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Research Article a Combined Convection Carreau–Yasuda Nanofluid Model Over a Convective Heated Surface Near a Stagnation Point: a Numerical Study

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Research Article a Combined Convection Carreau–Yasuda Nanofluid Model Over a Convective Heated Surface Near a Stagnation Point: a Numerical Study Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 6665743, 14 pages https://doi.org/10.1155/2021/6665743 Research Article A Combined Convection Carreau–Yasuda Nanofluid Model over a Convective Heated Surface near a Stagnation Point: A Numerical Study Azad Hussain,1 Aysha Rehman ,1 Sohail Nadeem,2 M. Y. Malik ,3 Alibek Issakhov,4,5 Lubna Sarwar,1 and Shafiq Hussain6 1Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan 2Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan 3Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia 4Department of Mathematical and Computer Modeling, Al-Farabi Kazakh National University, Almaty, Kazakhstan 5Department of Mathematical and Computer Modeling, Kazakh British-Technical University, Almaty, Kazakhstan 6Department of Computer Science, University of Sahiwal, Sahiwal, Pakistan Correspondence should be addressed to Aysha Rehman; [email protected] Received 18 November 2020; Revised 1 March 2021; Accepted 22 March 2021; Published 5 April 2021 Academic Editor: Adrian Neagu Copyright © 2021 Azad Hussain et al. 'is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 'e focus of this manuscript is on two-dimensional mixed convection non-Newtonian nanofluid flow near stagnation point over a stretched surface with convectively heated boundary conditions. 'e modeled equation representing nonlinear flow is trans- formed into a system of ordinary differential equations by implementing appropriate similarity transformations. 'e generated structure is numerically solved by applying the bvp4c method. Consequences of various involved parameters, e.g., stretching parameter, mixed convection parameter, thermophoresis parameter, Brownian movement parameter, Lewis number, Weis- senberg number, Prandtl number, Biot number, buoyancy ratio parameter, mass and heat transport rates on temperature and velocity, the stretched surface, and nanoparticle concentration patterns are analyzed. Outcomes are shown graphically and displayed in tables. Velocity fluctuations are responded to by growing parameters of mixed convection and Weissenberg number. Concentration and thermal fields are also discovered for the Prandtl number. 'ere are also flow line diagrams to analyze the behavior. 1. Introduction properties of cooling fluids, and the manufacture of a steady solid-liquid suspension called nanofluid has op- Today, enhancement in the heat transfer mechanism using timized the thermal performance of industrial instru- cooling fluids has become very significant in industrial ments and heat exchangers [1]. 'erefore, several applications such as machining and electronics, energy researchers have reported enhancement in heat transfer production industries, petrochemical, aeronautics, and through nanofluids [2–21]. transportation industries. 'e use of irreversible strength 'e combination of forced and free convection is known as resources and the creation of environmental provisions mixed convection. Extensive cases of mixed-convection are have supported this improvement. 'e objective of characterized as internal mass forces, where the flow is de- manufacturing heat transfer devices requires reducing termined collectively by a few external forcing systems (i.e., cost and increasing heat transfer in each surface area unit external power supplied to the fluid-streamlined body system). to achieve high efficiency. In recent years, technical ad- 'is is a distribution of fluid in a gravity domain with differing vances have made improvements in the rheological densities. 'e acceleration of the Earth’s temperature stratified 2 Mathematical Problems in Engineering mass of air and water region is the realistic presentation of discussed pulsatile blood flow through the human carotid mixed convection that is conventionally studied in geophysics. artery by applying computational fluid dynamics. However, in many engineering gadgets, mixed convection is Inspired by the above research work, the ambition of the found within the system of much smaller scales. 'is will be current investigation is to study the impact of mixed con- encapsulated on account of certain cases alluded to in channel vection on Carreau–Yasuda stagnation point nanofluid flow flows. Mixed convection is often perceived in normal and over a vertical elastic surface by applying a mathematical traditional situations, i.e., heating or cooling of channel walls, nanofluid model introduced by Buongiorno [38]. In our limited velocities of a fluid flow that are characteristics of a point of view, the problem is new and original. 