Numerical simulation of variable thermal conductivity on 3D flow of nanofluid over a stretching sheet

Abstract The present investigation deals with the steady three-dimensional flow and heat transfer of nanofluids due to stretching sheet in the presence of magnetic field and heat source. Three types of water based nanoparticles namely, copper (Cu), aluminium oxide (Al2O3), and titanium dioxide (TiO2) are considered in this study. The temperature dependent variable thermal conductivity and thermal radiation has been introduced in the energy equation. Using suitable similarity transformations the dimensional non-linear expressions are converted into dimensionless system and are then solved numerically by Runge-Kutta-Fehlberg scheme along with well-known shooting technique. The impact of various flow parameters on axial and transverse velocities, temperature, surface frictional coefficients and rate of heat transfer coefficients are visualized both in qualitative and quantitative manners in the vicinity of stretching sheet. The results reviled that the temperature and velocity of the fluid rise with increasing values of variable thermal conductivity parameter. Also, the temperature and normal velocity of the fluid in case of Cu-water nanoparticles is more than that of Al2O3- water nanofluid. On the other hand, the axial velocity of the fluid in case of Al2O3- water nanofluid is more than that of TiO2nanoparticles. In addition, the current outcomes are matched with the previously published consequences and initiate to be a good contract as a limiting sense.


Introduction
Recently, the study of nano uid ows generated due to a stretching sheet has been attracted by the many researchers because of their fascinating engineering and industrial applications such as microelectronics, engine cooling, refrigerator and fuel cells. The fundamental point of using nano uids in such frameworks are to upgrade the thermal conductivity and enhancing the heat transfer capacity in order to accomplish better cooling. The reviews [1][2][3][4] reviled that the heat exchange performance of the nanoparticles (like Cu, Al O , TiO ) are more in comparison with the base uids (such as water, glycol, toluene and oil etc.). Rehman et al. [5] examined the effect of thermal radiation on a stretching sheet in the presence of nano uid. They originate that the Cunanoparticles enhancing heat transfer rate with minimal entropy generation. Recently, many researchers [6][7][8][9][10] have paid their interest on nano uid ows with di erent boundary conditions on di erent models. Melting e ect on heat transfer ow of nano uid with Buongiorno model was studied by Sheikholeslami et al. [11]. Umavathi et al. [12] considered the convective heat transfer ow of nano uid by using Darcy-Forchheimer-Brinkman model. They concluded that the ow and heat transfer characteristics enhance with increase of Darcy or Grashof or Brinkman numbers while it reduce with the inertial or viscosity ratio parameters. Iqbal et al. [13] presented the e ects of heat convection on nano uid ow over a moving Riga plate. They solved the governing equations numerically by usingnite di erence scheme Keller Box method. Ahmad Khan et al. [14] employed two di erent models for 3D rotating ow of e ective thermal conductivity of nano uids. Falkner-Skan ow of MHD Carreau nano uid was investigated by Masood Khan et al. [15]. They applied two numerical techniques namely shooting and Newton's Raphson methods in their study. Hayat et al. [16] developed the 3D rotating ow of Maxwell uid through submersion nanoparticle. Impact of MHD on heat transfer ow of a nano uid generated by a stretching sheet was studied by [17][18][19].
The in uence of temperature dependent thermal conductivity in the presence of nano uids is most important because it change with temperature. The Cattaneo-Christov heat ux theory with Je ery liquid was reported by Meraj et al. [20]. They initiate the thermal boundary layer thickness is high for heat conduction of Fourier's law with comparison of Cattaneo-Christov heat ux model. But the concentration plays an opposite behaviour with reaction parameters. Hayat et al. [21] examined the Cattaneo-Christov heat ux on Je rey uid ow with variable thermal conductivity. Ramzan et al. [22] analyzed the e ects of temperature dependent thermal conductivity on non-Newtonian nano uid ow past a moving surface. They found that the temperature increases with the rise of thermal radiation. Srinivas Reddy et al. [23] examined the heat transfer characteristics of Williamson nano uid over a stretching sheet. They solved the equations by spectral quasi-linearization method by considering the variable thermal conductivity. Dyugaev et al. [24] studied the in uence of thermal conductivity and viscosity of uid with ultra -ne particles. Some of the scientist's worked on the in uence of thermal radiation on free convection ow through di erent models [25][26][27][28][29].
Animasaun [30] explored the in uence of chemical reaction and thermophoresis on the nano uid boundary layer ow over an upper horizontal surface of paraboloid of revolution. They solved the buoyancy model by using R-K-4-S-T (Runge-Kutta fourth order along with shooting technique) with volume fraction (ϕ) de ned as % ≤ ϕ ≤ . %. Three kinds of water-based nano uids (Al O , Cu, and TiO nanoparticles) with volume fraction 10% and 20% in a cubical enclosure has been analysed by Boutra et al. [31]. They implemented a boundary element method to simulate the ow and estimate the thermal conductivity and viscosity of nano uids. The natural convection of Al O /water nano uids with volume fractions up to 3% in a cubic cavity was proposed by Saghir et al. [32]. Purusothaman et al. [33] presented the natural convection of nano uids in a cavity model by nite volume method. Later, Snoussi et al. [34], Sheikholeslami and Ellahi [35] and Kolsi et al. [36][37][38] established convection ow of Al O /water nano uids in a cube cavity with a maximum volume fraction up to 20%. Meng and Li [39] simulated the free convection of Al O /water nano uids with two volume concentrations of 1 and 4% in a horizontal cylinder. Ho et al. [40] developed the correlations for thermal conductivity and viscosity.
The literature reviles that the thermal conductivity of nano uids are not constant and it varies linearly with the temperature and they play an important role in many engineering and industrial applications. Hence, the main ob-jective of current analysis is ll the gap in the literature by studying the in uence of variable thermal conductivity on three dimensional ow of MHD nano uid (with different nanoparticles) caused by a stretching sheet with thermal radiation. Similarity transformations are applied to nonlinear partial di erential equations and the transformed system can be solved numerically by Runge-Kutta-Fehlberg scheme with shooting technique. Expressions for various values of parameters on the velocity and temperature as well as the Nusselt number are discussed graphically.

