Chapter Introduction

Transcription

1 Chapter 4 Mixed Convection MHD Flow and Heat Transfer of Nanofluid over an Exponentially Stretching Sheet with Effects of Thermal Radiation and Viscous Dissipation 4.1 Introduction The study of boundary layer flow and heat transfer of a viscous incompressible fluid has many industrial applications such as in manufacturing of sheeting materials through an extrusion process, aerodynamic extrusion of plastic sheets,drawing plastic films, crystal growing, hot rolling, manufacturing of foods, paper production, glass-fiber production, metal and fiber spinning. During the manufacture of these products the quality and the desired characteristics depends entirely on the rate of cooling or heating of the fluid and stretching rate of the boundary during the process. The rate of cooling and rate of stretching depends on the nature of the fluid, flow and heat transfer, internal and external factors. In view of this Sakiadis [1, 119] investigated the boundary-layer flow of a viscous fluid past a moving solid surface. Since then various studies on the boundary layer flow of Newtonian and non-newtonian fluids over a linear and non linear stretching surfaces has been conducted [[120] - [121]]. Magyari and Keller [122] analyzed the steady 77

2 Chapter 4. Mixed Convection MHD Flow of nanofluids 78 boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution. Partha et al.[123] investigated the effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. Sajid and Hayat [124] studied the influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet by solving the problem analytically via homotopy analysis method (HAM). Magnetohydrodynamics(MHD) is the study of the flow of electrically conducting fluids in a magnetic field. The study of magnetohydrodynamic (MHD) has many important applications, and may be used to deal with problems such as cooling of nuclear reactors by liquid sodium and induction flow meter, which depends on the potential difference in the fluid in the direction perpendicular to the motion and to the magnetic field [7]. At high operating temperature, radiation effect can be quite significant. Many processes in engineering areas occur at high temperatures and knowledge of radiation heat transfer becomes very important for the design of pertinent equipment [125]. Elbashbeshy [126] added new dimension to the study on exponentially continuous stretching surface. Khan and Sanjayan [127] studied the viscous-elastic boundary layer flow and heat transfer due to an exponentially stretching sheet. The numerical simulation of boundary layer flow over an exponentially stretching sheet with thermal radiation was given by Bidin and Nazar [128]. Wang [129] studied the free convection on a vertical stretching surface, also Reddy Gorla and Sidawi [130] investigated the free convection on a vertical stretching surface with suction and blowing. An adequate understanding of radiative heat transfer in flow processes is very important in engineering and industries, especially in the design of reliable equipments, nuclear plants, gas turbines and various propulsion devices for aircraft, missiles, satellites and space vehicles. Thermal radiation effects are extremely important in the context of flow processes involving high temperature. The effects of thermal radiation on the boundary layer flow have also been considerably researched [ ]. Dissipation is the process of converting mechanical energy of downward-flowing water into thermal and acoustical energy. Vajravelu and Hadjinicolaou [131] analyzed the heat transfer characteristics over a stretching surface with viscous dissipation in the presence of internal heat generation or absorption. Convective boundary layer flow has wide applications in engineering as post accidental heat removal in nuclear reactors, solar collectors, drying processes, heat exchangers, geothermal and oil recovery, building construction, and so forth. Cheng and Minkowycz [98] also studied free convection from a vertical flat plate with applications to heat transfer from a dick. Gorla and Tornabene [132] solved the non similar problem of free

