Numerical modelling of 3D magnetohydrodynamics chemically reacting permeable sheet influenced by fre

Abstract

Keywords: Chemical Reaction; Magnetohydrodynamics; Nanofluid; Non-Newtonian Casson Fluid; Permeable Sheet

Introduction

Researchers are paying closer attention to nanofluid because of its unique physical characteristics and wide range of uses, particularly in the manufacturing, industrial, and biomedical sectors. Nanofluids are the most appropriate coolants in the aforementioned specific uses due to their increased heat conductivity. Choi and Eastman [1] introduced nanofluids to address the aforementioned significant concern. Nanofluids are defined as a blend of conventional fluids with Nanoscale particles (metals, metal oxides, etc.) that have superior heat transmission properties. Because nanofluids have noticeably better thermal conductivities, they function well as coolants in different zones. Computers, nuclear reactors, surgery, treatment of chronic illnesses, various microelectronic devices, cars, and transformers are a few examples of these application sectors [2-7]. Nagaraja et al. [8] expresses impact of conveying ternary hybrid nanofluid radiative flow in presence of aiding flow conditions by applying shooting algorithm. Influence of heat radiation and thermophoretic deposition by following RKF (Runge Kutta Fehlberg) technique has been reported by Karthik et al. [9] across wedge and shows that mass transfer rate declines because of enhancement in solid volume fraction. Reddy et al. [10] uses FEM (Finite Element Method) technique to study slip effects in presence of suction utilizing Al2O3 & TiO2 hybrid nanoparticles and presented that with rise in volume fraction of Al2O3 & TiO2, temperature rises up. Asalekar et al. [11] discussed the influence of vegetable nanofluids induced by novel hybrid nanoparticles by utilizing ANN prediction. Mebarek-Oudina et al. [12] discussed the magnetite-water fluid flow. Sheikholeslami and Khalili [13] presented numerical simulation for spectral splitter induced by nanofluids. Jayadevamurthy et al. [14] studied the bioconvective flow induced by rotating disc utilizing unsteady fluid flow. Sepehrnia et al. [15] discussed dynamic viscosity & sensitivity analysis based on Sio2 / MWCNT. The extensive industrial and engineering uses of Casson fluids have drawn more attention to their study. All of these commonly used goods have one thing in common: they are indifferent to Newton’s law of viscosity. As a result, these liquids are well known for not being Newtonian. When β is the Casson parameter, the Casson fluid has limiting features (non-Newtonian fluidity) that reflect infinite viscosity at zero shear stress β → 0 and Newtonian viscosity at enormous stress β → ∞. This is the unique quality that Casson fluid has. When shear tension and small shear strain exceed a threshold, the behavior of Casson fluid is similar to that of a flexible solid. Alqahtani et al. [16] utilizing HAM (Homotopy Analysis Method) algorithm to study impact of Al2O3 & Cu hybrid nanoparticles in presence of Darcy effects induced via exponential sheet. Puneeth et al. [17] studied influence of non-Newtonian nanofluid flow along with gyrotactic microorganisms flowing between diverging channels utilizing FEM & by applying similarity transformation. Akaje et al. [18] utilizing Spectral Collocation Method to analyze

analytically influence of chemical reaction & MHD Casson nano liquids and shows that with increase in Soret effects, concentration distribution rises up. Seethamahalakshmi et al. [19] reported impact of convection heating and slip flow induced by Maxwell nanofluid over porous surface. The look at of the drift of electrically carrying out fluids in a regular or fluctuating magnetic Field (MF) is referred to as magnetohydrodynamics (MHD). Many uses for MHD have been discovered, including the transportation of subterranean species, improved oil recovery, medicine, biology, geothermal reservoirs, refrigeration systems, and encompassing power plants, among many more. Lund et al. [20] utilizing the bvp4c approach to study thermal radiation impact induced by Al2O3 - Cu/SA hybrid Casson nanofluids with MHD flow over permeable surface. Ali et al. [21] expressed impact of hydromagnetic flow in addition to thermal radiation induced via stretching surface and calculated results by utilizing Tecplot-360 software. Influence of activation energy in presence of MHD Casson nanofluid flow over exponential surface under the impact of Hall current by using MATLAB bvp4c package has been presented by Kumar et al. [22]. In recent years, MHD nanofluid flow has been addressed in [23, 24]. Gaseous modelling combustion utilizing species transport approach has been presented by Rajak et al. [25]. Rajak et al. [26] discussed diesel engine characteristics by following 3D CFD numerical approach. This model garnered significant attention from scientists because of distinctive characteristics and considerable potential applications. It has diverse applications, encompassing power plants, refrigeration systems, medical science, and other fields. The goal of the current work is to investigate how chemical reactions affect the flow of free stream MHD nanofluids when heat convection occurs across a permeable surface. The study extensively analyzes and discusses the distinct behaviors of flow velocity, fluid temperature, and nanoparticle concentration in response to various critical physical parameters makes this research novel one. The dimensionless basic differential equation systems were computationally solved using the MATLAB software and the computational RKF technique by using the ODE45 solver. The current study’s distinctive findings are beneficial and relevant for academic research as well as other fields.

