Influence of buoyancy forces in MHD non-Newtonian convective nanofluid utilizing Buongiornos Model i

Abstract

Keywords: Buongiorno's Model; Buoyancy forces; Casson Fluid; Chemical Reaction; Nanofluid; Shooting Method

Introduction

Stream conduct over extending surfaces has drawn attention of many researchers because of its wide space of modern and fabricating applications like fake strands, petrol

ventures, metal turning, polymer handling and so forth. Crane [1] concentrated on the stream towards an extending sheet. Sarma and Rao [2] explored the stream conduct in viscoelastic liquid over an extending sheet. Additionally, Radiation impact with hotness source was considered by

*Corresponding author. *E-mail address: saloni.20.jindal@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Manjunatha et al. [3] over permeable medium. In ongoing many years, heat move angles are substantially more significant for issues identified with extending sheet because of its warming and cooling essential variables for making superior grade of eventual outcome. Fourier [4] proposed law of heat conduction very first and revealed that this law gives premise to discover the hotness move conduct under various conditions. Yet, this model has significant disadvantage that it offers numerical expression of energy in illustrative structure. From that point onward, Cattaneo [5] has acquired new statures of Fourier’s law with warm unwinding time because of the conditions that are in illustrative structure which moved into exaggerated structure. Very recently, heat dissipation impact under different physical conditions has been analyzed by researchers [6-9]. Researchers are paying more attention to nanofluid because of its physical properties and uses in many areas, mainly production, industry, and medicine. These tiny fluids have lots of important properties for research, like low resistivity and unique thermophysical aspects. In the last few decades, the word “nanofluid” has been a topic of interest in a number of scientific groups because it is able to transfer heat quickly. Choi and Eastman [10] identified a brand-new class of fluids called nanofluids. Because they have a high rate of heat transmission, nanofluids are of interest to many scientists and academics. Compared to solids, fluids are less effective in transferring heat. The creation of ultrafine particles enhances the thermal conductivity of nanofluids, according to Xuan and Li [11]. Through Brownian motion and thermophoresis diffusion, Buongiorno [12] created a methodical strategy that boosted the heat exchange phenomenon. Moreover, influence of activation energy over porous surface in presence of Carreau nanofluid has been presented by Shahid et al. [13]. Din et al. [14] studied the significance of bio-convection and slip conditions over wedge by utilizing tangential hyperbolic and Carreau nanofluids. Three-dimensional second-class nanofluid flow across a stretched surface has been studied by Ahmad et al. [15] using the Optimal Homotopy Analysis approach. The investigation of non-Newtonian fluids has numerous real-world uses in technology and other fields of study. Non-Newtonian fluid mechanics has a lot of interesting and important uses in the fields of engineering, healthcare, and the applied sciences. Since both the surrounding air and the ocean are fluids, the studies of fluids are essential to metrology, oceanography, and hydrology. Identifying the flow of various biological fluids is important for successful medical treatment. When hybrid nanoliquids were subjected to nonlinear solar radiation, Acharya et al. [16] investigated how various solar thermal devices affected the flow fluctuations and heat exchange properties of the materials. Very recently, Rana et al. [17] uses ANN to predict the stable solutions and critical points in viscoelastic fluids induced by horizontal sheet. In recent years, nanofluid flow in presence of Magnetohydrodynamics under different physical aspects has been explored by authors [18-21].

