MHD upper-convected maxwell hybrid nano fluid over a stretching sheet A numerical approach

Abstract

Keywords: Boundary Layer; Hybrid Nano Fluid; Magnetic Parameter; Maxwell Parameter; Prandtl Number; Stretching Sheet; UCM

Introduction

The last several years have seen a significant amount of coverage from scientists and researchers about the development of enhanced heat transfer fluids. In industrial and technical applications, regular fluids including water, oil, and ethylene glycol are frequently utilized. However, because

of their poor thermal conductivity, these fluids’ capacity to transmit heat is constrained. Consequently, to remedy this shortcoming, a certain type of nanoparticles-referred to as “nanofluid”is added to the fluids. Nanofluids incorporate a small number of nanoparticles to enhance the thermal capacities of conventional fluids. In practice, the idea put out by Choi and Eastman [1] to incorporate nanoparticles

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

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into a base fluid has been successful. Some of the benefits of using nanofluids through a stretchable surface were studied by Kalidas [2] and Khanafer et al. [3]. Besides, the additional references on these topics can be found in the research papers [4-8]. The goal of developing hybrid nanofluid was to improve the standard nanofluid’s thermal characteristics. Researchers typically choose hybrid nanofluids because they suspend two or more dissimilar nanoparticles in a typical heat transfer liquid, which are frequently utilized in numerous manufacturing, industrial, and biomedical engineering procedures. It seems that Suresh et al. [9] is the earliest researcher who considered the hybrid nano-composite particles in their experimental studies. Takabi and Shokouhmand [10] discovered that mixed Nanofluids not only improved the rate of thermal transfer but also reduced friction and pressure loss. Researchers Zainal et al. [11], Khashie et al. [12], Waini et al. [13], and Algehyne et al. [14] looked into Hybrid Nanofluids in their research. The mixed convection flow in a hybrid nanofluid across an exponentially stretching/shrinking vertical surface was studied by Waini et al. [15]. They discovered that when the nanoparticle volume fractions for copper increased, the rate of heat transfer decreased. Many researchers in the fields of biomechanics, industry, and engineering have focused on extensive applications of Newtonian and non-Newtonian fluids in the boundary layer flow across a stretching surface. Magneto hydrodynamic (MHD) flow studies are important for various industries and find applications in metallurgical and petroleum-related processes. The first researcherto investigate the MHD flow of a non-Newtonian fluid was Sarpakaya [16]. Boundary layer theory has proven to be very useful in the case of Newtonian fluids, as it allows for the transformation of Navier-Stokes equations into more manageable boundary layer differential equations.Some application ofmodeling of differential equation can seen from [17-19]. Accordingly, Mahanta and Shaw[20], Ravichandra Nayakar

et al. [21], Ahmed M. Megahed [22], Ram Prakash Sharma and Sachin Shaw[23], and Vishalakshi et al. [24] examined with different parameters on the non-Newtonian fluid flow past various stretching surfaces. Recently, numerous researchers explored the flow of different fluids with the impact of maragoni convection with magnetic effect and other influential factors [25–28]. One kind of viscoelastic or rate-type fluid is the upper-convected Maxwell fluid. Because it predicts the relaxation time impact and eliminates the complicated effects of shear-dependent viscosity, this model is highly significant. A common topic of study for numerous researchers is upper-convected Maxwell fluid flow. The effect of MHD flow and energy transmission on a stretched sheet was studied by Subhas et al. [29] using UCM fluid. It was shown that the velocity falls as the Maxwell parameter increases. Ishak et al. [30] have taken into consideration the problem of the MHD flow and heat transfer within a boundary layer of upper-convected Maxwell fluid over a stretching/shrinking sheet with prescribed heat flux. They infer that rising skin friction coefficient values are a direct result of rising magnetic parameter values. Non-Newtonian Maxwell fluids under various physical conditions, including porous media, transpiration, first-order chemical reactions, thermal radiation, heat source, and stretching surfaces, were examined by Amir and Kayva [31], Swati [32], Swati et al. [33], Vajravelu et al. [34], Gireesha et al. [35], and Ibrahim and Mekonnen [36]. Their findings demonstrate that temperature and heat transfer rate decreased as Prandtl number increased. Based on the research and an extensive literature review, it appears that there has been no previous investigation into the boundary layer flow and thermal transfer of UCM hybrid fluid over a stretched sheet. The aim of studying current research is to investigate the magnetohydrodynamics (MHD) behavior of upper-convected Maxwell hybrid nanofluids over a stretching sheet using numerical techniques. The scope includes analyzing fluid

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flow, heat transfer, and nanoparticle behavior under various parameters. The data collected from the study will be analyzed using the bvp4c procedure in MATLAB to obtain the most optimal solution.

