ a Ahmad ALSAEDI b and Muhammad AYUB a a Department o Mathematics, Quaid-I-Azam University 4530, Islamabad 44000, Pakistan b Nonlinear Analysis and Applied Mathematics (NAAM) Research Group Faculty o Science, King Abdulaziz University P. O. Box 8007, Jeddah 589, Saudi Arabia Abstract: his paper addresses double stratiied mixed convection boundary layer low o Jerey luid due to an impermeable inclined stretching cylinder. Heat transer analysis is carried out with heat generation/absorption. Variable temperature and concentration are assumed at the surace o cylinder and ambient luid. Nonlinear partial dierential equations are reduced into the nonlinear ordinary dierential equations ater using the suitable transormations. Convergent series solutions are computed. Eects o various pertinent parameters on the velocity, temperature and concentration distributions are analyzed graphically. Numerical values o skin riction coeicient, Nusselt and Sherwood numbers are also computed and discussed. Keywords: Jerey luid model; Stretching cylinder; Mixed convection; Heat generation/absorption; Double stratiication. Introduction Analysis o non-newtonian luids is still a topic o great interest. Scientists have stimulated in this ield o research due to numerous applications o non-newtonian luids in pharmaceuticals, physiology, iber technology, ood products, coating o wires, crystal growth etc. Characteristics o non-newtonian luids cannot be described by a single constitutive relationship. Hence various models o non-newtonain luids have been proposed. Generally the non-newtonian luids are divided into three main types i.e., (i) Rate type (ii) Dierential type and (iii) Integral type. Rate type luids describe the behavior o relaxation and retardation times. Maxwell luid is a subclass o rate type material which exhibits the behavior o relaxation time only. his model does not present the behavior o retardation time. hus Jerey luid model [-5] is proposed to ill this void. Jerey luid model characterizes the linear viscoelastic properties o luids which has wide spread applications in the polymer industries. he characteristics o low over a permeable and impermeable stretching suraces attained lot o interest and inspiration o researchers and scientists due to its numerous applications in the advanced industrial and technological processes. Further such lows with heat and mass transer are more signiicant since the quality o inal product greatly depends upon the two actors (i) cooling liquid (ii) rate o stretching phenomenon. Applications o such phenomenon include chemical processing equipment, ood-stu processing, extrusion process, paper production, cooling o continuous strips or ilaments, design o heat exchangers, wire and iber coating. Researchers have explored the low behavior due to stretching phenomenon in various directions. Freidoonimehr et al. [6] presented three dimensional rotating squeezing nanoluid low in a channel with stretching wall. Mukhopadhyay [7] examined magnetohydrodynamic low induced by a porous stretching sheet with slip eects and thermal radiation. Hayat et al. [8] studied magnetohydrodynamic stagnation point low o second grade luid past a stretching cylinder. hree-dimensional magnetohydrodynamic low o viscoelastic luid past a stretching/shrinking surace with various physical eects was analyzed by urkyilmazoglu [9]. Mukhopadhyay [0] investigated chemically reactive boundary layer low past a stretching cylinder saturated with porous medium. Characteristics o heat and mass transer in magnetohydrodynamic low o viscous luid over a stretching surace were explored by Sheikholeslami et al. []. In recent years the deposition o aerosol has a key role in the advanced technological processes. Explicitly the deposition o contaminant particle on the surace o inal products has a pivotal role in the electronic industry. Mixed convection (which is a combination o natural and orce convections) is one o the main actors which aects the particle deposition. Mixed convection lows appear in many natural, industrial and engineering processes. Such lows occur in drying o porous solid, nuclear reactors cooled during emergency shutdown, electronic devices cooled by ans, solar power collectors, lows in the atmosphere and ocean etc. Bhattacharyya et al. [] analyzed slip eect in mixed convection low past a vertical plate. Hayat et al. [3] studied melting heat transer characteristics in the stagnation point low o Maxwell luid with mixed