The field of MHD has important applications in the medical and industrial sciences. MHD has a wide range of potential uses, including endoscopy, cell separation, cancer tumor treatment, drug targeting and surgical blood flow control. MHD utilized in medication i.e., growth or disease treatment. A stagnation point MHD nanofluid stream was mathematically examined by Anwar et al. [22] induced by stretching surface. Then Shawky et al. [23] discussed Williamson nanofluid towards an extending plot. Vajravelu and Cannon [24] inspected the stream conduct of liquid on non-direct extending sheet. Additionally, a blended MHD nanofluid stream alongside entropy and convective investigation on a non-straight extending sheet was addressed by Matin et al. [25]. Jain and Choudhary [26] examined the Soret and Dufour impacts with MHD stream utilizing compound responses. Siddheshwar and Mahabaleshwar [27] concentrated on the issue of hotness and stream transportation on non-straight extending sheet with pull/infusion. By using hybrid nanofluids flow in presence of Stefan blowing bio-convection has been analyzed by Rana et al. [28]. The opinions of a few different specialists [29-36], considered a different aspect of those kinds of problems under diverse circumstances. Motivated by the already existing literature uncovers that no such study has been made to the till now. Current research focuses on the effect of transport phenomenon by creating outcome of MHD nanofluid flow and free stream in presence of compound response towards an extending surface. The philosophy embraced addresses whole framework through Runge Kutta Fehlberg strategy by following shooting procedure utilizing ODE45 solver. The numerical troubles showing up in the nanofluid conditions drove us to utilize the mathematical methodology. Utilizing the concept of buoyancy forces and nanoparticles, current research is relevant to novel microbial fuel cell technologies in exponential surface. The current research’s results assist the biological, technological, and manufacturing domains in producing desired products that make this study to novel one.

Materials And Methods

3D non-Newtonian nanofluid model with magnetohydrodynamic flow induced by stretching surface is considered in the current research (Figure 1). In this model, thermophoresis and Brownian motion impact of nanofluid has been incorporated among stretched velocities ,

direction serially. Buoyancy forces along with free stream velocity are incorporated in the current study. Modeled equations are shown as ( [14], [37-39]): (1)

(10) (11) Reduced B.C’s are: (12) Cfx, Cfy, Nux and Shx are defined as:

Now, a defined new set of variables is provided below to convert the equations previously defined into the first-order O.D.E’s:

Where Rex shows Reynolds’s number. Numerical Solution The system of D.E’s (8)-(11) and B.C’s (12) has been solved with Runge-Kutta-Fehlberg approach by following shooting procedure, as indicated in a flowchart in Figure

2. To get a numerical solution, the step size is set to 0.01

and the maximum value to 10. Main advantage of this technique is that it has 5th order truncation error over other numerical techniques. Additionally, as compared to other numerical techniques, the computation of the solution is simpler and easier. The well-known programme MATLAB is used in the current analysis to do computations utilizing the ODE45 solver. 1st order D.E’s have been created from the coupled D.E’s (8)-(11) in this case. The controlling nonlinear O.D.E’s are reconfigured as follows for this purpose: (15)

Equations (15)-(18) are changed into the following system of D.E’s by using Equation (19) to them: (20) (21)

parameter, L2 = 0.3; solutal buoyancy parameter, M = 0.1; magnetic parameter, c = 0.1; stretching ratio parameter, β = 0.1; Casson Fluid parameter, Pr = 0.72; Prandtl number, Nb = 2.5; Brownian motion, Nt = 0.5; thermophoresis, Le = 2; Lewis number, Cr = 0.5; chemical reaction parameter are shown in Table 2-4. Influence of free stream velocity λ over velocity profile f '(ξ) has been presented in Figure 3(a) and it shows that velocity profile enhances with enhancement in free stream velocity ranging from 0.00 ≤ λ ≤ 0.20. Figure 3(b) shows variety in liquid speed against attractive boundary M. Presence of attractive boundary M opposes the liquid molecule to move uninhibitedly and fundamental purpose for the obstruction is that attractive boundary M produces Lorentz power and this attraction conduct can be taken on for controlling the smooth motion. Accordingly, improvement in the worth of attractive boundary M causes the declination of speed dissemination. Figure 4(a) inspects temperature appropriation variety against the liquid boundary Brownian movement boundary Nb. Liquid temperature upgrades for higher Nb and in result neighborhood Nusselt number declines. Figure 4(b) ponders the effect of liquid temperature under the result of Nt. Temperature slope tumbles down for higher upsides of

