Influence of Magnetic Field in Three-Dimensional Flow of Couple Stress Nanofluid over a Nonlinearly Stretching Surface with Convective Condition

This article investigates the magnetohydrodynamic (MHD) three-dimensional flow of couple stress nanofluid subject to the convective boundary condition. Flow is generated due to a nonlinear stretching of the surface in two lateral directions. Temperature and nanoparticles concentration distributions are studied through the Brownian motion and thermophoresis effects. Couple stress fluid is considered electrically conducting through a non-uniform applied magnetic field. Mathematical formulation is developed via boundary layer approach. Nonlinear ordinary differential systems are constructed by employing suitable transformations. The resulting systems have been solved for the convergent series solutions of velocities, temperature and nanoparticles concentration profiles. Graphs are sketched to see the effects of different interesting flow parameters on the temperature and nanoparticles concentration distributions. Numerical values are computed to analyze the values of skin-friction coefficients and Nusselt number.


Introduction
Boundary layer flow over a continuous stretching surface has various industrial and engineering applications. Such flow commonly involves in the paper production, wire drawing, glass fiber production, extrusion of plastic sheets, hot rolling, cooling of a metallic plate in a cooling bath and many others. Many researchers have discussed the different problems through the linear stretching of the surface but there are various situations in the industrial and technological processes where the stretching of the surface is not necessarily linear. Particularly the flow induced by a nonlinear stretching surface has played important role in the polymer extrusion process. With this viewpoint Vajravelu [1] provided a study to examine the flow and heat transfer properties of viscous fluid induced by a nonlinear stretching surface. Cortell [2] performed a numerical study to investigate the flow of viscous fluid over a nonlinear stretching surface. He studied the two cases of heat transfer namely the constant surface temperature and the prescribed surface temperature. Cortell [3] also explored the flow of viscous fluid over a nonlinearly stretching surface in the presence of viscous dissipation and radiation effects. Hayat et al. [4] addressed the magnetohydrodynamic (MHD) flow over a nonlinear stretching surface by using the modified Adomian decomposition and Padé approximation techniques. Flow and heat transfer properties of nanofluid over a nonlinear stretching surface is reported by Rana and Bhargava [5]. Mukhopadhyay [6] discussed the boundary layer flow over a permeable nonlinear stretching surface subject to partial slip condition. Mabood et al. [7] studied the MHD flow of water-based nanofluid over a nonlinear stretching surface in the presence of viscous dissipation. Recently Mustafa et al. [8] investigated the flow of nanofluid over a nonlinearly stretching surface subject to the convective surface boundary condition.
The insertion of ultrafine nanoparticles (<100nm) in the base liquid is termed as nanofluid. The nanoparticles utilized in nanofluids are basically made of metals (Cu, Al, Ag), oxides (Al 2 O 3 ) carbides (SiC), nitrides (SiN, AlN) or nonmetals (carbon nanotubes, graphite) and the base liquids like water, oil or ethylene glycol. Addition of nanoparticles in the base liquids greatly enhances the thermal characteristics of the base liquids. Due to such interesting properties, nanofluids are useful in various industrial and technological processes such as the cooling of electronic devices, transformer cooling, vehicle cooling, heat exchanger, nuclear reactor, biomedicine and many others. Especially the magneto nanofluids are useful in MHD power generators, removal of blockage in the arteries, hyperthermia, cancer tumor treatment, magnetic resonance imaging etc. The term nanofluid was first introduced by Choi and Eastman [9] and they illustrated that the thermal properties of base liquids are enhanced when we add up the nanoparticles in it. Boungiorno [10] constructed a mathematical expression to investigate the thermal characteristics of base fluids. Here the effects of thermophoresis and Brownian motion are utilized to enhance the thermal properties of base liquids. Khan and Pop [11] employed the Boungiorno model [10] to analyze the boundary layer flow of nanofluid over a stretching surface. Pujari et al. [12] studied the orientation state of multiwalled carbon nanotubes (MWNTs) dispersions in steady and transient shear flows. Dong and Cao [13] examined the anomalous orientations of rigid carbon nanotube in a sheared fluid. Zhao et al. [14] reported both theoretical and experimental studies of collective effects on the Soret coefficient of particles in deionized (DI) water. Translational thermophoresis and rotational movement of peanut-like colloids under the influence of temperature gradient is addressed by Dong et al. [15]. Wang et al. [16] discussed the thermal diffusion behavior of dilute solutions of very long and thin charged colloidal rods by using the holographic grating technique. Few more recent studies in this direction can be quoted through the investigations [17][18][19][20][21][22][23][24][25][26][27] and several refs. therein.
Most of the studies in the literature explain viscous materials by the classical Navier-Stokes relations. There are several complex rheological materials such as paints, shampoos, slurries, toothpastes, polymer solutions, ketchup, paper pulp, blood, greases, drilling muds, lubricating oils and many others that cannot be characterized through the classical Navier-Stokes expressions. Such materials are known as the non-Newtonian fluids. However, there is no single relation that can present the characteristics of all non-Newtonian fluids. Hence various models of non-Newtonian fluids are developed in the literature. The couple stress fluid model [28][29][30][31][32][33] is one of such materials. This model has important features due to the presence of body couples, couple stresses and non-symmetric stress tensor. Some interesting examples of the couple stress fluid are blood, suspension fluids, lubricants and electro rheological fluids.
The main aim of the present communication is to generalize the analysis of ref. [8] into three directions. Firstly to consider the three-dimensional flow of couple stress nanofluid. Effects of Brownian motion and thermophoresis are present. We imposed the thermal convective [34,35] and zero nanoparticles mass flux conditions at the surface [36,37]. Secondly to analyze the influence of variable magnetic field under low magnetic Reynolds number assumption. Thirdly to compute the convergent series solutions through the homotopy analysis technique (HAM) [38][39][40][41][42][43][44][45]. Effects of various emerging parameters on the temperature and nanoparticles concentration distributions are sketched and discussed. Skin-friction coefficients and Nusselt number are computed numerically.

