NUMERICAL SOLUTION ON MHD FLOW AND HEAT TRANSFER FOR THE UPPER-CONVECTED MAXWELL FLUID OVER A STRETCHING/SHRINKING SHEET

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1 Jurnal Kara Asli Lorekan Ahli Matematik Vol. 10 No.1 (017) Page Jurnal Kara Asli Lorekan Ahli Matematik NUMERICAL SOLUION ON MHD FLOW AND HEA RANSFER FOR HE UPPER-CONVECED MAXWELL FLUID OVER A SRECHING/SHRINKING SHEE Nazila Ishak a Iskandar Waini b Muhammad Khairul Anuar Mohamed c Norhaati Rosli c and Mohd Zuki Salleh c* a Mathematics Department MRSM un Abdul Razak 6600 Pekan Pahang Malasia b Facult o Engineering echnolog Universiti eknikal Malasia Melaka Durian unggal Melaka c Facult o Industrial Science & echnolog Universiti Malasia Pahang 600 UMP Kuantan Pahang Abstract : In this paper e ill discuss the eect o magnetohdrodnamic (MHD) toards the lo and heat transer or the Upper-Convected Maell (UCM) luid over a stretching/shrinking sheet ith prescribed all temperature (PW). A similarit transormation is used to reduce the governing equations to a set o ordinar dierential equations (ODEs) hich are then solved numericall using Shooting method. he eect o various parameter such as the Prandtl number the magnetic parameter the suction parameter the stretching or shrinking parameter and the Maell parameter are considered. he results obtained under the special cases are illustrated graphicall and compared ith previous results to support their validit. Keords: Magnetohdrodnamic (MHD); Upper-Convected Maell luid (UCM); Heat transer; Stretching/Shrinking Sheet. 1.0 Introduction he concept o boundar laer lo on a continuous solid surace moving at constant speed as irst investigated b Sakiadis [1]. Due to the entrainment o the ambient luid this boundar laer lo is quite dierent rom the Blasius lo past a lat plate. Sakiadis s theoretical predictions or Netonian luids ere later corroborated eperimentall b sou et al. []. he luid dnamics to a stretching sheet is important in etrusion processes. he production o sheeting material arises in number o industrial manuacturing processes including both metal and polmer sheets like the cooling o an ininite metallic plate in a cooling bath paper production glass bloing and iber spinning. Since the pioneering stud b Crane [] ho presents an eact analtical solution or the stead to-dimensional lo due to a stretching surace in a quiescent luid man authors have considered various aspects o this problem and obtained similarit solutions. Gupta and Gupta [4] investigate the heat and mass transer corresponding to the similarit solutions or the boundar laer over a stretching sheet subjected to suction or a bloing. Chen and Char [5] analze the heat transer characteristics over continuous stretching surace ith variable surace temperature. Researches toards stead to-dimensional lo due to a stretching surace in a quiescent luid have been considered b man authors. It as investigated in dierent aspects b various authors and the results obtained sho the similarit solutions o the aorementioned problem. Amongst them are Wang [6] and Gorla and Sidai [7]. Ali [8 9] studied the heat transer characteristics o a continuous stretching surace and on thermal boundar laer on a poer-la stretched surace ith suction or injection. Elbashbesh [10] has studied the heat transer over a stretching surace ith variable surace heat lu. Recentl analtical solution or suction and injection lo o a viscoplastic Casson luid past a stretching surace in the presence o viscous dissipation is investigate b Hussanan et al. [11]. 017 Jurnal Kara Asli Lorekan Ahli Matematik Published b Pustaka Aman Press Sdn. Bhd.

