Non-Uniform Heat Source/Sink and Thermal Radiation Effects on the Stretched Flow of Cylinder in a Thermally Stratified Medium

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1 Journal of Applied Fluid Mechanics, Vol. 10, No., pp , 016. Available online at ISSN , EISSN DOI: /acadpub.jaf Non-Unifor Heat Source/Sink and Theral Radiation Effects on the Stretched Flow of Cylinder in a Therally Stratified Mediu T. Hayat 1,, Sadia Asad and A. Alsaedi 1 Departent of Matheatics, Quaid-i-Aza University 450, Islaabad 44000, Pakistan COMSATS Institute of Inforation Technology, Wah Cantt 47040, Pakistan. Nonlinear Analysis and Applied Matheatics (NAAM) Research Group, Departent of Matheatics, Faculty of Science, King Abdulaziz University,Jeddah 1589, Saudi Arabia Corresponding Author Eail: asadsadia@yail.co (Received August 1, 014; accepted October 7, 015) ABSTRACT The paper addresses the influence of non-unifor heat source/sink in flow of couple stress fluid by a stretching cylinder in a therally stratified ediu. Theral radiation effect in heat transfer analysis is also accounted. Conservation laws of ass, linear oentu and energy leads to nonlinear situation. Use of adequate transforations coverts the partial differential equations into the ordinary differential equations. Series solutions of the resulting equations are obtained for the velocity and teperature. Convergence of the solutions is explicitly checked. Ipacts of various sundry variable son the velocity, teperature, wall shear stress and Nusselt nuber are exained through graphical illustrations and nuerical values. The effect of β and Re on velocity field is qualitatively siilar. For larger values of curvature paraeter γ velocity enhances. Influences of S and R on teperature on the teperature distribution are opposite. Heat transfer at the surface decays when A and B increase. Keywords: Couple stress fluid; Theral radiation; Non-unifor heat source/sink and therally stratified ediu. 1. INTRODUCTION The stretched flow in presence of heat transfer is quite prevalent in several engineering processes such as in extrusion, elt spinning, hot rolling, wire drawing, anufacture of plastic and rubber sheets, glass fiber production, paper production, food processing, anufacture of polyer sheets and in the oveent of biological fluids. It is now recognized that fluids in such engineering applications are not viscous. Hence a new stage about non-newtonian aterials in the evolution of fluid dynaic theory is in progress during the past few decades. The flow behavior in shearing for such aterials cannot be characterized by Newtons' law of viscosity. An intensive research effort, both theoretical and experiental, has been devoted in the past to probles of non-newtonian fluids via different aspects. Different fro the viscous fluids, the non-newtonian liquids cannot be described by eploying one constitutive relationship between the stress and rate of strain. This is because of diverse properties of non-newtonian fluids in nature. Due to this fact several constitutive equations of Newtonian fluids exist. The additional rheological paraeters in such constitutive equations lead to the coplexities in the resulting differential equations. Generally the differential equations in non- Newtonian fluids are ore subtle, higher order and coplicated than the Navier-Stokes equations. The corresponding differential systes involving non- Newtonian fluids offer interesting challenges to the odelers, coputer scientists and atheaticians fro different quarters. One ay ention few relevant recent studies on this topic by Khan et al. (014). Although the flow and heat transfer due to a stretching cylinder is iportant in wire drawing, hot rolling and fiber production but very less ephasis is given in this direction. The steady lainar flow caused by a stretching cylinder iersed in an incopressible viscous fluid with prescribed surface heat flux is investigated by Bachok and Ishak (010) The authors coputed the nuerical solution in this study. Hydroagnetic flow by a pereable cylinder with heat and ass transfer has been reported by Chakha (011). He considered the proble in the presence of heat generation/absorption, cheical reaction and unifor transverse agnetic field. Mukhopadhyay