Mathematical Formulation

The 3D model under consideration includes nanofluid MHD flow, heat production, and chemical reaction embedded in a permeable sheet with concentration and thermal Biot numbers (Fig. 1). Where a and b are constants, flow is constrained at z ≥ 0 and positioned between stretching velocities Uw=ax & Vw=by at z = 0 at z = 0. Considered physical problem is analyzed with the below adopted assumptions: v Free stream velocity flow v Boundary layer Problem v Reynolds number v Heat radiation v Thermal Diffusion

ρ denotes density of fluid, U∞ denotes stream velocity, k1 denotes thermal conductivity, T denotes temperature, (ρCp) p denotes heat capacity at constant pressure, C denotes concentration, C∞ denotes ambient concentration, DT denotes thermophoretic diffusion coefficient, Kr denotes dimensionalized chemical reaction, (ρCp)f denotes heat capacity of the fluid, T∞ denotes ambient temperature, DB denotes Brownian diffusion coefficient.. Using similarity variables [29]:

Where, ζ denotes similarity variables, a & b denotes constants, Θ denotes temperature distribution, Φ denotes concentration distribution, Tw & Cw are the temperature and concentration at wall serially and c denotes stretching ratio parameter. Associated B.C’s are:

(7) (5) Where u, v and w denotes the velocity components along x, y and z direction respectively. Also, σ denotes electrical conductivity, β denotes Casson fluid parameter, ν denotes kinematic viscosity, B0 denotes magnetic field intensity,

(10) (11) Where, M denotes magnetic parameter, PR denotes permeability parameter, λ denotes free stream velocity, Le denotes Lewis number, Pr denotes Prandtl number, Nb denotes Brownian motion, CR denotes chemical reaction, Q denotes heat source parameter and Nt denotes thermophoresis parameter. Reduced B.C’s are:

Where, TB & CB denotes thermal and concentration Biot numbers respectively. Also non-dimensionalized fluid parameters are symbolically represented as follows:

(21) Non-dimensionalized physical quantities are as follows: (22) (14) (23) (24)

Here Rex is local Reynolds number. Numerical Scheme The MATLAB R2014a ODE45 solver and the RKF method are used to mathematically handle the arrangement of D.E.’s (8)–(11) with B.C.’s (12) (Fig. 2). Next, ten first-order D.E.s are created from these circumstances as (16)

Results And Discussion

In current analysis, results are compared along with residual error for different Pr as shown via Table 1. Table 2-4 presents values for f ’’(0) & g’’(0), -θ’(0) and Φ’(0) for fixed entries of controlling fluid parameters as λ = 0.1 (Free stream velocity), CB = 0.3 (Concentration Biot number), Q = 0.5 (Heat generation parameter), Cr = 0.6 (Chemical reaction parameter), c = 2.0 (Stretching ratio parameter), PR = 0.1 (Permeability parameter), β = 0.1 (Casson fluid parameter), M = 2.5 (Magnetic parameter), Pr = 0.73 (Prandtl number), Nb = 0.2 (Brownian motion parameter), Nt = 0.5 (Thermophoresis parameter), Le = 2.0 (Lewis number), TB = 0.1 (Thermal Biot number). The velocity distribution variation g’(ζ) against the parameter PR in the range of 0.05-0.10 is shown via Figure 3(a). This graph demonstrates how the velocity augmented somewhat when PR increases because there is more nanofluid present farther from the boundary, indicating a

decrease in the thickness of the boundary layer and consequently, g’’(0) drops down from -0.819938652677649 to -0.849485482650904. The velocity distribution under the influence of λ (0.10 ≤ λ ≤ 0.15) is visualized in Figure 3(b). It was demonstrated that the velocity distribution and the free stream velocity parameter were closely proportional. As a result, when the free stream’s velocity increases, the velocity distribution also increases and eventually decreases as it gets closer to the sheet’s surface. The velocity distribution variation g’(ζ) against parameter M in the range of 0.10.5 as visualized in Figure 3(c). This graph demonstrates how the generated Lorentz drag force causes the velocity profile to fall gradually with little variation as the values of M increase. Furthermore, as Figure 3(c) illustrates, it has been observed that the width and magnitude of the boundary layer of g’(ζ)) decrease with an increase in M and simultaneously g’’(0) declines from -0.574546600126926 to -0.628849145452216.