Mathematical formulation
The steady three-dimensional ow of an electrical conducting magnetohydrodynamic nanoliquid through a stretching surface is considered in this model. The variable thermal conductivity and heat transfer of nano uids have been studied in view of free convection. Choose a cartesian coordinates system (x, y, z) in which x− and y−axes are taken along the sheet surface in the direction of the uid ow and z−axis is perpendicular to it. The stretching sheet is coincident at z = and ow dwell in the region z > as shown in Figure 1. u * w (x) = a * x and v * w (y) = b * y are stretching velocities along the uid ow direction. Furthermore, w * = −W * is mass ux velocity, considered on the sheet. Where w * > is the suction and w * < is the injection. A uniform magnetic eld of strength B is assumed in the direction of z and normal to the surface (i.e. xy-plane). The induced Lorentz force and impressed electric eld are neglected due to the small magnetic Reynolds number. The dynamical equations of the three dimensional nano uid ow along with heat transfer can be expressed (see Ref. [41,42]) as The boundary conditions are de ned as Where ∆T * = T * w − T * ∞ and T * w is the sheet temperature, ω∞ is the conductivity of the uid far away from the sheet.
By Rosseland approximation, the radiative heat ux q * r (see Ref. [43,44]) is given by Substituting equation (7) in the energy equation (4) and it becomes Where The e ective dynamic viscosity of the nano uid is described as [see Ref. [5,6,41]) The similarity variables are With the help of the above relations, the governing equations nally reduce to The corresponding boundary conditions can be written as Where λ is the stretching ratio parameter, S is a constant mass wall transfer with S > for suction, S < for injunction and impermeable plate S = and The surface frictional coe cients C fx , C fy and rate of heat transfer Nux are respectively, de ned as follows is the wall ux from the stretching surface.
The non-dimensional form of surface friction coe cients and heat transfer coe cients are de ned as are local Reynolds number. It is observed that for λ = , present problem reduces to the case of two dimensional linear stretching work, while λ = , sheet is axisymmetric case where sheet is stretched in x and y directions with the same values and if λ is neither zero or one then the ow behaviour along both the directions will be di erent.