2 T. Hayat et al. / JAFM, Vol. 10, No., pp , 016. (01) presented ixed convection flow of viscous incopressible fluid and heat transfer towards a stretching cylinder ebedded in porous ediu. An unsteady ixed convection boundary layer flow over a pereable non-linearly stretching vertical cylinder has been considered by Patil et al. (01). They considered cobined effects of buoyancy force and theral diffusion in the presence of surface ass transfer. Effect of unifor agnetic field on slip flow by a stretching cylinder is exained by Mukhopadhyay (01). Moreover, the study of heat transfer in the presences of theral stratification is of considerable iportance in geological transport, geotheral syste, power plant condensation systes and volcanic flows. Stratification of the ediu ay arise due to a teperature difference which gives rise to a density variation in the ediu. This is known as theral stratification and it usually arises due to theral energy input into the ediu fro heated bodies and theral sources. Theral stratification is a characteristic of all fluid bodies surrounded by differentially heated side walls. Analysis for the axisyetric lainar boundary layer ixed convection flow of an incopressible viscous fluid towards a stretching cylinder iersed in a therally stratified ediu has been given by Mukhopadhyay and Ishak (01). MHD boundary layer flow and heat transfer by an exponentially stretching sheet ebedded in a therally stratified ediu are studied by Mukhopadhyay (01). Murthy et al. (01) investigated the flow, heat and ass transfer in a therally stratified nanofluid with convective boundary condition. Paolucci and Zikoski (01) investigated the free convective flow along a heated vertical wall iersed in a therally stratified environent. Areas arises at high teperatures and acquaintance of radiative heat transfer becoes very vital for design of relevant equipent. For production of plastic sheets, gas turbines, issiles, space vehicles aircraft, nuclear power plants, satellites and foils. Slip flow of electrically conducting fluid with theral radiation has been analyzed by Mukhopadhyay (015). Flow and heat transfer analysis of MHD fluid in the presence of a unifor agnetic field with theral radiation are investigated by Chaudhary et al. (015). Sahu and Mishra (015) reported radiative flow of a dusty fluid past a vertical stretching surface. Nuerical study of boundary layer flows with heat transfer has been presented by Gireesha et al. (015). They also take the effects of theral radiation, Darcy porous ediu and non-unifor heat source/sink. Theoretical influence of buoyancy and theral radiation on agnetohydrodynaic flow past a stretching porous sheet has been studies by Daniel and Daniel (015). The ain purpose of present attept is to investigate the flow and heat transfer characteristics by a stretching cylinder iersed in a therally stratified ediu. Effects of theral radiation and non-unifor heat generation are also considered. The flow equations are odeled for the constitutive relations of couple stress fluid. In fact the couple stress fluid shows the size dependent effect in the presence of couple stresses, body couples and nonsyetric stress tensor. The consideration of couple stress fluids has relevance in a nuber of processes that occur in industry such as the extrusion of polyer liquids, cooling of etallic plate in a bath, colloidal solutions, solidification of liquid crystals etc. The resulting nonlinear flow proble is solved by Hootopy analysis ethod (HAM) (Liao, 009; Hayat, 014; Hayat, 01, Maleki, 014; Rostai, 014; Sheikholeslai, 014; Farooq, 015, Hayat, 015). Results for velocity and teperature are graphically analyzed while the skin friction and local Nusselt nuber are exained through nuerical values.. FORMULATION We consider the steady two-diensional flow of couple stress fluid ebedded in a therally stratified ediu. Non-unifor heat source/sink and theral radiation effects are present. The stretched cylinder is placed along the x axis and r axis is taken noral to it. The boundary layer equations coprising the conservation laws of ass, linear oentu and energy can be written as follows: ru rv x r 0, u u u 1 u u v x r r r r 4 u u 4 r r r 1 u 1 u, r r r r T T 1 T cp u v k x r r r r 1 rq r q, r r (1) () () in which is the kineatic viscosity, is the fluid density, T is the fluid teperature, c p is the specific heat, q r 16 T T k r is the radiative heat flux, k is the ean absorption coefficient, is the Stefan-Boltzan constant, k is the theral conductivity of the fluid and q is the rate of internal heat generations/absorption coefficient. The diensionless fro of q can be put in the for entioned below [14] kb q A ( Tw T) B ( T T), (4) where A and B are paraeters of space and teperature dependent internal heat generation/absorption respectively. It is noted that 916

