# Analysis of Metallic Nanoparticles (Cu, Al2O3, and SWCNTs) on Magnetohydrodynamics Water-Based Nanofluid through a Porous Medium

Authors: P. K. Pattnaik; S. K. Parida; S. R. Mishra; M. Ali Abbas; M. M. Bhatti
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this communication, the effect of the addition of the copper (Cu), aluminum oxide (Al2O3), and single-wall carbon nanotubes (SWCNTs) metallic nanoparticles on the magnetohydrodynamics (MHD) water-based flow over a porous elastic surface is explored. The objective of the work is to include the radiative effect that interacts with the metallic nanoparticles due to permeability of the surface. The significance of this study stems from the fact that the design of various equipment, such as nuclear power plants, gas turbines, propulsion devices for aircraft, and missiles, is dependent on radiative heat transfer. To formulate the mathematical modelling, similarity transformations were used, and nonlinear differential equations were obtained. To solve the formulated nonlinear differential equations, the Runge–Kutta fourth-order numerical scheme is used in conjunction with the shooting technique. The behavior of velocity profile and temperature profile has been discussed in detail and also engineering quantities such as Nusselt and Sherwood number which are calculated. Furthermore, the addition of metallic nanoparticles enhanced the nanofluid properties for energy transfer enrichment and found many applications in various fields of science and technology.


