Significance of Thermal Slip and Convective Boundary Conditions in Three Dimensional Rotating Darcy-Forchheimer Nanofluid Flow

This article is concerned with the nanofluid flow in a rotating frame under the simultaneous effects of thermal slip and convective boundary conditions. Arrhenius activation energy is another important aspect of the present study. Flow phenomena solely rely on the Darcy–Forchheimer-type porous medium in three-dimensional space to tackle the symmetric behavior of viscous terms. The stretching sheet is assumed to drive the fluid. Buongiorno’s model is adopted to see the features of Brownian diffusion and thermophoresis on the basis of symmetry fundamentals. Governing equations are modeled and transformed into ordinary differential equations by suitable transformations. Solutions are obtained through the numerical RK45-scheme, reporting the important findings graphically. The outputs indicate that larger values of stretching reduce the fluid velocity. Both the axial and transverse velocity fields undergo much decline due to strong retardation produced by the Forchheimer number. The thermal radiation parameter greatly raises the thermal state of the field. The temperature field rises for a stronger reaction within the fluid flow, however reducing for an intensive quantity of activation energy. A declination in the concentration profile is noticed for stronger thermophoresis. The Forchheimer number and porosity factors result in the enhancement of the skin friction, while both slip parameters result in a decline of skin friction. The thermal slip factor results in decreasing both the heat and mass flux rates. The study is important in various industrial applications of nanofluids including the electro-chemical industry, the polymer industry, geophysical setups, geothermal setups, catalytic reactors, and many others.


Introduction
Flow analysis comprised of the slip boundary is of utmost interest; especially in the recent past, it has gained importance as compared to no-slip boundary conditions because no-slip is no more beneficial for the procedures that involve suspensions, polymer solutions, foams in which fluid behaves as a particulate, emulsions, etc. Therefore, the importance of partial slip and full slip boundary flow using mixed convective conditions. Rasool et al. [36][37][38][39] reported recent findings on Darcy-type nanofluid flow bounded by various surfaces including MHD and the Riga pattern. The results have been obtained by using HAM and numerical RK45 methods.
Models based on oil storage and geothermal procedures involve many typical and extended chemical reaction systems. The creation and consumption of chemical reactants is classified under various situations both within the liquid and mass transport mechanism. The pioneer study was reported by Bestman [40] considering the effect of activation energy on the Darcy-type model in fluid flow analysis. Later on, Makinde et al. [41] reported some good results considering time-dependent heat and mass convection together with activation energy and resistance force. Mustafa et al. [42] reported some good results on MHD nanofluid flow subject to activation energy and chemical reaction. Recently, Rasool et al. [43,44] reported some good findings on the nanofluids' flow subject to chemical reaction and convective boundary conditions using the Riga pattern. The solutions were obtained using HAM [45][46][47][48][49][50][51][52][53][54].
The thermal radiation effect is naturally accounted for in fluid flow analysis due to the temperature difference between the ambient fluid and continuum because it varies the structure of the boundary layer associated with the temperature distribution attribute. Thus, a decent number of applications are found in many engineering and industrial setups that involve the radiative heat transfer mechanism. Procedures such as gas turbines, nuclear power plants, nuclear reactors, and many satellites. The pioneer study was given by Nayak [55], showing that as long as the radiative effect was minimum, the more the cooling achieved due to viscoelastic MHD-type fluid flow. Furthermore, Nayak et al. [56] regarded the influence of thermal radiation as an important factor while considering a flow over a plate surface. Bhatti et al. [57] gave the thermal radiation impact on MHD fluid flow analysis in a metachronal wavy channel.
In the above literature, one can see that most emphasis was given to studies based on nanofluids' flow through different channels. However, the literature lacks articles based on the formulations that involve thermal slip, convective conditions, and the Darcy channel all together. In this study, for the first time, the model involves the Darcy-Forchheimer relation, thermal and velocity slip, together with the convective boundary. The impact of Arrhenius activation energy is yet another novel aspect of this study. Overall, the article is organized as follows: Firstly, we assumed a non-Newtonian, incompressible Darcy-Forchheimer MHD nanofluid flow bounded in a rotating frame via a stretching surface. Secondly, the modeled problems were converted into ordinary problems using transformations. Thirdly, the numerical RK45 scheme was applied to solve the problems, and finally, the solutions were plotted graphically for better understanding of the audience.

Formulation
Here, we investigated a Darcy-Forchheimer nanofluid flow in a three-dimensional rotating frame. Thermal radiation, thermal slip, velocity slip, and chemical reaction with Arrhenius activation energy were the important factors involved in this study. The concentration of nanoparticles was associated with the Brownian motion parameter and thermophoresis. The stretching surface was considered adjacent to the plane z ≥ 0. Ω = Ωk is the rotation velocity where k is the unit vector in the direction parallel to the z-axis. Assuming a small Reynolds number and a transverse magnetic field having strength B 0 applied alongside/parallel to the z-axis with partial slip, thermal radiation, and activation energy, the governing equations (see for example [4,42,53]) take the following form: ∂v ∂y with: Define, Using (7) in (1)-(6), we get, along with: where Physical quantities are given below: (1) Skin friction: (2) Local Nusselt and Sherwood numbers: where,

Methodology
The numerical RK45 scheme using the shooting technique was applied for final solutions of the nonlinear problems. An initial guess was made carefully to solve the problem numerically. The iterations was performed repeatedly unless a difference up to or less than 10 −5 was achieved for the most suitable convergence. These results were quite efficient and accurate compared to the analytic solutions reported in previous literature.

