Navier's slip effect on Carreau nanouid flow past a convectively heated wedge in the presence of nonlinear thermal radiation and magnetic field

Abstract

In this article, an investigation is carried out for Carreau nanofluid flow past a convectively heated stretching wedge with Navier's velocity slip condition along with magnetic field effect. The presence of nonlinear thermal radiation, viscous dissipation, Joule heating and suction influences are also taken into account. Rosseland's approximation is employed for approximating the radiative heat flux. The dimensional model equations are transformed into non-dimensional forms using some appropriate non-dimensional transformation and the resulting nonlinear boundary value problem is solved numerically using Runge-Kutta Cash-Karp method based shooting technique. The variations in velocity, temperature and concentration profiles with respect to various pertinent physical parameters are analyzed with the help of figures. Numerical values of skin friction coefficient, Nusselt and Sherwood numbers are explained and shown in the form of tables. Also, the obtained results are validated with the results existing in literature for some reduced case and an admirable agreement is noticed among the results.

Introduction

The study of hydromagnetic fluid flow with heat and mass transfer has become a significant topic of research in fluid dynamics. It has found huge applications [1] in the fields of petroleum engineering, geothermal energy abstractions, plasma studies, aerodynamics and to name a few. Nowadays, non-Newtonian fluids are considered more useful in comparison to the viscous fluid. Non-Newtonian fluids has found many applications [2] in industrial and engineering processes such as bubble absorptions, fermentation, plastic foam processing, extrusion process, composite processing, heat exchangers, wire and fiber coating, chemical processing equipment etc. Some of the non-Newtonian fluids [3] are multi-grade oil, blood transported in micro-circulatory system, paints, printer ink, liquid detergents, mud, sauce etc. Since past two decades various non-Newtonian fluid flow model have been proposed by the researchers. The Carreau nanofluid model is one of those non-Newtonian fluid flow models for which constitutive relationship holds at gooey, low and high shear rates. The constitutive model related with Carreau nanofluid flow [4] is written as $\overline{\mu }\left(\dot{\gamma }\right)={\overline{\mu }}_0{\left\{1+{\mathrm{\Gamma }}^2{\left(\dot{\alpha }\right)}^2\right\}}^{\frac{n-1}{2}}+{\overline{\mu }}_{\infty }\left[1-{\left\{1+{\mathrm{\Gamma }}^2{\left(\dot{\alpha }\right)}^2\right\}}^{\frac{n-1}{2}}\right]$, where ${\overline{\mu }}_0$, ${\overline{\mu }}_{\infty }$, $\dot{\alpha }$, n and Γ are the zero shear stress viscosity, infinite shear stress viscosity, shear rate, power-law-index and material parameter respectively. The expression of $\dot{\alpha }$ is defined as $\dot{\alpha }=\frac{\sqrt{D_1^2}}{\sqrt{2}}$ with D₁ =  ∇ V + (∇V)^T, where V is the velocity vector and ∇V and (∇V)^T are velocity gradient and its transpose respectively. Based on numerous applications of Carreau nanofluid flow, Hashim and Khan [5] studied the Carreau nanofluid flow over a stretching surface with Brownian motion and thermophoresis impacts. 2-dimensional hydromagnetic Carreau nanofluid flow over a stretching surface with variable thickness was scrutinized by Khan et al. [6]. Carreau Nanofluid flow through a stretching cylinder considering homogeneous and heterogeneous effects was interpreted by Khan et al. [7]. Khan et al. [8] examined hydromagnetic Carreau nanofluid flow towards a paraboloid surface of revolution based on the Cattaneo-Christov model of heat and mass fluxes. Subsequently, Khan et al. [9] presented hydromagnetic Carreau nanofluid flow through a stretching cylinder considering chemical reaction effect and zero normal flux condition.

The convective heat transfer is of great importance in procedures where high temperature exists. For example, nuclear plants, gas turbine, storage of thermal energy etc. It has found many applications [10] in engineering and industrial processes like material drying, transpiration cooling process and many more. Based on such applications, Khan et al. [11] presented the effect of convective heating condition on hydromagnetic Carreau nanofluid flow generated by a stretched surface taking presence of magnetic field into account. Ramzan et al. [12] highlighted hydromagnetic Williamson nanofluid flow near a stagnation point over a convectively heated stretching surface in the presence of homogeneous and heterogeneous reactions and Cattaneo-Christov heat flux. Maxwell fluid flow over a vertical slit in the presence of convective heating condition was discussed by Zaidi and Mohyud-Din [13]. Mamatha et al. [14] deliberated convecting heating impact on hydromagnetic Carreau dusty fluid flow towards a stretching sheet with heat generation influence. Convective heating boundary condition impact on gyrotactic mixed bioconvection nanofluid flow towards a circular cylinder was illustrated by Rashad and Nabwey [15].

Thermal radiation depends on the factors such as temperature, surface properties of the material that are generating or absorbing heat and solid geometric arrangement. The radiation heat transfer between two bodies is increased/reduced when the temperature difference between the bodies is more/less. Consideration of linear thermal radiation is not appropriate for processes with higher temperature differences. So, the investigators recently proposed the concept of nonlinear thermal radiation with the addition of an extra parameter as compared to the linear thermal radiation. This additional parameter exhibits the difference between surface temperature and uniform temperature. It has diverse usage [16] in industries, reactor cooling, vehicles in the space, electrical power generators, plasmas etc. In view of such applications, the nonlinear thermal radiation influence on non-aligned MHD nanofluid stagnation point flow towards a stretching sheet was discussed by Babu and Sandeep [17]. Thermal radiation aspect on Eyring-Powell nanofluid flow generated by a stretching cylinder in the presence of heat generation and magnetic field was illustrated by Rehman et al. [18]. Kumar et al. [19] explained the impacts of magnetic field, nonlinear thermal radiation and temperature jump on Williamson dusty nanofluid flow past a stretching sheet. Entropy generation on hydromagnetic viscous nanofluid flow over a stretching sheet considering nonlinear thermal radiation and heat generation influences was investigated by Hayat et al. [20]. Furthermore, Mohammadein et al. [21] examined hydromagnetic CuO-water nanofluid flow past a stretching sheet in the presence of nonlinear thermal radiation, magnetic field, suction/injection and Brownian motion effects.

