Thermally Radiative Rotating Magneto-Nanofluid Flow over an Exponential Sheet with Heat Generation and Viscous Dissipation: A Comparative Study

Commun. Theor. Phys. 69 (2018 317 328 Vol. 69, No. 3, March 1, 2018 Thermally Radiative Rotating Magneto-Nanofluid Flow over an Exponential Sheet with Heat Generation and Viscous Dissipation: A Comparative Study M. Sagheer, M. Bilal, S. Hussain, and R. N. Ahmed Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan (Received September 27, 2017; revised manuscript received November 21, 2017 Abstract This article examines a mathematical model to analyze the rotating flow of three-dimensional water based nanofluid over a convectively heated exponentially stretching sheet in the presence of transverse magnetic field with additional effects of thermal radiation, Joule heating and viscous dissipation. Silver (Ag, copper (Cu, copper oxide (CuO, aluminum oxide (Al 2 O 3 and titanium dioxide (TiO 2 have been taken under consideration as the nanoparticles and water (H 2 O as the base fluid. Using suitable similarity transformations, the governing partial differential equations (PDEs of the modeled problem are transformed to the ordinary differential equations (ODEs. These ODEs are then solved numerically by applying the shooting method. For the particular situation, the results are compared with the available literature. The effects of different nanoparticles on the temperature distribution are also discussed graphically and numerically. It is witnessed that the skin friction coefficient is maximum for silver based nanofluid. Also, the velocity profile is found to diminish for the increasing values of the magnetic parameter. PACS numbers: 47.15.Cb, 47.50.Cd DOI: 10.1088/0253-6102/69/3/317 Key words: magnetohydrodynamics, nanofluid, viscous dissipation, thermal radiation, rotating flow Nomenclature B variable magnetic field (u, v, w velocity components B 0 constant magnetic field (x, y, z cartesian coordinates Bi Biot number α nf thermal diffusivity of nanofluid C fx skin friction coefficient along x-axis η dimensionless similarity variable C fy skin friction coefficient along y-axis θ dimensionless temperature c p specific heat λ rotation parameter Ec Eckert number µ dynamic viscosity h convective heat flux µ f dynamic viscosity of base fluid h f convective heat transfer coefficient µ nf dynamic viscosity of nanofluid k thermal conductivity ν kinematic viscosity k f thermal conductivity of base fluid ν f kinematic viscosity of base fluid k nf thermal conductivity of nanofluid ν nf kinematic viscosity of nanofluid k s thermal conductivity of nanoparticles ρ density of fluid M magnetic parameter ρ f density of base fluid Nu Nusselt number ρ s density of nanoparticles P r Prandtl number ρ nf density of nanofluid Q h heat generation parameter σ nf electrical conductivity of nanofluid q r radiative heat flux σ Stefan-Boltzmann constant q w wall heat flux τ stress tensor R radiation parameter τ wx wall shear stress along x-axis Re Reynolds number τ wy wall shear stress along y-axis T temperature ϕ volume fraction of nanoparticles T f convective surface temperature χ mean absorption coefficient T ambient fluid temperature Ω angular velocity U w velocity of sheet in x direction Ω 0 reference angular velocity U free stream velocity σ s electrical conductivity of nanoparticles E-mail: m.bilal@cust.edu.pk c 2018 Chinese Physical Society and IOP Publishing Ltd http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn

