Assisting and Opposing Stagnation Point Pseudoplastic Nano Liquid Flow towards a Flexible Riga Sheet: A Computational Approach

Nanofluids are used as coolants in heat transport devices like heat exchangers, radiators, and electronic cooling systems (like a flat plate) because of their improved thermal properties. *e preeminent perspective of this study is to highlight the influence of combined convection on heat transfer and pseudoplastic non-Newtonian nanofluid flow towards an extendable Riga surface. Buongiorno model is incorporated in the present study to tackle a diverse range of Reynolds numbers and to analyze the behavior of the pseudoplastic nanofluid flow. Nanofluid features are scrutinized through Brownian motion and thermophoresis diffusion. By the use of the boundary layer principle, the compact form of flow equations is transformed into component forms. *e modeled system is numerically simulated. *e effects of various physical parameters on skin friction, mass transfer, and thermal energy are numerically computed. Fluctuations of velocity increased when modified Hartmann number and mixed convection parameter are boosted, where it collapses for Weissenberg number and width parameter. It can be revealed that the temperature curve gets down if modified Hartmann number, mixed convection, and buoyancy ratio parameters upgrade. Concentration patterns diminish when there is an incline in width parameter and Lewis number; on the other hand, it went upward for Brownian motion parameter, modified Hartmann, and Prandtl number.


Introduction
Mixed convection flow or a combination of free and forced convections exists in numerous electrify operations; both certainly occur in many engineering applications. Such applications mostly appear in heat exchangers, nuclear reactors, nanotechnology, atmospheric boundary layer flow, electronic equipment, and so on. ese operations arise at some stage in the outcomes of buoyancy forces in combined convections or the effects of forced flow in free convection become significant [1]. e unsteady mixed convection flow closer to a stretching sheet throughout the years is an important kind of flow of functional substantial in engineering and industries [2]. Patel and Singh [3] investigated mixed convection micropolar fluid flow in a permeable medium with a magnetic field towards a nonlinearly extendable surface. Arifin et al. [4] discussed the dusty Williamson mixed convection fluid flow towards a stretchable surface with the impact of an aligned magnetic field. Ishak et al. [5] investigated the stagnation-point mixed convection viscous fluid flow towards a stretchable surface in its plane. e non-Newtonian fluid form is essential to conceive the fluid flow in the latest industrial materials, so that the work productiveness could be improved. Shafiq et al. [6] studied the bioconvective second-grade nano liquid flow in the presence of chemical reaction and buoyancy effect. Naganthran et al. [7] demonstrated a computational approach that inspects the behavior of mixed convection steady stagnation-point Powell-Eyring fluid flow on a vertical stretching/shrinking surface. Azeman and Ishak [8] discussed numerically the problem of the combined convection stagnation-point flow on a vertical sheet. Many researchers had also studied the mixed convectional flow [9][10][11][12][13]. e research of heat transmission triggered employing boundary layer flow of an incompressible fluid towards a stretched surface had received extended interest from the scientists and researchers due to its advantages in industry and engineering. Presently, huge exertion has been made to concentrate on this subject to the frame of reference of its different industrial and engineering applications. e effects of combined convection MHD on the boundary layer flow in the aspect of heat transport of Hematite-water nanofluid on a stretchable surface are demonstrated in [14]. Ahmad et al. [15] demonstrated the boundary layer flow by a curved stretched surface enclosed in a porous material.
Nowadays, to improve the abilities of heat transport of habitual fluids like water, glycerin, and engine oil because of the decay of the thermal effects, nanofluid is the classic bestowal to enhance the thermal conductivity. Nano liquid is a liquid that was invented by nanometer-sized particles and microfibers having a diameter less than 100 nm. Nanofluids are used in many operations to fulfill industrial requirements, like propellant combustion, drug delivery, cooling of automotive engines, extraction of geothermal forces, and heat transfer. Al-Hossainy and Eid [16] demonstrated the heat transfer enhancement and DFT Structure and calculations in hybrid nanofluid flow as an application of potential solar cell coolant in a double tube. Zaib et al. [17] discussed the partial slip impacts on micropolar mixed convection fluid flow holding kerosene/water-based TiO 2 nanoparticle over a Riga sheet; results were obtained for aiding and opposing flows. Choi was the first who introduced the idea of nanofluid. He suggested that a new category of heat exchanger liquids could be engineered by suspending metallic nanoparticles in conventional heat exchanger liquids [18]. Ahmed et al. [19] discovered the impact of variable viscosity on the MHD unsteady flow of carbon nanotubes on a shrinking surface. Saleem et al. [20] explored the theoretical treatment for the unsteady Walter's B nanofluid flow over a rotating cone in the magnetic regime. Many researchers had discussed the flow of nanofluids [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].
Lorentz force theoretically [40] and experimentally [41] is an efficient agent to reduce skin friction. Nadeem et al. [42] demonstrated that flow is induced through exponentially stretching surface within the time-dependent thermal conductivity. Mahdy and Hoshoudy [43] explored the electromagnetohydrodynamic (EMHD) unsteady non-Newtonian nanotangent hyperbolic fluid that is electrically conducting towards a Riga surface in the presence of slip conditions and variable thickness. e current examination adds a novel era for scientists to find the characteristics of pseudoplastic nanofluid. Here, we inspect heat transfer analysis and mixed convection stagnation-point flow of a pseudoplastic non-Newtonian nanofluid over a convectively heated flexible Riga plate. Obtained equations are solved by bvp4c numerically. e consequences of the given problem are demonstrated by parameters such as modified Hartmann number, mixed convection parameter, Brownian motion parameter, buoyancy ratio parameter, thermophoresis parameter, width parameter, stretching parameter, Weissenberg, Biot, Prandtl, and Lewis numbers to the velocity, thermal energy, and mass transmission towards the vertical elastic Riga plate.

