- Email: [email protected]
A numerical study for three-dimensional viscoelastic flow inspired by non-linear radiative heat flux
International Journal of Non-Linear Mechanics 79 (2016) 83–87
Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
A numerical study for three-dimensional viscoelastic flow inspired by non-linear radiative heat flux Ammar Mushtaq a, M. Mustafa b,n, T. Hayat c,d, A. Alsaedi d a
Research Centre for Modeling and Simulation (RCMS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan c Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan d Non-linear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia b
art ic l e i nf o
a b s t r a c t
Article history: Received 9 April 2015 Received in revised form 18 September 2015 Accepted 15 November 2015 Available online 22 November 2015
Present article examines the three-dimensional flow of upper-convected Maxwell (UCM) fluid over a radiative bi-directional stretching surface. Novel non-linear Rosseland formula for thermal radiation is utilized in the formulation of energy equation. The conventional transformations lead to a strongly nonlinear differential system which is treated numerically through Runge–Kutta integration procedure together with the shooting approach. We found that heat transfer rate from the sheet has inverse as well as non-linear relationship with wall to ambient temperature ratio. Moreover an increase in viscoelastic fluid parameter (Deborah number) corresponds to a decrease in the fluid velocity and the boundary layer thickness. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Upper-convected Maxwell (UCM) fluid Three-dimensional flow Non-linear radiation Numerical solution
1. Introduction Wang [1] was probably the first to explore the threedimensional flow induced by a bi-directionally stretching surface. He successfully developed a self-similar solution of the threedimensional Navier–Stokes equations. Ariel [2] found the series solution for Wang's problem by using the homotopy perturbation method (HPM). Liu and Andersson [3] investigated the heat transfer over a bidirectional stretching sheet with variable thermal conditions. Homotopy analysis method (HAM) based analytical study of three-dimensional flow and heat transfer above an implusively stretching surface was presented by Xu et al. [4]. Sajid et al. [5] also derived homotopy solutions for three-dimensional flow of Walters B liquid over a linearly stretching surface. Timedependent flow of Maxwell fluid bounded by an unsteady stretching sheet was addressed by Awais et al. [6]. Khan et al. [7] discussed the three-dimensional flow and heat transfer caused by a stretching surface subject to general power-law surface velocity and temperature distribution. Xu and Pop [8] discussed the impact of bio-convection on nanofluid flow through a vertical channel containing gyrotactic microorganisms by optimal homotopy analysis approach. Very recently, three-dimensional flow of nanofluids n
Corresponding author. Tel.: þ 92 51 90855596. E-mail addresses: [email protected] (A. Mushtaq), [email protected], [email protected] (M. Mustafa), [email protected] (T. Hayat), [email protected] (A. Alsaedi). http://dx.doi.org/10.1016/j.ijnonlinmec.2015.11.006 0020-7462/& 2015 Elsevier Ltd. All rights reserved.
induced by a non-linearly stretching sheet was investigated by Khan et al. [9]. Revolving flow over a stretching disk with heat transfer was numerically examined by Turkyilmazoglu [10]. The phenomenon of radiative heat transfer has relevance in numerous industrial applications including power generation, combustion applications, nuclear reactor cooling etc. Raptis [11] investigated a boundary layer flow problem considering thermal radiation effect. He linearized the Rosseland formula for thermal radiation by assuming small temperature differences within the flow. In a recent article, Magyari and Pantokratoras [12] showed that linear radiation heat transfer problem reduces to a simple rescaling of Prandtl number by a factor containing the radiation parameter. Keeping this in view, Rahman and El-tayeb [13] considered the radiation effects on the flow of an electrically conducting nanofluid by using the exact Rosseland formula. Pantokratoras and Fang [14] investigated the Sakiadis and Blasius flow problems considering the non-linear radiation. Recent studies pertaining to the non-linear radiation heat transfer in the boundary layer flows can be found in the Refs. [15–19]. The purpose of current work is to investigate the non-linear radiative heat transfer in the three-dimensional flow of UCM fluid bounded by a bi-directional stretching surface. Upper-convected Maxwell (UCM) fluid is a subclass of rate type fluids which is important in describing the influence of fluid relaxation time. It has gained special attention of the researchers in the past due to its simplicity. Recently, various papers involving the flow analysis of Maxwell fluid have appeared (see for instance [20–26]). It will
84
A. Mushtaq et al. / International Journal of Non-Linear Mechanics 79 (2016) 83–87
be seen later that consideration of non-linear radiative heat flux produces a strongly non-linear but interesting energy equation for the temperature field. Shooting method together with fifth-order Runge–Kutta integration and Newton method is employed for the development of numerical solution. Computational results for both viscous and Maxwell fluids are presented in a tabular form. Graphical results for the velocity and temperature distributions are also presented and analyzed.