318 Communications in Theoretical Physics Vol. 69 1 Introduction In the present fast growing and developing computer age, the transportation, communication, heavy mechanical industries, electronics industries and house hold appliances, all are running by some mechanical and electronic devices. Almost in all such devices, according to the requirements of devices, a system of cooling or heating is built-in, by which a fluid flows through or around the device to prevent these devices from overheating or cooling down from certain temperature threshold. To meet the human requirements and demand of the market, it is essential that these devices work round the clock. To keep the devices at a constant temperature, the heat dissipated must be equal to the heat generated. The conventional fluids with low thermal conductivity do not meet the temperature requirements of many mechanical and electronics devices, which results in poor performance of these devices and reduces their efficiency and working age. Therefore, it is imperative to improve the thermal conductivity of the conventional fluids. The conventional fluids used for the transfer of heat energy were first time replaced by the nanofluids by Choi [1] followed by many researchers. A nanofluid is a mixture of nanoparticles in a conventional heat transfer fluid. The nanoparticles (1 100 nm in size are usually metals, metallic oxides, nanofibers, etc. Choi [1] experimentally found that the nanoparticles when added to the base fluids, considerably improve the thermal conductivity of the base fluid. Magyari and Keller [2] focused on the heat and mass transfer analysis in the boundary layer flow due to an exponentially continuous stretching sheet. Eastman et al. [3] observed that the thermal conductivity of pure ethylene glycol is much increased when copper nanoparticles are added to it. Li et al. [4] investigated the MHD nanofluid flow in a thin film through unsteady stretching sheet with additional effects of thermal radiation, heat generation, Brownian motion, and thermophoresis. They used the MATLAB built-in bvp4c solver to solve their ODEs. It is found form their investigation that temperature and nanoparticle concentration have opposite behavior for the thermophoresis parameter. Nadeem et al. [5] analyzed the influence of nanoparticles on the two-dimensional flow of Maxwell nanofluid over a stretching sheet for the heat and mass transfer effects. By applying the boundarylayer approximation, they also incorporated the effects of MHD and elasticity parameter. Sheikholeslami et al. [6] presented an analysis focusing on the unsteady squeezing flow of electrically conducting nanofluid using the homotopy perturbation method. Two phase simulation model for the nanofluid is considered along with the magnetohydrodynamics effects. They concluded that the Nusselt number is a decreasing function of the squeezing parameter. Another useful contribution of Sheikholeslami and Ganji, [7] is a review work addressing both the single and the double phase models for the nanofluids. They describe briefly the various attempts of different scientists on heat transfer of convective nanofluids. It is further analyzed that while increasing the Reynolds number and Rayleigh number, the rate of heat transfer is increased. Sheikholeslami et al. [8] discussed the thermal radiation on MHD free convection of Al 2 O 3 -water nanofluid. Chopkar et al. [9] studied the effect of the size of the nanoparticles on the thermal conductivity of nanofluid and found that the thermal conductivity decreases by increasing the size of the particles. In last half decade, many articles related to the nanofluid dynamics are published in literature. [10 24] A reasonable number of applications emphasizing the role of steady and unsteady rotating flows may be found in chemical and geophysical fluid mechanics. These all are of applied nature like in the thermal power generating systems, food processing, the skins of high speed air crafts and in rotor stator systems. The pioneering work highlighting the rotating flow was done by Wang. [25] Takhar et al. [26] discussed the effects of magnetohydrodynamic in a rotating flow. They concluded that the skin friction along the x-axis increases for the higher values of the magnetic parameter and has a reverse relation for the y-axis skin friction coefficient. Zaimi et al. [27] applied