convection. MHD mixed convection low o viscoelastic luid past a porous stretching surace was examined by urkyilmazoglu [4]. Double stratiied mixed convection low o micropolar luid with chemical reaction was explored by Rashad et al. [5]. Rashidi et al. [6] studied radiative mixed convection low o viscoelastic luid over a porous wedge. Hayat et al. [7] presented three dimensional radiative mixed convection low o viscoelastic luid in the presence o convective boundary condition. Ellahi et al. [8] examined mixed convection boundary layer low over a vertical slender cylinder. Singh and Makinde [9] explored slip eects in mixed convection low o viscous luid past a moving plate with ree stream. Stratiication is a phenomenon which plays a key role in many natural, engineering and industrial processes. It arises due to the variations in temperature and concentration or by combining the luids o dierent densities. Such phenomenon includes thermal stratiication in oceans and reserviors, heterogeneous mixtures in atmosphere, ground water reservoirs and energy storage. Concentration o the oxygen level becomes low in the lower bottom o the reservoirs due to biological processes. his diiculty can be handled with the implication o thermal stratiication. Double stratiied low o nanoluid over a vertical plate was studied by Ibrahim and Makinde [0]. Hayat et al. [] examined radiative low o Jerey luid past a stretching sheet with double stratiication eects. Mukhopadhyay [] presented thermally stratiied magnetohydrodynamic low induced by an exponentially stretching sheet. Inluence o double stratiication in MHD low o micropolar luid was examined by Srinivasacharya and Upendar [3]. Hayat et al. [4] studied the stagnation point low o an Oldroyd-B luid with thermally stratiied medium. Literature survey indicates that most o the researchers examined the low behavior o non-newtonian luids over the stretching sheet. It appears that the behavior o non-newtonian luids due to a stretching cylinder is not investigated widely. hereore the objective o present analysis is to explore the characteristics o double stratiied mixed convection low o Jerey luid past an inclined stretching cylinder. Heat and mass transer is also considered. emperature and concentration at the surace and away rom cylinder are assumed variable. Convergent series solutions are developed by homotopy analysis method [5-3]. Behaviors o various pertinent parameters on the velocity, temperature and concentration distributions are shown graphically. Skin riction coeicient, Nusselt and Sherwood numbers are computed numerically or dierent involved parameters. Mathematical modeling We consider the steady and incompressible mixed convection low o Jerey luid past an inclined stretching cylinder. Flow analysis is carried out with double stratiication and heat generation/absorption. emperature and concentration at the surace o cylinder are assumed higher than the ambient luid. Stretching velocity is due to two orces on the cylinder which are equal in magnitude but opposite in direction when origin is kept constant. he conservation laws ater using the boundary layer approximations are given as ollows: Fig.. Physical low problem. ( ru) ( rv) 0 x r 3 u u u u v u r r xr u v x r ( ) r r r ( ) u u r v u r x r ( g g ( C C ))cos, C 3 3 u v u u u u r r xr 0 u v x r cp r r r cp Q ( ) () () (3)
C C C C x r r r r u v D( ) with the boundary conditions ux o ax u( x, r) uwx, v( x, r) 0, ( x, r) w( x) 0, l l dx C( x, r) Cw( x) C0, at r R, l bx ex u( x, r) 0, ( x, r) 0, C( x, r) C C0, l l In the above expressions u and v are the velocity components in the x and r directions respectively, ( / ) is the ratio is the kinematic viscosity, is the luid density, is the dynamic viscosity, o relaxation to retardation times, is the retardation time, g is the acceleration due to gravity, is the thermal expansion coeicient, is the concentration expansion coeicient, is angle o inclination, C c p is the speciic heat at constant pressure, is the thermal conductivity, Q 0 is the heat generation / absorption coeicient, uw( x ) is the linear stretching velocity, u 0 is the reerence velocity, w ( x ) and C ( ) w x are the variable temperature and concentration at the surace o cylinder,, 0 and are the luid, reerence and ambient temperatures respectively, D is the mass diusivity, C, C 0 and C are the luid, reerence and ambient concentrations respectively, l is the characteristics length, a, b, d and e are the dimensional constants. Using the transormations u 0 r R C C,,, l R w 0 Cw C0 u0x R u0 u0x u, v, ( ) R ( ) l r l l equation () is identically satisied while Eqs. (-5) are reduced as ollows: ( )( ) 3 iv N S Sc P 0, (0) 0, (0), 0 S, (0) P, ( ) 0, 0, ( ) 0, as, ( ) ( )cos 0, Pr 0, where is the curvature parameter, is the Deborah number in terms o retardation time, is the ratio o relaxation to retardation times, is the retardation