Impact of Internal Heat Source on Mixed Convective Transverse Transport of Viscoplastic Material under Viscosity Variation

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1 Commun. Theor. Phys Vol. 70, No. 4, Otober 1, 2018 Impat of Internal Heat Soure on Mixed Convetive Transverse Transport of Visoplasti Material under Visosity Variation R. Tabassum, 1, R. Mehmood, 2 and E. N. Maraj 2 1 Department of Mathematis, Faulty of Basi and Applied Sienes, Air University, Islamabad, Pakistan 2 Department of Mathematis, Faulty of Natural Sienes, HITEC University, Taxila Cantt, Pakistan Reeived January 23, 2018; revised manusript reeived June 19, 2018 Abstrat This ommuniation addresses the impat of heat soure/sink along with mixed onvetion on oblique flow of Casson fluid having variable visosity. Similarity analysis has been utilized to model governing equations, whih are simplified to set of nonlinear differential equations. Computational proedure of shooting algorithm along with 4th order Range-Kutta-Fehlberg sheme is opted to attain the veloity and temperature distributions. Impat of imperative parameters on Casson fluid flow, temperature, signifiant physial quantities suh as skin frition, loal heat flux and streamlines are displayed via graphs. DOI: / /70/4/423 Key words: oblique stagnation point flow, variable visosity, partial slip, mix onvetion, heat generation/absorption, Runge-Kutta Fehlberg sheme 1 Introdution Stagnation point flows are the most ommon fluid flow studied and examined in field of fluid dynamis beause of its frequent ourrene in many industrial and manufaturing proedures. The most general ase for fluid striking on a solid rigid surfae is when fluid strikes the surfae at any random angle. Most of the researh had been performed for the speial ase when fluid partiles strike the surfae orthogonally. In the field of aerodynamis, aeronautis and marine engineering problems oblique stagnation point flows are usually enountered. These flows have gained attention by many researhers and engineers during past few deades due to the above mentioned primary reasons. Stagnation point appears whenever a flow enroahes on a solid surfae. For stagnated flows, the veloities approah to zero along with the highest pressure on the surfae. [1] The boundary layer flow striking obliquely on a rigid plane has many engineering appliations espeially in aeronautis. These flows usually arise when a spurt of visous fluid obliquely strikes on the rigid plane beause of surfae silhouette or physial onstraints on nozzle. [2] In early twenties researhers have made good investigations in this ontext. Investigation on steady, nonorthogonal stagnation point flow was performed by Reza et al. [3] They reported the existene of boundary layer for the ase where the surfae strethed with veloity less than free stream fluid veloity. Moreover, upturned boundary layer appeared when a fluid far away from strethed surfae flows with veloity less than strething surfae veloity. Li et al. [4] investigated fored onvetion influene on heat transfer of visoelasti fluid transport towards an infinite planar surfae. They found that visoelastiity of the fluid ontributed in deelerating fluid flow and momentum boundary layer thikness. Rahman et al. [5] explored suh flow for nanofluid towards a shrinking surfae. They onluded that thiknesses of momentum, thermal and nanopartiles volume fration dereased with an inrease in shrinking parameter, for the upper branh solution and reversed trend was notied for the lower branh solution. Moreover, flow obliquity toward the surfae is inreased as strain rate intensifies. Influene of applied magneti field along with thermal radiation on heat transfer phenomenon was examined by Lv and Zheng. [6] Notable findings inluded that veloity slip affets the fluid flow signifiantly. Shahmohamadi [7] employed Casson model for steady free onvetive boundary layer flow where wall temperature was taken variable on horizontal plate. Another investigation on Casson model was performed by Nadeem et al. [8] They onsidered hydro magneti flow towards