3 T. Hayat et al. / JAFM, Vol. 10, No., pp , 016. the case A 0 and B 0 corresponds to internal heat generation while A 0 and B 0 holds for internal heat absorption, Tw is the teperature of sheet and T is the constant teperature for away fro the sheet. The subjected boundary conditions are u U( x), v 0, T T at r a, (5) u u u 0, 0 0 T T as r. r r w (6) In above equationsu( x) U0x / l is the stretching velocity, Tw ( x) T0bx / l is the prescribed surface teperature and T( x) T0cx / l is the variable abient teperature. Further U0 is the reference velocity, T 0 is the reference teperature, l is the characteristic length and b and c are the positive constants. Considering (Bachok and Ishak, 010; Chakha, 011; Mukhopadhyay, 01; Patil, et al. 01; Mukhopadhyay, 01; Mukhopadhyay and Ishak, 01). = (), () =, Equation 6 becoe = ʋ (). = ʋ / (7) (1) is identically satisfied and Eqs. (1 +) (1 +) (1 +) 4 (1 R ) 1 (8) Pr f f Sf Af B 0, (9) f 0, f 1, 1 S at 0, f f f 0, 0 as, (10) Where β=η / μa ^, diensionaless couple stress paraeter R=(4 σ^* T ^ _ )/(kk^* ),Radiation paraeter S=c/b, Stratification paraeter γ= (lʋ/(a^ U_0 ),) Pr= μc _p/k, Re=(U_0 a^)/ʋ, Curvature paraeter Prandtl nuber Reynold nuber The wall shear stress is defined by C f u, w r w 1 U w ra 1/ x or Re C f (0). (11) Nusselt nuber Nu x f Nux xq w, kt ( ) w T is given by 16 T T qw k 1 k k r ra 1/ 4 x Nux or Re (1 R) (0). (1). SOLUTIONS Initial guesses and auxiliary linear operators for the velocity and teperature fields are f 0( ) 1 e, 0( ) (1 S) e, (1) f f f,. (14) The properties satisfied by the operators are f C1Ce Ce (15) ( ) 0, ( Ce 4 Ce 5 ) 0, (16) where C i ( i 1 5) are the constants. The zeroth order deforation probles are represented by the following equations ˆ 0 1 p f f ( ; p) f ( ) pf N f fˆ( ; p), (17) ˆ 0 1 p ( ; ) ( ) ˆ( ; ), ˆ p p f p (, p) N, (18) ˆ ˆ (0; ) 0, (0; ) 1, ˆ ( ; ) ˆ f p f p f p f ( ; p) 0, (19) ˆ f ( ; p) 0, ˆ (0, p) 1 S, ˆ (, p) 0, (0) where p [0,1] indicates an ebedding paraeter, h f and h are the non-zero auxiliary paraeters and the nonlinear operators N f and N are ˆ fˆ(, p ) N f [ f(, p)] 1 ˆ f (, p) ˆ ˆ f(, p) f(, p) fˆ(, p ) fˆ(, p ) Re8 917

4 T. Hayat et al. / JAFM, Vol. 10, No., pp , fˆ(, p ) fˆ(, p ) 1, 5 ˆ ˆ 4 N [ (, p), f (, p)] 1 R d 1 ˆ (, p) ˆ (, p) ˆ ˆ(, p ) Pr f (, p) fˆ(, p ) ˆ(, p ) fˆ(, p ) S fˆ(, p ) A B ˆ(, p ). (1) () For p 0 and p 1 we have the following equations fˆ( ;0) f ˆ 0( ), (,0) 0( ), fˆ( ;1) f ( ), ˆ (,1) ( ). () When p varies fro 0 to 1then f (, p) and (, p) approach fro f 0 ( ), 0 ( ) to f ( ) and ( ). The series of the velocity and teperature fields through Taylor's expansion are chosen convergent for p 1 and thus f f ( ) f ( ) f ( ), f ( ; p ) ( ),! 0 ( ) ( ) ( ), 1 1 ( ; p) ( ).! p 0 p 0 (4) (5) R R f ( ) 1 f 1f 1 1 f 1 kf k f 1 kf k k 0 Re 8 f 1 81 f 1 1 f 1, 4 ( ) 1 Rd 1 1( ) 1 1 1kf k k 0 1 kk PrSf 1 1B 1, Pr f (0) (1) Af 0 1 = 1 1. The general solutions ( f, ) coprising the special solutions ( f, ) are f ( ) f ( ) C1Ce Ce, () ( ) ( ) Ce 4 Ce 5. () 4. CONVERGENCE OF THE SOLUTIONS Convergence of series solutions depends upon the non-zero auxiliary paraeter. The auxiliary paraeter in Eqs. (4 and 5) can be regarded as an iteration factor that is widely used in nuerical coputations. It is well known that a properly chosen iteration factor can ensure the convergence of iteration. As pointed by Liao (009), the valid region of is a horizontal line segent parallel to horizontal axis. Figs. and 4 witness that the adissible values of f and are 1.1 f 0. and respectively. th The resulting probles at order can be presented in the following fors f f f1 f f [ ( ) ( )] R ( ), (6) [ ( ) 1( )] R ( ), (7) f (0) f (0) f ( ) f ( ) 0 (8) f ( ) (0) ( ) 0, (9) Fig. 1. Residual error for velocity field. 918