Mathematical Modeling
We inspected the incompressible, two-dimensional, steady pseudoplastic nanofluid flow and heat transfer improvement examination over the vertical extendable Riga surface as shown in Figure 1. u w (x) � ax is the sheet stretched velocity and u ∞ (x) � bx is the ambient fluid velocity, while the origin is fixed. It is assumed that the plate is heated by convection from a hot fluid at the temperature T f and the plate is heated by convection fluid at a temperature T ∞ which is by heat transport coefficient h f . Considering the Cartesian coordinates system, fluid flow velocity will change through x -axis and y -axis in a manner where the x − axis is taken vertically and the y − axis is taken horizontally as shown below. e elemental equation of pseudoplastic fluid is [44] τ where zero shear rate viscosity is μ 0 , infinite shear rate viscosity is μ ∞ , Γ and d are Carreau-Yasuda fluid parameters, A 1 is the Rivlin-Ericksen tensor, τ is the extra stress tensor, and c · is expressed as c · � ���������� tr(A 2 1 )(1/2); here, A 1 � [(gradv) t + gradv]. Take into account the fact that infinite shear rate viscosity μ ∞ � 0; then, equation (1) takes the form and, under these considerations, governing equations can be written as follows: In the above equations, u and v are the velocity components in the x -axis and y -axis, respectively, ] is the viscosity coefficient, ((ρc p ) p /(ρc p ) f ) is the proportion of nanoparticles heat capacity to that of the based fluid heat capacity, α is the thermal diffusivity of the based fluid, ρ f is the density of the base fluid, c p is the constant pressure specific heat, D T is thermophoresis diffusion coefficient, D B is Brownian diffusion coefficient, and the subscripts ∞, f, and np denote the values very far away from the solid surface, base fluid, and the nanoparticles, respectively. e subjected boundary conditions are Assisting flow Invoking similarity variables are defined as, By using the upper relationships, the continuity equation (3) is satisfied automatically. Equation (3) becomes the form of ordinary differential equations: with boundary conditions Here, prime denotes derivative for η, and other dimensionless characteristics are demonstrated as Here, Le, N r , λ, Bi, and W e denote Lewis number, buoyancy ratio parameter, mixed convection parameter, Biot number, and Weissenberg number, respectively, the symbol Z represents modified Hartmann number, c denotes width parameter, Pr denotes Prandtl number, r is the stretching parameter, Nb is the Brownian motion characteristic, and Nt denotes the thermophoresis characteristics. e skin friction coefficient C f and local Nusselt number where q w represents surface heat flux, τ w represents surface shear stress, and q m represents surface mass flux, where pseudoplastic fluid is and, by using appropriate similarity variable equation (5), the expressions for skin friction and local Nusselt number become,