2. Basic equations Consider three-dimensional flow of upper-convected Maxwell (UCM) fluid induced by a stretching surface occupying the xy-plane (see Fig. 1). The velocities of the sheet along the x- and y-directions are assumed to be U w ðxÞ ¼ ax and V w ðyÞ ¼ by respectively in which a; b 4 0 are constants. Let T w be the constant temperature at the sheet whereas T 1 denotes the fluid temperature outside the thermal boundary layer. The equations governing the three-dimensional flow of UCM fluid with radiative heat transfer are expressed below (see Liu and Andersson [3] and Awais et al. [6] for details): ∂u ∂v ∂w þ þ ¼ 0; ∂x ∂y ∂z
ð1Þ
0 1 2 2 2 u2∂∂xu2 þ v2∂∂yu2 þ w2∂∂zu2 2 ∂u ∂u ∂u @ A ¼ ν ∂ u; u þ v þ w þ λ1 ∂2 u ∂2 u ∂2 u ∂x ∂y ∂z ∂z2 þ 2uv∂x∂y þ2uw∂x∂z þ 2vw∂y∂z
ð2Þ
0 1 2 2 2 u2∂∂xv2 þ v2∂∂yv2 þ w2∂∂z2v 2 ∂v ∂v ∂v @ A ¼ ν∂ v; u þ v þ w þ λ1 ∂2 v ∂2 v ∂2 v ∂x ∂y ∂z ∂z2 þ 2uv∂x∂y þ2uw∂x∂z þ 2vw∂y∂z
ð3Þ
u
∂T ∂T ∂T ∂2 T 1 ∂qr þ v þw ¼ α 2 ; ∂x ∂y ∂z ρcp ∂z ∂z
ð4Þ
where u; v and w are the velocity components along the x-, y- and z-directions respectively, λ1 is the fluid relaxation time, ν is the kinematic viscosity, α is thermal diffusivity of the fluid, ρ is the fluid density, C p is the specific heat, qr ¼ 4σ =3k ∂T 4 =∂z is the Rosseland radiative heat flux in which σ is the Stefan–Boltzman con stant and k is the mean absorption coefficient respectively. The boundary conditions in the present problem are: u ¼ U w ¼ ax; v ¼ V w ¼ by; w ¼ 0; T ¼ T w
at z ¼ 0;
u; v-0; T-T 1
as z-1:
ð5Þ
z Vw
by
o
x
Tw
Uw
ax
y Fig. 1. A schematic diagram showing the development of boundary layer.
We now introduce the following non-dimensional quantities rffiffiffi pffiffiffiffiffiffi a df dg T T1 α ¼ z ; u ¼ ax : ; v ¼ ay ; w ¼ ν νaðf þ g Þ; θ ¼ dη dη ν Tw T1 ð6Þ In view of the above variables, the continuity Eq. (1) is automatically satisfied and the Eqs. (2)–(5) become " # 2 3 2 2 3 d f df d f df d f 2d f þ ðf þ g Þ 2 þ K 2ðf þ g Þ ðf þ g Þ ¼ 0; ð7Þ dη dη dη2 dη3 dη dη3 " # 2 3 2 2 3 d g dg d g dg d g 2d g þ ð f þ g Þ þ K 2 ð f þ g Þ ð f þ g Þ ¼ 0; dη dη dη2 dη3 dη2 dη3
ð8Þ
3 dθ 1 d dθ þ ðf þ g Þ ¼ 0; 1 þRd 1 þ θw 1 θ Pr dη dη dη
ð9Þ
f ¼ g ¼ 0; ddfη ¼ 1; ddgη ¼ c; θ ¼ 1 df -0; ddgη-0; dη
θ-0
at
η ¼ 0;
as
η-1:
ð10Þ
In the above equations K ¼ λ1 a is the Deborah number, c ¼ b=a is the stretching rates ratio, θw ¼ T w =T 1 is the temperature ratio parameter, Pr ¼ ν=α is the Prandtl number and Rd ¼ 16σ T 31 =3kk is the radiation parameter. Note that for c ¼ 0 the above model corresponds to the two-dimensional flow whereas axisymmetric flow case is achieved by setting c ¼ 1. The quantity of practical interest here is the local Nusselt number Nux defined by Nux ¼
xqw ; kðT w T 1 Þ
ð11Þ
where qw ¼ kð∂T=∂zÞz ¼ 0 þ qr is the wall heat flux. Now using dimensionless quantities from Eq. (6) into Eq. (11) we obtain 1=2
Rex
0
Nux ¼ ½1 þ Rdθw θ ð0Þ; 3
ð12Þ
where Rex ¼ U w x=ν is the local Reynolds number.