Materials And Methods

Mathematical Formulation We consider a laminar 2D boundary-layer flow and the Heat and mass transfer of an incompressible Maxwell Hybrid nano fluid (which are meant to have alumina and copper as two small ingredients and water as the host fluid) over stretching porous sheet.The flow is restricted to z =0 and the stretching velocity for a linear sheet is uw = Bx.When a non-uniform magnetic field of starting intensity B0 is applied in the direction normal to the surface (see in Fig. 1). Table 1 shows the thermo physical properties of base solutions and nanoparticles. Under these conditions, our system’s main equations are as follows:

Concentration Equation: (4) In this, x and y are the directions that are parallel and perpendicular to the sheet. The velocity components u and v are taken along the x and y directions. T stands for Maxwell hnf ’ s temperature, ρhnf designates Maxwell Hybrid Nanofluid’s density, σhnf indicates for electrical conductivity of hybrid nano suspension, κhnf stands for Maxwell Hybrid Nanofluids heat/thermal conductivity, (ρcP)hnf is the Maxwell Hybrid Nanofluid’s specific heat, B0stands magnetic field’s amplitude, B >0 indicates the rate at which it is stretching, λ is the Maxwell parameter, µ is the fluid’s kinematic viscosity. Boundary conditions: The appropriate flow boundary conditions are

(5) (1) The subscripts hnf, f stand for “hybrid nano liquid,” “base fluid”, 1 and 2 stand for “Cu” and “Al2O3 nanoparticles,” and ϕ 1, ϕ 2 stand for “volume fraction”, respectively.

(2) Table 1. Thermo-physical properties of the nano particles and water (Suresh et al.[9])

Table 2. The correlations of single(NF) and hybrid nano fluids (HNF) (Takabi et al [10], Zainal et al [11]) Properties Density Thermal capacity Dynamic viscosity Thermal conductivity

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Similarity transformation: We have chosen the following similarity functions to transform the first (1-4) equations into their dimensionless form:

Equations (1-4) yield the following results using the similarity approach described above: (7)

(9) Here, the prime indicates the derivative with respect to η and

Subsequently the boundary conditions are taken in the form, (10)

(13) Where Rex, Nux, and Shx are, respectively, local Reynolds, Nusselt, and Sherwood numbers.

Numerical Solution And Methodology

It is worth mentioning that the Prandtl number was fixed at 6.2 (representing water) throughout the analysis of this study (zainal et al [11]). Obeying the remarkable work of Suresh et al. [9], the nanoparticles’ volume concentration of this study was set within the range of 0.005 0.015 to ensure the stability of the hybrid nanofluid. Meanwhile, the other parameters were used between these ranges (excluding the validation part): 0 < M2 <2 (magnetic parameter), 0 < β,kp < 1 (Maxwell and porosity parameter) and 0 < Le, Nt, Nb < 1.5. Further, the bvp4c in Matlab was fully utilized to solve the ordinary and reduced differential equation systems of Equations (7) and (9) together with the boundary equations (see Equation (10)). Thus, utilizing the RK-4-based shooting technique, the necessary velocity profiles, temperature fields, and concentration profiles with boundary conditions (10) have been drawn. First, the primary equations governing the flow are transformed into 1storder ODEs, and then an RK-4 procedure with shooting criteria is used to execute stepwise integration. All of the solution profiles have been generated using MATLAB. The Program and Algorithm The analytic solution of the boundary value problem Eqns. (7)-(9) cannot be found because these equations are non-linear and coupled. The system of nonlinear ODEs Eqns. (7)-(9) along with boundary condition Eqn. (10) are converted into first order ODEs. The first order systems of ODEs with appropriate boundary condition are solved by using shooting method. We adopt the following procedure: (14)

The Nusselt number, a conventional dimensionless expression of the rate of heat transmission between a surface and a fluid,is given by

(11) Skin friction coefficient and Sherwood number are given by (12)