5 T. Hayat et al. / JAFM, Vol. 10, No., pp , 016. Fig.. Residual error for teperature fields. These series solutions converge for the whole region of when f 0.7 and 0.8. Table 1 depicts the convergence of series solutions. It is confired here that 0 th -order approxiations are enough for the convergence of velocity and teperature profiles. Fig. 1 and show the residual error for velocity and teperature field respectively. Fig.. Ћf curve for velocity field. Fig. 4. ћθ curve for teperature field. Fig. 6. Influence of Re on velocity field. 5. DISCUSSION Study of different pertinent paraeters on physical quantities of interest is presented in this section. Hence for this purpose we plot Figs. (5-15) for the velocity f ( ) and teperature fields ( ). Fig. 5 shows the influence of couple stress paraeter on the velocity field. It is clearly seen fro this Fig. that the velocity field is decreasing function of. In fact couple stress paraeter depends upon the couple stress viscosity and this couple stress viscosity acts as a retarding agent which akes the fluid ore denser resulting into a decrease in the velocity of the fluid. Fig. 6 illustrates the effect of Reynolds nuber on f ( ). As Reynolds nuber is the ratio of the inertia force due to wall stretching to the viscous force. Therefore for larger values of Reynolds nuber the velocity decreases to zero as increases and flow decays very slowly to the abient. Variation of the velocity field f ( ) for different values of curvature paraeter is displayed in Fig.7. Curvature paraeter increases the velocity in the region 1 6. Fig.8 depicts the effect of couple stress paraeter on the teperature field ( ). Teperature profile and theral boundary layer thickness are enhanced for larger values of. Influence of Reynolds nuber on the teperature field is displayed in Fig. 9. Fig. 5. Influence of β on velocity field. Fig. 7. Influence of γ on velocity field. 919

6 T. Hayat et al. / JAFM, Vol. 10, No., pp , 016. Fig. 8. Influence of β on teperature field. teperature field is shown in Fig1. Teperature profile is increasing function of R. Increase in the theral radiation paraeter enhances the theral boundary layer thickness. Fig.1 plots for the teperature field for different values of A. By analyzing the graphs it reveals that saller values of A decrease the teperature field. Fig. 14 shows the effect of teperature dependent heat source/sink paraeter B on ( ). Teperature profile is decreasing function of B when B 0. Here ore energy is absorbed and in result the teperature significantly drops within the boundary layer. It is evident fro Fig. 15 that the teperature field increases when A and B are positive (in case of internal heat generation). This is due to the fact that for larger values of A and B the theral boundary generates the energy and this causes an increases in the teperature. Fig. 9. Influence of Re on teperature field. Table 1 Convergence of series solutions for different order of approxiations when 0.6, S Re 0.1, Pr 1.0, R 0., A B 0.1, f 0.7 and 0.8. Order of approxiations (0) (0) Fig. 10. Influence of γ on teperature field. Fig. 11. Influence of S on teperature field. It is noticed that enhanceent in Reynolds nuber leads to increase in teperature and theral boundary layer thickness. Fig. 10 shows the influence of curvature paraeter on ( ). Teperature rises significantly after soe larger distance fro the wall when increases. Fig. 11 plots the influence of S on the teperature field ( ). It is observed that the teperature in the boundary layer decreases for larger values of stratification paraeter. Physically this is due to the fact that buoyancy factor reduces within the boundary layer when we increase stratification paraeter S. Also abient theral stratification causes a significant decrease in the teperature. The effect of theral radiation paraeter R on the Table shows the nuerical values of wall shear stress. This table indicates that the surface exerts drag force on the fluid. Enhanceent in couple stress paraeter ( ), curvature paraeter ( ) and Reynolds nuber (Re) lead to increase the agnitude of wall shear stress. Table illustrates the effect of different physical paraeter of interest on the Nusselt nuber. The fluid paraeter ( ) and stratification paraeter ( S ) decrease the value of Nusselt nuber. Furtherore the Nusselt nuber for negative values of A and B (i.e. internal heat absorption) increases when copared with the positive values of A and B (i.e. internal heat generation). Also see Fig. (16-19) 90