time, Pr is the Prandtl number, Sc is the Schmidt number, is the heat generation/absorption parameter, S is the thermal stratiication parameter, P is the solutal stratiication parameter, is the thermal buoyancy (or mixed convection) parameter and N is the ratio o concentration to thermal buoyancy orces, Gr is the Grasho number due to temperature, Gr is the Grasho number due to concentration, hese parameters are deined as ollows: l u c 0 p lq0 b e,, Pr, Sc,, S, P, u R l D c u a d 0 p 0 3 3 Gr g w 0 x Gr gc( Cw C0) x, Gr, N, Gr. Re Gr Skin riction coeicient, local Nusselt and sherwood numbers are deined as ollows: w x qw x jw C, Nu,, x Sh u k( ) D( C C ) w w 0 w 0 (4) (5) (6) (7) (8) (9) (0) () ()
u u u C w v u q w k jw D ( ) r r xr r r R rr r rr,,. In dimensionless orm these quantities can be expressed as ollows: C Re x ( (0) ( (0) (0) (0) (0) (0) (0))), ( ) where Re x ux o l Nu Re x Sh 0, (0), Re is the local Reynolds number. x Series solutions Homotopy analysis method was irst proposed by Liao [3] in 99 which is used or the construction o series solution o highly nonlinear problems. It is preerred over the other methods due to the ollowing advantages. (i) It does not depend upon the small or large parameters. (ii) It ensures the convergence o series solutions. (iii) It provides us great choice to select the base unction and linear operator. o proceed with such method, it is essential to deine the initial guess and linear operator. So initial guesses,, L, L, L or the momentum, energy and concentration equations are 0 0 0 and linear operators expressed in the orms exp, ( S)exp and ( ) ( P)exp( ), with in which i 0 0 0 3 d d d d L, L 3 and L, d d d d L A A exp( ) A3 exp( ) 0, L A4 A5 exp( ) exp( ) 0, L A6exp( ) A7exp( ) 0, A i 7 are the arbitrary constants. (3) (4) (5) (6) (7) (8) (9) Convergence analysis he series solutions by homotopy analysis method depend upon the auxiliary parameter. his auxiliary parameter provides us great reedom to adjust and control the convergence region o the series solutions. hereore we have plotted the -curves at the 5 th order o approximations in the Figs. ( 4). It is seen that permissible values o, and are 0.95 0.3, 0.95 0.5 and 0.95 0.35. Fig.. -curve or. Fig. 3. -curve or θ.
Fig. 4. -curve or ϕ. able : Convergence o the series solutions or dierent order o approximations when 0.,., 0., 0., N 0., / 4, Pr.5, S 0., 0., Sc.5, and P 0.. Order o approximation '' (0) ' (0) ' (0) -.666 -.063 -.0950 5 -.3974 -.497 -.93 0 -.3990 -.496 -.66 5 -.3984 -.484 -.79 0 -.398 -.480 -.86 6 -.3979 -.479 -.9 8 -.3979 -.478 -.94 35 -.3979 -.478 -.94 Results and Discussion he main objective o this section is to explore the impacts o various parameters on the velocity, temperature and concentration distributions. Fig. 5 shows the eect o curvature parameter on the velocity proile. Velocity distribution decreases near the surace o cylinder while it increases away rom the surace. Velocity curves vanish asymptotically at some large values o. It is also noted that boundary layer thickness increases. In act or higher values o curvature parameter the radius o cylinder decreases. So contact surace area o cylinder with the luid decreases which oers less resistance to the luid motion. hereore velocity proile increases. Inluence o on velocity distribution is plotted in Fig. 6. It is shown that velocity proile decreases or larger values o. Since is the ratio o relaxation to retardation times so or larger values o the relaxation time increases which produces more resistance to the luid motion. Hence velocity proile decreases. Variation o mixed convection parameter on velocity proile is sketched in Fig. 7. It is analyzed that velocity proile is higher or larger values o mixed convection parameter. It is due to the act that larger values o mixed convection parameter corresponds to the higher thermal buoyancy orce which is responsible in the enhancement o velocity proile. Eect o Deborah number (in terms o retardation time) on velocity distribution is displayed in Fig. 8. Velocity and momentum boundary thickness are higher or larger values o. With the increase o the retardation time increases (or elasticity o the material increases) which is responsible in the enhancement o velocity proile. Fig. 9 shows the inluence o ratio o buoyancy orces N on the velocity distribution. It is noted that velocity proile and momentum boundary layer thickness are higher or larger values o N. Since N is the ratio o concentration to thermal buoyancy orces so with the increase o N the concentration buoyancy orce increases which results in the enhancement o velocity proile. Fig. 0 displays the eect o angle o inclination on the velocity distribution. It is observed that the velocity proile decreases with an increase in. hrough increase o the gravity aect decreases which results in the reduction o velocity proile. Variation o thermal stratiication parameter S on velocity distribution is sketched in Fig.. It is analyzed that the velocity and
