International Journal of Pure and Applied Mathematics

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Volume 117 No. 11 2017, 317-325 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.

1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu MHD Flow of a Nanofluid and Heat transfer over an Exponentially Shrinking Sheet with Viscous Dissipation and Heat Source with Suction: Alumina and Silver water M. Thiagarajan 1,M. Dinesh Kumar 2,C. Periasamy 3 1,2 Department of Mathematics, PSG College of Arts and Science, Coimbatore thiyagu2665@gmail.com, 2 dineshmdkc.111@gmail.com 3 Department of Mechanical Engineering, BITS Pilani, Dubai Campus, Dubai, UAE. Abstract A mathematical analysis has been carried out to investigate the effects of magnetohydrodynamic flow of a nanofluid and heat transfer over an exponentially shrinking sheet with viscous dissipation and heat source in the presence of a transverse uniform magnetic field. The fluid is viscous assumed to be viscous, incompressible and electrically conducting. Governing nonlinear Partial differential equations are transformed into nonlinear ordinary differential equations by using similarity transformation and numerical solution is obtained by using Runge-Kutta-Merson method with shooting technique. These numerical solutions are shown graphically by means of graphs. The effects of nanoparticle volume fraction, magnetic interaction parameter, heat source parameter, nonlinearly stretching sheet parameter, suction/injection parameter, eckert number and Prandtl number on velocity, temperature, skin friction and rate of heat transfer are thoroughly discussed. Key Words: MHD, nanofluid, shrinking sheet, heat transfer, viscous dissipation, heat source, suction AMS Subject Classification: 70W05, 80A20, 74G15. 1 Introduction Nanofluid is a significant factor affecting the next major industrial revolution of the current century. Many researchers have focused on modeling the thermal conductivity and examined different viscosities of nanofluid because the thermal conductivity of these fluids plays an important role on the heat transfer coefficient 317

2 between the heat transfer medium and the heat transfer surface. To all the numerous applications must be added that nanofluids could be used in major process industries, including materials and chemicals, food and drink, oil and gas, paper and printing and textiles. The term nanofluid was initially utilized by Choi [1] to describe the mixtures of conventional fluids and nanometer-sized particles. Nanofluids are suspensions of submicronic solid particles (nanoparticles) in common fluids. The study of flow and heat transfer past a stretching/shrinking surface in a nanofluid has attracted significant interest from researchers over the past few years. Heat transfer performance of the conventional fluid is expected to enhance with the addition of small nanoparticles since the nanofluid has high thermophysical properties in thermal conductivity and convective heat transfer coefficient compared to conventional fluids by Murshed [2]. A large number of experimental and theoretical studies have been carried out by numerous researchers on thermal conductivity of nanofluids [3]. Nonlinear Magnetohydrodynamic Stagnation-Point Flow and Heat Transfer of Diamond, Platinum-Mineral Oil Based Nanofluid over a Flat Plate with Viscous Dissipation was discussed by Thiagarajan and Selvaraj [4]. The boundary layer flow induced by an exponentially stretching/shrinking sheet is not studied much, though it is very important and realistic flow frequently appears in many engineering processes. Heat transfer characteristics of MHD viscoelastic fluid flow over the nonlinear stretching sheet in the presence of radiation and heat generation/absorption was presented by Rafael Cortell [5]. Magnetohydrodynamic (MHD) flow when especially associated with heat transfer has received considerable attention in the recent years because of their wide variety of applications in engineering areas, such as crystal growth in liquid, cooling of a nuclear reactor and electronic package. Anjali Devi and Thiyagarajan [6] studied the steady nonlinear hydromagnetic flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a power-law velocity in the presence of a variable transverse magnetic field. The heat source effects in thermal convection, are significant where there may exist high-temperature differences between the surface and the ambient fluid. Heat generation is also important in the context of exothermic or endothermic chemical reactions. MHD heat and mass transfer free convection flow along a vertical stretching sheet in presence of magnetic field with heat generation are studied by Samad et al. [7]. Undoubtedly the viscous dissipation yields an appreciable rise in fluid temperature. This is because of the conversion of kinetic motion of fluid to thermal energy and characteristics of the source term in the fluid flow. When the flow field is of extreme size or in the high gravitational field, viscous dissipation is inevitable. Recently, Makinde and Mutuku [8] investigated the hydromagnetic thermal boundary layer of nanofluids over a convectively heated flat plate with viscous dissipation and Ohmic heating effects. Due to these practical importance the two-dimensional, steady, viscous incompressible, nonlinear hydromagnetic flow of a nanofluid and heat transfer over an exponentially shrinking sheet with heat source is considered in this paper. 318