7 T. Hayat et al. / JAFM, Vol. 10, No., pp , 016. Fig. 1. Influence of R on teperature field. Fig. 16. Influence of S on Nusselt nuber. Fig. 1. Influence of A on teperature field. Fig. 17. Influence of R on Nusselt nuber. Fig. 14. Influence of B on teperature field. Fig. 18. Influence of A on Nusselt nuber. Fig. 15. Influence of A and B on teperature field. Fig. 19. Influence of B on Nusselt nuber. 91

8 T. Hayat et al. / JAFM, Vol. 10, No., pp , 016. Table Values of wall shear stress for different paraeters β γ Re (0) Table Values of Nusselt nuber for different eerging paraeters β γ Re R Pr S A B / CONCLUSIONS Effects of theral radiation and non-unifor heat source/sink in flow of couple stress fluid by a stretching cylinder ebedded in a therally stratified ediu are exained. Main findings of the presented analysis are entioned below. Velocity f ( ) decays for larger values of couple stress paraeter. Reynolds nuber Re increases the velocity and associated boundary layer thickness. Velocity and boundary layer thickness increase for larger curvature paraeter. Effects of curvature paraeter, couple stress paraeter and Reynolds nuber Re on the teperature field are qualitatively siilar. Stratification paraeter S decays the teperature and teperature gradient. Teperature and theral boundary layer thickness increase when theral radiation paraeter R increases. Magnitude of wall shear stress increases through larger values of, and Re. Nusselt nuber decays considerably when A and B increase. REFERENCES Bachok, N. and A. Ishak (010). Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malaysian Journal of Matheatical Sciences 4, Chakha, J. A. (011). Heat and ass transfer fro MHD flow over a oving pereable cylinder with heat generation or absorption and cheical reaction. Counication and Nuerical Analysis 011, 0. Chaudhary, S., S. S. Singh and S. Chaudhary (015). Theral radiation effects on MHD boundary layer flow over an exponentially stretching surface. Applied Matheatics 6, Daniel, S. Y. and K. S. Daniel (015). Effects of buoyancy and theral radiation on MHD flow over a stretching porous sheet using hootopy analysis ethod. Alexandria Engineering Journal 54, Ellahi, R, T Hayat, M. F. Mahoed and S. Asghar (010). Effects of slip on the non-linear flows of a third grade fluid. Nonlinear Analysis: Real World Applications 11, Ellahi, R. and A. Riaz (010). Analytical solutions for MHD flow in a third-grade fluid with variable viscosity. Matheatics and Coputational. Modelling 5, Farooq, U., Y. L. Zhao, T. Hayat, A. Alsaedi and S. J. Liao (015). Application of the HAM-based atheatica package BVPh.0 on MHD Falkner-Skan flow of nanofluid, Coputers & Fluids 111, Gireesha, J., B. B. Mahanthesh, R. S. R. Gorla and T. P. Manjuntha (015). Theral radiation and 9

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10 T. Hayat et al. / JAFM, Vol. 10, No., pp , 016. Analysis 11. Sahu, R. and K. S. Mishra (015). Boundary layer flow and heat transfer of a dusty fluid over a vertical pereable stretching surface. International JournalResearch Engineering Technology 4, Sheikholeslai, M., R. Ellahi, H. R. Ashorynejad, G. Doairry and T. Hayat (014). Effect of heat transfer in flow of nanofluids over a pereable stretching wall in a porous ediu. Journal Coputational and Theoretical Nanosciences11, Turkyilazoglu, M. and I. Pop (01). Exact analytical solutions for the flow and heat transfer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid. International Journal Heat Mass Transfer 57,