Scrutiny of flow and heat transfer characteristics of hybrid nanofluid passing through a squeezing c

Abstract

Keywords: Graphene; Heat Absorption; MHD; Nanofluid

Introduction

The increasing utilization of nanofluids in real-world industrial applications has made them a major focus of research. In 1995, Choi and Eastman [1] discovered that the addition of metallic nanoparticles enhances heat transfer. Buongiorno [2] identified seven methods to improve the heat transfer rate of base fluids, with Brownian motion and thermophoresis playing significant roles. Nanofluids find important applications in energy conversion, microsystems cooling, and the medical industry. Numerous researchers

have investigated the heat transfer characteristics of nanofluids [3-7]. Graphene, with its notable properties such as thermal conductivity, flexibility, and high elasticity [8], has found various applications in batteries, supercapacitors, and miniaturized solar devices [9]. Upadhya et al. [10] conducted a study on enhancing the heat transfer rate using ethylene glycol with graphene nanoparticles. Bhattacharyya et al. [11] carried out the statistical analysis of the Ethylene glycol and graphene based nanofluid flow over a stretching sheet.

*Corresponding author. *E-mail address: manojmishra.iitg@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic 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/).

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In recent years, researchers have focused on the use of hybrid nanofluids, which involve mixing multiple nanoparticles in a fluid, to further improve heat transfer rates. Devi and Devi [12] investigated the flow of a hybrid nanofluid (Cu-Al2O3/water) over a stretching sheet, presenting a novel mathematical model. Suresh et al. [13] validated this new model using experimental data. Devi and Devi [14] extended their work to study 3-dimensional flow under Newtonian heating conditions, finding higher heat transfer rates in hybrid nanofluids compared to regular nanofluids. Prakash and Devi [15] examined the influence of Al2O3– Cu/water nanofluid passing through a slender stretching sheet. Bahiraei and Mazaheri [16] explored the application of graphene-platinum hybrid nanofluids in miniature devices. Aziz et al. [17] investigated entropy generation due to hybrid nanofluids with Maxwell as the base fluid, considering radiation, magnetic fields, and joule heating. Yashkun et al. [18] studied hybrid nanofluid flow past an exponentially stretching/shrinking sheet, taking mixed convection into consideration. Rafique et al. [19] shows that heat transfer performance in hybrid nanofluid is better than mono-nanofluid. Joule dissipation, a significant heat source in fluid flow under the influence of a magnetic field, plays a vital role in heat-treated materials. Rashid et al. [20] studied the impact of Joule heating on MHD flow over a stretchable wall of graphene−carboxymethyl cellulose (CMC) with water as a base fluid. Hence, several studies have examined the effect of Joule dissipation [21-23]. Flow within the channel is often observed in many engineering and industrial processes, for example, lubrication systems, moving pistons, hydraulic lifts, injection moulding, flow inside the nasogastric tubes syringes. Due to its nobility, many researchers analyzed the flow within the channel under various conditions [24, 25]. The flow induced by the compression of two parallel plates due to external applied stress is known as squeezing flow. The

squeezing flow situation is often observed in many engineering and industrial processes. Noor and Shafie [26] discovered nature of hydromagnetic flow of hybrid nanofluid within squeeze channel. Due to its immense applications, the flow through a squeezing channel has attracted many researchers [27-30]. A careful review of the literature suggests that, no investigation has been conducted on comparative analysis for flow and heat transfer characteristics among three different fluids namely base fluid (Ethelene Glycol), mono-nanofluid (Ethelene Glycol + Graphene), and hybrid nanofluid (Ethelene Glycol + Graphene + Copper) passing through a squeezing channel taking into account of heat absorption and magnetic field effects. Thus, the present study analyses the impact of various physical parameters on base fluid, mono-nanofluid and hybrid nanofluid taking into account of these effects. Modeling of the Problem Consider 2-dimensinal electrically conducting, heat absorbing, radiating, incompressible, electrically conducting hybrid nanofluid flow within squeezing channel taking Joule dissipation into account. Our mathematical model is related to 2 parallel plates which is along x and z-axis and y is normal to the plates. The gap between two plates is at time t. Direction and speed of plates and the gap is controlled by α. The velocity of plates is . Negative value will be 0 between two plates at of α denotes that distance between plates are increasing. Magnetic field is applied perpendicular to the surface of . It is also plates and strength is given by assumed that there is no chemical reaction. Geometry of the problem express in Figure 1. Under these conditions, the mathematical model governing the boundary layer flow are given by [27]:

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Here 𝜇𝑓 presents the dynamic viscosity of C2H6O2. 𝐾𝑓, 𝐾𝑛1 and 𝐾𝑛2 denote the thermal conductivity of base fluid, graphene and copper respectively. 𝜙1 and 𝜙2 indicate the volume fraction of graphene nanoparticle and copper nanoparticle respectively. 𝜎𝑓, 𝜎𝑛1 and 𝜎𝑛2 is the notation of electrical conductivity of C2H6O2, Graphene and Copper. 𝜌𝑓, 𝜌𝑛1 and 𝜌𝑛2 is the notation of density of C2H6O2, Graphene and Copper. (𝜌𝑐𝑝)𝑓, (𝜌𝑐𝑝)𝑛1 and (𝜌𝑐𝑝)𝑛2 is the notation of heat capacity of C2H6O2, Graphene and Copper. Boundary conditions are taken as:

. (4) Where, 𝑢 and 𝑣 are the velocity of hybrid nanofluid in x and y- directions. 𝑇 indicates the temperature of fluid, 𝑝, 𝜌ℎ𝑛𝑓, 𝜇ℎ𝑛𝑓 , 𝜗ℎ𝑛𝑓, (𝜌𝐶𝑃)ℎ𝑛𝑓, 𝐾ℎ𝑛𝑓 , 𝑄 and 𝜎ℎ𝑛𝑓 denote the fluid pressure, density of hybrid nanofluid, kinematic viscosity of hybrid nanofluid, heat capacity of hybrid nanofluid, thermal conductivity of hybrid nanofluid, heat absorption coefficient and electrical conductivity of the hybrid nanofluid respectively. Properties of hybrid nanofluid are noted as [25]: (5)

(11) Similarity transformations are given as [27] (12) Using the similarity transformations given in equation (12) into the PDEs (2) to (4) and eliminating the pressure terms by cross- differentiations of equations (2) and (3), the resultant ordinary differential equation is mentioned from equation (13) to (14) as follows:

The base fluid Ethylene glycol (C2H6O2) and nanoparticle’s thermophysical properties are mentioned in Table 1 [23].

Table 1. Thermophysical properties of Ethylene glycol and nanoparticles [23] Thermo-physical properties Graphene

Numerical Method Implementation

The majority of natural processes are characterized by highly intricate nonlinear dynamics, typically described by nonlinear differential equations. Attaining a closed-form exact solution under such circumstances proves to be unattainable. In these situations, researchers seek approximate solutions through either numerical methods or analytical techniques. Numerical solutions can be obtained employing diverse methods like the finite difference method [31], finite volume method [32], finite element method (FEM) [33], reproducing kernel algorithm [34-37] and shooting method [38]. The solution of the present problem is obtained by using shooting method along with RungeKutta fourth order technique. In order to find the solution of transformed ordinary differential equations (13) and (14) subject to boundary conditions (15) and (16), the following steps are used: • To solve the equations (13) and (14), first converted into a system of six first-order differential equations. • To find the solution of the system of ordinary differential equations, the initial values of 𝑓”(0) and 𝜃′(0) are found using the shooting method along with RungeKutta fourth order technique.

After the grid Independence analysis, the step size selected is 0.001 throughout the computation. A tolerance error of 10-6 is chosen. The entire process is repeated until the desired accurate result is obtained.