Problem Formulation
Let us consider the steady three-dimensional flow of an incompressible couple stress nanofluid by a bidirectional nonlinear stretching surface. The couple stress fluid is assumed an electrically conducting through a non-uniform magnetic field applied in the z−direction. Effects of electric field and Hall current are neglected. The induced magnetic field is not considered subject to the small magnetic Reynolds number. Brownian motion and thermophoresis effects are present. Consider the Cartesian coordinate system in such a way that the x− and y−axes are taken along the stretched sheet and z−axis is perpendicular to it. Let U w (x,y) = a(x+y) n and V w (x,y) = b(x+y) n denote the surface stretching velocities along the x− and y−directions respectively with a, b, n>0 as the constants. The temperature at the surface is controlled through a convective heating mechanism which is denoted via heat transfer coefficient h f and temperature of the hot fluid T f below the surface. The boundary layer expressions governing the flow of couple stress nanofluid are given by The subjected boundary conditions are Note that u, v and w are the velocity components in the x−, y− and z−directions respectively, v (= μ/ρ f ) represents the kinematic viscosity, μ stands for dynamic viscosity, ρ f represents the density of base fluid, v 0 (= n Ã /ρ f ) denotes the couple stress viscosity, n Ã stands for couple stress viscosity parameter, σ denotes the electrical conductivity, Bðx; yÞ ¼ B 0 ðx þ yÞ nÀ1 2 stands for non-uniform magnetic field, T denotes the temperature, α = k/(ρc) f represents the thermal diffusivity of fluid, k stands for thermal conductivity of fluid, (ρc) f represents the heat capacity of fluid, (ρc) p denotes the effective heat capacity of nanoparticles, D B stands for Brownian diffusion coefficient, C denotes the nanoparticles concentration, D T represents the thermophoretic diffusion coefficient, h f ¼ hðx þ yÞ nÀ1 2 denotes the non-uniform heat transfer coefficient, T 1 represents the temperature far away from the surface and C 1 represents the nanoparticles concentration far away from the surface. We now use the following transformations Eq (1) is now satisfied and Eqs (2)-(7) have the following forms In above expressions K denotes the couple stress parameter, M represents the magnetic number, c stands for ratio parameter, Pr denotes the Prandtl number, Nb represents the Brownian motion parameter, Nt stands for thermophoresis parameter, γ denotes the Biot number, Le stands for Lewis number and prime denotes the differentiation with respect to η. These variables are defined by Skin-friction coefficients and local Nusselt number are given by ðf 00 ð0Þ À Kf iv ð0ÞÞ; ðg 00 ð0Þ À Kg iv ð0ÞÞ; Re À1=2 x It is seen that the dimensionless mass flux denoted by a Sherwood number Sh x is now identically zero and Re x = U w (x+y)/v and Re y = V w (x+y)/v represent the local Reynolds numbers.

