Faculty of Science, King Abdulaziz University, Jeddah 21598, Saudi Arabia

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1 HEAT TRANSFER ANALYSIS BASED ON CATTANEO CHRISTOV HEAT FLUX MODEL AND CONVECTIVE BOUNDARY CONDITIONS FOR FLOW OVER AN OSCILLATORY STRETCHING SURFACE b Sai ULLAH KHAN a,*, Nasir ALI b, Tasawar HAYAT c,d and Zaheer ABBAS e a,* Departent o Matheatics, COMSATS Institute o Inoration Technolog, Sahiwal 57000, Pakistan b Departent o Matheatics and Statistics, International Islaic Universit, Islaabad 44000, Pakistan c Departent o Matheatics, Quaid-i-Aza Universit 4530, Islaabad 44000, Pakistan d Nonlinear Analsis and Applied Matheatics (NAAM) Research Group, Departent o Matheatics, Facult o Science, King Abdulaziz Universit, Jeddah 1598, Saudi Arabia e Departent o Matheatics, The Islaia Universit o Bahawalpur, Bahawalpur 63100, Pakistan Abstract: In this stud, we investigate the heat transer characteristics in unstead boundar laer low o Maxwell luid b using Cattaneo-Christov heat lux odel and convective boundar conditions. The low is caused b a sheet which is stretched periodicall back and orth in its own plane. The phsical odel that takes into account the eects o constant applied agnetic ield is transored into highl nonlinear partial dierential equations under boundar laer approxiations. The solution o diensionless version o these equations is developed using hootop analsis ethod. The siulations are presented in the or o teperature and velocit proiles or suitable range o phsical paraeters. The obtained results illustrate that an increase in Deborah nuber and Hartann nuber suppress the velocit proiles. It is urther observed that Cattaneo-Christov heat lux odel predicts the suppression o theral boundar laer thickness as copared to Fourier law. Kewords: Maxwell luid, Cattaneo-Christov heat lux odel, oscillator stretching sheet, hootop analsis ethod. 1. Introduction The stud o convective o heat transer gained great attention o investigators because o its nuerous applications in industrial and cheical processes like oil and gas processing, annealing o etal and plastic sheets, glass tepering, paper and textile dring petrocheical, preiu theral oil, reining industr etc. The phenoenon o heat transer pla ke role in several cheical engineering phenoena like cooling o cheical equipent, anuacturing o cheical aterials etc. In view o the practical applications, various researchers are engaged to investigate heat transer phenoenon in various luids odels. The analsis o heat transer in viscous low caused b stretching surace was conducted b Gupta [1].Another iportant stud is due to Lawrence and Rao [] which is concerned with the inluence o various paraeters on low o non-newtonian luid causes b pereable heated stretching surace. The obtained closed or solution o their proble and claied that the solution is not unique. Rollins and Vajravelu [3] adopted analtic technique to obtained solution o a proble regarding heat transer * Corresponding author: sk_iiu@ahoo.co 1