the numerical technique to examine the rotating flow of viscoelastic fluid. Turkyilmazoglu [28] applied the spectral numerical integration method for the problem related to the shrinking rotating disk with the effect of magnetohydrodynamic. Some recent attempts emphasizing the rotating flow can be found in Refs. [29 31]. During the study of nanofluid, the thermal radiative properties of Newtonian and non-newtonian fluids for the heat transfer phenomenon have got much attention. Because of the insertion of the nanoparticles in the base fluid, the thermal properties are enhanced which resultantly rises the temperature of the nanofluids and for the higher temperature differences, the effects of the thermal radiation cannot be neglected. The operating systems performing the energy conversion at high temperature show a comparable effect of the thermal radiation. In other engineering and chemical processes such as solar water technology, fossil fuel combustion, astrophysical flows, hypersonic flights, gas turbines, space vehicles, nuclear reactors etc., the effects of thermal radiations are quite phenomenal. Many researchers have considered the influence of thermal radiation on the boundary layer flow of Newtonian and non-newtonian fluids. Mushtaq et al. [32] considered the nonlinear thermal radiation in the two-dimensional stagnation point flow with additional effects of Joule heating and viscous dissipation over a convectively heated surface. They deduced that both the temperature and its gradient are increasing functions of thermal radiation parameter. Pourmehran et al. [33] numerically investigated the MHD boundary layer flow of nanofluid through convectively heated vertical stretching sheet. During the study, the influence of thermal radiation and buoyancy effects got special attention. Pourmehran et al. considered three

No. 3 Communications in Theoretical Physics 319 different types of base fluid i.e., pure water, ethylene glycol 30% and ethylene glycol 50% while the four types of nanoparticles i.e., copper, silver, alumina, and titanium oxide. Motivated by the above mentioned literature, the primary objective of the present study is to examine the effects of thermal radiation and viscous dissipation on heat transfer flow over a bi-directional convectively heated exponentially stretching sheet in the presence of transverse magnetic field and volumetric rate of heat generation. Five different nanoparticles (silver, copper, copper oxide, titanium oxide, alumina are assumed to be suspended in the pure water. A detailed comparative study of these nanofluids for the flow and heat transfer is presented and discussed graphically and numerically. Boundary layer approximations are used to govern the partial differential equations, which are then transformed to the ordinary differential equations with the help of transformations. The modeled problem is solved numerically by the shooting method using Runga-Kutta integration scheme of order 4. Effects of emerging parameters on velocity and temperature profiles are discussed in detail. Nusselt number and skin friction coefficient are also calculated. In the limiting case, the results are verified by reproducing the results of previously published article [34 35] 2 Mathematical Formulation A laminar, incompressible and steady water-based electrically conducted nanofluid flow over an exponentially bidirectional stretching sheet is considered. Sheet temperature T f is controlled via convection by considering hot fluid below it. The temperature faraway from the surface where its difference is negligible is known as the ambient temperature and is denoted by T. The fluid is assumed to rotate with angular velocity Ω = Ω 0 e x/l along z-axis having the coriolis effect. A transverse variable magnetic field B(x = B 0 e x/2l is applied along z-axis with the assumption of small Reynolds number, ignoring the induced magnetic field. The fluid has