Since Eq. (14) is a function of f and its derivatives, which can be solved numerically by shooting method. The solution of Eq. (14) can be used in Eq. (15) and Eq. (16) as a known input. For further proceeding, use the following notations: (17)

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The coupled nonlinear flow equations are turned into the subsequent system of seven first order ODEs together with the initial conditions:

Results And Discussion

The RK method has been taken into consideration for solving the above initial value problem. In the above system of equations, the missing conditions are to be chosen such that

This part describes how the parameters affect the flow features. The required parameters effectson velocity, concentration, and temperature have been explored and described. Figures 2, 3, and 4 show how the magnetic field parameter affects flow velocity, temperature, and concentration. The figures indicate that the fluid’s velocity dropped as the magnetic field increased, whereas the temperature and concentration profiles showed a rising pattern.This is because the magnetic field acts as a retarding body force, or Lorentz

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Figure 18. ϕ(η) for distinct values of Nt. force, acting in a direction perpendicular to the direction of the applied magnetic field. This body force restricts both the momentum boundary layer’s thickness and the boundary layer’s flow. Similarly, heat is produced by the Lorentz force, a fractional resistive force that resists fluid motion. This means that the concentration and thermal boundary layers are thicker under stronger magnetic fields.Figures 5, 6, and 7 depict the influences of the Maxwell constant on the profiles of velocity, concentration, and temperature over the sheet. An increase in the Maxwell parameter tumbles fluid velocity while increasing the fluid’s temperature and concentration Profiles. When the Maxwell parameter increases, it suggests that the viscous-elastic effects in the fluid become more pronounced. These effects can manifest as a boost in fluid’s elastic properties, resulting in a reduction in the fluid’s ability to flow easily. This increased resistance to flow causes the fluid velocity above the sheet to decrease. The association between the Maxwell parameter and thermal boundary layer thickening is presumably mediated by the interaction between these non-Newtonian properties and the heat transfer process. As the Maxwell parameter increases, it likely influences the fluid’s viscosity or flow behavior, causing changes in the properties of heat transfer. These changes can lead to a thicker thermal boundary layer.The effects of the porosity measure on the temperature, concentration, and velocity profiles are shown in Figures 8, 9, and 10. It has been noted that when the porosity parameter value grows, the concentration and temperature rise and the velocity gradient decreases. As Kp increases, it implies that the porous medium becomes more resistant to fluid flow. However, in porous media, increased resistance to fluid flow (as indicated by higher) can result in greater heat transfer between the fluid and the porous medium. This enhanced heat transfer can lead to higher temperatures in the fluid, causing the temperature graph to increase. Like the temperature effect, the increased resistance to fluid flow in the porous medium

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(higher) can influence mass transport processes. This can lead to enhanced concentration gradients and increased concentration in the fluid as it passes through the porous medium. Figures 11 and 12 depict how the graphs of temperature and concentration change in relation to the Prandtl number, respectively. As shown in Figure 11, when the Prandtl number goes up, the temperature and the thickness of the thermal boundary go down. This is because a fluid with a high Pr value has a relatively low thermal conductivity, which reduces conduction and, as a result, diminishes the width of the thermal boundary layer. As evidenced in Figure 12, the thickness of the concentration boundary layer increases as Pr values rise. Figures 13 and 14 depict temperature and concentration graphs with respect to the Lewis number (Le). The temperature profile increased for high Le values. The concentration profile decreased for high Le values, resulting in a small molecular diffusivity. Typically, as the Lewis number rises, the concentration profile falls. Furthermore, as the Lewis number rises, the concentration boundary layer becomes thinner. Most likely, this is because the mass transfer rate goes up as the Lewis number goes up. The concentration gradient within the sheet is also increased. Temperature and concentration graphs vary with Brownian motion parameter in Figures 15 and 16, respectively. As Nb values rise, the temperature graph rises. Physically, Nb is associated with fluid particle movement. The kinetic energy of the particles in a fluid rises as rises. The thermal boundary layer thickens as increases, as shown in the graph. Figure 16 indicates that concentration distribution decreases as Nb increases. Thermophoresis parameter Nt affects temperature and concentration profiles, as shown in Figures 17 and 18. As depicted in graph 17, the fluid’s temperature profile rises as Nt values increase. In the presence of Nt, the nanoparticles on the hot boundary side have been shifted to the cold boundary side, and the thermal boundary layer has thickened. As shown in Figure 18, the thickness of the concentration boundary layer increases as Nt rises. Increasing is observed to increase the concentration distribution progressively. Nusselt Number, Sherwood Number, and Skin Friction Coefficient Tables 1 and 2 describe the computed numerical results of the Skin friction coefficient, Nusselt number, and Sherwood number using various physical parameters listed in the tables. The skin friction coefficient quantifies the quantity of friction encountered by a fluid as it flows along a solid boundary. A greater value for the skin friction coefficient indicates increased fluid flow resistance. The coefficient of skin friction is displayed against applied magnetic field strength M is portrayed in figure