2 ro oving surace iersed in non-newtonian luid odel. Bhattachara [4] ipleented shooting technique to highlight the low o Newtonian luid caused b shrinking surace. Sahoo[5] discussed the Heienz low o third grade luid over a heated surace. He reported that the velocit at wall is suppressed b increasing third grade luid paraeter.abbas et al. [6] discussed siilar solution or Maxwell luid low generated bheated stretching surace. The heat transer in peristaltic Newtonian low has been studied b Ali et al. [7]. Mahantesh et al. [8] exained heat absorption/generation eects in lainar low o Walters-B luid odel inluenced b pereable surace. Oztop and Dagtekin [9] considered three dierent cases or ixed convection low in dierentiall heated square cavit. Karii- Fard et al. [10] perored nuerical coputations or double-diusive natural convection low in a porous square cavit. According to the, the boundar and inertia eects have signiicant role in doublediusive convection. The inluence o heat source/sink in low o icropolar luid over linearl stretched surace was addressed b Eldaha and Aziz [11]. Mandal and Mukhopadha [1] presented nuerical coputations regarding stead low o linear luid odel with heat transer. Malik et al. [13] used Sisko luid odel and convective boundar conditions to predict the heat transer phenoena. Shara et al. [14] presented nuerical solution to stud heat transer in non-newtonian low caused b exponential stretching surace. Khan et al. [15] used Von Karan Pohlhausen ethod to solve the integral equations to predict heat transer phenoenon in viscous luid low over ininite circular clinder. Goaa and Taweel [16] analzed the oscillator low o viscous luid generated b heated vertical lat surace. Further contribution on this topic can be ound in [16-0]. The literature surve indicates that aoreentioned studies ade use o siple Fourier law o heat lux [1]which states that heat lux is proportional to teperature gradient. However, it is observed that this odel is applicable to acroscopic sstes where tie scale o the sste is higher than average relaxation tie. The atheatical odeling based on this law shows that heat equation results in parabolic or which shows that the whole sste is instantl inluenced b the initial disturbance. To the best o our knowledge, Cattaneo []was irst who proposed an extension in the Fourier law b introducing a relaxation tie expression and derived a single equation or teperature ield. One o the iportant eature o this law is that it allows the heat transportation via propagation o inite speed theral waves. The work o [] was extended b Christov [3]b using Oldrod upper convective derivative[4]. Subbaraa et al. [5] introduced Maxwell-Cattaneo heat conduction law to discuss the Raleigh-Bénard agneto convection in a viscoelastic luid. Straughan [6] presented the analsis o stead low o linear luid odel using Cattaneo heat lux odelin presence o theral relaxation eects. Haddad [7] eploed this odel to stud theral instabilit in viscous low through porous edia. Han et al. [8] used Cattaneo-Christov expression to stud the heat transer eects in Maxwell luid over a stretching plate. Haat et al. [9] studied the heat transer analsis in stagnation-point low based on Cattaneo Christov heat lux. Khan et al. [30] applied a nuerical schee based on shooting ethod to evaluate the inluence o heat transer b using this law in low due to bi-directional stretching surace. Li et al. [31] obtained sel-siilar solution or proble regarding stead low o Maxwell luid and heat transer b using Cattaneo-Christov heat lux odel. Mustaa [3] exained the rotating low o Maxwell luid caused b oving surace with the Cattaneo-Christov heat lux odel. More recent studies regarding low o various luids based on this law can be ound in res. [33-36].

3 Bearing in ind the previous attepts, as well as the industrial and practical iportance o these probles, the ain ai o this stud is to analze the unstead low o Maxwell luid [37-40] due to oscillator stretching sheet b using Cattaneo-Christov heat lux odel proposed in [3] and convective boundar conditions [41-43]. Analtic expressions or both velocit and teperature proiles are obtained using hootop analsis ethod. This stud presents MHD low o non-newtonian luid over an oscillator stretching surace which have iense iportance in an industrial and cheical engineering processes. Particularl, the proble presented here with considered geoetr has an industrial applications like hot rolling, ibers spinning, anuacturing o rubber sheets. Moreover, theagnetohdrodnaic (MHD) eects are useul in MHD power generating sstes, telephone sste, coputers, X-ras and scanning devices etc. The orulated proble is solved analticall b hootop analsis ethod. A detailed analsis or several iportant paraeters is presented.. Flow Analsis Consider two-diensional, unstead low o an electricall Maxwell luid over an oscillator stretching sheet. The luid occupies the region 0. Let us assued that sheet is stretched and oscillate periodicall along x -axis with velocit u bxsint where represents the requenc and b is a constant having the diension [ T] 1. We have also considered the eects o transverse agnetic ield o agnitude B 0 which is iposed noral to the sheet (see Fig. 1). Using low Renolds nuber assuptions, the eects o induced agnetic ield are neglected. The governing boundar laer equations or two-diensional Maxwell luid low are [39] u v x 0, Fig. 1: Geoetr o proble (1) u u u u v t x t t u u v u B0 u u u v 1 u 1v 1, t x t u u u u v uv x x () 3