Characteristics of Shooting Technique • Shooting method converts boundary value problem to initial value problem. Then initial value problem needs to be solved and initial guesses are to be found in each iteration. This process will continue until solution satisfies the given boundary conditions. • The convergence of shooting method depends on initial guess. Poor initial guess may decrease the rate of convergence or possibly solution may diverge. Characteristics of Runge-Kutta Fourth Order Method • Runge-Kutta fourth order method is used to solve initial value problem with high accuracy. • In Runge-Kutta fourth order method, step size can be adjusted during the solving process. Validation of Present Finding In order to validate the numerical solution obtained using the method described in above section, a comparison of the Nusselt number at the upper plate is performed with the earlier published results and are presented Table [2]. It is noted from Table 2 that this manuscript is in excellent agreement with the earlier published manuscripts ([39-40]).

Results And Discussions

In this section, all the results for velocity temperature, skin friction coefficient and Nusselt number, computed using the numerical method as described in previous section are presented graphically in Figures 2 to 12. Impact of various physical parameter M = 0.5, Pr =11, Ec = 0.05, S = 0.5, G = 0.1 and different values of nano particles volume fraction on hybrid nanofluid velocity (from Figures 2 and 3) and temperature (from Figures 4 to 6) is discussed and analyzed. In all graphical representations the results are displayed for three different cases of fluid flow: (1) Base fluid (Ethylene

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Glycol- C2H6O2) only using solid lines, (2) mono-nanofluid (C2H6O2 + Graphene) using dashed lines and (3) Hybrid nanofluid (C2H6O2 + Graphene + Copper) using dotted lines. Throughout the manuscript 𝜙1 and 𝜙2 represents graphene and Copper nanoparticle volume fractions.

Temperature Profile The temperature profile for flow problem is represented in Figures 4 to 6 against varying values of magnetic parameter, Eckert number and absorption parameter.

Figure 3. Varation in S values cause various velocity profiles. It is clearly noted from Figure 4 that the hybrid nanofluid temperature is enhanced due to the increase in the magnetic field. This is because the fluid will face more difficulties to flow against the magnetic field. Figure 5 shows that enhancing the Eckert number has the nature to enhance the temperature of the hybrid nanofluid. The Eckert number represents the ratio of kinetic energy to enthalpy. Some part of energy converts to kinetic energy; therefore, in the boundary layer, viscous dissipation can enhance the temperature. It is also noted that the hybrid nanofluid temperature increases when the nanoparticle volume is increased. It also shows that in the case of the mono-nanofluid (𝜙1 = 0.05 and 𝜙2 = 0), the temperature is higher than the base fluid and lower than the hybrid nanofluid (𝜙1 = 0.025 and 𝜙2 = 0.025). Figure 6 shows 1.8

Velocity Profile The velocity profiles of the flow problem are presented in Figures 2 and 3, against the magnetic field and squeezing number. Figure 2 illustrates the impact of magnetic field nanoparticles volume faction on hybrid nanofluid velocity. It is notice that due to an increase in the magnetic field, a very small reduction in velocity of the hybrid nanofluid up to central region while the opposite nature is observed after that. This phenomenon is happening due to Lorentz force up to central region and after that due to the continuity equation. It also shows that in case of mono-nanofluid (𝜙1 = 0.05 and 𝜙2 = 0), velocity is higher than the base fluid and lower than hybrid nanofluid (𝜙1 = 0.025 and 𝜙2 = 0.025) up to central region. Figure 3 demonstrates the influence of the Squeeze number on hybrid nanofluid velocity. It is evident that the Squeeze number decreases the hybrid nanofluid velocity, but the opposite nature is visible as the distance between the channels increases. Further, increase in squeezing number does not seem to affect the velocity abnormally as we can see a very small variations in velocity against S. It also shows that in the case of the mono- nanofluid (𝜙1 = 0.05 and 𝜙2 = 0), the velocity is higher than the base fluid and lower than the hybrid nanofluid (𝜙1 = 0.025 and 𝜙2 = 0.025) up to the central region.

Figure 2. Varation in M values causes various velocity profile.