2 Jurnal KALAM Vol. 10 No. 1 Page During the manuacturing o these sheets the melt issues rom a slit and is subsequentl stretched to achieve the desired thickness. he investigation o magnetohdrodnamic (MHD) lo is ver important or industrial and other dierent areas o researches such as petroleum production and metallurgical processes [1]. he heat and mass transer or a hdromagnetic lo over a stretching sheet has been ell-studied b Liu [1]. Chakrabarti and Gupta [14] discovered that the temperature distribution in this lo hen a uniorm suction is applied at the stretching surace. It is also considered b Chiam [15] ith a poerla velocit. Anjali and hiagarajan [16] studied the stead o nonlinear hdromagnetic lo and heat mass transer over a stretching surace o variable temperature. Ai [17] discovered the similarit solution in MHD about the eects o thermal diusion and diusion thermo on ree convection heat and mass transer over a stretching surace b considering the suction or injection. Recent stud regarding this topic include the orks b Hussanan et al. [1819] and Alavi et al. [0] ho investigated the unstead ree and mied convection lo in porous medium ith Netonian heating and stagnation point lo toards an eponentiall stretching sheet. Birds et al. [1] suggest that the viscoelastic luid models used in previous orks are simple models. hese include second-order model and/or Watler s B model hich are knon to be suitable onl or eakl elastic luids subject to slo and slol-varing los. Hoever these to luid models are knon to violate certain rules o thermodnamics. Moreover a non-netonian second grade is less applicable or highl elastic luids (polmer melts) hich occur at high Deborah numbers. hereore the signiicance o the results reported in the above orks are limited at least as ar as polmer industr is concerned. Sarpkaa [] as the irst researcher to stud the MHD lo o non-netonian luid. Indeed the viscoelastic luid model has recentl been used b Alizadeh-Pahlavan et al. [] to stud the lo o viscoelastic luids above porous stretching sheets ith no heat transer eects involved. Sadegh et al. [4] Haat et al. [5] and Aliakbar et al. [6] have studied the UCM luid b using numerical methods ith no heat transer also. Alizadeh-Pahlavan and Sadegh [7] have done the ork related to UCM luid using homotop analsis method (HAM). MHD stagnation point lo o upper-convected Maell luid over a stretching sheet has been studied b Haat et al. [8]. he boundar laer lo and heat transer due to stretching or shrinking sheet or the upper-convected Maell luid have been considered and etended b man researches (see [9] [5]) in various as. he aim o this paper is to stud the MHD lo and heat transer or the upper-convected Maell luid over a stretching/shrinking sheet. hermal boundar condition is considered in this research hich is prescribed all temperature (PW). he results obtained under special cases are then compared ith those in the literatures to support the validit o our result. he governing o nonlinear partial dierential equations is irst transormed into ordinar dierential equations and is solved using Shooting method..0 he Problem Formulation Consider a stead to-dimensional boundar laer lo o an upper-convected Maell luid past a stretching/shrinking sheet at 0. Here the magnetic ield B 0 is applied in the momentum equation here it is imposed along the -direction produces magnetic eect in the -direction. he phsical model and coordinate sstem or stretching surace o this problem is shon in Figure 1. he induced magnetic ield hich is due to the motion o the electricall conducting luid is negligible. We also assumed that the eternal electric ield is zero and the polarization charges created b the electric ield are negligible

3 Nazila Ishak et al. [15]. he -ais runs along the continuous surace in the direction o the motion and the - ais is perpendicular to it. boundar laer slot B 0 stretching sheet U Figure 1: Phsical model and coordinate sstem or stretching/shrinking surace. hereore under these assumptions the governing equations or the upper-convected Maell luid can be ritten as ollos [6]: u v 0 (1) u u u u u u B u v u v uv v 0 u u v k c subject to the boundar conditions: u U ( ) a v v ( ) u 0 ( ) b as at 0 here is a constant rate o stretching/shrinking u and v are the velocit components along the and aes respectivel. Meanhile is the temperature o the luid is the temperature o the ambient is the luid densit is the electrical conductivit o the luid is the relaation time parameter o the luid hile v is the kinematic viscosit o the luid c is the speciic heat at a constant pressure k is the thermal conductivit o the luid p v is the all mass suction velocit and u U a v is the stretching/ shrinking velocit o the sheet here 1 1 is respectivel or stretching and shrinking sheet. Introduce the stream unction or similarit solution is deined as u and v (5) so that the continuit Equation (1) is automaticall satisied. Equations () and () can then be ritten as () () (4)

4 Jurnal KALAM Vol. 10 No. 1 Page B v 0 (6) c p k (7) he similarit solution is obtained b assuming av v a ) ( ) ( 1/ ) ( 1/ (PW) (8) On substituting (8) into Equations (6) and (7) e obtained the olloing nonlinear ordinar dierential equations 0 ) ( M (9) 0 Pr 1 (10) and the boundar conditions (4) become as at S 0 ) ( 0 ) ( 0 (PW) 1 (0) (0) (0) (11) here primes denote dierentiation ith respect to. he dimensionless Maell parameter Magnetic parameter M Prandtl number Pr and Suction parameter S are given b 1 0 Pr av v S k c a B M a p (1) It is orth mentioning that the problem discussed b Abel et al. [1] can be retrieved hen 1 hich is or the stretching sheet problem. he phsical quantities o interest are the skin riction coeicient C and local Nusselt number Nu hich are deined as ) ( k q Nu U C (1) here is the luid densit hile the surace shear stress and the surace heat lu q are respectivel given b 0 u and 0 k q (14) ith and being the dnamic viscosit and thermal conductivit. Using the similarit variables (8) e obtain (0) 1/ Re C (15) and