3 Chapter 4. Mixed Convection MHD Flow of nanofluids 79 convective heat transfer from a vertical plate embedded in a saturated porous medium with an arbitrary varying surface temperature. Fluid heating and cooling are important in many industries such as power, manufacturing, transportation, and electronics. Effective cooling techniques are greatly needed for cooling any sort of high-energy device. Common heat transfer fluids such as water, ethylene glycol, and engine oil have limited/poor heat transfer capabilities due to their low heat transfer properties. In contrast, metals have thermal conductivities up to three times higher than these fluids, so it is natural that it would be desired to combine the two substances to produce a heat transfer medium that behaves like a fluid, but has the thermal conductivity of a metal. A lot of experimental and theoretical researches has been made to improve the thermal conductivity of these fluids. In 1993, during an investigation of new coolants and cooling technologies at Argonne National Laboratory in U.S, Choi invented a new type of fluid called Nanofluid [93]. Nanofluids are fluids that contain small volumetric quantities of nanometer-sized particles, called nanoparticles. These fluids are engineered colloidal suspensions of nanoparticles in a base fluid [30]. The nanoparticles used in nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. Common base fluids include water, ethylene glycol and oil. This kind of fluids exhibit anomalous heat transfer characteristics and their use as advanced coolants along with the benefits over their conventional counterparts (pure fluids or micron-sized suspensions/slurries) is investigated. After the pioneer investigation of Choi, thriving experimental and theoretical researches were undertaken to discover and understand the mechanisms of heat transfer in nanofluids. The knowledge of the physical mechanisms of heat transfer in nanofluids is of vital importance as it will enable the exploitation of their full heat transfer potential. Masuda et al. [4] observed the characteristic feature of nanofluid is thermal conductivity enhancement. This observation suggests the possibility of using nanofluids in advanced nuclear systems [15]. A comprehensive survey of convective transport in nanofluids was made by Buongiorno [30], who says that a satisfactory explanation for the abnormal increase of the thermal conductivity and viscosity is yet to be found. He focused on further heat transfer enhancement observed in convective situations. Khan and Pop [16] presented a similarity solution for the free convection boundary layer flow past a horizontal flat plate embedded in a porous medium filled with a nanofluid. Makinde and Aziz [117] studied MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition. Very recently, Kuznetsov and Nield [95] have examined the influence of nanoparticles on natural convection boundary-layer flow past a vertical plate using a model in which Brownian motion and thermophoresis are accounted for. The authors have assumed the simplest possible boundary conditions, namely those in which both the temperature and the nanoparticle fraction are constant

4 Chapter 4. Mixed Convection MHD Flow of nanofluids 80 along the wall. Furthermore, Nield and Kuznetsov [96, 97] have studied the Cheng and Minkowycz [133] problem of natural convection past a vertical plate in a porous medium saturated by a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis for the porous medium. However, all these studies are restricted to the linear or non linear stretching sheet and to a natural convection or a forced convection means of heat transfer. The flow of nanofluids over an exponentially stretching sheet has been given less attention. Moreover, the effects of the magnetic field, thermal radiation, viscous dissipation, Bronian motion and thermophoresis are neglected in many of the studies conducted on flow and heat transfer of nanofluids. A few recent studies focused on an exponentially stretching sheet. Recently, Nadeem and Lee [99] used the homotopy analysis method to study the boundary layer flow of nanofluid over an exponentially stretching sheet. The magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate has been discussed by Hamad et al. [11]. Very recently, Haddad et al. [100] experimentally investigated natural convection in nanofluds by considering the role of thermophoresis and Brownian motion in heat transfer enhancement. They indicated that neglecting the role of Brownian motion and thermophoresis deteriorate the heat transfer and this deterioration elevates by increasing the volume fraction of nanoparticles. Habibi et al. [134] studied the mixed convection MHD flow of nanofluid over a non-linear stretching sheet with effects of viscous dissipation and variable magnetic field. The aim of the present paper is therefore to study the mixed convection magnetohydrodynamic flow and heat transfer of nanofluids over an exponentially stretching sheet with effects of thermal radiation and viscous dissipation by taking the the effects of Brownian motion and thermophoresis into consideration. The combined effect of all the above mentioned parameters has not been reported so far in the literature, which makes the present paper unique. The governing highly nonlinear partial differential equation of momentum, energy and nano particle volume fraction has been simplified by using a suitable similarity transformations and then solved numerically with the help of a powerful, easy to use method called the Keller box method. This method has already been successfully applied to several non linear problems corresponding to a parabolic partial differential equations. As discussed in [19] the exact discrete calculus associated with the Keller-Box Scheme is shown to be fundamentally different from all other mimic numerical methods. The box-scheme of Keller [19], is basically a mixed finite volume method, which consists in taking the average of a conservation law and of the associated constitutive law at the level of the same mesh cell. The paper is outlined as follows