a nonlinearly shrinking porous planar sheet. Another innovation onsidering the Casson nanofluid was reported by Nadeem et al. [9] Ellahi et al. [10] derived homotopi analytial series solution of MHD third grade fluid in whih the effets of variable visosity were onsidered. They depited that inrease in pressure gradient deelerated fluid flow and third grade fluid parameter ontributed in reduing temperature and veloity distributions. Elbashbeshy and Bazid [11] used Runge-Kutta numerial integration sheme to examine heat transfer towards an extending surfae influened by variable internal heat generation and visosity having inverse linear relationship with temperature. Umavathi [12] applied a non-dary model to numerially investigate the ombined effets of fluid thermo physial harateristis and variable visosity on free onvetive Corresponding author, rabail.tabasum@mail.au.edu.pk; rashid.mehmood@hiteuni.edu.pk 2018 Chinese Physial Soiety and IOP Publishing Ltd

2 424 Communiations in Theoretial Physis Vol. 70 flow. Lin et al. [13 15] onsidered a Marangoni boundary layer flow of nanoliquid ontaining opper nanopartiles over a permeable disk with MHD and different nanopartiles shapes effets. No slip ondition between base fluid and nanopartiles was assumed. In some other investigations Lin et al. [16 17] studied the influene of film momentum, internal heat soure and thermal transport harateristis of thin power law liquids upon a strethed surfae plaed horizontally with influene of visous dissipation and variable thermal ondutivity. Lin et al. [18] also examined the heat transport harateristis of nanofluid in a rotating irular groove. Two types of thermal ondutivity models were onsidered. Reently Manjunatha et al. [19] arried out a numerial investigation on eletrially onduting dusty fluid over an unsteady extending planar surfae. In this problem both ondutivity and visosity were taken variable. Influene of slip ondition on nanofluid transport towards an elongating sheet was inspeted by Noghrehabadi et al. [20] Thermal radiation effets along with partial slip on a boundary layer flow was explored by Mukhopadhyay and Golra. [21] Das [22] inorporated variable internal heat soure/sink, thermal buoyany and partial slip in a onvetive heat transfer enhanement of nanofluid passing over the porous elongating surfae. Gorder and Vajravelu [23] made a omparative analysis of analytial and numerial solution of onvetive flow towards a permeable strething sheet. Sution and internal heat soure/sink onsequenes were also taken into aount. Alsaedi et al. [24] extended it by onsidering nanofluid with onvetive boundary ondition. Coalese outomes of mixed onvetion and internal heat generation or absorption in lid-driven avity under the influene of magneti field was investigated by Kumar et al. [25] Reent ontributions in this regard inlude Refs. [26 29]. In the light of above disussion, this is an attempt to examine influene of partial slip ondition and heat generation/ absorption on an oblique stagnation point flow in presene of mixed onvetion and variable visosity. No suh attempt has been reported in literature yet. Our formulation ontains nine parameters, namely, slip parameter ω, heat generation onstant δ, mix onvetion parameter λ, variable visosity parameter α, Casson fluid parameter, Prandtl number P r, Biot number Bi, strething ratio a/, and obliqueness of flow γ. Influene of above mentioned parameters on veloity and temperature distribution in addition to signifiant measurements like skin frition, loal heat flux and flow patterns are examined through graphs. Present novel finding may be benefiial and useful in aademi researh, aerodynamis and marine engineering. 