3. Numerical results and discussion The solutions of Eqs. (7)–(9) with the boundary conditions (10) are obtained numerically by employing shooting approach with fifth-order Runge–Kutta method. The unknown initial conditions 0 00 f ð0Þ; g″ð0Þ and θ ð0Þ are estimated iteratively through Newton method. All computations are successfully performed in MATLAB with error tolerance of 10 6 . Our main focus in this section is to explore the behaviors of embedded flow parameters on the velocity and temperature distributions. For this purpose we plot Figs. 2–8 and prepare Tables 1 and 2 for the computational results. For the validity of present simulations, we compared the 0 numerical results of f ″ð0Þ; g″ð0Þ and θ ð0Þ with Liu and Andersson [3] in a limiting sense. The results appear to be almost identical in all the cases as can be seen through Table 1. Influence of Deborah number K on the x- and y-components of velocity is sketched in the Fig. 2. The Deborah number is defined as the ratio of fluid relaxation time to its characteristic time scale. Relaxation time is the time taken by the fluid to gain equilibrium once the shear stress is imposed. The relaxation time is expected to be larger for the fluids having higher viscosity. Therefore an increase in K may be regarded as an increase in the fluid viscosity which restricts the fluid motion and hence the velocity decreases. Due to this reason the hydrodynamic boundary layer thins when K is increased. We 0 also noted that change in the velocity fields f and g 0 is larger in the three-dimensional flow when compared with the twodimensional and axisymmetric flows.
A. Mushtaq et al. / International Journal of Non-Linear Mechanics 79 (2016) 83–87
0
Fig. 2. Effect of K on f and g0 .
85
Fig. 5. Effects of θw and K on θ.
Fig. 6. Effect of Rd on θ when θw ¼ 1:1. Fig. 3. Effects of K and c on θ.
Fig. 7. Effect of Rd on θ when θw ¼ 2. Fig. 4. Effects of Pr and Rd on θ.
Fig. 3 displays the variation in temperature θ when the Deborah number K is increased. The coupled differential system indicates that heat flux is also related with the velocity. Temperature θ appears to increase with an increase in the fluid relaxation time. This means that elastic force promotes the heat transfer in the viscoelastic fluids. Fig. 4 shows that temperature θ and the thermal boundary layer thickness decrease when Prandtl
number Pr is increased. This decrease infact results due to a reduction in the thermal diffusivity which allows for shorter penetration depth of temperature. Fig. 5 includes the behavior of temperature ratio parameter θw on the temperature distribution. Bigger values of θw corresponds to a larger temperature and thicker thermal boundary layer. This is explained as follows. It is clear from energy Eq. (4) that effective thermal diffusivity is the sum of classical thermal diffusivity (α) and thermal diffusivity due
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A. Mushtaq et al. / International Journal of Non-Linear Mechanics 79 (2016) 83–87
Fig. 8. Effects of Pr; K and c on θ0 ð0Þ. Table 1 Comparison of present results with Liu and Andersson [3] when K ¼ 0; Pr ¼ 1 and Rd ¼ 0. c
Liu and Andersson [3] 00
0 0.25 0.50 0.75 1
Present
f ð0Þ
g″ð0Þ
θ0 ð0Þ
f ð0Þ
00
g″ð0Þ
θ0 ð0Þ
4. Summary
1 1.048813 1.093096 1.134486 1.173721
0 0.194565 0.465206 0.794619 1.173721
– 0.665933 0.735334 0.796472 –
1 1.048811 1.093095 1.134486 1.173721
0 0.194564 0.465205 0.794618 1.173721
0.581978 0.665926 0.735333 0.796472 0.851992
Numerical solutions for flow and non-linear radiative heat transfer in upper-convected Maxwell (UCM) fluid are presented and analyzed. The main observations of this study are outlined below:
Table 2 Numerical results of wall temperature gradient θ0 ð0Þ for different values of K; c; Pr and Rd. K
c
Pr
Rd ¼0
Rd ¼1 Linear radiation
1
0.5
1
0 0.3 0.6 1 0.5
0 0.5 1 1.5
deviate from the corresponding profiles of the linear radiation case (see Fig. 7). This leads to the conclusion that linear and non-linear radiation results would be identical only when θw is nearly equal to unity and Rd is sufficiently small (say Rd o0:1). In Fig. 8 the effects of Pr on the local Nusselt number are presented at different values of Deborah number K. Prandtl number compares the convective heat transfer to the conductive heat transfer. Larger Pradntl number fluids are favorable in transferring energy from the unit area compared to pure conduction. We already observed in Fig. 4 that profiles become steeper when Pr is gradually increased. This means that heat transfer rate from the surface (which is proportional to the wall slope of the temperature) escalates with an increase in Pr: On the other hand magni0 tude of θ ð0Þ has inverse and non-linear relationship with the Deborah number K. 0 Table 2 gives the values of wall slope of temperature θ ð0Þ for different values of K, c, Pr and Rd. In absence of thermal radiation, there is approximately 6% reduction in the heat transfer when K varies from K ¼ 0 to K ¼ 1:5 with Pr ¼ 7 and c ¼ 0:5. This reduction increases to almost 10% when non-linear radiation with θw ¼ 1:5 is considered. Similarly heat transfer rate grows sharply with an increase in Pr in the presence non-linear radiation case.