4 For the present low coniguration, the initial and boundar conditions are u u bx sin t, v 0, at 0, t 0, u 0, as, In above equation u and v denotes the velocit coponents along x and directions, respectivel, is the kineatic viscosit, is the densit, is the electric conductivit and 1 denotes the relaxation paraeter. 3. Heat Transer Analsis In this section, we are going to orulate heat transer proble. Unlike tpical studies, we derive governing equation o heat transer using Cattaneo-Christov heat lux odel [16]. According to this odel the heat lux and teperature gradient are related through ollowing expression q q V q q V Vq kt, t (4) Where q represents the heat lux, denotes the relaxation tie o the heat lux, V is the velocit vector,t is the Maxwell luid teperature, k is the theral conductivit. Eq. (4) reduces to well-known Fourier law or 0.. For incopressible luid V 0 and thereore Eq.(4)becoes q q V q q V kt. t (5) The energ equation or incopressible luid ater neglecting viscous dissipation eects is T cp T, t V q (6) where c p is the speciic heat. Eliination oq ro Eq. (5) and Eq. (6) ields to the ollowing single equation or the teperature ield [16] T T T v T u T v T u v u v T T T t x t t t x x T u v, (7) t x v T u T T T T u v uv u v x x x x where k/ cp is theral diusivel. Eq. (7) is subjected to the convective boundar conditions given b T k ht T, at 0, t 0, T T as, (8) where h denotes the heat transer coeicient. Moreover, below the surace and abient luid teperature, respectivel. 3.1 Diensionless orulation T and T are the convective luid teperatures Beore going to the solution o proble, it is better to reduce the nuber o independent variables in Eqs. () and (7). We introduce ollowing diensionless quantities [44] (3) 4

5 b, t, u bx,, v b,, (9) T T (, ). T T Using Eqs. (9) and (10), the continuit equation is identicall satisied and Eqs.() and (7)are transored into ollowing ors S S S 1 M 1 0, M M 1 S S S S 0. (1) Pr Siilarl, the boundar conditions o proble under consideration becoe 0, sin, 0, 0, (0, ) 1 1 0,, (10) (11), 0, (, ) 0, (13) where M B0 / brepresents the Hartann nuber, 1b denotes Deborah nuber, b is the diensionless relaxation tie o heat lux, S / b is ratio o oscillation requenc to stretching rate, 1 ( h / k) / b represent the Biot nuber and Pr / is the Prandtl nuber. Eq. (1) represents the energ equation based on Fourier law or 0. For stead low, Eq. (11) reduces to M 1 M 0. (7) Moreover, or 0 it reduces to corresponding equation or low o hdroagnetic viscous luid. The corresponding equation or low o hdrodnaic viscous luid can be recovered b taking M 0 [44]. 4. Hootop analsis ethod The diensionless partial dierential equations (11)-(1) are highl nonlinear in nature and thereore exact solution is diicult to obtain. Thereore, we ipleent hootop analsis ethod to copute series solution o these partial dierential equations subject to boundar conditions (13). This ethod was originall proposed b Liao [45] and then successull applied b an researchers in various disciplines o science and engineering [46-49]. To proceed with the solution, we suggest ollowing initial approxiations exp( ) 1 0(, ) sin (1 exp( )), 0( ). 11 The auxiliar linear operators are 3 ( ), ( ), 3 satising [ A A exp( ) A exp( )] 0, (16) 1 3 (14) (15) 5

6 [ A exp( ) A exp( )] 0, 4 5 where A i ( i 1,,...5) represent constants. The zeroth-order deoration probles or give proble is (1 p ) [ ˆ(, ; p) (, ) ph N [ ˆ(, ; p)], (18) 0 (1 p ) [ ˆ (, ; p) (, ) ph N [ ˆ (, ; p), ˆ (, ; p)], (19) 0 (17) ˆ(, ; p) ˆ(, ; p) ˆ(, ; p ) 0, sin, (0) ˆ(0, ; p ) 1 1 ˆ (0, ; p), ˆ (, ; p) 0, (1) The nonlinear operators are 3 ˆ ˆ ˆ ˆ (, ; p) (, ; p) (, ; p) N [ (, ; p)] S 3 1 M M (, ; ) ˆ (, ; ) 1 ˆ M (, ; p) ˆ p p () 3 ˆ ˆ ˆ 3 ˆ (, ; p) (, ; p) (, ; p) ˆ (, ; p) S S (, ; p), 3 ˆ ˆ ˆ ˆ (, ; ) ˆ (, ; ) (, ; ) (, ; p) p (, ; p) p p 3 ˆ ˆ ˆ N [ ˆ (, ; p), ˆ(, ; p)] (, ; p) S 1 (, ; p) ˆ (, ; p) (, ; p) Pr ˆ (, ; p) ˆ(, ; ) ˆ(, ; ) ˆ ˆ p p (, ; p) S S (, ; p) S (3). ˆ ˆ (, ; ) (, ; ) ˆ ˆ p p ˆ (, ; p) (, ; p) (, ; p) The solution o zeroth-order deoration probles at p 0 and p 1 is (, ;0) (, ), (, ;1) (, ), (4) 0 0 ˆ (, ;0) (, ), ˆ (, ;1) (, ). (5) Using Talor's series expansion, we have 1 ˆ(, ; ) ˆ(, ; ) 0 (, ) (, ) p p p, (, ), (6)! p 1 1 ˆ(, ; ) ˆ(, ; ) 0 (, ) (, ) p p p, (, ), (7)! p 1 6