internal volumetric rate of heat generation Q 0. In the x direction, the velocity of the sheet is taken as U w = U e x/l as shown in Fig. 1. Fig. 1 Geometry of the Problem. Further the effects of the thermal radiation, Joule heating and viscous dissipation are considered in the formulation of energy equation. For the nanofluid model, the Tiwari and Das model. [36] has been utilized. Applying the boundary layer by incorporating the Boussinesq approximations, the conservation equations of mass, momentum and energy in the mathematical form can be expressed as u x + v y + w z = 0, (1 ( ρ nf u u x + v u y + w u z 2Ωv ( 2 u = µ nf z 2 σ nf B 2 u, (2 ( ρ nf u v x + v v y + w v z + 2Ωu = µ nf ( 2 v z 2 σ nf B 2 v, (3 u T x + v T y + w T z = α nf 2 T z 2 1 q r (ρc p nf z + σ nfb 2 (u 2 + v 2 + Q 0(T T (ρc p nf (ρc p nf + µ (( nf u (ρc p nf z 2 + ( v z 2. (4 The corresponding boundary conditions for the velocity and temperature expressions are u = U w (x = U e x/l, v = 0, w = 0, k nf T z = h f (T f T at z = 0, u 0, v 0, T T as z, (5 where, h f is the convective heat transfer coefficient and µ nf is the dynamic viscosity of nanofluid proposed by Brinkman [37] and these are given by the relations h f = h e x/2l µ f, µ nf =. (6 (1 ϕ 2.5 The effective heat capacity (ρc p nf, the effective density ρ nf and the thermal diffusivity α nf of the nanofluid are formulated as [38 40] ρ nf = (1 ϕρ f + ϕρ s, (ρc p nf = (1 ϕ(ρc p f + ϕ(ρc p s, α nf = k nf, (ρc p nf σ nf 3(σ s /σ f 1ϕ = 1 + σ f (σ s /σ f + 2 (σ s /σ f 1ϕ. (7 The effective thermal conductivity modeled by Hamilton- Crosser (H-C [41] is given by k nf = k s + (n 1k f (n 1ϕ(k f k s. (8 k f k s + (n 1k f + ϕ(k f k s In Table 1, the thermo-physical properties of different nanoparticles and pure water are shown. The following

320 Communications in Theoretical Physics Vol. 69 dimensionless variables are used to convert the system of the non-linear PDEs to the system of ODEs. U η = z 2ν f L ex/2l, u = U e x/l f (η, v = U e x/l g(η, νf U w = 2L ex/2l (f(η + ηf (η, θ(η = T T T f T. (9 1 ( (1 ϕ 2.5 f σnf [ Mf 1 ϕ + ϕ (ρ ] s σ f (ρ f 1 (1 ϕ 2.5 g ( knf + R θ + 1 P r k f ( σnf σ f Mg In Eq. (4, q r is a radiative heat flux which is defined as q r = 4σ 3χ T 4 r. (10 Furthermore, the temperature difference within the flow is assumed such that T 4 may be expanded in a Taylor series. Hence, expanding T 4 about T and neglecting the higher order terms, we get T 4 = 4T 3 T 3T 4. (11 Using the similarity transformation defined in Eq. (9, Eq. (1 is identically satisfied while Eqs. (2 (5 are converted into the following nonlinear ordinary differential equations: (2f 2 ff 4λg = 0, (12 [ 1 ϕ + ϕ (ρ ] s (2f g fg + 4λf = 0, (13 (ρ f [ 1 ϕ + ϕ (ρc ] p s fθ + Q h θ + (ρc p f Table 1 Ec ( (1 ϕ 2.5 (f 2 + g 2 σnf + M Ec(f 2 + g 2 = 0. (14 σ f Thermo-physical properties of H 2O and nanoparticles. Quantities ρ/(kg/m 3 c p /(J/kg K k/(w/m K σ/(ω m 1 Pure water (H 2 O 997.1 4179 0.613 0.05 Silver (Ag 10 500 235 429 6.30 10 7 Copper (Cu 8954 383 400 5.96 10 7 Copperoxide (CuO 6320 531.5 76.5 10 10 Titanium oxide (TiO 2 4250 686.2 8.9538 2.6 10 6 Aluminium oxide (Al 2 O 3 3970 765 40 3.5 10 7 The transformed boundary conditions are: f(0 = 0, f (0 = 1, g(0 = 0, θ (0 = k f k nf Bi(1 θ(0, f (η 0, g(η 0, θ(η 0, as η. (15 Different dimensionless parameters appearing in Eqs. (11 (15 are defined as Ec = U 2 w (T f T (c p f, Ω = Ω 0 e x/l, M = σb2 02L Uρ f, Q h = 2LQ 0 (ρc p f U w, Bi = h K f 2νf L U, P r = ν f (ρc p f k f, R = 16σ T 3 3χk f, λ = Ω 0L U. (16 The important quantities of interest are the skin friction coefficient and the local Nusselt number which are defined as τ wx C fx = ρ f Uw 2, C fy = τ wy ρ f Uw 2, Nu x = xq w k f (T f T, (17 where τ wx, τ wy and q w denote the wall shear stress along x-axis, y-axis and wall heat flux respectively and are given as under: ( u ( v τ wx = µ nf z, τ wy = µ nf z=0 z, z=0 ( T q w = k nf + (q r z=0. (18 z z=0 The skin friction coefficient and the local Nusselt number, in the dimensionless form, are as follows C fx 2Rex = f (0 (1 ϕ 2.5, C fy 2Rex = g (0, (19 (1 ϕ 2.5 ( Nu x Re 1/2 knf x = + RP r θ (0, (20 where Re x = U 0 z 2 /νl. 3 Solution Methodology An efficient numerical technique, namely the shooting method has been employed to solve the transformed ordinary differential equations along with the boundary con- k f