2 4 7 10 7
7
1.01695 1.60165 2.24393 2.75508 1.82603 2.09403 2.31293 2.55918 2.35436 2.29665 2.24393 2.19501
0.61177 1.01695 1.47271 1.83692 1.20254 1.37824 1.51544 1.66440 1.59321 1.53021 1.47271 1.41998
Non-linear radiation θw ¼ 1:1
θw ¼ 1:5
0.53932 0.90348 1.31435 1.64284 1.07372 1.23051 1.35219 1.48361 1.42760 1.36840 1.31435 1.26482
0.31686 0.55107 0.82083 1.03739 0.67255 0.77013 0.84341 0.92038 0.91088 0.86380 0.82083 0.78170
to the radiation effect (16σ T 3 =3ρC p k ). Thus one anticipates that parameter θw , being the coefficient of the later term, would support the thermal boundary layer thickness. It can be further noticed that the profiles attain special S-shaped form when θw enlarges which dictates the existence of adiabatic case. In other words, the wall temperature gradient approaches zero value when wall to ambient temperature ratio is sufficiently large. Fig. 6 shows the effect of radiation parameter Rd on temperature profile in linear and non-linear radiation cases when θw ¼ 1:1. It is noticeable that profiles in both the cases merge only at small values of radiation parameter Rd. Difference between the linear and non-linear radiation results continue to grow when Rd is incremented. As temperature ratio increases from θw ¼ 1:1 to θw ¼ 2, the profiles in non-linear radiation case significantly
1. Boundary layer thickness in the Newtonian fluid is greater than that in the UCM fluid. 2. In contrast to the linear radiation heat transfer, the thermal boundary layer thickness is controlledbyvariable (non-linear) thermal diffusivity of the form α þ 16σ T 3 =3ρC p k : 3. The vertical component of velocity at the far field boundary is negative and its magnitude increases with an increase in the stretching rates ratio c. 4. The heat transfer rate from the sheet in a viscous fluid is greater than that in the UCM fluid. 5. Temperature profiles exhibit an interesting S-shaped pattern when the wall to ambient temperature ratioθw is sufficiently large. 6. Variation in the temperature distribution with an increase in the radiation parameter Rd is significant when larger values of θw are accounted. 7. Present consideration in the case of Newtonian fluid can be obtained by substituting K ¼ 0.
References [1] C.Y. Wang, The three-dimensional flow due to a stretching flat surface, Phys. Fluids 27 (1984) 1915–1917. [2] P.D. Ariel, Three-dimensional flow past a stretching sheet and the homotopy perturbation method, Comput. Math. Appl. 54 (2007) 920–925. [3] I.C. Liu, H.I. Andersson, Heat transfer over a bidirectional stretching sheet with variable thermal conditions, Int. J. Heat Mass Transf. 51 (2008) 4018–4024. [4] H. Xu, S.J. Liao, I. Pop, Series solutions of unsteady three-dimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate, Eur. J. Mech. B/Fluids 26 (2007) 15–27. [5] M. Sajid, T. Hayat, I. Pop, Three dimensional flow over a stretching surface in a viscoelastic fluid, Nonlinear Anal.: RWA 9 (2008) 1811–1822. [6] M. Awais, T. Hayat, A. Alsaedi, S. Asghar, Time-dependent three-dimensional boundary layer flow of a Maxwell fluid, Comput. Fluids 91 (2014) 21–27. [7] J.A. Khan, M. Mustafa, T. Hayat, A. Alsaedi, On three-dimensional flow and heat transfer over a non-linearly stretching sheet: analytical and numerical solutions, PLoS One 9 (2014) e107287, http://dx.doi.org/10.1371/journal. pone.0107287.
A. Mushtaq et al. / International Journal of Non-Linear Mechanics 79 (2016) 83–87 [8] H. Xu, I. Pop, Fully developed mixed convection flow in a horizontal channel filled by a nanofluid containing both nanoparticles and gyrotactic microorganisms, Eur. J. Mech. B/Fluids 46 (2014) 37–45. [9] J.A. Khan, M. Mustafa, T. Hayat, A. Alsaedi, Three-dimensional flow of nanofluid over a non-linearly stretching sheet: an application to solar energy, Int. J. Heat Mass Transf. 86 (2015) 158–164. [10] M. Turkyilmazoglu, Bodewadt flow and heat transfer over a stretching stationary disk, Int. J. Mech. Sci. 90 (2015) 246–250. [11] A. Rapits, C. Perdikis, Viscoelastic flow by the presence of radiation, ZAMP 78 (1998) 277–279. [12] E. Magyari, A. Pantokratoras, Note on the effect of thermal radiation in the linearized Rosseland approximation on the heat transfer characteristics of various boundary layer flows, Int. Commun. Heat Mass Transf. 38 (2011) 554–556. [13] M.M. Rahman, I.A. Eltayeb, Radiative heat transfer in a hydromagneticnanofluid past a non-linear stretching surface with convective boundary condition, Meccanica 48 (2013) 601–615. [14] A. Pantokratoras, T. Fang, Blasius flow with non-linear Rosseland thermal radiation, Meccanica 49 (2014) 1539–1545. [15] A. Mushtaq, M. Mustafa, T. Hayat, A. Alsaedi, Nonlinear radiative heat transfer in the flow of nanofluid due to solar energy: a numerical study, J. Taiwan Inst. Chem. Eng. 45 (2014) 1176–1183. [16] M. Mustafa, A. Mushtaq, T. Hayat, B. Ahmad, Nonlinear radiation heat transfer effects in the natural convective boundary layer flow of nanofluid past a vertical plate: a numerical study, PLoS One (2014), http://dx.doi.org/10.1371/ journal.pone.0103946. [17] R. Cortell, MHD (magneto-hydrodynamic) flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/ absorption, Energy 74 (2014) 896–905.