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## Body

## 1. Introduction

Choi introduced a new type of fluid called nanofluid in 1995, which has amazing thermal conductivity properties. The goal of the concept is to saturate nanosized particles in conventional fluids known as base fluids. Nanofluids are extremely important in thermal conductivity, heat transfer enhancement, energy, and other thermos-physical properties for industrial applications [1–5]. The heat transfer capacity of a nanofluid after the addition of metallic and nonmetallic nanoparticles in a conventional base fluid was of particular interest to the researchers. Mohebbi et al. [6] investigated the mathematical model of the Heat Transfer Augmentation Associated with Cu/Water Nanofluid in a Channel with Surface Mounted Blocks by using Lattice Boltzmann Method. The numerical method is applied for the forced convection flow and heat transfer of a nanofluid flowing inside a straight circular pipe by Saryazdi et al. [7]. Moreover, Baag and Mishra [8] discussed heat and mass transfer analysis on MHD 3D water-based nanofluid. The prominent examples of nanofluids are ethylene glycols, kerosene, and water. It has been observed that conducting nanofluids presents their special attention because of their use in diversified areas such as biomedical solicitation as tuneable optical filters, drug delivery, and cancer therapy. Watanabe and Pop [9] deliberately presented the magnetohydrodynamic flow of particular fluid for the occurrence of applied magnetic field through a flat plate. Numerical treatment is depicted by Armaghani et al. [10] for the mixed convective flow phenomena of nanofluid within open C-shaped enclosures. For the enhanced properties, they have used CuO nanoparticles which are dispersed within the base fluid water and the enclosure is imposed with constant magnetic field. Furthermore, the influential behavior of the characterizing parameters such as Richardson number and volume concentration affects the flow phenomena as well. The work of Ibrahim and Terbeche [11] leads to bring out the effective properties of the non-Newtonian power-law fluid with due occurrence of the magnetic field. Analytical approach is employed for the solution of the designed problem and numerical methods are useful for the validation of the current result and the convergence criterion.Fluid flow and heat transfer with non-Newtonian fluids, for example, are a challenge in the modern revolution, particularly in the oil industry, bubble columns and absorption, zymosis, boiling, plastic foam processing [12], etc. However, the possible applications relating to this type of flow can be observed in various industries. The generation of electric power in the corresponding electric power industry is one of the examples that uses the extraction of energy. The governing equations for different non-Newtonian fluid models are amid the utmost complex equations so that the development in mathematical modelling is of great interest nowadays. A time-dependent flow characterized by the several parameters for the nanofluids past an expanding sheet is presented by Andersson et al. [13]. Furthermore, similarity approach for the complex unsteady flow problem past over an expanding sheet is carried out by Elbashbeshy and Bazid [14]. Thermophoresis and Brownian motion effect on the flow of nanofluid through a vertical plate has been studied by Kuznetsov and Nield [15]. They pointed out that the cooling rate of the plate decreases due to decrement in strengthens of thermophoresis and Brownian motion. Heidary and Kermani [16] studied the effect of solid volume fraction of nanofluid and magnetic strength. They examined that existence of magnetic field and nanofluid could significantly enhance heat transfers properties of the flow phenomena. The thermal properties of the base fluids change appreciable after addition of the metallic nanoparticles and calculate the thermos-physical parameters [17]. Masuda et al. [18] reported that, after addition of ultrafine nanoparticles, there is an alteration in the thermal conductivities and viscosities. Mishra et al. [19] recently studied a chemically reactive nano-micropolar fluid with variable heat sink/source and slip conditions. Shutaywi and Shah [20] proposed a numerical and mathematical model of a nanofluid that includes entropy formation.The application of electrically conductive fluid currents is encircled in the field of nanocomposite and metallurgy. The flows of several fluids under the action of magnetic field such as MHD generators, oil exploration, energy extraction, and boundary layer control have attracted many researchers. Metallurgical requirements consist of continuous cooling belts or filaments such as hardening, disperse, and sketching processes for copper wires. It has been noticed that the effects of Coriolis force are larger than those of viscosity and inertia forces in the hydro-magnetic equations of motion in a rotating environment. Several researchers have been investigated on MHD with various kinds of fluid and geometries. For example, Ibrahim and Negera [21] investigated the upper-convected Maxwell nanofluid flow with slip and MHD effects through a stretching sheet and chemical reaction. Abdal et al. [22] examine the thermo-diffusion with magnetized mixed convection unsteady nanofluid flow through stretching/shrinking surface with heat source and thermal radiation. Ghasemi and Hatami [23] described the solar radiation effects on magnetized stagnation point nanofluid flow through a stretching surface. Some important references related to the proposed topic can be found in [24–28] and several therein. Recently, Upreti et al. [29, 30] considered carbon nanotube nanofluids for the behavior of various physical quantities in different geometries. They have projected the effect of drag force with an interaction of Joule heating and nonuniform heat source/sink. Also, binary chemical reaction with the impact of radiative heat on the flow phenomena over an expanding surface is considered. Sabu et al. [31] investigated the enhancement of heat transfer caused by a thermal and space dependent heat source, magnetic field, and nanoparticles propagating over an elastic spinning disk. Mahanthesh et al. [32] investigated Reiner–Rivlin nanofluid flow through a rotating disk with multiple slips and a distinct heat source.Therefore, the primary goal of this research is to determine the presence of three metallic nanoparticles (Cu, Al2O3, and SWCNTs) in an electrically conducting water-based nanofluid propagating through a porous medium. Thermal radiation is important in industrial applications. As a matter of fact, the study’s novelty stems from the incorporation of thermal radiation as well as an additional heat source/sink within a permeable medium. The mathematical modelling was developed using similarity transformations. The nonlinear differential equations are solved using the Runge–Kutta and shooting techniques. When compared to other similar methods used for nonlinear problems, the current numerical method yields promising results [33, 34]. The graphical interpretation of the velocity and temperature profiles have been discussed in detail and expected results show the excellent industrial applications.