5 Chapter 4. Mixed Convection MHD Flow of nanofluids 81 In section 2, the the problem is formulated and the similarity transformation has been used to reduce the formulated partial differential equations in to a non linear ordinary differential equations. In section 3, numerical solution has been presented. In section 4, the results are discussed in detail both numerically and graphically and compared to previously reported data. Finally, in 5, concluding remarks are given. Nomenclature 4.2 Mathematical Formulation Consider a steady two-dimensional mixed convection boundary layer flow of nanofluid over an exponentially stretching sheet with variable magnetic field, thermal radiation and viscous dissipation effect. The fluid is electrically conducting under the influence of an applied magnetic field B(x) normal to the exponentially stretching surface. Since the magnetic Reynolds number is very small for most fluid used in industrial applications the induced magnetic field is assumed to be small compared to the applied magnetic field and is neglected. If we consider the cartesian coordinate system in such a way that x - axis is taken along the exponentially stretching surface in the direction of the motion and y - axis is normal to the sheet. The flow is generated due to an exponential stretching of the surface. Keeping the origin fixed the sheet is then stretched with a velocity u w = u o exp ( x L) at y = 0, where uo is the reference velocity and L is the reference length. A schematic representation of the physical flow model and the coordinate system is presented in Figure 1. The continuity, momentum, energy and concentration equations describing the flow can be written as u x + v y = 0. (4.1) u u x + v u y = ν 2 u y 2 σb2 o(x) ρ u + gβ(t T ) + gβ (C C ). (4.2) u T x + v T y = α 2 T y 2 + ν c p ( ) { u 2 C + τ D B y y T y + D ( ) } T T 2 ν q r T y c p y (4.3) u C x + v C y = D 2 C B y 2 + D T T ( 2 ) T y 2 (4.4) Where u and v are the velocity components in the x and y directions respectively, ν is the kinematic viscosity, σ is the electrical conductivity, ρ is density of the fluid, c p is the

6 Chapter 4. Mixed Convection MHD Flow of nanofluids 82 Figure 4.1: Physical model and coordinate system. specific heat at constant pressure, B o is uniform magnetic field strength, β and β are the thermal and concentration expansion coefficients respectively, T is the temperature, C is the nanoparticle volume fraction, α is the thermal diffusivity of the fluid, q r is the radiative heat flux, D B is the Brownian diffusion coefficient and D T is the thermophoresis diffusion coefficient. Using the Rosseland approximation for radiation, the radiative heat flux is simplified as: q r = 4σ T 4 3k y (4.5) where σ and k are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively.the temperature differences within the flow are assumed to be sufficiently small so that T 4 may be expressed as a linear function of temperature T using a truncated Taylor series about the free stream temperature T and and neglecting higher-order terms, we get: If we take N r = kk 4σ T 3 (4.6), equation (7.3) becomes: T 4 = 4T 3 T 3T 4 (4.6) as a radiation parameter, then making use of equation (4.5) and

7 Chapter 4. Mixed Convection MHD Flow of nanofluids 83 u T x + v T y = α k o 2 T y 2 + ν c p ( ) { u 2 C + τ D B y y T y + D ( ) } T T 2. (4.7) T y where k o = 3Nr 3N. It is worth mentioning here that as N r+4 r (i.e, k o 1), we get the classical solution to the energy equation (7.3) with out the influence of thermal radiation. The associated boundary conditions to the flow problem can be written as u = u w = u o e x/l, v = 0, T = T w = T + T o e x/2l, C = C w = C + C o e x/2l at y = 0 u = 0, T T, C C, as y. (4.8) In which u o, T o and C o are the reference velocity, temperature and concentration respectively. T w and T are the temperatures of the sheet and the ambient fluid, C w and C are the nanoparticles volume fraction of the plate and the fluid, respectively. To obtain similarity solutions, it is assumed that the magnetic field B(x) is of the form: B(x) = B o exp ( ) x 2L where B o is the constant magnetic field. Introducing the following similarity transformations uo η = y 2νL ex/2l, u = u 0 e x/l f νuo (η), v = [ 2L ex/2l f(η) + ηf (η) ] θ = T T T w T, φ = C C C w C. (4.9) Making use of Eq(7.6), the continuity equation (7.1) is automatically satisfied and equations (7.2),(4.7), (6.4) and (7.4) reduces to f + ff 2f 2 2Mf + 2λ T θ + 2λ M φ = 0. (4.10) ( ) θ P rfθ + P recf 2 + P rnbθ φ + P rntθ 2 = 0. (4.11) 3N r