momentum boundary layer thickness decrease with an increase in thermal stratiication parameter S. In act convective potential between the surace o cylinder and ambient luid decreases. Hence velocity proile decreases. Inluence o curvature parameter on the temperature proile is displayed in Fig.. emperature proile decreases near the surace o cylinder and it increases away rom the surace. Fig. 3 presents the behavior o mixed convection parameter on temperature distribution. With the increase o mixed convection parameter the thermal buoyancy orce increases which is responsible or high rate o heat transer. hereore temperature proile decreases. Fig. 4 shows the characteristics o Deborah number on the temperature proile. Both temperature and thermal boundary layer thickness decrease or higher values o Deborah number. Variation o angle o inclination on temperature distribution is expressed in Fig. 5. Increase in shows the increasing behavior o temperature proile. Due to the increase in the gravity aect decreases which results in the reduction o rate o heat transer. hereore temperature proile increases. Fig. 6 provides the analysis or the variation o Prandtl number Pr on the temperature proile. It is noticed that a decrease in the temperature proile and thermal boundary layer thickness is observed when Prandtl number Pr increases. Prandtl number is the ratio o momentum diusivity to thermal diusivity. So with the increase o Prandtl number Pr the thermal diusivity decreases which results in the reduction o temperature proile. Fluids with high Prandtl number Pr corresponds to low thermal diusivity. Behavior o thermal stratiication parameter S on the temperature distribution is sketched in Fig. 7. Higher values o thermal stratiication parameter reduce the temperature and thermal boundary layer thickness. his is due to the act that the temperature dierence gradually decreases between the surace o cylinder and ambient luid which causes a reduction in the temperature proile. Eect o heat generation/absorption parameter on temperate proile is shown in Fig. 8. Both temperature and thermal boundary layer thickness increase when generation/absorption parameter is increased. Here more heat is produced during the heat generation process which increases the temperature proile. Fig. 9 presents the eect o curvature parameter on concentration distribution. It is observed that concentration proile decreases near the surace o cylinder and it increases away rom the surace. Inluence o Schmidt number Sc on concentration proiles is sketched in Fig. 0. Concentration proile decreases or larger values o Schmidt number Sc. It is the ratio o momentum diusivity to mass diusivity. Higher values o Schmidt number Sc corresponds to lower mass diusivity which results in the reduction o concentration proile. Characteristic o solutal stratiication parameter P on concentration proile is displayed in Fig.. Concentration proile decreases when solutal stratiication parameter P is increased. Further it is also noted that concentration boundary layer thickness decreases. Figs. and 3 show the impacts o various parameters on skin riction coeicient. It is analyzed that skin riction coeicient is higher or larger values o α, β and γ while it decreases with an increase in. able shows the convergence o series solutions or momentum, energy and concentration equations. It is noted that 6th order o approximation is suicient or momentum equation and 8th order o approximation is suicient or energy and concentration equations. able presents the eects o various parameters on skin riction coeicient. It is noted that skin riction coeicient increases or larger,, and S while it decreases with increasing the values o, and N. Negative values o skin riction physically mean that cylinder exerts a drag orce on the luid particles. able 3 shows the behavior o dierent parameters on Nusselt number. It is analyzed that Nusselt number increases or higher values o,, and Pr while it decreases with an increase in,, S and. able 4 is constructed to examine the behavior o various parameters on Sherwood number. It is observed that Sherwood number increases with the increase in,,, N and Sc while it decreases when and P are increased. It is also noted that negative values o Nusselt and Sherwood numbers represent heat and mass transer rom cylinder surace to the luid i.e., normal to the surace.