2 Problem Development Here we onsider a non-orthogonal steady flow of a visoelasti fluid towards the planar strething sheet. Planar surfae is plae along x-axis. Surfae is strethed in suh a way that origin remains unaltered as shown in Fig. 1. Physial flow problem is onsidered to be influened by partial slip ondition and mix onvetion in presene of heat soure or sink. Moreover, visous dissipative effet is ignored in present study. Furthermore, all the fluid physial harateristis are taken to be onstant exept visosity. Model equations of the flow an be written as: [9] ū x + v ȳ = 0, 1 ū ū ū + v x ȳ + 1 p ρ x = µ b T ū ρ ȳ ū µ b T T ρ ȳ T ȳ + ρ T T T, 2 ū v v + v x ȳ + 1 p ρ ȳ = µ b T v ρ ȳ ρ v ȳ µ b T T T ȳ, 3 ū T T + v x ȳ = Q 0 α 2 T + ρ T T, 4 p where, µ b T = µ 0 e d T T is temperature dependent visosity, [30] T stands for temperature, µ0 is referene visosity and T denotes ambient temperature. ū, v represent veloity omponents in x, ȳ diretions, pressure is p, ρ f denotes fluid density, Casson fluid parameter is = µ B 2π /p y, T symbolizes oeffiient of thermal expansion, α denotes thermal diffusivity, Q 0 stands for oeffiient of heat generation/absorption and speifi heat is represented by p. Assoiated boundary onditions are: [9] ū = x + Nµ b T ū ȳ + v, v = 0, x k T ȳ = ht f T at ȳ = 0, 5 ū = a x + bȳ, T = T as ȳ. 6 In whih a, b, and are dimensional onstants and N is slip onstant. Fig. 1 Desription of the flow. Utilizing similarity analysis and employing following relations as defined in Ref. [9] x = x ν, y = ȳ ν, u = ū 1, ν

3 No. 4 Communiations in Theoretial Physis 425 v = v 1 ν, p = p µ, T = T T T f T, 7 where ν is the effetive kinemati visosity. Invoking Eq. 7 into Eqs. 1 to 6, following non-dimensional form is attained u x + v = 0, 8 u u x + v u + p x = e αt 2 u 2 α u u v x + v v + p T + λt, 9 1 = + 1 e αt 2 v 2 α v T, 10 P r u T x + v T = 2 T + P rδt, 11 u = x + ω e αt u + v x v = 0, T, = Bi1 T at y = 0, 12 u = a x + b y, T = 0, at y, 13 where α = dt f T represents variable visosity parameter, γ = b/ haraterizes obliqueness of the flow, Bi = h/k ν/ is Biot number, P r = ν/α is the Prandtl number, λ = ρ T T f T / ν is the mix onvetion parameter, ω = Nρ ν is the slip parameter and δ = Q 0 /ρ p is the heat soure δ > 0 or sink δ < 0 parameter. By invoking well established stream funtion relations [9] u = ψ, v = ψ x. 14 Inorporating above relations in Eqs. 8 to 11 and elimination of pressure term p by means of the equality p xy = p yx in Eqs. 9 and 10, gives e αt α T 2 ψ ψ + α 2 2 ψ T 2 3 ψ T α 2 3 T 2 ψ α α T x 3 ψ T 2 + α2 x x 2 ψ T x α 2 ψ x 2 T x + ψ, 2 ψ + λ T x, y = 0, 15 ψ T P r x ψ T = 2 T + P rδt. x 16 Following assoiated boundary onditions are yield: ψ = 0, ψ = x + ω 2 ψ e αt 2 2 ψ x 2, T = Bi1 T as y = 0, 17 ψ = a xy γy2, T = 0 as y. 18 Rewriting the stream funtion as defined in Ref. [9] ψx, y = xfy + gy, T x, y = θy. Here fy and gy represent normal and tangential flow omponents. Employing Eq. 19 into Eqs. 15 to 18 and integrating one with respet to y, one reahes to following system of non-linear ordinary differential equations: e αθ αθ f + f + ff f 2 + C 1 = 0, e αθ αθ g +g +fg f g +λθ+c 2 = 0, 20 θ + P rfθ + δθ = Here the differentiation with respet to y is denoted by primes, C 1 and C 2 are integration onstants. Consequently, orresponding boundary onditions take the following form: f0 = 0, f 0 = 1 + ω e αθ0 f 0, g 0 = ω e αθ0 g 0, θ 0 = Bi1 θ0, f = a, g = γ, θ = Constant C 1 is omputed by applying the limit y on Eq. 20 and using boundary ondition f = a/. Preisely, we get C 1 = a/ 2. From Eq. 20, one an depit that normal flow omponent is of the form a/y +A as y, here A is onstant, whih is responsible for boundary layer shift. Value of arbitrary onstant C 2 is omputed by applying the limit y on Eq. 21 and using the boundary ondition g = γ. Preisely, we get C 2 = Aγ. Aordingly, Eqs. 20 and 21 take the following form: 1+ 1 a 2 e αθ αθ f +f +ff f 2 + = 0, e αθ αθ g +g +fg f g +λθ Aγ = 0.24 Introduing g y = γhy. 25 Using Eq. 26 in Eq e αθ αθ h + h + fh f h + λ θ A = 0, 26 γ along with boundary onditions h0 = ω e αθ0 h 0, h = Numerial Solution The simplified system of Eqs. 22, 24, 27 along with boundary onditions 23 and 28 are takled numerially by utilizing fourth order Range-Kutta Fehlberg sheme embedded with shooting algorithm. [30] Firstly,