The viscous dissipation has significant impact on temperature distribution because it serves as an internal heat source due to the work done by shearing stresses on fluid layers. Viscous dissipation and Joule heating influences are more effective in plate colling and heating process. These effects have profound applications in power generation systems, electronic chips, metallic sheet cooling, food processing, electronic cigarette, nuclear reactor cooling and many more. Based on such applications, viscous dissipation and Joule heating aspects on hydromagnetic Cu-water nanofluid flow over a stretched surface was investigated by Hayat et al. [22]. Khan et al. [23] contemplated the Carreau nanofluid flow which is generated by a stretching cylinder in the presence of Joule heating and convecting heating conditions. Entropy generation in a hydromagnetic nanofluid flow over a stretching sheet with nonlinear thermal radiation, convective heating condition and viscous dissipation was explored by Sithole et al. [24]. Kumar et al. [25] considered viscous dissipation and Joule heating influences on Oldroyd B nanofluid flow with magnetic field and thermal radiation impacts. Patel and Singh [26] analyzed viscous dissipation and Joule heating aspects on hydromagnetic microploar fluid flow past a convectively heated stretching sheet which is placed in a porous medium considering nonlinear thermal radiation into account.

One of the central tenets of the Navier-Stokes theory is no-slip boundary condition. The idea of no-slip boundary condition at the boundary is invalid for fluid flows in micro-electromechanical systems and must be changed by slip boundary condition. Between shear stress and shear strain, a linear association exists in the slip flow model. Slip condition at the surface has attracted many researchers because of its various applications in nano-channels or micro-channels. Keeping in mind such applications, velocity slip and magnetic field effects on gyrotactic microorganisms based nanofluid flow near a vertical plate was demonstrated by Khan et al. [27]. Hayat et al. [28] investigated Navier slip impact on hydromagnetic 3D nanofluid flow near a porous stretching surface with nonlinear thermal radiation. Ibrahim and Makinde [29] examined velocity slip influence on hydromagnetic power-law nanofluid flow past a stretching sheet with convective heating condition. Magnetic field and slip effect on nanofluid flow over a nonlinearly stretching permeable sheet in the presence of thermal radiation was demonstrated by Tausif Sk et al. [30]. Velocity slip aspect on hydromagnetic Casson fluid flow towards a convectively heated stretching surface was analyzed by Ibrahim and Makinde [31]. Makinde et al. [32] presented Navier's slip and heat generation effects on nanofluid flow generated by a thermally radiated stretching surface in the presence of magnetic field, chemical reaction and buoyancy force. Reddy et al. [33] deliberated Navier's slip impact on hydromagnetic Maxwell nanofluid flow towards a convectively heated exponentially stretching surface. Multiple slips and Joule heating influences on hydromagnetic fluid flow towards a stretching surface which is embedded in a permeable medium with thermal radiation aspect was investigated by Sekhar et al. [34]. Navier's slip effect on MHD nanofluid flow past a radiating sheet in the presence of zero mass flux boundary conditions and Newtonian heating effect was premeditated by Uddin et al. [35]. Makinde et al. [36] considered velocity slip aspect on hydromagnetic fluid flow past a thermally radiated stretched surface in the presence of heat generation and thermal buoyancy force.

Fluid flow past a wedge shape geometry has been gained extensive attention of the researchers due to its several practical applications in the field of chemical industry and engineering. Some of the applications of flows towards a wedge could be found in crude oil extraction, liquid metal flows in heat exchangers, polymer processing, flow of molten metals over ramped surfaces, nuclear power plants, ground water pollution, packed bed reactors, geothermal industries, modeling of warships etc. Due to these significant impacts in real life problems, Rahman et al. [37] inspected numerically the transverse magnetic filed, heat generation and Navier slip influences on MHD water based nanofluid flow past a wedge shape geometry with convective heating condition. Hydromagnetic Falkner-Skan Casson nanofluid flow which is generated by a convectively heated wedge in the presence of Brownian motion, thermal radiation and thermophoresis effects was elaborated by Raju and Sandeep [38]. Hydromagnetic nanofluid flow over a convectively heated wedge which is placed in a porous medium considering viscous dissipation, velocity slip and suction influences into account was premeditated by Pandey and Kumar [39]. The Williamson nanofluid flow near a convectively heated wedge shape geometry in the presence of magnetic field and enhance heat flux effects was illustrated by Hashim et al. [40]. Subsequently, Hashim et al. [41] studied the nonlinear thermal radiation impact on heat and mass transfer in Carreau nanofluid flow past a wedge shape geometry.

The objective of the present investigation is to analyze Navier's slip and magnetic field effects on Carreau nanofluid flow past a convectively heated porous stretching wedge in the presence of nonlinear thermal radiation, viscous dissipation and Joule heating. Non-dimensional transformation is used to convert the governing model equations into non-dimensional form and then the nonlinear boundary value problem is solved by using Runge-Kutta Cash-Karp method based shooting technique. The influences of various physical parameters are analyzed and shown through figures and tables.