3 2 Numerical Simulation Methodology 2.1 Governing Equation The steady two-dimensional, laminar, nonlinear hydromagnetic flow of an incompressible, viscous, electrically conducting nanofluid over a nonlinear porous shrinking sheet has been considered. The x- axis is along the continuous shrinking surface, and y- axis is normal to the surface. A variable magnetic field of strength B(x) is applied normal to the sheet. The sheet is shrinking in the x- direction with a velocity u w (x) = ce x/l defined at y = 0. The magnetic Reynolds number of the flow is taken to be sufficiently small enough so that the induced magnetic field can be neglected in comparison with the applied magnetic field. Since the flow is steady, Curl E= 0 Also div E= 0 in the absence of surface change density and hence E= 0 is assumed. A uniform heat source Q(x) is considered in this study. The viscous dissipation terms in the energy equation are combined. The flow and heat transfer characteristics under the boundary layer approximations are governed by equations of motion and the energy equation may be written in usual notation as u T x + v T y = α 2 T nf y + 2 u x + v y = 0 (1) u u x + v u y = µ nf ρ nf 2 u y 2 σb2 (x)u ρ nf (2) µ nf (ρc p ) nf ( ) 2 u + Q(x) (T T ) (3) y (ρc p ) nf where u and v are the velocity components along the x and y directions respectively, µ nf is the viscosity of the nanofluid, ρ nf is the density of the nanofluid, α nf is the thermal diffusivity of the nanofluid, σ is the electrical conductivity and T is the temperature of the nanofluid, which are given by Oztop and Abu-Nada [9], ρ nf = (1 φ)ρ f + φρ s (4) (ρc p ) nf = (1 φ)(ρc p ) f + φ(ρc p ) s (5) µ nf = µ f (1 φ) 2.5 (6) k nf = (k s + 2k f ) 2φ(k f k s ) k f (k s + 2k f ) + 2φ(k f k s ) Here, φ is the nanoparticle volume fraction, ρ f and ρ s are the densities of the fluid and of the solid fractions, respectively. (ρc p ) nf is the heat capacity of the nanofluid, k nf is the thermal conductivity of the nanofluid k f and k s are the thermal conductivities of the fluid and of the solid fractions, respectively. It should be mentioned that the use of the above expression for k nf is restricted to spherical nanoparticles where it does not account for other shapes of nanoparticles (Abu- Nada [10]). Also, the viscosity of the nanofluid µ nf has been approximated by Brinkman [11] as the viscosity of a base fluid µ f containing a dilute suspension of (7) 319

4 fine spherical particles and Q(x) = Q 0 e x/l is the heat source parameter, where Q 0 is any constant. The boundary conditions are given by u = u w (x), v = v w = v 0 e x/2l, T = T w = T + T 0 e x/2l at y = 0 (8) The shrinking velocity u w is given by u 0, T T as y (9) u w (x) = ce x/l, (10) where c > 0 is shrinking constant and B(x) = B 0 e x/2l is the magnetic field of constant strength, where B 0 is a constant. 2.2 Similarity Transformations The governing Equations (1) - (3) subject to the boundary conditions (8) and (9) can be expressed in a simpler form by introducing the following transformation: ( ) 1/2 c η = y e x/2l (11) 2v f L ψ = (2v f Lc) 1/2 e x/2l f (η) (12) θ(η) = T T T w T (13) where η is the similarity variable and ψ is the stream function defined as u = ψ/ y and v = ψ/ x, which identically satisfies Equation (1). Using (4) - (9), Eq. (2) and (3) reduce to the following ordinary differential equations: ( f + (1 φ) φ + φρ ) s (ff 2f 2 ) (1 φ) 2.5 M 2 f = 0 (14) ρ f ( ) ( ) ( ) ( 1 knf f 2 θ + Ec + 1 φ + φ(ρc ) p) s (fθ f θ) + Q P r k f (1 φ) 2.5 H θ = 0 (ρc p ) f (15) Subjected to the boundary conditions (8) become = 2σB2 0 L f(0) = S, f (0) = 1, θ(0) = 1 (16) f ( ) 0, θ( ) 0 (17) vf c 2L where M 2 ρ f is the Magnetic interaction parameter, Ec = is c (C p) f (T w T ) Eckert number, P r = ν f α f is the Prandtl number, Q H = 2LQ 0 c(ρc p) f is the Heat source parameter and S = vw is the mass transfer parameter with S > 0 (v w < 0) corresponds mass suction and S < 0 (v w > 0) corresponds to the mass injection. u 2 w 320