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[18] A. Pantokratoras, Natural convection along a vertical isothermal plate with linear and non-linear Rosseland thermal radiation, Int. J. Therm. Sci. 84 (2014) 151–157. [19] M. Mustafa, A. Mushtaq, T. Hayat, A. Alsaedi, Radiation effects in threedimensional flow over a bi-directional exponentially stretching sheet, J. Taiwan Inst. Chem. Eng. 47 (2015) 403–409. [20] M. Kumari, G. Nath, Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field, Int. J. Non-linear Mech. 44 (2009) 1048–1055. [21] C. Fetecau, M. Jamil, C. Fetecau, I. Siddiqui, A note on the second problem of Stokes for Maxwell fluids, Int. J. Non-linear Mech. 44 (2009) 1085–1090. [22] K.L. Hsiao, MHD mixed convection for viscoelastic fluid past a porous wedge, Int. J. Non-linear Mech. 46 (2011) 1–8. [23] A.D. Polyanin, A.I. Zhurov, Integration of linear and some model non-linear equations of motion of incompressible fluids, Int. J. Non-linear Mech. 49 (2013) 77–83. [24] K. Bhattacharyya, T. Hayat, A. Alsaedi, Dual solutions in boundary layer flow of Maxwell fluid over a porous shrinking sheet, Chin. Phys. B 23 (2014) 124701. [25] B.S. Bhadauria, P. Kiran, Weak non-linear oscillatory convection in a viscoelastic fluid layer under gravity modulation, Int. J. Non-linear Mech. 65 (2014) 133–140. [26] M. Mustafa, J.A. Khan, T. Hayat, A. Alsaedi, Sakiadis flow of Maxwell fluid considering magnetic field and convective boundary conditions, AIP Adv. 5 (2015) 27106, http://dx.doi.org/10.1063/1.4907927.
Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
A numerical study for three-dimensional viscoelastic flow inspired by non-linear radiative heat flux Ammar Mushtaq a, M. Mustafa b,n, T. Hayat c,d, A. Alsaedi d a
Research Centre for Modeling and Simulation (RCMS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan c Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan d Non-linear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia b
art ic l e i nf o
a b s t r a c t
Article history: Received 9 April 2015 Received in revised form 18 September 2015 Accepted 15 November 2015 Available online 22 November 2015
Present article examines the three-dimensional flow of upper-convected Maxwell (UCM) fluid over a radiative bi-directional stretching surface. Novel non-linear Rosseland formula for thermal radiation is utilized in the formulation of energy equation. The conventional transformations lead to a strongly nonlinear differential system which is treated numerically through Runge–Kutta integration procedure together with the shooting approach. We found that heat transfer rate from the sheet has inverse as well as non-linear relationship with wall to ambient temperature ratio. Moreover an increase in viscoelastic fluid parameter (Deborah number) corresponds to a decrease in the fluid velocity and the boundary layer thickness. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Upper-convected Maxwell (UCM) fluid Three-dimensional flow Non-linear radiation Numerical solution
1. Introduction Wang [1] was probably the first to explore the threedimensional flow induced by a bi-directionally stretching surface. He successfully developed a self-similar solution of the threedimensional Navier–Stokes equations. Ariel [2] found the series solution for Wang's problem by using the homotopy perturbation method (HPM). Liu and Andersson [3] investigated the heat transfer over a bidirectional stretching sheet with variable thermal conditions. Homotopy analysis method (HAM) based analytical study of three-dimensional flow and heat transfer above an implusively stretching surface was presented by Xu et al. [4]. Sajid et al. [5] also derived homotopy solutions for three-dimensional flow of Walters B liquid over a linearly stretching surface. Timedependent flow of Maxwell fluid bounded by an unsteady stretching sheet was addressed by Awais et al. [6]. Khan et al. [7] discussed the three-dimensional flow and heat transfer caused by a stretching surface subject to general power-law surface velocity and temperature distribution. Xu and Pop [8] discussed the impact of bio-convection on nanofluid flow through a vertical channel containing gyrotactic microorganisms by optimal homotopy analysis approach. Very recently, three-dimensional flow of nanofluids n
Corresponding author. Tel.: þ 92 51 90855596. E-mail addresses: [email protected] (A. Mushtaq), [email protected], [email protected] (M. Mustafa), [email protected] (T. Hayat), [email protected] (A. Alsaedi). http://dx.doi.org/10.1016/j.ijnonlinmec.2015.11.006 0020-7462/& 2015 Elsevier Ltd. All rights reserved.