Table 1. A comparison for local Nusselt number with Hussain et al. [30], Wang [31] and Mabood et al. [32] along with residual error for different Pr Pr

Table 2. Values of f ’’(0) & g’’(0) for PR (0.05 ≤ Pr ≤ 0.10), λ(0.10 ≤ λ ≤ 0.15) and M (0.1 ≤ M ≤ 0.5) PR

Figure 3. Velocity profile g’(ξ) in direction of y-axis against (a) PR (0.05 ≤ PR ≤ 0.10), (b) λ (0.10 ≤ λ ≤ 0.15) and (c) M (0.1 ≤ M ≤ 0.5).

The temperature distribution’s influence on the fluid parameters TB, c, Nb, Nt, Pr and Q are shown serially in Figures 4(a)-4(f). The temperature profile fluctuation against TB (0.1 ≤ TB ≤ 0.9) is explained in Figure 4(a), where it is observed that temperature and convection heating rising for higher TB. Additionally, graphs diverge for

Conclusion

The transformation of partial differential equations into ordinary differential equations is accomplished by the use of similarity conversion. Final results include the temperature distribution, volume fraction of nanoparticles, Magnetohydrodynamics Casson nanofluid flow and the importance of Biot numbers. Final outcomes are as follows:

1. Intensification in concentration has been noticed out

for higher concentration Biot number while concentration falls down for greater chemical reaction, Lewis number as well as Brownian motion.

2. Augmentation in the values of thermal Biot number,

Brownian motion, thermophoresis and heat generation helps to increases temperature distribution. On

the other hand, temperature declines with the increase in values of stretching ratio parameter and Prandtl number. 3. Higher stretching ratio parameter ranging from 2.010.0 leads to falls down heat transfer rate by 7.24%.

5. Velocity declines against higher magnetic parameter

due to Lorentz drag force. 6. Inclination in velocity is observed for greater permeability parameter and free stream velocity parameter. 7. Skin friction coefficient rises up with simultaneous augmentation in thermal Biot number and Prandtl number. The results of this study are useful for controlling the heat transfer rate and fluid velocity in various industrial

applications, biochemical systems, and manufacturing processes so that the final product satisfies quality requirements. Current study can be extended for various non-Newtonian fluids by utilizing entropy generation, porous medium, joule heating, etc.

Nomenclature

x, y, z Cartesian Coordinates u, v, w Velocity Components ν Kinematic Viscosity Free Stream U∞ σ Electrical Conductivity Magnetic Field Intensity B0 ρ Density Thermal Conductivity k1 β Casson Fluid T Temperature (ρCp)p Specific Heat at Constant Pressure Thermophoretic Diffusion Coefficient DT Ambient Temperature T∞ Brownian Diffusion Coefficient DB C Concentration Chemical Reaction Kr Ambient Concentration C∞ ζ Similarity Variable a,b Constants Θ Dimensionless Temperature Wall Temperature Tw Φ Dimensionless Concentration Wall Concentration Cw f Dimensionless Velocity in x-direction g Dimensionless Velocity in y-direction M Magnetic Parameter λ Free Stream Velocity PR Permeability Parameter Pr Prandtl Number Nb Brownian Motion Nt Thermophoresis Parameter Q Heat Generation Le Lewis Number CR Dimensionless Chemical Reaction TB Thermal Biot Number CB Concentration Biot Number Sherwood Number Shx Nusselt Number Nux Skin Friction in y-direction Cfy Skin Friction in x-direction Cfx Reynolds Number Rex c Stretching Ratio Parameter (ρCp)f Heat capacity of the fluid

Abbreviations

O.D.E’s Ordinary Differential Equations P.D.E’s Partial Differential Equations

Mwcnt

CFD Computational Fluid Dynamics MHD Magnetohydrodynamics RKF45 Runge Kutta Fehlberg forth-fifth

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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