7 The convergence o HAM depends upon solution converges at p 1. Thereore 0 1 h and h. We assue that 7 h and h are selected so that series (, ; p) (, ) (, ), (8) 0 1 (, ) (, ) (, ), (9) where and can be coputed through the th-order o deoration problegiven b, 1, h R,, (30),, h R,, 1 (31) (0, ) (, ) (0, ) 0, 0, 0, (0, ) 1 (0, ) (, ) 0, k 1k k R (, ) S 3 1 M M S 1 M 1 k k0 k0 (34) 1 3 k 3 1 k k k k l l S S l k k kl, k 0 l k k 1k k R (, ) S S 1k S 1k Pr k0 k0 (35) l k kl l l lk k l, k 0 l 0 0, 1, 1, 1. The general solution is o the or (, ) (, ) A A exp( ) A exp( ), (37) 1 3 (, ) (, ) A exp( ) A exp( ). (38) 4 5 where (, ) and (, ) represents the particular solution. Using (0) and (1), the constants A i (i=1,,5) are eliinated as 0, 1 0, A A4 0, A3, A1 A3 0,, A5 1 0,. 1 1 (3) (33) (36) (39)

8 5. Discussion The auxiliar paraeters involved in the analtic expressions are h and h. The convergence region as well as rate o approxiations can be estiated or solutions b these paraeters. The plots o 0, versus h and 0, versus h have been shown in Fig. at 6th order o approxiations to estiate suitable range o these paraeter or convergent solutions. It is observed that or chosen set o involved paraeters the adissible ranges or h and h are: h 0.3, 1 h 0, respectivel. The plots o residual error or and or a particular set o involved paraeters at 6 th order o approxiation is shown in Fig. 3(a) when h 0.6. Clearl, the axiu error over the whole doain is less than Siilarl, Fig. 3(b) testiies that residual error or is in acceptable range when h 0.6 and h 0.8. Fig..h-curvesor (a) velocit (b) teperature proiles. Fig. 3. Residual error or (a) velocit (b) teperature proiles. Now we coe to the discussion o graphical results concerning the velocit and teperature distribution or diverse values o various low paraeters like Hartann nuber M, Deborah nuber, relaxation tie o the heat lux, Biot nuber 1 and Prandtl nuber Pr. Fig.4(a) explains the eects o Deborah nuber on the transverse velocit coponent ' b keeping S 0., M 0.5 and /. This igure shows that velocit decreases b increasing Deborah 8