Fig. 5. Eect o γ on velocity proile. Fig. 6. Eect o on velocity proile. Fig. 7. Eect o on velocity proile. Fig. 8. Eect o on velocity proile. Fig. 9. Eect o N on velocity proile. Fig. 0. Eect o on velocity proile. Fig.. Eect o S on velocity proile. Fig.. Eect o on temperature proile.
Fig. 3. Eect o on temperature proile. Fig. 4. Eect o on temperature proile. Fig. 5. Eect o on temperature proile. Fig. 6. Eect o Pr on temperature proile. Fig. 7. Eect o S on temperature proile. Fig. 8. Eect o on temperature proile. Fig. 9. Eect o on concentration proile. Fig. 0. Eect o Sc on concentration proile.
Fig.. Eect o P on concentration proile. Fig.. Eects o and on skin riction. Fig. 3. Eects o and on skin riction. able : Eects o various parameters on the skin riction coeicient when Pr=., Sc=., P=0. N S 0.5Re 0.5 x C 0.. 0. 0. 0. /4 0. 0.693 0. 0.7304 0.5 0.7850 0.. 0. 0. 0. /4 0. 0.7304. 0.7.5 0.6643 0.. 0. 0. 0. /4 0. 0.7304 0. 0.6999 0.5 0.659 0.. 0. 0. 0. /4 0. 0.7304 0. 0.7655 0.5 0.8643 0.. 0. 0. 0. /4 0. 0.7304 0. 0.773 0.5 0.798 0.. 0. 0. 0. 0.0 0. 0.774 /4 0.7304 /3 0.7394 0.. 0. 0. 0. /4 0. 0.7304 0. 0.7345 0.5 0.749
able 3: Eect o various involved parameters on the local Nusselt number when N=0., Sc=., P=0. Pr S ' (0) 0.0. 0. 0. /4. 0. 0. 0.973 0. 0.9954 0.3.096 0.. 0. 0. /4. 0. 0. 0.9954. 0.9873.5 0.9657 0.. 0. 0. /4. 0. 0. 0.9954 0..099 0.5.0649 0.. 0. 0. /4. 0. 0. 0.9954 0..006 0.5.053 0.. 0. 0. 0.0. 0. 0..0064 /4 0.9954 /3 0.9857 0.. 0. 0. /4 0.8 0. 0. 0.7687.0 0.8857. 0.9954 0.. 0. 0. /4. 0. 0. 0.9954 0. 0.9640 0.5 0.8584 0.. 0. 0. /4. 0. 0. 0.9954 0. 0.8649 0.3 0.677 able 4: Eect o various involved parameters on the Sherwood number when., Pr=., S=0. N Sc P ' (0) 0. 0. 0. 0. /4. 0..0385 0..0750 0.5.795 0. 0. 0. 0. /4. 0..0750 0..0948 0.5.308 N Sc P ' (0) 0. 0. 0. 0. /4. 0..0750 0..0866 0.5.80 0. 0. 0. 0. /4. 0..0750 0..0768 0.5.0798 0. 0. 0. 0. 0.0. 0..080
/4.0750 /3.0707 0. 0. 0. 0. /4 0.8 0. 0.8348.0 0.9600..0750 0. 0. 0. 0. /4. 0..0750 0..0383 0.5 0.933 Concluding remarks Here we investigated the double stratiied mixed convection low o Jerey luid induced by an impermeable inclined stretching cylinder. Heat transer characteristics are explored with heat generation/absorption. he key points are summarized as ollows: Velocity, temperature and concentration proiles increase or larger curvature parameter away rom the cylinder. hermal and solutal stratiication parameters reduce the temperature and concentration respectively. Higher values o Deborah number result in the enhancement o velocity distribution. emperature proile enhances with the increase o heat generation/absorption parameter. Skin riction coeicient, Nusselt and sherwood numbers increase via curvature parameter. Nomenclature (r,x) Space coordinates, C Fluid temperature and concentration (u,v) Velocity components, C Ambient temperature and concentration Dynamic viscosity Curvature parameter Kinematic viscosity Deborah number in terms o retardation time Density Pr Prandtl number λ₁ Ratio o relaxation to retardation times g Gravitational acceleration λ₂ Retardation time β hermal expansion coeicient C p Heat genearation/absorption parameter hermal conductivity w, C w Wall temperature and concentration α Angle o inclination C Skin riction β C Solutal expansion coeicient w Surace shear stress D Mass diusivity Nu x Nusselt