5 The physical quantities of interest are the skin friction coefficient C f local Nusselt number Nu x which is defined as and the C f = τ w, Nu ρ f u 2 x = w xq w k f (T w T ) (18) where the surface shear stress τ w and the surface heat flux q w are given by ( ) ( ) u T τ w = µ nf, q w = k nf y y=0 y y=0 with µ f and k nf being the dynamic viscosity and thermal conductivity of the nanofluids, respectively. Using the similarity variables, we obtain 2Cf Re 1/2 x = (19) 1 f (0) (20) (1 φ) 2.5 where Re x = uwx ν f Nu x /Re 1/2 x is the local Reynolds number. = k nf k f θ (0) (21) 3 Solution of the Problem Numerical solution of the governing ordinary differential equations (14) and (15) with the boundary conditions (16-17) is obtained by using iteration technique along with Runge-Kutta-Merson Shooting method. The solutions were obtained by setting different initial guesses for the missing values of f (0) and θ (0) where all the profiles satisfy the boundary conditions (16-17) asymptotically but with different shapes are illustrated in graphs. Following Oztop and Abu-Nada [9], the value of Prandtl number P r is taken as 6.2 (water) and the volume fraction of the nanoparticle φ is from 0.0 to 0.1 (0.0 φ 0.1) in which φ = 0 corresponds to the regular Newtonian fluid. It is worth mentioning that we have used the data related to the thermophysical properties of the fluid and the nanoparticle of Alumina and Silver as listed in Table 1. Physical properties H 2 O Al 2 O 3 Ag ρ(kg/m 3 ) C p (J/kgK) k(w/m K) Table 1: Thermo-physical properties of base fluid and nanoparticles 4 Results and Discussion Figures 1 and 2 describe the effect of the suction parameter on the flow field when the magnetic parameter is fixed. It is noted that a steady raise in the 321

6 Figure 1: Velocity profiles for different values of S (For Alumina-water Figure 2: Velocity profiles for different values of S (For Silver-water Figure 3: Velocity profiles for different values of M 2 (For Alumina-water Figure 4: Velocity profiles for different values of M 2 (For Silver-water Figure 5: Temperature profiles for different values of Q H (For Alumina-water Figure 6: Temperature profiles for different values of Q H (For Silver-water Figure 7: Temperature profiles for different values of Ec (For Alumina-water Figure 8: Temperature profiles for different values of Ec (For Silver-water velocity accompanies arise in the suction parameter. In Figure 1, the dimensionless velocity for different values of the suction parameter S for P r = 6.2 is portrayed. It is inferred that as S increases, the velocity of the Alumina water nanofluid flow increases. The same trend follows for Silver water nanofluid also which is shown in Figure 2 Figure 9: Velocity profiles for different values of φ (For Alumina-water Figure 10: Velocity profiles for different values of φ (For Silver-water Figures 3 and 4 depict the effect of magnetic interaction parameter M 2 on dimensionless velocity for alumina and silver water nanofluid respectively. It is observed that a growing M 2 accelerates the velocity of the flow field at all points due to the magnetic pull of the Lorentz force acting on the flow field. The momentum boundary layer thickness is enlarged due to enhancing values of magnetic interaction parameter M 2. The velocity for Alumina water nanofluid and Silver water nanofluid accelerates respectively for increasing M 2. In Figure 5 shows the effect of the heat source parameter Q H on the temperature profiles for alumina-water nanofluid. The temperature profiles significantly increase as the heat source parameter increases. The same trend follows for silver water nanofluid also which is shown in Figure 6. The effect of Eckert number Ec over the dimensionless temperature is visualized in Figure 7. It is obviously seen that the temperature is enhanced due to the dis- 322