4 426 Communiations in Theoretial Physis Vol. 70 higher order boundary value problem is simplified into system of initial value problem by introduing additional onditions in terms of unknown parameters termed as shooting parameters as a substitute of boundary onditions as y. Seondly, this system of initial value problem is solved iteratively and the unknown shooting parameters are determined suh that boundary onditions as y are satisfied. Following the above mentioned proedure new variables y 1, y 2, y 3, y 4, y 5, y 6, and y 7 are introdued as: f = y 1, f = y 1 = y 2, f = y 2 = y 3, h = y 4, h = y 4 = y 5, θ = y 6, θ = y 6 = y By invoking above mentioned substitutions in set of Eqs following system is yield: y 1 = y 2, 29 y 2 = y 3, 30 y 3 = eαy 6 a 2 ηα e αy6 y 7 y 3 y 1 y 3 + y2 2, η 31 y 4 = y 5, 32 y 5 = eαy6 ηα e αy 6 y 7 y 5 y 1 y 5 + y 2 y 4 λ η γ y 6 + A, 33 y 6 = y 7, 34 y 7 = P ry 1 y 7 + δy 6, 35 where, η = 1 + 1/. Along with Initial onditions y 1 0 = 0, y 2 0 = 1 + ω e αy60 y 3 0, y 3 0 = b 1, y 4 0 = ω e αy60 y 5 0, y 5 0 = b 2, y 6 0 = b 3, y 7 0 = Bib Here the shooting parameters b 1, b 2, and b 3 are initially guessed and afterward determined by means of Newton Raphson s method for eah set of parameter value. The onverted initial value problem is numerially dealt by applying integration sheme of fourth order Runge- Kutta-Fehlberg method. Iterative steps are performed till auray of ten deimal plaes is ahieved. Computational proedure is performed in omputational software MAT- LAB. 4 Results and Disussion Present setion fouses on examining flow harateristis along with temperature distribution, skin frition and loal surfae heat flux against signifiant emerging physial fators. For this purpose Figs. 2 to 16 are plotted, whih provide graphial illustrations for distint parameters suh as slip parameter ω, heat generation onstant δ, variable visosity parameter α, Prandtl number P r, Biot number Bi and mix onvetion parameter λ on normal f y, tangential h y veloity omponents, and temperature θy. Streamlines plots for slip parameter ω are also shown to desribe the flow pattern in Figs Fig. 2 of ω. Normal veloity variation for inreasing values Fig. 3 Normal veloity distribution for inreasing values Fig. 4 of ω. Tangential veloity variation for distint values Figures 2 and 3 reveal the behavior of normal omponent of veloity. Figure 2 desribes the behavior of veloity profile f y for various values of slip parameter ω. Graph shows that f y dereases by inreasing slip parameter ω. Figure 3 shows that normal veloity f y dereases with rise in variable visosity parameter α. Effets of sundry parameters on tangential veloity h y are displayed in Figs. 4 to 6.

5 No. 4 Communiations in Theoretial Physis 427 α on temperature distribution θy. It is onluded that Prandtl number P r ontributes in lowering temperature as shown in Fig. 10. This happens beause P r being the ratio of visous to thermal diffusivity leads to lessen fluid temperature. Fig. 5 Tangential veloity distribution for distint values of λ. Fig. 8 Temperature variation for inreasing values of δ. Fig. 6 Tangential veloity variation for distint values Fig. 9 Temperature distribution for inreasing values of Bi. Fig. 7 of ω. Temperature distribution for inreasing values From these figures it is witnessed that tangential veloity omponent aelerates with inrease in slip parameter ω, mix onvetion parameter λ, and variable visosity parameter α. However, away from the strething surfae this trend altered. Figures 7 to 11 illustrate the influene of slip parameter ω, heat generation onstant δ, Biot number Bi, Prandtl number P r and variable visosity parameter Fig. 10 of P r. Temperature distribution for inreasing values Figures 7, 8, 9, and 11 illustrate that temperature inreases by inreasing slip parameter ω, heat generation onstant δ, Biot number Bi, and variable visosity parameter α respetively. Influene of variable visosity parameter α on normal and tangential skin frition oeffiients