Series Solutions
Our purpose now is to compute the series solutions via homotopy analysis technique (HAM). The appropriate initial guesses (f 0 ,g 0 ,θ 0 ,ϕ 0 ) and the corresponding auxiliary linear operators (L f ,L g ,L θ ,L ϕ ) for homotopic solutions are selected as follows: L f ¼ f 000 À f 0 ; L g ¼ g 000 À g 0 ; L y ¼ y 00 À y; L ¼ 00 À : The above operators have the following properties in which B i (i = 1 − 10) depict the arbitrary constants.

Convergence Analysis
No doubt the auxiliary parameters in the series solutions have key role regarding convergence.
The proper values of these parameters play a key role to develop the convergent series solutions. For such interest, the ħcurves for the velocities, temperature and nanoparticles concentration profiles are plotted at 15th order of deformations.   Order of approximations -f''(0) -g''(0) -θ'(0) ϕ'(0)    Table 2 depicts the values of skin-friction coefficients ÀRe 1=2 x C fx and ÀRe 1=2 y C fy for various values of power-law index n, couple stress parameter K, magnetic number M and ratio parameter c. Here the skin-friction coefficients are increasing functions of power-law index n. It is also seen that the skin-friction coefficients are enhanced for the larger values of magnetic number M. Values of local Nusselt number Re À1=2

Results and Discussion
x Nu x for various values of n, K, M, c, γ, Nt, Nb, Le and Pr are computed in Table 3. We noticed that the larger values of couple stress parameter K, magnetic number M and thermophoresis parameter Nt correspond to a lower local Nusselt number while opposite behavior is observed for Biot number γ. Table 4 presents the values of temperature and nanoparticles concentration profiles for various values of n, K, M, c, γ, Nt, Nb, Le and Pr when η = 1.0. From this Table, it is clearly shown that the Biot number γ and Prandtl number Pr influences the temperature and nanoparticles concentration profiles the most.

Main findings
Three-dimensional flow of couple stress nanofluid over a nonlinear stretching surface with the convective surface boundary condition and non-uniform magnetic field is analyzed. The main findings of the present research are given below: • An increase in the couple stress parameter K shows an enhancement in the temperature and nanoparticles concentration profiles.
• Temperature and nanoparticles concentration profiles are enhanced with an increase in the values of magnetic number M.
• Effects of Biot number γ on the temperature and nanoparticles concentration profiles are qualitatively similar.
• Temperature θ(η) and thickness of the thermal boundary layer are lower for the larger values of Prandtl number Pr.
• Larger values of thermophoresis parameter Nt present similar behavior for temperature and nanoparticles concentration profiles.
• An enhancement in the Brownian motion parameter corresponds a weaker nanoparticles concentration profile.
• Coefficients of skin-friction are higher for the larger values of the magnetic number M.
• Rate of heat transfer at the wall is constant for Brownian motion parameter while it is lower for the thermophoresis parameter.