number Q₀ Heat q w Surace heat lux R generation/absorption Radius o cylinder coeicient Re x Local Reynold number l Characteristics length Stream untion U₀ Reerence velocity Dimensionless variable U w Stretching surace velocity Dimensionless stream unction, φ Dimensionless temperature and concentration Linear operator or momentum L L, Dimensionless coordinate Linear operators or energy and concentration Auxiliary parameter or momentum Auxiliary parameters or temperature and, concentration 0, C 0 Reerence temperature and concentraion a, b, d, Dimensional constants Sc Schmidt number e S, P hermal and solutal stratiied parameters λ Mixed convection parameter N Concentration to thermal buoyancy ratio Gr, Gr * j w emperature and concentration Grasho numbers Mass lux Sh Sherwood number
Higher Prandtl number results in the reduction o temperature proile while Nusselt number increases. Velocity proile increases while temperature proile decreases when mixed convection parameter increases. Reerences Hussain,., Shehzad, S. A., Hayat,., Alsaedi, A., Al-Solamy, F. and Ramzan, M., Radiative hydromagnetic low o Jerey nanoluid by an exponentially stretching sheet. Plos one., 9 (04), pp. e0379.. urkyilmazoglu, M. and Pop, I., Exact analytical solutions or the low and heat transer near the stagnation point on a stretching / shrinking sheet in a Jerey luid. Int. J. Heat Mass ranser., 57 (03), pp. 8-88.. Ellahi, R., Rahman, S. U. and Nadeem, S., Blood low o Jerey luid in a catherized tapered artery with the suspension o nanoparticles. Phys. Letter A., 3-78 (04), pp. 973-980. 3. Hayat,., Shehzad, S. A. and Alsaedi, A., hree dimensional stretched low o Jerey luid with variable thermal conductivity and thermal radiation. Appl. Math. Mech. Engl. Ed., 34 (03), pp. 83-83. 4. Hayat,., Qayyum, A., Alsaadi, F., Awais, M. and Dobaie, A. M., hermal radiation eects in squeezing low o a Jerey luid. Eur. Phys. J. Plus., 8 (03), pp. 85. 5. Freidoonimehr, N., Rostami, B., Rashidi, M. M. and Momoniat, E., Analytical modelling o threedimensional squeezing nanoluid low in a rotating channel on a lower stretching porous wall. Math. Prob. Eng., 04 (04), pp. 6978. 6. Mukhopadhyay, S., Slip eects on MHD boundary layer low over an exponentially stretching sheet with suction/blowing and thermal radiation. Ain Shams Eng. J., 4 (03), pp. 485-49. 7. Hayat.., Anwar, M. S., Farooq, M. and Alseadi, A., MHD stagnation point low o second grade luid over a stretching cylinder with heat and mass transer. Int. J. Nonlinear Sci. Number. Simul., 5 (04), pp. 365-376. 8. urkyilmazoglu, M., hree dimensional MHD low and heat transer over a stretching/ shrinking surace in a viscoelastic luid with various physical eects. Int. J. Heat Mass ranser, 78 (04), pp. 50-55. 9. Mukhopadhyay, S., Chemically reactive solute transer in boundary layer low along a stretching cylinder in porous medium. Ar. Mat., 5 (04), pp. -0. 0. Sheikholeslami, M., Ashorynedjad, H. R., Barari, A. and Soleimani, S., Investigation o heat and mass transer o rotating MHD viscous low between a stretching sheet and a porous surace. Eng. Comput., 30 (03), pp. 357-378.. Bhattacharya, K., Mukhopadhyay, S. and Layek, G. C., Similarity solution o mixed convection boundary layer slip low over a vertical plate. Ain Shams Eng. J., 4 (03), pp. 99-305.. Hayat,., Farooq, M. and Alsaedi, A., Melting heat transer in the stagnation-point low Maxwell luid with double-diusive convection., 4 (04), pp.760-774. 