9 nuber. For viscous luid 0 the oentu boundar laer is thicker as copared to non- Newtonian luid. Fro phsical and experiental point o view it is seen that at lower values o Deborah nuber, the luid behaves uch like liquid whereas the luid shows viscoelastic solid like behavior at high Deborah nuber due to an increase in viscous properties and thus the luid velocit starts to decelerate which is noted in Fig. 4(a).Moreover, the thickness o the boundar laer is suppressed or higher values o Deborah nuber. The transverse distributions o the low velocit ' or speciic values Hartann nuber M is shown in Fig. 4(b). Application o strong agnetic orce tends to resist the velocit o luid particles near the surace. This is in act due to the act that the presence o agnetic orce produces the Lorentz orce which resists the low produced b oscillating sheet. The velocit ' as unction o tie at a speciic location 0.5 is plotted or dierent values o Deborah nuber and Hartann nuber M in Fig. 5. A decrease in aplitude o low velocit is noted with an increase in the Deborah nuber (Fig. 5(a)). In act or larger β, the viscous orces are doinant which restrict the otion o luids particles and as a result aplitude is decreased. Fig.5(b) is sketched to exaine the behavior o Hartann nuber M on '. Here, it is observed that the aplitude o the velocit ' decreases with increasing the values o Hartann nuber M. Again this suppression in the aplitude is due to the resistive orce produced due to application o agnetic ield noral to the sheet. The dependence o the luid teperature on the Pr and is sketched in Fig. 6. Increasing the Prandtl nuber results in the thickening o theral boundar laer. It can be justiied because theral diusivit decreases or large values o Prandtl nuber which results in decrease o teperature and corresponding theral boundar laer thickness. However, or non-zero values o the theral boundar laer thickness decreases ore rapidl with increasing Prandtl nuber. In Fig. 7, we give the variation o teperature ield or various values o and or two dierent values o Prandtl nuber. It is interesting to note that the teperature and theral boundar laer thickness decreases with increasing. Further, it is observed that this decrease is aster or larger values o Prandtl nuber Pr. In Fig. 8 and 9, the eects o Deborah nuber and Hartann nuber M on teperature ield are shown, respectivel. Both the paraeters eect the teperature ield in a siilar anner i.e. the teperature ield increases b increasing either o and M. The variation o teperature proile or our dierent values o theral Biot nuber 1 1,1.5,.5,3.5 are displaed in Fig. 10. Theral Biot nuber is associated with heat transer coeicient h, thereore its higher values represent the case o enhanced heat transer ro stretching sheet to the luid strea. This enhanceent in heat transer is responsible or increase in the teperature o luid. Fig.11 relects the inluence o ratio o oscillation requenc to stretching rate S on teperature proile. One can easil observe that teperature ield is decreased b increasing S. In Table 1, the obtained solution is validated against existing results o Zheng et al. [19] and Ali et al. [0]. An excellent agreeent between both solutions is observed. 6. Concluding rearks The heat transer analsis in unstead low o Maxwell luid b using Cattaneo-Christov heat lux odel is presented when plate is stretched periodicall. Ater coputing the solution b hootop analsis 9

10 ethod, a coprehensive analsis has been presented to highlight the eects o various low paraeters. The ain indings o the analsis can be suarized as: B increasing Deborah nuber the luid velocit is suppressed in the vicinit o the surace in given doain. The sae is true with increasing Hartann nuber. A oscillations in low velocit at a speciic location are suppressed or larger values o Deborah nuber and Hartann nuber. It is noted that the heat transer rate ro sheet to the luid becoe slow or larger values o Prandtl nuber and relaxation tie o heat lux. The Cattaneo-Christov heat lux odel predicts lower values o teperature inside the theral boundar laer as copared heat lux odel based on Fourier law. The rate o heat transer enhanced b increasing Biot nuber 1. Moreover, there is no heat transer when 1 0. Fig. 4: The velocit proile or dierent values o (a) (b) M Fig. 5:Variationo velocit with tie(a) eects o and (b) eects o M 10

11 Fig. 6:Eects o Pr on Fig. 7: Eects o on Fig. 8: Eects o on Fig. 9: Eects o M on 11

12 Fig: 10: Eects o 1 on Fig. 11: Eects o S on Table 1: Coparison o '' 0, luid ( 0). or various values o when S 1, M 1 in case o Newtonian Zheng et al. [19] Ali et al. [0] Present Results HAM Results Nuerical Result Acknowledgents We are grateul to the reviewers or their useul coents. Reerences [1] Gupta, P. S. and Gupta, A. S., Heat and ass transer on a stretching sheet with suction and blowing. The Canadian Journal o Cheical Engineering, 55(6), (1977), pp [] Lawrence, P. S. and Rao, B. N., Heat transer in the low o a viscoelastic luid over a stretching sheet, Acta Mechanica, 93 (199) pp [3] Rollins, D. and Vajravelue, K., Heat transer in a viscoelastic order luid over a Continuous Stretching surace, Acta Mechanica,89 (1991) pp [4] Bhattachara, K.,Boundar laer low and heat transer over an exponentiall shrinking sheet. Chin.Phs.Lett.8,( 011) [5] Sahoo, B., Hieenz low and heat transer o a third grade luid, Counications in Nonlinear Science and Nuerical Siulation, pp [5] Cortell, R., Flow and heat transer o an electricall conducting luid o second grade over a stretching sheet subject to suction and to a agnetic ield, International Journal o Heat and Mass Transer, 49 (006) pp