8 Chapter 4. Mixed Convection MHD Flow of nanofluids 84 φ + Lefφ f φ + Nt Nb θ = 0. (4.12) With boundary conditions f(0) = 0, f (0) = 1, θ(0) = 1, φ(0) = 1, f (η) 0, θ(η) 0, φ(η) 0 as η. (4.13) Where f(η), θ(η) and φ(η) are the dimensionless velocity, temperature and nanoparticle concentration, respectively, primes denote differentiation with respect to the similarity variable η, and M = 2σB 0 2 L, (Magnetic parameter) ρu 0 λ T = Gr Re 2 x λ M = Gm Re 2 x = gβ(t w T ), (Thermal convective parameter) u 2 w = gβ(c w C ), (Mass convective parameter) u 2 w P r = ν, (Prandtl number) α N r = k k nf 4σ T 3, (Radiation parameter) Uw 2 Ec =, (Eckert number) (C p ) f (T w T ) (ρc) p Nb = D B (C w C ), (Brownian motion parameter) (ρc) f Nt = D T (ρc) p (T w T ), (Thermophoresis parameter) T (ρc) f ν Le = ν, (Lewis number) D B Note: λ T represents the ratio of the bouncy forces to inertia forces. Therefore, this parameter can be used as a criterion to establish the dominant regions of free and forced convection: Gr 2 Re x 1 : F ree convection dominant Gr 2 Re x = 1 : F ree and F orced convection are of comparable magnitude Gr 2 Re x 1 : F orced convection dominant

9 Chapter 4. Mixed Convection MHD Flow of nanofluids 85 The physical quantities of interest in this problem are the local skin friction coefficient C f, the Nusselt number Nu x, which represents the rate of heat transfer at the surface of the plate and the local Sherwood number Sh x, which represents the rate of mass transfer at the surface of the plate are defined as τ w y=0 C f = ρu 2 0 exp( 2x L ), Nu x = x T T Making use of Eq.(7.6) and (7.10), we get 2Rex C f = f (0), ( ) T y y=0 x, Sh x = φ w φ ( ) φ (4.14), y y=0 Nu x x Sh x x = 2Rex 2L θ (0), = 2Rex 2L φ (0). (4.15) where Re x = uwx ν is the local Reynolds number. 4.3 Numerical Solution The non linear boundary value problem represented by equations (7.7) - (6.11) and (7.9) is solved numerically using the Keller box method. In solving the system of non linear ordinary differential equations (7.7) - (6.11) together with the boundary condition (7.9) using the Keller box method the choice of an initial guess is very important. The success of the scheme depends greatly on how much good this guess is to give the most accurate solution. This choice has been made based on the convergence criteria together with the boundary conditions in consideration. As in Cebeci and Pradshaw [20], the values of the wall shear stress, in our case f (0) is commonly used as a convergence criteria. This is because in the boundary layer flow calculations the greatest error appears in the wall shear stress parameter. In the present study this convergence criteria is used. In this study a uniform grid of size η = 0.01 is chosen to satisfy the convergence criteria of 10 8, which gives about a four decimal places accuracy for most of the prescribed quantities. 4.4 Results and Discussion The transformed non linear equations (7.7) - (6.11) subjected to the boundary condition (7.9) was solved numerically using Keller box method, which is described in Cebeci and Bradshaw [20]. The velocity, temperature, and concentration profiles were obtained and utilized to compute the skin-friction coefficient, the local Nusselt number, and local Sherwood number in equation (7.12). The results of the velocity profiles, temperature profile and concentration profile for different values of the governing parameters viz.,

