Transportation of heat and mass transport in hydromagnetic stagnation point flow of Carreau nanomaterial: Dual simulations through Runge-Kutta Fehlberg technique

Abstract

In this research communication, MHD time dependent stagnation point flow of non-Newtonian fluid (Carreau fluid) is addressed subject to stretchable and shrinking surface of sheet. Stagnation point flow is considered. The novel features of slip mechanisms like Brownian and thermophoresis diffusions are accounted for the modeling of energy expression. The nonlinear partial differential expressions are converted to ordinary ones in the presence of similarity transformations and then dual solutions are obtained through implementation of Runge-Kutta Fehlberg Technique (RKFT) along with Shooting method (SM). The dual solutions are also computed which elaborate the Skin fraction, Nusselt number and Sherwood number and stress model number and its they're first and second solutions. A numerical compression is also prepared to justify our solutions with the available results as a limiting case.

Introduction

The analysis of convective heat transfer is of considerable theoretical and practical interest according to its huge number of operations in industry and technology. Since the last century researcher developed lot of ways to improve the heat transfer efficiency but not much succeed. In 1995, Choi [1] has given an idea named as nanofluid rascally increase the thermal conductivity of fluids. After his [1] idea many developments have been proposed to further improve the efficiency of nanofluids. A detailed study on the analysis of convective heat transfer increment with nanofluid was published by (pramuanjroenkij and kakac, [2]). Pop and Khan [3], introduced the boundary layer flow of a nanofluid through a stretching sheet. In the absence of surface heat flux and chemical reaction for MHD flow of nanofluid has been introduced by Zhang et al. [4]. In another studies [[5], [6], [7]] researchers addressed the motion of nanofluid in the absence of Lorentz forces along various flow geometries. The collection of articles [[8], [9], [10], [11], [12]] present various trials related to the ongoing effort involving porosity, MHD, nanofluid and slip.

Two dimensional flows along to stretching and shrinking surfaces have several employments in engineering sciences such as it is used in paper manufacturing, metal spinning extrusion of plastic sheets, drawing plastic films, glass blowing and many other [[13], [14], [15]]. Gupta and Mahapatra [16,17] have initially supposed the stagnation point flow through a stretching sheet. Fang et al. [18] observed slip MHD viscous flow against a stretching surface. Their result presented that the slip velocity enhances when the force of the wall friction reduces as the slip parameter raises. The study of permeable shrinking surface of MHD viscous liquid was studied by Fang et al. [19]. Recently, researchers have been discussed fluid flow over a stretching surface which see Refs. [[20], [21], [22]].

The given impartial is to intend a theoretical model of mass and heat transfer in Carreau fluid flow across a permeable stretching surface with convective slip effects in the absence of the induced Hartman field, nonlinear thermal radiation and annoyed absorption. The experiment is important for many metallurgical techniques, namely micro-circular blood flow, food processing and polymer, and magma flows etc. Similarity parameters are used to convert nonlinear partial differential equation into ODEs. The modified ordinary nonlinear differential equations are numerically decoded by using Newtons and Runge-Kutta techniques. We described dual solutions in case of suction and injection. Graphs for multiple important variables on the f(η), θ(η) and φ(η) are described and evaluated in depth. Also, a relation of the present works with the past ones is presented to justify our ideas.