induced by a non-linearly stretching sheet was investigated by Khan et al. [9]. Revolving flow over a stretching disk with heat transfer was numerically examined by Turkyilmazoglu [10]. The phenomenon of radiative heat transfer has relevance in numerous industrial applications including power generation, combustion applications, nuclear reactor cooling etc. Raptis [11] investigated a boundary layer flow problem considering thermal radiation effect. He linearized the Rosseland formula for thermal radiation by assuming small temperature differences within the flow. In a recent article, Magyari and Pantokratoras [12] showed that linear radiation heat transfer problem reduces to a simple rescaling of Prandtl number by a factor containing the radiation parameter. Keeping this in view, Rahman and El-tayeb [13] considered the radiation effects on the flow of an electrically conducting nanofluid by using the exact Rosseland formula. Pantokratoras and Fang [14] investigated the Sakiadis and Blasius flow problems considering the non-linear radiation. Recent studies pertaining to the non-linear radiation heat transfer in the boundary layer flows can be found in the Refs. [15–19]. The purpose of current work is to investigate the non-linear radiative heat transfer in the three-dimensional flow of UCM fluid bounded by a bi-directional stretching surface. Upper-convected Maxwell (UCM) fluid is a subclass of rate type fluids which is important in describing the influence of fluid relaxation time. It has gained special attention of the researchers in the past due to its simplicity. Recently, various papers involving the flow analysis of Maxwell fluid have appeared (see for instance [20–26]). It will
84
A. Mushtaq et al. / International Journal of Non-Linear Mechanics 79 (2016) 83–87
be seen later that consideration of non-linear radiative heat flux produces a strongly non-linear but interesting energy equation for the temperature field. Shooting method together with fifth-order Runge–Kutta integration and Newton method is employed for the development of numerical solution. Computational results for both viscous and Maxwell fluids are presented in a tabular form. Graphical results for the velocity and temperature distributions are also presented and analyzed.
2. Basic equations Consider three-dimensional flow of upper-convected Maxwell (UCM) fluid induced by a stretching surface occupying the xy-plane (see Fig. 1). The velocities of the sheet along the x- and y-directions are assumed to be U w ðxÞ ¼ ax and V w ðyÞ ¼ by respectively in which a; b 4 0 are constants. Let T w be the constant temperature at the sheet whereas T 1 denotes the fluid temperature outside the thermal boundary layer. The equations governing the three-dimensional flow of UCM fluid with radiative heat transfer are expressed below (see Liu and Andersson [3] and Awais et al. [6] for details): ∂u ∂v ∂w þ þ ¼ 0; ∂x ∂y ∂z
ð1Þ
0 1 2 2 2 u2∂∂xu2 þ v2∂∂yu2 þ w2∂∂zu2 2 ∂u ∂u ∂u @ A ¼ ν ∂ u; u þ v þ w þ λ1 ∂2 u ∂2 u ∂2 u ∂x ∂y ∂z ∂z2 þ 2uv∂x∂y þ2uw∂x∂z þ 2vw∂y∂z
ð2Þ
0 1 2 2 2 u2∂∂xv2 þ v2∂∂yv2 þ w2∂∂z2v 2 ∂v ∂v ∂v @ A ¼ ν∂ v; u þ v þ w þ λ1 ∂2 v ∂2 v ∂2 v ∂x ∂y ∂z ∂z2 þ 2uv∂x∂y þ2uw∂x∂z þ 2vw∂y∂z
ð3Þ
u
∂T ∂T ∂T ∂2 T 1 ∂qr þ v þw ¼ α 2 ; ∂x ∂y ∂z ρcp ∂z ∂z
ð4Þ
where u; v and w are the velocity components along the x-, y- and z-directions respectively, λ1 is the fluid relaxation time, ν is the kinematic viscosity, α is thermal diffusivity of the fluid, ρ is the fluid density, C p is the specific heat, qr ¼ 4σ =3k ∂T 4 =∂z is the Rosseland radiative heat flux in which σ is the Stefan–Boltzman con stant and k is the mean absorption coefficient respectively. The boundary conditions in the present problem are: u ¼ U w ¼ ax; v ¼ V w ¼ by; w ¼ 0; T ¼ T w
at z ¼ 0;
u; v-0; T-T 1
as z-1:
ð5Þ
z Vw
by
o
x
Tw
Uw
ax
y Fig. 1. A schematic diagram showing the development of boundary layer.