10 Chapter 4. Mixed Convection MHD Flow of nanofluids 86 magnetic parameter M, thermal convective parameter λ T, mass convective parameter λ M, Prandtl number Pr, radiation parameter N r, viscous dissipation parameter (Eckert number) Ec, Brownian motion parameter Nb, thermophoresis parameter Nt, and Lewis number Le are presented in graphs. In addition to this, the effects of the governing parameters on the skin friction coefficient, rate of heat and mass transfer is presented in graphs. To validate the accuracy of our results a comparison has been made with previously reported work by Bidin and Nazar [135] and Ferdows et al. [136]. The comparisons are found to be in an excellent agreement as shown in table 1 and 2. Table 1, contains the values of the local Nusselt number θ (0), which shows the heat transfer rate at the surface, for some values of the Prandtl number Pr and Eckert number Ec, when M = 0, λ T = 0, λ M = 0, Nr, Ec = 0, Nt = 0, Nb 0, and Le = 2.0. Table 4.1: Comparison of the local Nusselt number θ(0) at M = 0, λ T = 0, λ M = 0, Nr, Ec = 0, Nt = 0, Nb 0, and Le = 2.0 for different values of Pr and Ec with previously reported data. θ (0) Ec Pr Bidin and Nazar [135] Ferdows et al. [136] Present study Table 4.2: Comparison of the skin friction coefficient f (0) for different values of M, when λ T = 0, λ M = 0, Nr, Ec = 0, Nt = 0, Nb 0, and Le = 2.0. M HAM ADM Pade Exact Present study In Figures 2-18, we presented the highlights of the effects of the governing parameters on the nanofluid velocity, temperature and concentration profiles as well as the skinfriction, local Nusselt number and Sherwood number on the plate surface. Figures 2-4, show the effects of the magnetic parameter M, thermal convective parameter λ T and mass convective parameter λ M on the flow field velocity f (η). Figure 2, shows the effects of the magnetic parameter M on the flow field velocity f (η). The velocity field decreases with an increase in M. This would happen because the application

11 Chapter 4. Mixed Convection MHD Flow of nanofluids 87 of a transverse magnetic field sets up the Lorentz force, which retards the nanofluid fluid velocity. It is interesting to note from Fig.14 that an increase in M leads to increase in skin friction at the plate surface. Moreover, the boundary layer thickness decrease with an increase in M. Figure 3, shows the effect of the thermal convective parameter λ T on the flow field velocity f (η). It can be observed that, the flow velocity and the boundary layer thickness increases with an increase in λ T. This can be attributed to the fact that as the value of λ T increase, the skin friction on the plate decreases refer Fig.15. In Figure 4, it is observed that the velocity filed increases with increasing the values of the mass convective parameter λ M. Figures 5-10, demonstrate the effects of various parameters on the nanofluid temperature profiles θ(η). Figure 5 shows that the fluid temperature increases with an increase in magnetic field intensity. This is because the rate of heat transfer at the plate surface decrease with an increase M, see Fig.18. Figure 6, shows that as the thermal convective parameter increase the temperature of the nanofluid decrease. Figure 7, illustrates the effects of the Prandtl number Pr on temperature of the fluid. The temperature of the fluid at a fixed point decreases with an increase in the Prandtl number Pr. This is due to the fact that a fluid with higher Prandtl number has a weaker thermal diffusivity, which in turn reduces the thermal boundary layer thickness. Thus, the temperature and the thermal boundary layer thickness are decreasing functions of Pr. An increase in the radiation parameter N r causes a decrease in the temperature and the thermal boundary layer thickness as displayed in Figure 8. This is due to the fact that the thermal radiation enhances rate of heat transfer. As shown in Fig. 16, the rate of heat transfer increases as the thermal radiation parameter Nr increase which in turn causes the temperature of the fluid to decrease. However, the nanofluid temperature increases with an increase in the viscous dissipation parameter, Eckert number Ec, this is due to the action of viscous heating as shown in Figure 9. Figure 10, shows the effect of the change of the Brownian motion parameter Nb and thermophoresis parameter Nt on temperature profile provided that Nb = Nt. It is observed that the thermal boundary layer thickness and the temperature increases as Brownian motion parameter Nb increases. This is because of the fact that the magnitude of the temperature gradient at the surface decrease as both Nb and Nt increase, which can be seen from figure 17. Figures 11-14, show the effects of the magnetic parameter M, thermal convective parameter λ T, and Lewis number Le on the concentration profile. Figure 11, illustrates the effect of the magnetic parameter M on the concentration profile φ(η). The concentration field of the nanofluid increases as the magnetic parameter M increases. However, the concentration profile decrease as the thermal convective parameter λ T increases, see Fig. 12. Figure 13 illustrates the effects of mass convective parameter λ M on the concentration profile. It show that the concentration profile decreases with an increase