'e outcomes laminar flow. Turbulent channel flows studies with consider- are discussed through various parameters such as the able gravity area impacts have evolved since the 1960s after its Brownian motion parameter, mixed convection parameter, essential use in industrial applications in nature, e.g., electronic thermophoresis parameter, buoyancy ratio parameter, appliances, cooling through electric fans, solar panels exposed stretching parameter, Lewis, Weissenberg, Prandtl, and Biot to wind currents, atomic reactor cooled during emergency numbers on the mass and heat transfer. shutoff, flow in the ocean and the atmosphere, heat trans- mission placed in a low-velocity environment, and so on. 2. Modeling Cesarano [22] discussed several fractional generalizations of different types, starting from the heat equation, and suggested a We inspect the incompressible, 2D, steady flow on a stag- method useful for analytical/numerical solutions. Cesarano nation point attaining the stretched surface in a Carreau- [23] introduced a nonconventional approach of multi-di- –Yasuda nanofluid as shown in Figure 1. Sheet is stretched mensional Chebyshev polynomials. Mixed convection flow was with velocity Uw(x) � ax and ambient fluid velocity is also reported by many researchers [24–30]. UE(x) � bx while the origin is fixed at S. It is assumed that Commonly, fluids are categorized into two main classifi- the surface is warmed up by convection from a hot fluid at cations: non-Newtonian and Newtonian fluids; the main dif- the temperature Tf and the sheet is warmed up by con- T ference between Newtonian and non-Newtonian fluid is the vection fluid at the temperature ∞ which is by heat connection between the deformation rate tensor and the extra transport coefficient hf. 'e surface motion will source the stress tensor [31]. In a 3D free-form extrusion printing process, evolution of the boundary layer. Consider the Cartesian the rheological property plays an important role. But no coordinates system; fluid flow velocity will change through x rheological model was available in the open literature that and y axis in a manner that the x-axis is taken vertically and could accurately consider the impacts of both the concentration the y-axis is taken horizontally. of nano/microparticles and the non-Newtonian viscosity in the 'e elemental mathematical form of Carreau–Yasuda paste [32]. Zuo and Liu [32] introduced a fractal rheological fluid is [39] model for 3D print pastes by applying a fractional derivative d (n− 1)/d τ ��μ +μ − μ ��(1 + Γc_) � �A ; (1) and it can be also used for other non-Newtonian fluids. 'e ∞ 0 ∞ 1 experimental findings are in strong agreement with the the- oretical forecast, suggesting that the model is accurate and where Γ and d are Carreau–Yasuda fluid parameters, A1 is realistic. Defining the behaviors of melts and many polymer first Rivlin Ericksen tensor, μ∞ is infinite shear rate viscosity, _ solutions, Carreau–Yasuda model was established successfully. μ0 is zero shear rateq���������� viscosity, τ is extra stress tensor, andc is It was accurately applicable to many experimental viscosity expressed as c_ � tr(A2)(1/2) , in which A � [(gradv)t+ patterns with a wide range of shear rates [33]. Khan et al. [34] 1 1 ] investigated the dissimilar solutions of MHD Carreau–Yasuda gradv . Take into account that infinite shear rate viscosity � fluid with slip conditions flow past a rotating disk. Andrade μ∞ 0 and then equation (1) reduces to the form et al. [35] presented the turbulent flow of the non-Newtonian d (n− 1)/d τ ��μ �(1 + Γc_) � �A : (2) Carreau–Yasuda fluid. Salahuddin et al. [36] examined the 0 1 influence of transverse magnetic field on the squeezed two- dimensional flow of a free stream past a sensor surface of an Under these considerations, governing equations can be electrically conducting Carreau–Yasuda fluid. Kumar et al. [37] written as follows: zu zv + � 0; (3) zx zy d zu zu dU z2u (n − 1) zu z2u u + v � U E + ] + ]24 :(d + 1)Γd ! 35 zx zy E dx zy2 d zy zy2 (4) − (1 − C)βρfm ρp ρfm + g" T − T∞ � − !C − C∞ �#; ρf ρp Mathematical Problems in Engineering 3 x C∞ Vw Assisting flow T∞ UE = bx uw S g Tf y uw V w Opposing flow u v Tw x Figure 1: Geometry of the physical problem. 2 c 2 zT zT z T �ρ p �p zC zT D zT u + v � α + 24D + T ! ! 35; (5) zx zy zy2 c B zy zy T zy �ρ p �f ∞ zC zC z2C D z2T T ( ) u + v � DB 2 + ! 2 ; 6 zx zy zy T∞ zy p�� ψ � a]xf(η); Here, ] is the viscosity characteristics and u and v are the r�� velocity components in the x and y directions, respectively, a η � y; where (ρcp)p/(ρcp)f is the proportion of nanoparticles ] capacity of heat to that of the base fluid capacity of heat, ρf is the density of the base fluid, α � K/(ρcp)f is the thermal zψ u � � axf′(η); diffusibility of the base fluid, cp is the constant pressure zy specific heat, DB is the Brownian diffusion, DT is the (8) thermophoresis diffusion, and the subscripts np; ∞, and f zψ p�� v � − � a]f(η); denote nanoparticles, a value very far away from the solid zx surface, and the base fluid, respectively. 'e subjected boundary conditions are defined as T − T θ(η) � ∞ ; Tf − T∞ u � Uw(x) � ax; C − C v � 0; φ(η) � ∞: C∞ zT − k � h �T − T �; Applying the above relationship, the continuity equation zy f f (3) is satisfied.
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