Analysis of the Solutions
Here, we interpret the results and findings plotted graphically in Figures 1-28. In particular, Figure 1 gives the impact of the rotational parameter on the axial velocity field. Physically, the stretching rate is directly related to the rotational parameter. For larger values, stretching reduces, which effectively results in a declining change in the fluid velocity. Figure 2 presents the impact of Ω 1 on transverse velocity field. A similar, but more prominent impact is noticed in this case. The higher the value of Ω 1 , the lesser is the stretching rate, which certainly affects the fluid motion. Figures 3 and 4 present the influence received by the axial and transverse velocity field due to the magnetic parameter. Physically, a strong and intensive Lorentz force generated by the MHD results in sudden bumps and retardation in the fluid flow directions, which leads to a reducing trend in both the axial and transverse velocity fields. However, the impact is quite prominent in the case of transverse velocity. Figures 5 and 6 represent the influence of the Forchheimer parameter on the velocity field. Both the axial and transverse velocity fields decline greatly due to the strong retardation produced by the Forchheimer number. Physically, the Forchheimer number is directly related to the resistance offered to the fluid motion due to the porous media. A similar declination is noted in fluid motion due to the velocity slip parameters γ 1 and γ 2 , respectively, given in Figures 7 and 8. The impact of non-dimensional parameter λ 1 on the thermal distribution is given in Figure 9. An enhancement in the field is noticed for elevated values of λ 1 . Physically, the stretching rate is inversely related to the chemical reaction. For larger λ 1 , the stretching rate is less, which reduces the fluid flow, creating more relaxation to the fluid packets, leading to a rise in the temperature field. Figure 10 presents the impact of the thermal radiation parameter on the temperature field. It greatly raises the thermal state of the field due to the natural heat source attributes. The resistive force is enhanced due to the porous medium, and consequently, the temperature rises. Figures 11 and 12 are related to the chemical reaction part involved in the governing equations. An opposite trend in the temperature profile is found for both the chemical reaction parameter and the Arrhenius activation energy parameter, respectively. The temperature field rises for a stronger reaction within the fluid flow, however reducing for an intensive quantity of activation energy. Both the Brownian diffusion and thermophoresis increase for the thermal distribution. The unpredictable motion of particles due to Brownian diffusion increases strongly for the given strong thermophoretic force resulting in a clear rise in the temperature field, as displayed in Figure 13 and 14. Curiously, the rising values of the Biot number produce more and stronger convective heating at the surface given the greater gradient temperature at the wall. Thus, a rise in the thermal field is noticed as shown in Figure 15. A similar rising trend in the thermal field is noticed due to the stronger thermal slip parameter shown in Figure 16. Figures 17-20 are the influences offered by thermophoresis, Brownian diffusion, Arrhenius activation energy, and the Schmidt factor to the concentration of nanoparticles in the base fluid. Stronger Brownian diffusion results in a higher concentration of the nanoparticles due to enhanced motion as shown in Figure 17; however, a declination is noticed for stronger thermophoresis due to the unpredictable motion of the fluid particles, leaving more gaps within the fluid shown in Figure 18. The Arrhenius activation energy gives rise to the concentration field shown in Figure 19. The inverse relation between kinematic viscosity and Brownian diffusion gives rise to the higher concentration field given in Figure 20.       Table 2. Numerical data of local Nusselt and Sherwood numbers n 1 = 0.5.

Concluding Remarks
In this article, we chose a nanofluid flow in a rotating channel considering slip and convective boundary conditions simultaneously. The solutions were found using a numerical scheme. The salient findings are listed below:

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Lorentz force generated by the MHD resulted in reducing trend in both the axial and transverse velocity fields.

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Both the axial and transverse velocity fields greatly declined for larger values of the Forchheimer number.

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The thermal radiation parameter greatly raised the thermal state of the field.

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The chemical reaction part involved in the governing equations showed the opposite trend in the temperature profile for both the chemical reaction parameter and the Arrhenius activation energy parameter, respectively. • Both the Brownian diffusion and thermophoresis were rising factors for the thermal distribution.

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The augmented Biot number resulted in a rise in the thermal field.

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The augmented thermal slip parameter enhanced the temperature field. • Stronger Brownian diffusion resulted in a higher concentration of the nanoparticles.

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A declination was noticed for stronger thermophoresis.

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The Arrhenius activation energy gave rise to the concentration field.