12 Chapter 4. Mixed Convection MHD Flow of nanofluids 88 in λ M. Figure 14 shows the effects of the Lewis number on on the concentration profile, it illustrates that the concentration profile decreases as the Lewis number increases. This is because as the value of Lewis number gets larger the molecular diffusivity gets smaller, thereby causes a decrease in the concentration field. Moreover, as the values of Le increase, the concentration boundary layer thickness decreases. This is due to the fact that the mass transfer rate at the surface increases as the Lewis number Le increases, see Fig. 19. Thus, the higher the mass transfer rate at the surface, the lower the concentration field of the nanofluid. Figures demonstrate the effects of the various parameters on the local coefficient of friction f (0), local Nusselt number θ (0), and local sherwood number φ (0). Figure 15, illustrates the variation of the skin friction coefficient f (0) with respect to the magnetic field parameter M for different values of the thermal convective parameter λ T. The magnitude of skin friction coefficient f (0) increase with M, while decreasing with an increase in λ T. Figure 16, demonstrates the variation of the Nusselt number θ (0) with the Eckert number Ec, for different values of the radiation parameter Nr. It is noticed that, an increase in Nr increases the local Nusselt number. However, the local Nusselt number decreases as Ec increases. Generally, since the decrease in Ec and increase in N r causes an increase in the rate of heat transfer on the plate this leads to an increase in the temperature of the nanofluid. Figure 17, depicts the variation of the local Nusselt number θ (0) with the thermophoresis parameter Nt for different vales of the Brownian motion parameter Nb. It is observed that, an increase in both Nb and Nt causes a decrease in the magnitude of the local Nusselt number. Figure 18, shows the variation of the local Nusselt number θ (0) and local Sherwood number φ (0) with M for different values of λ T. It depicts that the local Nusselt number and local Sherwood number φ (0) decrease with M. Moreover, both θ (0) and φ (0) increase as λ T increase. Figure 19 depicts the variation of the local Sherwood number φ (0), which represents the mass transfer rate at the surface, with the magnetic parameter M for different values of the Lewis number Le. It is simple to understand from the figure that the local Sherwood number decreases with increasing the values of M and increase with an increase in Le.

13 Chapter 4. Mixed Convection MHD Flow of nanofluids 89 Figure 4.2: Velocity profiles for various values of M, with λ T = 5, λ M = 2, P r = 1, N r, Ec = 0.2, Nb = Nt = 0.1, Le = 5. Figure 4.3: Velocity profiles for various values of λ T, with M = 4, λ M = 2, P r = 1, N r, Ec = 0.2, Nb = Nt = 0.1, Le = 5.

14 Chapter 4. Mixed Convection MHD Flow of nanofluids 90 Figure 4.4: Velocity profiles for various values of λ M, with M = 4, λ T = 5, P r = 1, N r, Ec = 0.2, Nb = Nt = 0.1, Le = 5. Figure 4.5: Temperature profiles for various values of M, with λ T = 5, λ M = 2, P r = 1, N r, Ec = 0.2, Nb = Nt = 0.1, Le = 5.

15 Chapter 4. Mixed Convection MHD Flow of nanofluids 91 Figure 4.6: Temperature profiles for various values of λ T, with M = 4, λ M = 2, P r = 1, N r, Ec = 0.2, Nb = Nt = 0.1, Le = 5. Figure 4.7: Temperature profiles for various values of Pr, with M = 4, λ T = 5, λ M = 2, N r, Ec = 0.2, Nb = Nt = 0.1, Le = 5.