Figure 4. Varation in M values cause various temperature profiles.

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Figure 5. Varation invalues of Ec cause various temperature profiles.

Figure 7. Varation in values of M and ɸ cause various 𝐶𝑓 profiles.

Figure 6. Varation invalues of G cause various temperature profiles.

Figure 8. Varation in values of S and ɸ cause various 𝐶𝑓 profiles.

that due to the enhancement of the absorption parameter, the temperature of the hybrid nanofluid velocity is reduced because of the increment of the absorption capacity of the fluid. It also shows from Figure 4 to 6 that in case of mono-nanofluid (𝜙1 = 0.05 and 𝜙2 = 0), temperature is higher than base fluid and lower than hybrid nanofluid (𝜙1 = 0.025 and 𝜙2 = 0.025).

behavior of 𝐶𝑓 is observed due to increase in S. It is also noted that in case of mono-nanofluid (𝜙1 = 0.05 and 𝜙2 = 0), 𝐶𝑓 is higher than base fluid and lower than hybrid nanofluid (𝜙1 = 0.025 and 𝜙2 = 0.025). Figures 9 to 12 presents the impact of 𝑀, 𝐺, 𝑆 and 𝐸𝑐 on 𝑁𝑢 at the upper plate of the channel. From Figure 9 it is noted that when magnetic parameter is increased from 0 to 2, a 10.01% of relative increment in heat transfer rate is reported. Figure 10 which displays that the Nusselt number is a decreasing function of absorption parameter 𝐺, to be exact, we reported a relative decline of 115.24% in Nusselt number when 𝐺 is varied from 0 to 1. It is observed that there is a 34.748% of decrease in Nusselt number as the value of 𝑆 increases from 0 to 1, whereas an increase of 18.28% of heat transfer rate is noted for hybrid nanofluid as compared

Skin Friction and Nusselt Number Profile The quantities of engineering interests like skin friction coefficient and Nusselt number profiles against the pertinent flow parameter is displayed in figures 7 to 12. Figures 7 and 8 presents the impact of M and S on 𝐶𝑓 at the channel’s upper plate. From Figure 7, It is visible that 𝐶𝑓 is reducing at upper plate due to enhancement of M and opposite

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Figure 9. Varation in values of M and ɸ cause various Nu profiles.

Figure 10. Varation in values of G and ɸ cause various Nu profiles.

Figure 12. Varation in values of Ec and ɸ cause various Nu profiles. to base fluid. Similarly in Figure 12 the Eckert number 𝐸𝑐 demonstrated an increasing nature of Nusselt number against it. The investigation revealed a huge relative increase of 31.2% in Nusselt number when 𝐸𝑐 is varied from 0 to

Conclusion

Figure 11. Varation in values of S and ɸ cause various Nu profiles.

This article presents the comparative study of flow and heat transfer characteristics of three different type of fluids namely (1) Base fluid (Ethylene Glycol- C2H6O2) (2) Mono-nanofluid (C2H6O2 + Graphene) and (3) hybrid nanofluid (C2H6O2 + Graphene + Copper) passing through a squeezing channel. The study also takes into account the combined effects of transverse magnetic field, heat absorption and Joule dissipation. The behavior of fluid velocity, temperature, skin friction coefficient and Nusselt number are analyzed. The major findings of the present investigation are listed below: • The hybrid nanofluid velocity is going down due to increase in magnetic field till central region and after that opposite tendency is visible. • The nature of temperature is to grow up by enhancing due to magnetic field, Joule dissipation while opposite behavior due to enhancing of absorption parameter.

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Local skin friction at upper plate is enhancing due to enhancement of S and opposite behavior happened due to enhancement in M. • The local Nusselt number at the upper plate is enhancing due to enhancement of M and Ec and opposite behavior happened due to enhancement in G and S. The heat transfer rate as well as skin friction coefficient, for hybrid nanofluid found to be higher than that of both mono-nanofluid and base fluid.

Ethics

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

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