We now introduce the following non-dimensional quantities rffiffiffi pffiffiffiffiffiffi a df dg T T1 α ¼ z ; u ¼ ax : ; v ¼ ay ; w ¼ ν νaðf þ g Þ; θ ¼ dη dη ν Tw T1 ð6Þ In view of the above variables, the continuity Eq. (1) is automatically satisfied and the Eqs. (2)–(5) become " # 2 3 2 2 3 d f df d f df d f 2d f þ ðf þ g Þ 2 þ K 2ðf þ g Þ ðf þ g Þ ¼ 0; ð7Þ dη dη dη2 dη3 dη dη3 " # 2 3 2 2 3 d g dg d g dg d g 2d g þ ð f þ g Þ þ K 2 ð f þ g Þ ð f þ g Þ ¼ 0; dη dη dη2 dη3 dη2 dη3
ð8Þ
3 dθ 1 d dθ þ ðf þ g Þ ¼ 0; 1 þRd 1 þ θw 1 θ Pr dη dη dη
ð9Þ
f ¼ g ¼ 0; ddfη ¼ 1; ddgη ¼ c; θ ¼ 1 df -0; ddgη-0; dη
θ-0
at
η ¼ 0;
as
η-1:
ð10Þ
In the above equations K ¼ λ1 a is the Deborah number, c ¼ b=a is the stretching rates ratio, θw ¼ T w =T 1 is the temperature ratio parameter, Pr ¼ ν=α is the Prandtl number and Rd ¼ 16σ T 31 =3kk is the radiation parameter. Note that for c ¼ 0 the above model corresponds to the two-dimensional flow whereas axisymmetric flow case is achieved by setting c ¼ 1. The quantity of practical interest here is the local Nusselt number Nux defined by Nux ¼
xqw ; kðT w T 1 Þ
ð11Þ
where qw ¼ kð∂T=∂zÞz ¼ 0 þ qr is the wall heat flux. Now using dimensionless quantities from Eq. (6) into Eq. (11) we obtain 1=2
Rex
0
Nux ¼ ½1 þ Rdθw θ ð0Þ; 3
ð12Þ
where Rex ¼ U w x=ν is the local Reynolds number.
3. Numerical results and discussion The solutions of Eqs. (7)–(9) with the boundary conditions (10) are obtained numerically by employing shooting approach with fifth-order Runge–Kutta method. The unknown initial conditions 0 00 f ð0Þ; g″ð0Þ and θ ð0Þ are estimated iteratively through Newton method. All computations are successfully performed in MATLAB with error tolerance of 10 6 . Our main focus in this section is to explore the behaviors of embedded flow parameters on the velocity and temperature distributions. For this purpose we plot Figs. 2–8 and prepare Tables 1 and 2 for the computational results. For the validity of present simulations, we compared the 0 numerical results of f ″ð0Þ; g″ð0Þ and θ ð0Þ with Liu and Andersson [3] in a limiting sense. The results appear to be almost identical in all the cases as can be seen through Table 1. Influence of Deborah number K on the x- and y-components of velocity is sketched in the Fig. 2. The Deborah number is defined as the ratio of fluid relaxation time to its characteristic time scale. Relaxation time is the time taken by the fluid to gain equilibrium once the shear stress is imposed. The relaxation time is expected to be larger for the fluids having higher viscosity. Therefore an increase in K may be regarded as an increase in the fluid viscosity which restricts the fluid motion and hence the velocity decreases. Due to this reason the hydrodynamic boundary layer thins when K is increased. We 0 also noted that change in the velocity fields f and g 0 is larger in the three-dimensional flow when compared with the twodimensional and axisymmetric flows.
A. Mushtaq et al. / International Journal of Non-Linear Mechanics 79 (2016) 83–87
0
Fig. 2. Effect of K on f and g0 .
85
Fig. 5. Effects of θw and K on θ.
Fig. 6. Effect of Rd on θ when θw ¼ 1:1. Fig. 3. Effects of K and c on θ.
Fig. 7. Effect of Rd on θ when θw ¼ 2. Fig. 4. Effects of Pr and Rd on θ.
