Inquiry of inclined magnetic field effects on Walter –B nanofluid flow with heat generation / absorption

The article deals with Walter-B nanoliquid flow towards a extending surface with inclined magnetic field effects. Thermal relaxation analysis is made by non fourier heat flux model. Radiation, heat generation / absorption impacts are included. The non linear Partial governing systems are rebuild into nonlinear ordinary systems with the assist of proper similarity transformations. The graphical results are portrayed for velocity, concentration and temperature profile. The physical entitles of heat and mass transfer rates are graphically reported. The comparission with previous results notified the excellent agreement.


Introduction
Magnetohydrodynamic (MHD) flow has a limitless number of practical and theoretical researches in numerous geophysical, physical and industrial segments. Mostly the MHD flows associated with heat transmission have acquired substantial consideration up to now. It is due to the interest that usages of MHD flows occur in many manufacturing areas like nuclear reactor cooling, microelectronic tools, electronic packages, etc. To author's knowledge, MHD flow caused by a stretched wall is examined by Pavlov [1]. He obtained the exact analytic solution in the apperance of a uniform magnetic field. Moreover, some imperative impacts in MHD flow with heat transmission were done by ref's [2]- [10]. Mostly the cooling liquids have the constant physical properties that are considered from the previous data. In practical situations the variable characteristics with physical propertiesare important. Hayat et al. [11] explored the 3D flow of Jeffry fluid flow in the appearance of variable thermal conductivity. Bhattacharyya et al. [12] consider the temperature-dependent viscosity with the boundary layer slip flow due to stretchy sheet. The impact of the combined convective flow of viscous fluid embedded in a porous surface is explored by Hayat et al. [13]. Rashidi et al. [14] studied the MHD flow of a fluid over a rotating disk. They use a numerical way to obtain the results also examine the influence of magnetic interaction number. Pal and Mondal [15] presented the MHD forced convection through a wedge with thermal radiation temperature-dependent viscosity. Pal and Mondal [16] additionally described variable thermal conductivity and temperature-dependent viscosity effects in MHD stretchy non-Darcy combined the convective flow of species.
This study addressed the 2D flow of Walter-B nanofluid flow across a convective surface with radiation and heat generation effects. Nanomaterial includes Brownian motion and thermophoresis  [16][17][18][19][20][21] is applied to solve the nonlinear function of ODE. Results of velocity, nanoparticle concentration, and temperature are produced via graphical representation. Code validation with formerly published results is obtained good agreement.

Mathematical formulation of the problem
We examine the flow of 2-D incompressible laminar boundary layer flow of Walter-B nano liquid flow towards a shrinking plate. The non Fourier heat flux is taken into account. Radiation, heat genration and convective heating effects are included. The two tempertures ‫܂‬ ‫܅‬ and ‫܂‬ ∞ are on and apart from thesurface. The applied magnetic field is implimented in positive y-direction. The governing equations are defined with the help of Figure.

Results and discussion
In this section we examined the impacts of different flow parameters over velocity, nanoparticle concentration, and temperature. The homotopy technique (HAM) is implimented for solve the nonlinear systems and the results are presented through graphically. Figures.2&3 displays the impression of ߙ and ‫ܨ‬ ‫ݓ‬ on the velocity distribution. It is observed that the velocity distribution and its related boundary layer thickness diminishes by increasing ߙ and ‫ܨ‬ ‫ݓ‬ . Figure.4 represents the differences in the velocity field with an increase in Weissenberg number ܹ݁. It is examined that the velocity falls with a rise in Weissenberg number ܹ݁. Conversely, the momentum boundary layer also weakens when Weissenberg number ܹ݁ is enhances. Substantially, rising values of Weissenberg number ܹ݁ upsurges tensile stresses which oppose the momentum transport and therefore boundary layer thickness reduces.The influence of Hartmann number ‫ܽܪ‬ on the velocity profile ‫ܨ‬ ′ are portrayed in Figure. 5. The applied magnetic field has the movement to reduce speed of the liquid which diminishes the velocity profile.  The effect of ‫ݐܰ‬ and ܾܰ on ߶(ߞ) is exposed in Figures. 14 & 15, respectively. As it is displayed in these figures, nanofluid parameters have reverse impacts on the concentration profiles, that is, with enhancing thermophoresis parameter the nanoparticle concentration and their associated boundary thickness rises but higher rates of Brownian motion parameter tend to lesser the ߶(ߞ) values. Physically, rising the thermophoresis parameter results in augmentation of thermophoresis energy that indicates nanoparticles moving from hot to cold parts and as a result, size of nanoparticle volume fraction increases. Furthermore, increasing Brownian motion parameter results in diminishing the diffusion of nanoparticles into the liquid regime apart from the surface, then ߶(ߞ) decreases in the boundary layer.
From Figures. 16 & 17 shows the influence of local heat and mass transfer rates individually. Figure. 16 shows the heat transfer rate with the combination ܹ݁ and ߜ. Heat transfer rate reduces for higher values of Weissenberg number and thermal relaxation time constant. Figure. 17 depicts that the effect of mass transfer rate with the combination of ܾܰ, ‫ݐܰ‬ and ‫.ܽܪ‬ It conclude tha Brownian motion and thermophoratic parameters showes the inverse efffect on mass transfer rate.   Figure. 16 Impact of ܹ݁ ܽ݊݀ ߜ onܰ‫ݑ‬ ‫ݔ‬ Figure. 17 Impact of ܾܰ, ‫ݐܰ‬ and ‫ܽܪ‬ on ܵℎ ‫ݔ‬

Main outcomes
From the present examination the main observations are noted below: x Velocity profile has reducing performance for higher ܹ݁.
x Temperature profile reduces for higher thermal relaxation time constant.
x Larger values of Nt and Nb rises the temperature profile x Nt and Nb shows the opposite effect on the concentration and Sherwood number profiles. .
x Heat transfer rate reduces via Weissenberg number ܹ݁.