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14 [3] Christov, C.I., On rae indierent orulation o the Maxwell--Cattaneo odel o inite-speed heat conduction, Mechanics Research Counications, 36 (009) pp [4] Oldrod, J.G., On the orulation o rheological equations o state, Proceedings o the Roal Societ A, 00 (1949) pp [5] Pranesh, S. and Kiran, R., V., Stud o Raleigh-Bénard Magneto Convection in a Micropolar Fluid with Maxwell-Cattaneo Law, Applied Matheatics, (010), 1, pp [6] Straughan, B., Theral convection with the Cattaneo-Christov odel, International Journal o Heat and Mass Transer, 53 (010) pp [7] Haddad, S.A.M., Theral instabilit in Brinkan porous edia with Cattaneo--Christov heat lux, International Journal o Heat and Mass Transer, 68 (014) pp [8] Han, S., Zheng, L., Li, C. and Zhang, X., Coupled low and heat transer in viscoelastic luid with Cattaneo--Christov heat lux odel, Applied Matheatics Letters, 38, (014) pp [9] Haat T, Khan, I., Farooq, M, Yaseen T and Alsaedi A. Stagnation point low with Cattaneo Christov heat lux and hoogeneous heterogeneous reactions. Journal o Molecular Liquids, 0 (016), pp [30] Khan, M. Ahad, L. Khan, W.A. Alshoranic, A.S., Alzahranic, A.K. and Alghadi, M.S., A 3D Sisko luid low with Cattaneo-Christov heat lux odel and heterogeneous-hoogeneous reactions: A nuerical stud, Journal o Molecular Liquids, 38 (017) pp [31] Li, J., Zheng, L. and Liu, L., MHD viscoelastic low and heat transer over a vertical stretching sheet with Cattaneo-Christov heat lux eects, Journal o Molecular Liquids, 1 (016) pp [3] Mustaa, M., Cattaneo-Christove heat lux odel or rotating low and heat transer o upper convected Maxwell luid, AIP Advances, 5 (015), [33] Abbasi, F M, Mustaa, M, Shehzad, S A, Alhuthali, M S and Haat, T., Analtical stud o Cattaneo- Christov heat lux odel or boundar laer low o an Oldrod-B luid, Chinese Phsics B, 5 (016) [34] Waqas, M., Haat, T., Farooq, M., Shehzad, S.A. and Alsaedi, A., Cattaneo-Christov heat lux odel or low o variable theral conductivit generalized Burgers luid, Journal o Molecular Liquids 0, pp [35] Shehzad, S.A., Abbasi, F.M., Haat, T. and Ahad, B., Cattaneo-Christov heat lux odel or thirdgrade luid low towards exponentiall stretching sheet, Applied Matheatics and Mechanics, 37 (6), pp [36] Abbasi, F. M., Shehzad, S. A., Haat, T., Alsaedi, A. and Hegaz, A., Inluence o Cattaneo- Christov heat lux in low o an Oldrod-B luid with variable theral conductivit, International Journal o Nuerical Methods or Heat & Fluid Flow,pp [37] Haat T., Awais M., Sajid M., Mass Transer Eects on the Unstead Flow o UCM Fluid over a Stretching Sheet. International Journal o Modern Phsics B, 5 (011), pp. pp [38] Pahlavan, A. A., Aliakbar, V., Farahani, F. V., and Sadegh, K. MHD low o UCM Fluids above Stretching Sheet using Two-auxiliar-paraeter Hootop Analsis Method. Counication in Nonlinear Science and Nuerical Siulation, 14 (009), pp. pp [39] Awais, M. M, Haat, T., Alsaedi, A. and Asghar, S., Tie-Dependent Three-Diensional Boundar 14

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