No. 3 Communications in Theoretical Physics 321 ditions for different values of the emerging parameters. While applying the shooting method, [42] first the higher order boundary value problem is converted to a system of first order initial value problem (IVP. During the conversion, f is denoted by y 1, g by y 4 and θ by y 6. The missing initial conditions are supposed to be ς 1, ς 2 and ς 3. The converted first order IVP takes the following form y 1 = y 2, y 1 (0 = 0, y 2 = y 3, y 2 (0 = 1, y 3 = (1 ϕ 2.5( σ [ nf My 2 + 1 ϕ + ϕ (ρ ] s (2y2 2 y 1 y 3 4λy 4, y 3 (0 = ς 1, σ f (ρ f y 4 = y 5, y 4 (0 = 0, y 5 = (1 ϕ 2.5( σ [ nf My 4 + 1 ϕ + ϕ (ρ ] s (2y 2 y 4 y 1 y 5 + 4λy 2, y 5 (0 = ς 2, σ f (ρ f y 6 = y 7, y 6 (0 = ς 3, Ec (1 ϕ (y 2 2.5 3 + y5 2 + ( σ nf σ f M Ec(y2 2 + y4 2 y 7 = P r( [1 ϕ + ϕ (ρc p s (ρc p f ]y 1 y 7 + Q h y 6 + y 7 (0 = k f k nf Bi(1 y 6 (0. ( k nf k f + R, (21 Fourth order Runge-Kutta method is utilized to solve this IVP. The refinement of initial guesses is carried out by the Newton s method. Because the numerical solution cannot be computed on the unbounded domain [0,, a bounded domain [0, η ] has been considered, where η is an appropriate real number. After performing a number of computational experiments, η is set to 4, because there is no significant variation in the results for η > 4. The stoping criteria set for the Newton s iterative process is max{ y 2 (4, y 4 (4, y 6 (4 } < ϵ. Throughout this article, ϵ is chosen as 10 6. For the validation of the MATLAB code of the shooting method, it is affectively applied to reproduce the numerical results of Javed et al. [35] and Ahmad and Mustafa. [34] The successful comparison has been presented in Table 2. Table 2 Comparison of present results with those of Javed et al. [35] and Ahmad and Mustafa. [34] Javed et al. [35] Ahmad et al. [34] Present result λ f (0 g (0 f (0 g (0 f (0 g (0 0.2 1.347 416 9 0.370 152 23 1.347 420 4 0.370 152 5 1.347 420 4 0.370 152 3 0.5 1.519 413 1 0.762 514 09 1.519 419 5 0.762 514 3 1.519 419 4 0.762 514 1 2 2.282 796 6 1.848 504 4 2.282 812 7 1.848 503 2 2.282 812 6 1.848 503 2 5 3.344 433 8 3.060 919 2 3.344 460 6 3.060 916 4 3.344 460 6 3.060 916 4 10 4.601 722 4.399 064 4.601 761 4.399 058 4.601 760 9 4.399 50 10.058 172 9.966 809 9 10.058 26 9.966 799 10.058 26 9.966 798 9 100 14.183 223 14.118 628 14.183 358 14.118 612 14.183 357 68 14.118 612 26 4 Results and Discussions In this section, we discuss the influence of different parameters such as nanoparticles volume fraction ϕ, rotational parameter λ, magnetic parameter M, thermal radiation parameter R, Eckert number Ec, heat generation/absorption parameter Q h on the velocity, temperature, skin-friction and Nusselt number, both graphically and numerically in the tabular form. In Table 3, the influence of the nanoparticle volume fraction ϕ, rotational parameter λ and magnetic parameter M on the skin friction coefficient along x-axis is presented for different nanoparticles. It is observed that due to the addition of more nanoparticles in the base fluid, the skin-friction is enhanced. This enhancement is more rapid in Ag-H 2 O nanofluid whereas in case of Al 2 O 3 -H 2 O, the increase in the skin-friction is less as compared to the other nanolfuids. Quite similar behavior is noticed for the rotational parameter λ. When the magnetic field is intensified along the z-axis, the skin-friction escalates along the x-axis due to the presence of the Lorentz force. Again, Ag-H 2 O nanofluid has more frictional force as compared to the other nanofluids.