5 Nu (0) Re 1/ Nazila Ishak et al. Where Re U ( ) / is the local Renolds number based on the stretching velocit ( ) a hereas the skin riction and heat transer coeicients are represent b 0 U and 0 respectivel. (16).0 Results and Discussion he appropriate similarit transormation is adopted to transorm the governing partial dierential equations o lo and heat transer into a sstem o non-linear ordinar dierential equations. Equations (9) and (10) are subjected to the boundar conditions (11) are solved numericall b the Shooting method. In order to veri the accurac o the present method and results are compared ith those reported b Sadegh et al. [4] Mamaloukas et al. [] Abel et al. [1] Muhaimin et al. [4] and Bhattachara [5] as shon in ables 1 and respectivel. It has been ound that the are in good agreement. ABLE 1 he comparison o the skin riction coeicient 0 or a dierent values o Maell parameter hen Pr 1 M 0 and S 0 in the case o stretching sheet 1. 0 Sadegh et al. [4] Mamaloukas et al. [] Abel et al. [1] Present results ABLE Values o the skin riction coeicient S 0 or a dierent values o Suction parameter S hen Pr 1 M 1and 0 in the case o shrinking sheet 1. 0 Muhaimin et al. [4] Bhattachara [5] Present results

6 Jurnal KALAM Vol. 10 No. 1 Page ABLE : Values o the skin riction coeicient 0 0 and local heat transer coeicient or a dierent values o the Maell parameter hen Pr 1 M 1and S 0. 5 in the case o stretching sheet 1. 0 he value o the skin riction coeicient and the heat transer coeicient 0 or various values o in able. It is ound that hen increases both the values o 0 0 decrease. and ABLE 4: Values o the skin riction coeicient and local heat transer coeicient or a dierent values o Suction parameter S hen Pr 1 M 1and 0. 8 in the case o stretching sheet 1. S From able 4 it can be seen that the dierent value o suction parameter S or the skin riction coeicient 0 and heat transer 0. From this table note that hen the suction parameter S increases the 0 increase. 0 decrease and ABLE 5: Values o the skin riction coeicient 0 0 and local heat transer coeicient or a dierent values o the stretching parameter hen Pr 1 M 1 S and

7 Nazila Ishak et al. 0 and local heat transer coeicient ABLE 6: Values o the skin riction coeicient 0 or a dierent values o the Magnetic Parameter M hen Pr 1 S and 0. 8 in the case o stretching sheet 1. M From able 5 hen the stretching parameter increase the skin riction coeicient 0 decrease but the heat transer coeicient increase. he values o the skin riction coeicient 0 and local heat transer coeicient 0 or a dierent values o the Magnetic Parameter M hen Pr 1 S 0. 8 and 1presented in able 6. It is ound hen the magnetic parameters M increase the both values o 0 increase. 0 and Figure : he eect o magnetic parameter on the velocit proile or 0 7

8 Jurnal KALAM Vol. 10 No. 1 Page Figure : he eect o magnetic parameter on the temperature proile or 0 Figures 4 and 5 illustrates the eect o magnetic parameter M on the velocit and temperature proile or to value o Maell parameter 0 and 0 hen S 0.5 Pr 0. 7 and 0. 5 respectivel. It is ound that as the value o Magnetic parameter increases the velocit proile decrease but the temperature proile increase this due to a M change the proile to be less latter to be near the edge o the boundar laer and less steeper near to the edge o the surace. Figure 4: he eect o magnetic parameter on the velocit proile ( ) or 0 8

9 Nazila Ishak et al. Figure 5: he eect o magnetic parameter on temperature proile or 0 Figures 6 and 7 sho the eect o the suction parameter S on velocit and temperature proile hen Pr 0.7 M and 0. 5 respectivel. An increasing o suction parameter S lead to decreasing on velocit temperature proile and the thermal boundar laer thickness. Figures 8 and 9 sho the eect o Maell parameter on velocit and temperature proile hen Pr 0.7 M 1 1 and S 0. 5 respectivel. From these igure sho that hen. From these igure sho that hen Maell parameter increase the value or velocit and temperature proile become decrease. hereore hen the momentum boundar laer thickness decrease it can induces an increase in the absolute value o the velocit gradient at the surace. he value o the concentration gradient at the surace increases depend hen the value Maell parameter also increase. hen the mass o the transer rate at the surace also increases. Figure 6: he eect o suction parameter S on the velocit proile ( ) or 0 9

10 Jurnal KALAM Vol. 10 No. 1 Page Figure 7: he eect o suction parameter on the temperature proile or 0 = Figure 8: he eect o maell parameter on the velocit proile ( ) = Figure 9: he eect o Maell parameter on the temperature proile 10