19. As β values increase, −f˝(0) rises. This implies that with

increasing β, the skin friction experienced by the fluid near the boundary increases. This behavior suggests that the

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Table 1. Numerical values of −f˝(0) for Kp = 0.4; Le = 1; Pr = 2; Nt = 0.5; Nb = 0.5.

Table 2. Numerical values of −f˝(0) ,−θ´(0) and −ϕ´(0) for β = 2, Pr = 2, Kp = 3, Le = 2, M = 1.5.

flow is more impeded; leading to a greater resistance to flow and skin friction coefficient.Figure 20 illustrates the influence of the Brownian motion variable on the Nusselt number in comparison to the thermophoresis parameter. This suggests that increasing the Brownian motion parameter decreases heat transfer at the specific location. The influence of Brownian motion on the Nusselt number could be due to enhanced particle dispersion or reduced convective heat transfer, which can hinder heat transfer at the surface. Figure 21 illustrates how the thermophoresis parameter influences the local Sherwood number −ϕ´(0) in relation to the Brownian motion parameter. As Nt values increase, the local Sherwood number graph also increases. The thermophoresis parameter relates to the thermophoresis phenomenon, which is the motion of particles induced by temperature gradients. The impact of increasing the thermophoresis parameter on Sherwood number could be due

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to enhanced particle concentration or improved diffusion of species near the surface. The enhanced thermophoresis parameter influences species transport near the surface, thickening the concentration boundary layer.

the manuscript and formatted it to the journal specification; G. Nagaraju wrote introduction part and performed drafting in the manuscript; Santoshi Misrarevised the manuscript and added themissing contents.

Data Availability Statement

A numerical investigation of the boundary layer and MHD flow of UCM hybrid Nano fluidsover a stretching sheetwas provided.This work will undoubtedly be useful for many applications in (i) Heat transfer enhancement in industrial processes, (ii) Optimization of cooling systems in electronic devices, (iii) Designing efficient lubricants for high-speed machinery, (iv) Understanding fluid dynamics in biomedical applications, like drug delivery systems and (v) Improving the performance of aerospace propulsion systems. The primary equations governing the flow were converted into a set of ODEs using similarity variables, and numerical solutions for various governing parameters were provided. The mathematical model is solved using bvp5c technique. The substantial consequences of numerous physical factors on non-dimensional profiles of velocity, temperature, concentration, coefficient of skin friction, local Nusselt number, and local Sherwood number are graphically shown. The study yields the following conclusions: • When the Maxwell parameter values are increased, the velocity boundary layer thickness is observed to decrease. • As thehigh intensity of magnetic field is applied, the magnetic field boundary layer becomes thinner. • Increasing Maxwell parameter causes the high molecular velocity, which enhances temperature field, concentration profile, and boundary layer thickness. • The thickness of the concentration boundary layer decreases as the Lewis number Le and Brownian motion Nb increase, while it increases as Nt values rise. • The surface temperature increases for increasing values of porous parameter Kp, Nb and Nt, but it decreases as Pr values were increased during observations. • The present investigations are useful in industrial application as well as in the botanical segment limited to their experimental existence. Future Scope: We can analyse this work using advanced computational techniques such as machine learning algorithms for getting more accurate predictions of fluid behaviour. Additionally, investigating the application of this fluid model in industrial processes like nanotechnology manufacturing or biomedical engineering could open new avenues for technological innovation and development.

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.

Authorship Contributions

G. Thirupathi executed the numerical code and drawn all the graphs and tables; K. Govardhan did mathematical modelling of the problem andwrote the concerned parts of

Conflict Of Interest

The author 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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