322 Communications in Theoretical Physics Vol. 69 Table 3 Effect of ϕ, λ and M on the skin friction coefficient along x-axis when Q h = 0.1, Bi = 0.6, P r = 6.2, n = 3.0, R = 0.2, Ec = 0.01. f (0/(1 ϕ 2.5 ϕ λ M Ag-H 2 O Cu-H 2 O CuO-H 2 O TiO 2 -H 2 O Al 2 O 3 -H 2 O 0.1 0.2 0.3 2.225 518 0 2.141 561 3 1.990 350 8 1.862 970 0 1.845 074 4 0.00 1.441 490 6 1.441 490 6 1.441 490 6 1.441 490 6 1.441 490 6 0.03 1.673 673 8 1.646 231 4 1.598 405 9 1.559 810 5 1.554 517 8 0.05 1.828 627 7 1.784 326 8 1.706 219 8 1.642 267 0 1.633 427 8 0.1 2.225 518 0 2.141 561 3 1.990 350 8 1.862 970 0 1.845 074 4 0.1 2.163 647 1 2.083 152 0 1.938 430 5 1.816 767 2 1.799 691 6 0.2 2.225 518 0 2.141 561 3 1.990 350 8 1.862 970 0 1.845 074 4 0.3 2.304 801 8 2.217 199 7 2.059 272 9 1.925 920 5 1.907 152 2 0.1 2.173 820 6 2.087 594 1 1.931 651 3 1.799 509 6 1.780 880 1 0.2 2.199 659 9 2.114 573 5 1.961 019 8 1.831 297 8 1.813 042 6 0.3 2.225 518 0 2.141 561 3 1.990 350 8 1.862 970 0 1.845 074 4 Table 4 Numerical values of skin friction coefficient along y-axis for different values of parameters when Q h = 0.1, R = 0.2, Ec = 0.01. g (0 /(1 ϕ 2.5 ϕ λ M Ag-H 2 O Cu-H 2 O CuO-H 2 O TiO 2 -H 2 O Al 2 O 3 -H 2 O 0.1 0.2 0.3 0.558 281 6 0.533 153 6 0.487 279 7 0.447 851 4 0.442 244 1 0.00 0.332 495 4 0.332 495 4 0.332 495 4 0.332 495 4 0.332 495 4 0.03 0.401 020 6 0.392 457 3 0.377 424 8 0.365 185 1 0.363 498 7 0.05 0.445 787 8 0.432 172 6 0.407 923 5 0.387 806 5 0.385 005 7 0.1 0.558 281 6 0.533 153 6 0.487 279 7 0.447 851 4 0.442 244 1 0.1 0.293 022 5 0.279 164 1 0.253 919 6 0.232 336 8 0.229 279 1 0.2 0.558 281 6 0.533 153 6 0.487 279 7 0.447 851 4 0.442 244 1 0.3 0.789 310 1 0.754 683 0 0.691 835 5 0.638 154 3 0.630 537 0 0.1 0.579 802 1 0.555 471 6 0.511 053 9 0.472 791 5 0.467 337 7 0.2 0.568 784 3 0.544 016 6 0.498 789 3 0.459 865 6 0.454 323 6 0.3 0.558 281 6 0.533 153 6 0.487 279 7 0.447 851 4 0.442 244 1 Table 5 Numerical values of local Nusselt for different values of parameters when ϕ = 0.01, λ = 0.2, M = 0.3. (K nf /K f + Rθ (0 R Ec Q h Bi Ag-H 2 O Cu-H 2 O CuO-H 2 O TiO 2 -H 2 O Al 2 O 3 -H 2 O 0.2 0.01 0.1 0.6 0.459 119 1 0.465 618 7 0.471 725 2 0.475 690 3 0.476 566 6 0.1 0.435 361 5 0.441 208 6 0.446 541 2 0.449 103 5 0.450 729 8 0.2 0.459 119 1 0.465 618 7 0.471 725 2 0.475 690 3 0.476 566 6 0.3 0.482 207 5 0.489 394 1 0.496 297 1 0.501 645 6 0.501 817 4 0.01 0.459 119 1 0.465 618 7 0.471 725 2 0.475 690 3 0.476 566 6 0.05 0.379 583 8 0.391 549 4 0.404 874 0 0.413 785 7 0.415 999 9 0.1 0.280 164 6 0.298 962 7 0.321 310 1 0.336 404 9 0.340 291 6 0.15 0.180 745 5 0.206 376 1 0.237 746 2 0.259 024 1 0.264 583 3 0.2 0.081 326 4 0.113 789 4 0.154 182 2 0.181 643 3 0.188 875 0 0.00 0.485 766 9 0.490 