11 Nazila Ishak et al. Figures 10 and 11 sho the eect o the stretching sheet hen or Pr 0.7 M 1 S 0.5 and respectivel. From these igure it is ound that hen stretching sheet increase the velocit proile increase and the temperature proile decrease and the thermal boundar laer thickness also decrease. Figure 10: he eect o stretching sheet on the velocit proile ( ) 4.0 Conclusion Figure 11: he eect o stretching sheet on the temperature proile ( ) In this paper e have numericall studied the problem o the MHD lo and heat transer ithin a boundar laer o upper-convected Maell luid over a stretching/ shrinking sheet ith prescribed all temperature using the Shooting method. It is shon ho the magnetic parameter Maell parameter suction parameter stretching sheet aect the skin riction coeicient 0 and the temperature proile 0 local heat transer coeicient. We can conclude that velocit 11

12 Jurnal KALAM Vol. 10 No. 1 Page an increase in the value magnetic parameter leads to increase o the skin riction coeicient local heat transer coeicient and temperature proile and decrease on the values o velocit proile.. an increase in the Maell and suction parameter leads to decreases o the skin riction coeicient velocit and temperature proile and increase on the local heat transer coeicient.. an increase in the value stretching sheet leads to increases o the local heat transer coeicient and temperature proile and decreases skin riction coeicient and velocit proile. Acknoledgement. he author thanks to Universiti Malasia Pahang or inancial supports (RDU17058) and Ministr o Higher Education (RDU150101). Reerences (1) Sakiadis B.C Boundar-laer behaviour on continuous solid suraces: I. Boundar laer equations or todimensional and aismmetric lo. Am. Inst. Chem. Engineers J. 7: 6-8. () sou F. Sparro E. and Goldstein R Flo and heat transer in the boundar laer on a continuous moving suraces. Int. J. Heat Mass ranser. 10: () Crane L. J Flo past a stretching plate. Z. Ange. Math.Phs. 1: (4) Gupta P. S. and Gupta A. S Heat and mass transer on a stretching sheet ith suction or bloing. Can. J. Chem. Eng. 55: (5) Chen C. K. and Char M Heat transer o a continuous stretching surace ith suction or bloing. Journal o Mathematical Analsis and Applications. 15: (6) Wang C. Y Free convection on a vertical stretching surace. J. Appl. Math. Mech. (ZAMM). 69: (7) Gorla R. S. R. and Sidai I Free convection on a vertical stretching surace ith suction and bloing. Appl. Sci. Res. 5: (8) Ali M. E Heat transer characteristics o a continuous stretching surace. Warme Stoubertragung. 9: 7-4. (9) Ali M. E On thermal boundar laer on a poer-la stretched surace ith suction or injection. International Journal o Heat and Fluid Flo. 16: (10) Elbashbesh E. M. A Heat transer over a stretching surace ith variable surace heat lu. J. Phs. D: Appl. Phs. 1: (11) Hussanan A. Salleh M. Z. Khan I. and Shaie S Analtical solution or suction and injection lo o a viscoplastic Casson luid past a stretching surace in the presence o viscous dissipation. Neural Computing and Applications 1-9. (1) Abel M.S. aade J.V and Nandeppanavar M.M. 01. MHD lo and heat transer or the upper-convected Maell luid over a stretching sheet. Meccanica. 47: (1) Liu I. C A note on heat and mass transer or a hdromagnetic lo over a stretching sheet. Int. Commun. Heat Mass ranser. : (14) Chakrabarti A. and Gupta A. S Hdromagnetic lo and heat transer over a stretching sheet. Quart. Appl. Math. 7: (15) Chiam. C Hdromagnetic lo over a surace stretching ith a poer-la velocit. Int. J. Engng SCi. : (16) Anjali Devi S. P. and hiagarajan M Stead nonlinear hdromagnetic lo and heat transer over a stretching surace o variable temperature. Heat and Mass ranser. 4: (17) Ai A. A Similarit solution in MHD: Eects o thermal diusion and diusion thermo on ree convective heat and mass transer over a stretching surace considering suction or injection. Commun Nonlinear Sci Numer Simulat. 14: (18) Hussanan A. Ismail Z. Khan I. Hussein A. G. and Shaie S Unstead boundar laer MHD ree convection lo in a porous medium ith constant mass diusion and Netonian heating. he European Phsical Journal Plus 19: 46. (19) Hussanan A. Salleh M. Z. Khan I. ahar R. M. and Ismail Z Soret eects on unstead magnetohdrodnamic mied-convection heat-and-mass-transer lo in a porous medium ith Netonian heating. Maejo International Journal o Science and echnolog. 9:

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