3. urkyilmazoglu, M., he analytical solution o mixed convection heat transer and luid low o a MHD viscoelastic luid over a permeable stretching surace. Int. J. Mech. Sci., 77 (03), pp. 63-68. 4. Rashad, A. M., Abbasbandy, S. and Chamkha, A. J., Mixed convection low o a micropolar luid over a continuously moving vertical surace immersed in a thermally and solutally stratiied medium with chemical reaction. J. aiwan Inst. Chem. Eng., 45 (04), pp. 63-69. 5. Rashidi, M. M., Ali, M., Freidoonimehr, N., Rostami, B. and Hossain, M. A., Mixed convection heat transer or MHD viscoelastic luid low over a porous wedge with thermal radiation. Adv. Mech. Eng., 04 (04), pp. 735939. 6. Hayat,., Ashra, M. B., Alsulami, H. H. and Alhuthali, M. S., hree-dimensional mixed convection low o viscoelastic luid with thermal radiation and convective conditions. Plos one., 9 (04), pp. e90038. 7. Ellahi, R., Riaz, A., Abbasbandy, S., Hayat,., and Vaai, K., A study on the mixed convection boundary layer low and heat transer over a vertical slender cylinder. hermal Sci. 8 (04) 47-58. 8. Singh, G., and Makinde, O. D., Mixed convection slip low with temperature jump along a moving plate in presence o ree stream. hermal Sci. 9 (05) 9-8. 9. Ibrahim, W. and Makinde, O. D., he eect o double stratiication on boundary layer low and heat transer o nanoluid over a vertical plate. Computers and Fluids., 86 (03), pp. 433-44. 0. Hayat.., Hussain,., Shehzad, S. A. and Alsaedi, A., hermal and concentration stratiications eects in radiative low o Jerey luid over a stretching sheet. Plos one., 9 (04), pp. e07858.. Mukhopadhyay, S., MHD boundary layer low and heat transer over an exponentially stretching sheet embedded in a thermally stratiied medium. Alex. Eng. J., 5 (03), pp. 59-65.. Srinivasacharya, D. and Upendar, M., Eect o double stratiication on MHD ree convection in a micropolar luid. J. Egyp. Math. Soc., (03), pp. 370-378.
3. Hayat.., Hussain, Z., Farooq. M., Alsaedi, A. and Obaid, M., hermally stratiied stagnation point low o an Oldroyd-B luid. Int. J. Nonlinear Sci. Numer. Simulat., 5 (04), pp. 77-86. 4. Liao, S. J., Homotopy analysis method in non-linear dierential equations, Springer and Higher Education Press, Heidelberg (0). 5. Rashidi, M. M., Kavyani. N. and Abelman, S., Investigation o entropy generation in MHD and slip low over a rotating porous disk with variable properties. Int. J. Heat Mass rans., 70 (04), pp. 89-97. 6. Abbasbandy, S. and Jalili, M., Determination o optimal convergence-control parameter value in homotopy analysis method. Numer. Algor., 64 (03), pp. 593-605. 7. Hayat,., Farooq, M., Alsaedi, A. and Iqbal, Z., Melting heat transer in the stagnation point low o Powell-Eyring luid. J. hermophys. Heat rans., 7 (03), pp. 76-766. 8. urkyilmazoglu, M., Solution o the homas-fermi equation with a convergent approach. Commun. Nonlinear Sci. Numer. Simulat., 7 (0), pp. 4097-403. 9. Hayat,., Qayyum, A. and Alseadi, A., MHD unsteady squeezing low over a porous stretching plate. Eur. Phys. J. Plus., 8 (03), pp. 57. 30. Hayat,., Imtiaz, M., and Alsaedi, A., MHD low o nanoluid over permeable stretching sheet with convective boundary conditions. hermal Sci. (05) DOI: 0.98/SCI408939H.