471 7 0.494 489 3 0.497 025 6 0.497 659 7 0.10 0.459 119 1 0.465 618 7 0.471 725 2 0.475 690 3 0.476 566 6 0.20 0.414 639 5 0.425 712 3 0.436 913 1 0.444 395 9 0.445 521 3 0.25 0.376 643 0 0.393 484 6 0.410 636 8 0.421 990 5 0.423 209 5 0.2 0.192 244 8 0.193 761 6 0.195 368 2 0.197 044 4 0.196 690 7 0.3 0.271 010 6 0.273 648 7 0.276 303 5 0.278 660 0 0.278 456 0 0.4 0.340 832 9 0.344 709 9 0.348 487 7 0.351 443 9 0.351 520 3 0.5 0.403 153 1 0.408 331 4 0.413 267 1 0.416 756 0 0.417 202 4

No. 3 Communications in Theoretical Physics 323 Table 3 shows the effect of variation of nanoparticles volume fraction ϕ, rotational parameter λ, and magnetic parameter M on the skin friction coefficient along y- axis. From the table, it is highlighted that fractional force between the fluid and the solid surface in the y- direction is enhanced when the nanoparticles volume fraction is increased. However this increase in skin friction is very small. Again it is observed that Al 2 O 3 water base nanoflluid has least increase when compared with the other nanofluids. By escalating the angular velocity of the nanoparticles, the skin friction rises along the y-axis. The effect of magnetic field M on the skin friction along the y-axis is quite similar as already shown in Table 3. By enhancing the magnetic parameter, the surface fractional force also increased. Fig. 2 Influence of ϕ, λ, and M on f (η for Al 2 O 3 -H 2 O nanofluid. Fig. 3 Influence of ϕ, λ, and M on f (η for CuO-H 2 O nanofluid. Fig. 4 Influence of ϕ, λ and M on f (η for TiO 2 -H 2 O nanofluid. The effect of thermal radiation parameter R, Eckert number Ec, heat generation/absorption parameter Q h and Biot number on Nusselt number is shown in Table 5. From this table, it is observed that these parameters have increasing effect on the Nusselt number. The comparison among different nanoparticles exhibits that Al 2 O 3 -H 2 O nanofluid possesses the highest value of Nusselt number for thermal radiation parameter. It is also perceived that Ag-H 2 O nanofluid has more heat transfer rate as compared to the other nanofluids for the increasing values of Eckert number and heat generation/absorption parameter Q h. Nusselt number is also enhanced for the higher

324 Communications in Theoretical Physics Vol. 69 values of the Biot number Bi. It happens because for the higher values of the Biot number, a stronger convection is produced which causes higher rate of change in the temperature. Fig. 5 Influence of ϕ, λ and M on f (η for Cu-H 2O nanofluid. Fig. 6 Influence of ϕ, λ and M on f (η for Ag-H 2O nanofluid. Fig. 7 Influence of Bi, Ec and Q h on θ(η for Al 2 O 3 -H 2 O nanofluid. Fig. 8 Influence of Bi, Ec and Q h on θ(η for CuO-H 2 O nanofluid.