7 sipation effects. Further, the thermal boundary layer thickness is broadened as a result of step up in the values of Eckert number, which converges the fact that the dissipative energy becomes more important with an enhancement in temperature for Alumina-water nanofluid. It is also interesting to note that the same trend is noticed due to the effect of Eckert number Ec for Silver-water nanofluid and all these are obviously seen from Figure 8. The effect of volume fraction on the velocity for alumina-water nanofluid is portrayed in Figure 9. It is noted that increasing volume fraction φ reduces the temperature. The velocity of silver nanofluid gets decelerated respectively for increasing values of volume fraction. It is vivid from the figure that there is shrinkage in momentum boundary layer thickness due to volume fraction φ in Figure 10. Table 2 inferred the skin friction coefficient for different values of magnetic interaction parameter, suction parameter and volume fraction for Alumina and Silverwater nanofluid respectively. It is seen that the skin friction coefficient increases with increasing values of all these parameters. The numerical values of non-dimensional rate of heat transfer for different values of M 2, φ, S, Q H and Ec for both aluminawater nanofluid and silver water nanofluid when P r = 6.2 is provided in Table 3. It is noted that for increasing values of both the magnetic interaction parameter and suction parameter, the Non-dimensional rate of heat transfer increases in case of Alumina and Silver-water nanofluid. The non-dimensional rate of heat transfer decreases for increasing volume fraction, Heat Source parameter and Eckert number for both the nanofluids. 5 Conclusion For both the types of nanofluids namely Alumina-water and Silver-water nanofluids, Skin friction increases with increasing the magnetic interaction parameter. A fixed value of the nanoparticle volume fraction φ the velocity of fluid increases by increasing Magnetic Interaction parameter. The non-dimensional rate of heat transfer decreases with increasing the nanoparticle volume fraction and also the Non-dimensional rate of heat transfer increases with increasing the suction parameter. Velocity increases with increases suction parameter. 323

8 References [1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, edited by D. A. Singer and H. P. Wang, Amer. Soci. of Mech. Engineers, New York USA. 231 (1995), [2] S.M.S. Murshed, C.A.N.De Castro, M.J.V Lourenco, A review of boiling and convective heat transfer with nanofluids, Renew. Sust. Energy. Rev., 15 (5) (2011), [3] N. Bachok, A. Ishak, I. Pop, Boundary layer flow of nanofluid over a moving surface in a flowing fluid, Int. J. Therm. Sci., 49 (2010), [4] M. Thiagarajan, M. Selvaraj, Nonlinear Magnetohydrodynamic Stagnation-Point Flow and Heat Transfer of Diamond, Platinum-Mineral Oil Based Nanofluid over a Flat Plate with Viscous Dissipation, Journal of nanofluids., 5(2) (2016), [5] R. Cortell, MHD flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption, Energy., 74 (2014), [6] S.P. Anjali Devi, M. Thiyagarajan, Steady nonlinear Hydromagnetic flow and heat transfer over a stretching surface of variable temperature,heat Mass Transfer, 42 (2006), [7] M.A. Samad, M. Mohebujjaman, MHD Heat and Mass Transfer Free Convection Flow along a Vertical Stretching Sheet in Presence of Magnetic Field with Heat Generation, Res. J. of Applied Sci., Eng. and Tech., 1(3)(2009), [8] O.D Makinde, WN Mutuku, Hydromagnetic thermal boundary layer of nanofluids over a convectively heated flat plate with viscous dissipation. UPB Sci. Bull Ser. A., 76 (2014), [9] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow, 29 (2008), [10] E. Abu-Nada, Application of nanofluids for heat transfer enhancement of separated flow encountered in a backward facing step, Int. J. Heat Fluid Flow, 29 (2008), [11] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys., 20 (1952),

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*Corresponding Author: Surajit Dutta, Department of Mathematics, C N B College, Bokakhat, Golaghat, Assam, India

*Corresponding Author: Surajit Dutta, Department of Mathematics, C N B College, Bokakhat, Golaghat, Assam, India International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 6, Issue, 8, PP -6 ISSN 347-37X (Print) & ISSN 347-34 (Online) DOI: http://dx.doi.org/.43/347-34.6 www.arcjournals.org