Results and discussion
The transformed equations (12), (13) and (14) with boundary conditions (15) have been solved numerically by Runge-Kutta-Fehlberg method along with shooting technique. The in uence of di erent ow parameters on the axial and normal velocities, temperature, surface frictional coe cients and rate of heat transfer are discussed graphically in Figures  2-11. The thermophysical properties of di erent nano uids are de ned in Table 1. The validity of the current work outcomes are compared with those of Magyari and Keller [45] and Liu et al. [46] in Table 2 and Table 3 respectively. It is observed that the current results are in good agreement with those existing results.    ) and temperature (θ(η)) pro les respectively for two distinct nanoparticles, namely Cu and TiO . It is noticed that the velocity pro les declines in both the directions and temperature increases by increasing values of M. This is because, M is inversely proportional to the density by M = σB aρ f and hence the temperature of the uid rises with rising values of M. Also noticed that the heat transfer rate of Cu is more than that of TiO nanoparticles. Therefore, the two distinct nanoparticles boundary layer is vary with magnetic eld parameter M.
The variations of the velocity and temperature pro les for buoyancy parameter γ are presented in Figures 3(a)-3(c), respectively with (i.e.Cu, Al O and TiO ) three different nanoparticles. It is obvious that both the velocity pro lesf (η) and g (η) are growing and θ(η) demolish with rising values of γ. This is due to the high heat source on temperature pro le. Figures 4(a)-4(c) illustrate the in uence of stretching ration parameter λ on both velocity components and temperature distributions. The velocity increases along the normal direction and temperature, velocity along axial di- rection decrease with increasing values of λ. Therefore, the Coriolis force in the uid motion is due to stretching ratio parameter λ which is responsible for the acceleration of the uid motion and hence the momentum boundary layer thickness. Increasing values of stretching ratio parameter λ causes the less heat transfer of the uid from sheet which leads to decline in the temperature and thermal boundary layer thickness. The variations of Pr (Prandtl number) on the velocity and temperature distributions are exhibited in Figure  5 for di erent Cu, TiO nanoparticles. It is noticed that the pro le θ(η) diminishing for ascending values of Pr. Physically, less Pr values indicate the high thermal di usivity which causes a reduction in the uid temperature. Hence, the thermal boundary layer thickness of Pr is proportional to the thermal di usivity.
The nano uid volume fraction ϕ characterization on both velocities and temperature pro les are elucidated respectively in Figures 6(a)-6(c). It is noticed that both the velocity pro les along axial and transverse directions increase with distinct ascending values of ϕ and intensies the resistance force within the uid and the temperature pro les has opposite behaviour. This is due to the fact that the thermal conductivity and thermal boundary layer thickness decreases with growing values of ϕ. ascending values of ε along axial and normal directions. Also, the temperature of the uid and thermal boundary layer thickness increases with ascending values of ε.
Well-strategy pro le θ(η) is illustrated in Figure 9 for distinct numerical values of R d . It is observed that θ(η) increases with ascending numerical values R d . This means that the uid absorbs more heat from the radiation, with this both temperature and its association thermal boundary layer thickness increases. Figures 10 displayed the e ect of ε against λ on Re − / x Nux. It is clear that the heat transfer rate increase with ascending values of ε for titanium oxide nanoparticles. This is due to the fact that, the surface heat transfer rate enhance signi cantly for xed large values of ϕ. Figure 11 displays the in uence of S on heat transfer rate (Re − / x Nux) against λ. It is observed that the Re − / x Nux increases for increasing values of S. Physically, the principal of suction is associate of the heated uid particles through stretching sheet where the buoyancy forces will slow down due to the high viscosity e ect of the uid particles.
The e ect of Prandtl number on skin friction coecient against volume fraction parameter along y and x axis is displayed respectively in Figures 12 (a) and (b). The local skin friction value increases with increasing values of Pr and ϕin the presence of Copper nanoparticles.

Concluding remarks
The major outputs of the current research work are listed below