No. 3 Communications in Theoretical Physics 325 Fig. 9 Influence of Bi, Ec and Q h on θ(η for TiO 2 -H 2 O nanofluid. Fig. 10 Influence of Bi, Ec and Q h on θ(η for Cu-H 2 O nanofluid. Fig. 11 Influence of Bi, Ec and Q h on θ(η for Ag-H 2O nanofluid. Fig. 12 Influence of R, M and ϕ on θ(η for Al 2 O 3 -H 2 O nanofluid.

326 Communications in Theoretical Physics Vol. 69 Fig. 13 Influence of R, M and ϕ on θ(η for CuO-H 2 O nanofluid. Fig. 14 Influence of R, M and ϕ on θ(η for TiO 2 -H 2 O nanofluid. Fig. 15 Influence of R, M and ϕ on θ(η for Cu-H 2O nanofluid. Fig. 16 Influence of R, M and ϕ on θ(η for Ag-H 2O nanofluid.

No. 3 Communications in Theoretical Physics 327 To visualize the effect of different physical parameters on the velocity f (η and the temperature profile θ(η, Figs. 2 6 are plotted. In Figs. 2 6, the effects of nanoparticle volume fraction ϕ, rotational parameter λ and magnetic parameter M on the velocity profile for alumina, copper oxide, titanium oxide, copper and silver based nanofluids are displayed. For all the nanofluids, it is observed that the velocity as well as the boundary layer thickness of the nanofluid decreases when the quantity of the nanoparticles in the base fluid is increased. Velocity distribution is dominant at the surface of the sheet. The effect of rotational parameter λ which is the associated with the angular velocity of the fluid, on velocity profile is displayed in Figs. 2 6. From these figures, it is noticed that the velocity profile and its momentum boundary layer thickness is reduced for the increasing values of λ. Hence, the rotational effects resist the fluid flow in the x-direction. For higher values of rotational parameter, the velocity becomes negative in some part of the boundary layer thickness and an interesting phenomenon of oscillatory decaying profile are also observed. The Lorentz forces, which are resistive in nature are produced when the magnetic field is applied across the fluid flow. These forces are responsible for the reduction in the fluid particle s motion for the higher values of magnetic parameter M. Hence for all the nanofluids, the speed of the fluid decreases for the increasing values of magnetic parameter. To observe the effect of the variation in the Biot number Bi, Eckert number Ec, and heat generation parameter Q h on the temperature distribution Figs. 7 11 are plotted. It is observed that the higher values of the Biot number escalate the temperature distribution and the thermal boundary layer thickness. The same observation is preserved for all the nanofluids. The strength of the convected heating is signified for the higher values of the Biot number which resultantly rise the temperature distribution. The temperature is enhanced when the Eckert number is increased. Eckert number appears in the energy equation because of the consideration of the viscous dissipation effects in the fluid motion. It is inversely proportional to the difference between the fluid temperature on the surface and the ambient temperature. An increase in the Eckert number means there is a slight temperature difference between the surface and the thermal boundary layer and hence the rate of heat transfer is reduced. This reduction in heat transfer rate leads to escalate the temperature of the nanofluid as shown in Figs. 7 11. The effect of heat generation parameter Q h is also displayed in the same figures. It is quite obvious that if heat is generated from any external or internal source the temperature of the fluid is increased. In Figs. 12 16, the influence of the thermal radiation parameter R, magnetic parameter M and nanoparticle volume fraction ϕ is displayed for the temperature distribution. Higher values of thermal radiation produces more heat in the working fluid which rises the temperature and the thermal boundary layer thickness of the nanofluid as shown in these figures. By increasing the magnetic field across the fluid, the resistive forces are enhanced. Temperature is increased due to these resistive forces. Lastly, by inserting the more quantity of nanopartices in the base fluid, the thermal properties of the fluid go up and hence the temperature of the fluid is increased. 5 Concluding Remarks This article encompasses the three-dimensional MHD rotating flow of electrically conducting nanofluid over an exponentially stretching sheet. The effect of heat generation, viscous dissipation and thermal radiation for five different nanoparticles is analyzed graphically and numerically. 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