16 Chapter 4. Mixed Convection MHD Flow of nanofluids 92 Figure 4.8: Temperature profiles for various values of N r, with M = 4, λ T = 5, λ M = 2, P r = 1, Ec = 0.2, Nb = Nt = 0.1, Le = 5. Figure 4.9: Temperature profiles for various values of Ec, with M = 4.0, λ T = 5.0, λ M = 2.0, P r = 3.0, N r, Nb = Nt = 0.1, Le = 5.0.

17 Chapter 4. Mixed Convection MHD Flow of nanofluids 93 Figure 4.10: Temperature profiles for various values of Nb = Nt, with M = 4, λ T = 5, λ M = 2, P r = 1, N r, Ec = 0.2, Le = 5. Figure 4.11: Concentration profiles for various values of M, with λ T = 5, λ M = 2, P r = 1, N r, Ec = 0.2, Nb = Nt = 0.1, Le = 5.

18 Chapter 4. Mixed Convection MHD Flow of nanofluids 94 Figure 4.12: Concentration profiles for various values of λ T, with M = 4, λ M = 2, P r = 1, N r, Ec = 0.2, Nb = Nt = 0.1, Le = 5. Figure 4.13: Variation of the concentration profile g(η) for different values of Le, when M = 1.0, V w = 0.0, P r = 2.0, Nb = 0.5, Nt = 0.5.

19 Chapter 4. Mixed Convection MHD Flow of nanofluids 95 Figure 4.14: Concentration profiles for various values of Le, with M = 4, λ T = 5, λ M = 2, P r = 1, N r, Ec = 0.2, Nb = Nt = 0.1. Figure 4.15: Effects of M on the skin friction for different values of λ T, with λ M = 2, P r = 1, N r = 5, Ec = 1, Nb = Nt = 0.1, Le = 5.

20 Chapter 4. Mixed Convection MHD Flow of nanofluids 96 Figure 4.16: Effects of Ec on the plate surface heat transfer rate for different values of N r, with M = 4, λ T = 5, λ M = 2, P r = 1, Nb = Nt = 0.1, Le = 5. Figure 4.17: Effects of Nt on the plate surface heat transfer rate for different values of Nb, with M = 4, λ T = 5, λ M = 2, P r = 1, Ec = 1, N r = 5, Le = 5.

21 Chapter 4. Mixed Convection MHD Flow of nanofluids 97 Figure 4.18: Effects of M on the plate surface heat(with circle marker) and mass transfer rate for different values of λ T, with λ M = 2, P r = 1, Ec = 1, N r = 5, Nb = Nt = 0.1, Le = 5. Figure 4.19: Effects of M on the plate surface mass transfer rate for different values of Le with, λ T = 5, λ M = 2, P r = 1, Ec = 1, N r = 5, Nb = Nt = 0.1.

22 Chapter 4. Mixed Convection MHD Flow of nanofluids Conclusion The effects of the thermal radiation and viscous dissipation on the mixed convection MHD flow and heat transfer is investigated. After reducing the governing boundary value problem to a non linear problem, it is solved numerically using Keller box method. The influence of the governing parameters on flow, heat and mass transfer characteristics is determined, discussed and presented in graphs. It has been noted that 1. The skin friction increases with an increase in the magnetic field parameter M, while it decreases with an increase in the thermal convective parameter λ T. 2. The velocity field of the nanofluid increases with an increase in the thermal convective parameter λ T and mass convective parameter λ M, while it decreases with an increase in the magnetic parameter M. 3. The rate of heat transfer at the plate surface inreases with an in crease in the thermal radiation parameter N r and thermal convective parameter λ T, while it decreases with an increase in the Eckert number Ec, Brownian motion parameter Nb and thermophoresis parameter Nt. 4. The temperature of the nanofluid increases with an increase in M, Ec, Nb and Nt, while it decreases with an increase in λ T, Pr, and N r. 5. The rate of mass transfer on the plate surface increases with an increase in thermal convective parameter λ T and Lewis number, while it decreases with an increase in the magnetic parameter M. 6. The concentration profile increases with an increase in M, while it decreases with an increase in λ T, λ M and Le. A comparison with previously reported data is made and an excellent agreement is noted.