Fig. 3 displays the variation in temperature θ when the Deborah number K is increased. The coupled differential system indicates that heat flux is also related with the velocity. Temperature θ appears to increase with an increase in the fluid relaxation time. This means that elastic force promotes the heat transfer in the viscoelastic fluids. Fig. 4 shows that temperature θ and the thermal boundary layer thickness decrease when Prandtl
number Pr is increased. This decrease infact results due to a reduction in the thermal diffusivity which allows for shorter penetration depth of temperature. Fig. 5 includes the behavior of temperature ratio parameter θw on the temperature distribution. Bigger values of θw corresponds to a larger temperature and thicker thermal boundary layer. This is explained as follows. It is clear from energy Eq. (4) that effective thermal diffusivity is the sum of classical thermal diffusivity (α) and thermal diffusivity due
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A. Mushtaq et al. / International Journal of Non-Linear Mechanics 79 (2016) 83–87
Fig. 8. Effects of Pr; K and c on θ0 ð0Þ. Table 1 Comparison of present results with Liu and Andersson [3] when K ¼ 0; Pr ¼ 1 and Rd ¼ 0. c
Liu and Andersson [3] 00
0 0.25 0.50 0.75 1
Present
f ð0Þ
g″ð0Þ
θ0 ð0Þ
f ð0Þ
00
g″ð0Þ
θ0 ð0Þ
4. Summary
1 1.048813 1.093096 1.134486 1.173721
0 0.194565 0.465206 0.794619 1.173721
– 0.665933 0.735334 0.796472 –
1 1.048811 1.093095 1.134486 1.173721
0 0.194564 0.465205 0.794618 1.173721
0.581978 0.665926 0.735333 0.796472 0.851992
Numerical solutions for flow and non-linear radiative heat transfer in upper-convected Maxwell (UCM) fluid are presented and analyzed. The main observations of this study are outlined below:
Table 2 Numerical results of wall temperature gradient θ0 ð0Þ for different values of K; c; Pr and Rd. K
c
Pr
Rd ¼0
Rd ¼1 Linear radiation
1
0.5
1
0 0.3 0.6 1 0.5
0 0.5 1 1.5
deviate from the corresponding profiles of the linear radiation case (see Fig. 7). This leads to the conclusion that linear and non-linear radiation results would be identical only when θw is nearly equal to unity and Rd is sufficiently small (say Rd o0:1). In Fig. 8 the effects of Pr on the local Nusselt number are presented at different values of Deborah number K. Prandtl number compares the convective heat transfer to the conductive heat transfer. Larger Pradntl number fluids are favorable in transferring energy from the unit area compared to pure conduction. We already observed in Fig. 4 that profiles become steeper when Pr is gradually increased. This means that heat transfer rate from the surface (which is proportional to the wall slope of the temperature) escalates with an increase in Pr: On the other hand magni0 tude of θ ð0Þ has inverse and non-linear relationship with the Deborah number K. 0 Table 2 gives the values of wall slope of temperature θ ð0Þ for different values of K, c, Pr and Rd. In absence of thermal radiation, there is approximately 6% reduction in the heat transfer when K varies from K ¼ 0 to K ¼ 1:5 with Pr ¼ 7 and c ¼ 0:5. This reduction increases to almost 10% when non-linear radiation with θw ¼ 1:5 is considered. Similarly heat transfer rate grows sharply with an increase in Pr in the presence non-linear radiation case.
2 4 7 10 7
7
1.01695 1.60165 2.24393 2.75508 1.82603 2.09403 2.31293 2.55918 2.35436 2.29665 2.24393 2.19501
0.61177 1.01695 1.47271 1.83692 1.20254 1.37824 1.51544 1.66440 1.59321 1.53021 1.47271 1.41998
Non-linear radiation θw ¼ 1:1
θw ¼ 1:5
0.53932 0.90348 1.31435 1.64284 1.07372 1.23051 1.35219 1.48361 1.42760 1.36840 1.31435 1.26482
0.31686 0.55107 0.82083 1.03739 0.67255 0.77013 0.84341 0.92038 0.91088 0.86380 0.82083 0.78170
to the radiation effect (16σ T 3 =3ρC p k ). Thus one anticipates that parameter θw , being the coefficient of the later term, would support the thermal boundary layer thickness. It can be further noticed that the profiles attain special S-shaped form when θw enlarges which dictates the existence of adiabatic case. In other words, the wall temperature gradient approaches zero value when wall to ambient temperature ratio is sufficiently large. Fig. 6 shows the effect of radiation parameter Rd on temperature profile in linear and non-linear radiation cases when θw ¼ 1:1. It is noticeable that profiles in both the cases merge only at small values of radiation parameter Rd. Difference between the linear and non-linear radiation results continue to grow when Rd is incremented. As temperature ratio increases from θw ¼ 1:1 to θw ¼ 2, the profiles in non-linear radiation case significantly
1. Boundary layer thickness in the Newtonian fluid is greater than that in the UCM fluid. 2. In contrast to the linear radiation heat transfer, the thermal boundary layer thickness is controlledbyvariable (non-linear) thermal diffusivity of the form α þ 16σ T 3 =3ρC p k : 3. The vertical component of velocity at the far field boundary is negative and its magnitude increases with an increase in the stretching rates ratio c. 4. The heat transfer rate from the sheet in a viscous fluid is greater than that in the UCM fluid. 5. Temperature profiles exhibit an interesting S-shaped pattern when the wall to ambient temperature ratioθw is sufficiently large. 6. Variation in the temperature distribution with an increase in the radiation parameter Rd is significant when larger values of θw are accounted. 7. Present consideration in the case of Newtonian fluid can be obtained by substituting K ¼ 0.
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