Results And Discussion

For adjusting fluid parameter Pr in the absence of magnetohydrodynamic flow, the estimated findings for Nusselt number are compared with those from Abolbashari et al. [40] and Khan and Pop [41] in the current analysis (Table 1). When compared to the conclusions of the current literature, it is found that the results are very accurate. The results (f "(0), g"(0)), -θ’(0), -ϕ’(0), respectively, for fixed values of governing fluid parameters as e = 0.1; thermal Biot number, h = 0.1; concentration Biot number, λ = 0.1; free stream velocity parameter, L1 = 0.3; thermal buoyancy

Table 1. Comparison of Nusselt number along with residual error against Prandtl number Pr

Table 2. Values of f '' (0) and g'' (0) for variables β, λ and M β

thermophoresis boundary Nt that decrease of conduction of nanoparticles. Thus, temperature ascends for higher thermophoresis boundary Nt as illustrated in Figure 4(b). Figure 4(c) describes the variation in temperature profile

as a function of e; (0.15 ≤ e ≤ 0.75) and it can be seen that temperature and convective heating increase as e increases. Additionally, plots diverge for around 0.0 to 3.0 and convergent with ξ → ∞. Additionally, when e changes between

Conclusion

Current investigation deals with the study of 3D magnetohydrodynamic flow with buoyancy forces induced by stretching surface. Major outcomes of the study are: • Augmentation in concentration is for larger thermophoresis parameter Nt. • Velocity lessens down for larger β and magnetic parameter M due to produced Lorentz drag force that ultimately enhances the quality of final product. • Temperature and concentration rises with rise in thermal and concentration Biot numbers serially.

Larger Prandtl number Pr declines temperature and concentration distribution. • Skin friction coefficient declines with rise in values of Le and Pr and falls down with rise in both the values of nanofluid parameters Nb &Nt. • Comparative study has been performed under [40, 41] to get higher accuracy and validation of the current research. Results of the current survey have important implications for the regulation of transport phenomena and fluid velocity in a wide variety of formulation processes, which in turn benefits production, implements of technology, and •

Figure 7. Skin friction coefficient for (a) Lewis number Le & Chemical reaction Cr, (b) Prandtl number Pr & Lewis number Le, (c) Brownian motion Nb & Thermophoresis parameter Nt and (d) Prandtl number Pr & Thermophoresis parameter Nt.

biological networks in their pursuit of the ideal product standard.

Nomenclature

Cartesian coordinates Casson fluid parameter Concentration Biot number Free stream velocity parameter Thermophoresis parameter Chemical reaction parameter Magnetic parameter Thermal Biot number

Thermal buoyancy parameter Temperature profile Gravitational acceleration Skin friction coefficient along x-axis Solutal buoyancy parameter Prandtl number Horizontal velocity profile Concentration profile Skin friction coefficient along y-axis Sherwood number Wall shear along x- direction Nusselt number Convective mass transfer coefficient

T τyz C∞ Tw T∞ Cw ρ Uw, Vw u, v, w hf C a L U0, V0 ν ξ k DT (ρCp)nf Nb DB (ρCp)p σ B0 βC ' U∞ βT Le

Temperature Wall shear along y- direction Ambient concentration Wall temperature Ambient temperature Wall concentration density Stretching velocity in x- and y- direction Velocity components Convective heat transfer coefficient Concentration Constant Characteristic Length Constant Kinematic viscosity Similarity variable Thermal conductivity Thermophoretic diffusion coefficient Specific heat capacity of nanofluid Brownian motion parameter Brownian diffusion coefficient Specific heat capacity of particle Electrical conductivity Magnetic field intensity Volumetric coefficient of solutal expansion Prime denotes derivative w.r.t. ξ Stream velocity Volumetric coefficient of thermal expansion Lewis number

Abbreviations

B.C’s Boundary Conditions D.E’s Differential Equations O.D.E’s Ordinary Differential Equations P.D.E’s Partial Differential Equations MHD Magnetohydrodynamics ANN Artificial Neural Network 3D Three Dimensional

Ethics

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

References

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