6 428 Communiations in Theoretial Physis Vol. 70 is shown through Figs. 12 and 13. Normal skin frition oeffiient f 0 dereases with a rise in variable visosity parameter α as shown in Fig. 12, on the other hand, Fig. 13 desribes that tangential skin frition oeffiient h 0 rises when variable visosity parameter α inreases. and γ = 10. Figure 15 depits that flow with ω = 2 is more tilted towards the left as ompared to the flow with ω = 0.2 and γ = 10. It is observed in Fig. 16 that flow pattern is more tilted towards the right with slip parameter ω = 2 and γ = 10. Fig. 14 Variation in loal heat flux for distint values Fig. 11 Temperature distribution for inreasing values Fig. 12 Variation in normal skin frition oeffiient for distint values Fig. 15 Streamlines for slip parameter ω with obliqueness γ = 10. Fig. 13 Variation in tangential skin frition oeffiient for distint values Figure 14 is skethed to visualize the loal heat flux θ 0 for distint values of variable visosity parameter α. From this figure it is depited that loal heat flux drops with a rise in variable visosity parameter α. Figures 15 and 16 present streamlines of the flow for different values of slip parameter ω with obliqueness parameter γ = 10 Fig. 16 Streamlines for slip parameter ω with obliqueness γ = Conluding Remarks Present artile examined heat transfer and flow phenomena of a fluid having variable visosity influened by

7 No. 4 Communiations in Theoretial Physis 429 mixed onvetion, partial slip ondition and heat generation or absorption. Here fluid was onsidered to be striking the strething surfae obliquely. Moreover, visous dissipation effet was ignored and Casson fluid model was inorporated to study visoelasti fluid rheologial harateristis. Governing non-linear ODE s of physial problem were numerially dealt by means of Range-Kutta Fehlberg sheme along with shooting algorithm. [30] Computational results were extrated out by keeping auray up to ten deimals. Influene of effetive parameters was disussed through graphs. Core findings of above study are: i Normal veloity profile f y dereases while tangential veloity h y inreases with an inreases in slip parameter ω. ii Temperature profile θy rises with visosity variation parameter α, slip parameter ω and heat generation onstant δ. iii A derease is found in normal skin frition oeffiient f 0 with variable visosity parameter α, while tangential skin frition oeffiients h 0 enhaned with α. iv Loal heat flux θ 0 against slip parameter ω dropped with an inrease in variable visosity parameter α. Present finding may be benefiial and useful in aademi researh, aerodynamis and marine engineering. Referenes [1] F. Labropulu, D. Li, and I. Pop, Int. J. Therm. Si [2] C. Y. Wang, Phys. Fluids [3] M. Reza and A. S. Gupta, Fluid Dyn. Res [4] D. Li, F. Labropulu, and I. Pop, Int. J. NonLinear Meh [5] M. M. Rahman, T. Grosan, and I. Pop, Int. J. Numer. Methods Heat Fluid Flow [6] Y. Lv and L. Zheng, Int. J. Engg. Si. Innov. Teh [7] S. Shahmohamadi, Mehania [8] S. Nadeem and R. UlHaq, Si. Iran [9] S. Nadeem, R. Mehmood, and N. S. Akbar, J. Comput. Theor. Nanosi [10] R. Ellahi and A. Arshad, Math. Comput. Model [11] E. M. A. Elbashbeshy and M. A. A. Bazid, Can. J. Phys [12] J. C. Umavathi, Transp. Porous Media [13] Y. Lin and L. Zheng, AIP Advanes [14] Y. Lin, L. Zheng, and X. Zhang, Meh. Time-Depend. Materials [15] Y. Lin, B. Li, L. Zheng, and G. Chen, Powder Tehnology [16] Y. Lin, L. Zheng, and G. Chen, Powder Tehnology [17] Y. Lin, L. Zheng, and L. Ma, Appl. Math. Meh [18] Y. Lin and Y. Jiang, Int. J. Heat Mass Trans [19] S. Manjunatha and B. J. Gireesha, Ain Shams Engg. J [20] A. Noghrehabadi, R. Pourrajab, and M. Ghalambaz, Int. J. Therm. Si [21] S. Mukhopadhyay and R. S. R. Gorla, Heat Mass Transf [22] K. Das, Comput. Fluids [23] R. A. Van Gorder and K. Vajravelu, Appl. Math. Comput [24] A. Alsaedi, M. Awais, and T. Hayat, Commun. Nonl. Si. Numer. Simul [25] K. Saha, K. M. Salahuddin, and M. A. Taher, Amer. J. Appl. Math [26] Z. Mehmood, R. Mehmood, and Z. Iqbal, Commun. Theor. Phys [27] B. Ahmad, Z. Iqbal, and E. Azhar, Int. J. Appl. Eletrom. Meh [28] S. Rana, R. Mehmood, and N. S. Akbar, J. Mol. Liq [29] Z. Iqbal, R. Mehmood, E. Azhar, and Z. Mehmood, Eur. Phys. J. Plus [30] W. A. Khan, O. D. Makinde, and Z. H. Khan, Int. J. Heat Mass Transf