Results and Discussion
e obtained flow of nonlinear differential equation (6) with boundary conditions (7) was numerically worked out by MATLAB program bvp4c method. In the present study, there is a strong focus on the importance of the behavior of physical characteristics associated with the fluid temperature, velocity, and concentration, and those are presented in the form of graphs and have been demonstrated splendidly. Figure 2 describes the behavior of the modified Hartmann number in the form of adding flow and opposing flow. Field of velocity inclined by increased values of Z and declined whenever the value of (Z < 0). Figure 3 demonstrates the performance of λ on f ′ (η). Here, λ behaves increments when the values of λ rise, and a contrary behavior was shown for reverse flow. In Figure 4, boundary layer thickness expands immediately, with the value of N r growing. It is also noticeable that the conflicting flow N r declined. Figure 5 defines the variations of Nb on the velocity field, and it is revealed that thickness goes down in assisting flow, while it moves upward in case of increased values of Nb. Figure 6 shows the flow of Bi on the velocity pattern. With the increment of Bi, distribution of f ′ (η) ascends. Figure 7 discusses the affection of c on velocity flow; in this case, the value of c grows and the boundary layer thickness diminishes. Figure 8 shows the structure of Nt velocity model. It can be seen that field of velocity gets larger with the values of Nt addition. Figures 9-11 show the exhibited impression of Pr and W e on f ′ (η); the profile of f ′ (η) gets larger with the increment in the values of Pr and W e . Figure 10 shows that while r is increased, boundary layer width increases. Variation of temperature flow is demonstrated in Figures 12-19. Figure 12 discusses the influence of Bi on temperature field, whether the value of Biot number grow temperature sketched expand. It is noted that when the value of Le becomes larger, the profile of θ(η) goes down, in Figure 13. It is revealed that impact of c on θ(η) is that when we increase the value of c temperature curve enhance for aiding flow and for opposing flow curve moves down in Figure 14. Figure 15 examines the performance of r on θ(η).

Mathematical Problems in Engineering
In opposing and assisting flow, r shrinks on θ(η). Figure 16 obtaines the implication of Nb. It shows that θ(η) becomes larger for increased values of Nb. Figures 17 and 18 show the behaviors of Nt and Pr, respectively; both are sketched towards down when the values of Nt and Pr grow. It can be easily observed that assisted flow falls and conflicting flow moves up for bigger values of Z on θ(η) in Figure 19. Deviation of concentration field can be revealed in Figures 20-27. Figure 20 describes the effect of Z on φ(η); the value of Z grows and then the boundary layer increases for aiding flow and decreases in opponent flow. Figure 21 shows that when we increase the values of c, the concentration curve falls in assisting flow and moves up in opposing flow. It is noticeable that Figures 22 and 23  e boundary layer width falls whenever the value of one and the other increases. Figures 26 and 27 depict the behavior of r and Pr one by one. e impact of these two parameters on φ(η) is that the growth of these parameters caused boundary layer increment. Figures 28-33 are created to examine the influence of λ and Z on streamlines. Table 1           Brownian motion parameter (− ) η: Similarity variable (− ) K: ermal conductivity u w : Stretching sheet velocity (ms − 1 ) T f : Hot fluid temperature (K)