## 2. Problem Formulation

The time-dependent electrically conducting flow of nanofluids through a permeable medium is presented in this article. For the enhanced feature in heat transfer attempt is made to consider SWCNTs in the water-based nanofluid along with Cu and Al2O3 nanoparticles. Moreover, the novelty of the study arises for the inclusion of radiative heat transfer with additional external heat source/sink that enriches the energy profile. The flow through porous elastic surface along the x-direction and the transverse magnetic field of uniform strength B0 is proposed along the normal direction of the surface, i.e., y-direction, as given in Figure 1. Due to permeability of the surface, the occurrence of suction/injection has its immense use on the flow phenomena. Following Zhang et al. [35], the proposed assumptions lead to design the model with the boundary conditions as(1)∂u∂x+∂v∂y=0,ρnf∂u∂t+u∂u∂x+v∂u∂y=−∂p∂x+μnf∂2u∂y2−μnfK+σnfB02u,ρcpnf∂T∂t+u∂T∂x+v∂T∂y=knf∂2T∂y2−∂qr∂y+Q0T−T∞,with boundary conditions,(2)ux,0,t=0,vx,0,t=v0t,−knf∂Tx,0,t∂t=qx,ux,∞,t=Ux,t,Tx,∞,t=T∞.Figure 1
Flow configuration.Here,u and v, are the components of velocities along x− and y− direction, T is the temperature of the nanofluid, t is the time taken, p is the fluid pressure, v0 is a constant, and σs and σf are the electrical conductivity of the base and nanofluid, respectively.The physical properties relating to nanofluid such as viscosity, specific heat, density, and conductivity are presented as follows [36]:(3)μnf=μf1−ϕ2.5,ρnf=1−ϕρf+ϕρs,ρcpnf=1−ϕρcpf+ϕρcps,σnf=1−ϕσf+ϕσs,knfkf=ks+2kf−2ϕkf−ksks+2kf+ϕkf−ks,where ϕ is the particle concentration, μf is the dynamics viscosity, ρf and ρs are the densities, kf and ks are the thermal conductivities, and the subscripts f and s are for the base fluid and the solid nanoparticles.(4)ρnf∂u∂t+u∂u∂x+v∂u∂y=∂U∂t+U∂U∂x+μnf∂2u∂y2−μnfK+σnfB02u−U,where(5)−∂p∂x=∂U∂t+U∂U∂x−υnfK+σnfρnfB02U,Here,K is the permeability of the given medium and B0 is an external magnetic field strength.The stream function as well as variables for the problem is expressed as (see [36])(6)u=∂ψ∂y,v=−∂ψ∂x,η=yavf,ψ=avfxfη,a=1s+bt,T=T∞+q0xkfvfaθη,qx=q0x,v0t=vit.Using the aforesaid functional expressions, the governing equations are presented as(7)f‴+A1f+bη2f″−f′2+bf′2−1+1+MA2+Da1−f′=0,(8)1A3Prknfkf+Nrθ″+f+bη2θ′−f′+b2+δA3θ=0,where M=σf/ρfbB02 is the magnetic parameter, Pr=υf/κ is the Prandtl number, Nr=16σ1T∞3/3κkf is the thermal radiation parameter, Da=υf/aK is the Darcy number with boundary condition, b is the unsteadiness parameter, and δ=q0x/ρcpfU is the heat source parameter.(9)f0=fw,f′0=0,θ′0=−kfknf,f′∞⟶1,θ∞⟶0,where(10)A1=1−ϕ2.51−ϕ+ϕρsρf,A2=1−ϕ2.51−ϕ+ϕσsσf,A3=1−ϕ+ϕρcpsρcpf.The physical quantities are as follows:Cf=τw/ρfU2 is called as skin friction coefficient and Nux=qwx/kfTw−T∞ is called local Nusselt number:(11)CfRex0.5=f″01−ϕ2.5,NuxRex−0.5=−knfkf+Nrθ′0.


## 3. Numerical Methodology

For solving equations (10)–(12), a multistep integration method, i.e., the Runge–Kutta method, with shooting technique has been deployed. In this process, equations (7) and (8) are reduced to a set of ordinary differential equations as defined below:(12)f=y1,f′=y2,f″=y3,f‴=−A1y1+bη2y3−y22+by12−1+1−MA2+Da1−y1,θ=y4,θ′=y5,θ″=A3Pr−y1+bη2y5+y2+b2+δA3y4knfkf+Nr−1,under the boundary condition,(13)y1=fw,y2=0,y5=−kfknf,y2⟶1,y4⟶0..Now, it uses only the initial conditions, i.e., forη=0. However, due to the unavailability of initial conditions, the assumed initial conditions y3 and y4 are to be determined. Therefore, some initial guesses are incorporated to both these values in order to satisfy the boundary conditions at η⟶∞η=4. These corrections are taken care by a corrective procedure that follows a self-iterative process. This procedure is used to implement a more accurate method, i.e., the RK method, with shooting numerical technique. For the computational purpose, the step size is assumed as h=0.01. Therefore, the accuracy of computation and the convergence criteria are followed.


## 4. Results and Discussion

An unsteady two-dimensional flow of metallic water-based nanofluids is considered which past a permeable medium for the action of transverse magnetic field is presented. Interaction of Cu and Al2O3 nanoparticles along with SWCNTs in base fluid water is dispersed to prepare nanofluid. Incorporation of radiative heat energy enriches the profile in conjunction to the permeable surface. Numerical technique is used to find the solution of the set of equations for the suitable choice of the pertinent parameters. Table 1 displays all the physical properties of both the particles as well as the base fluid. Table 2 present the validation of the present outcomes for the shear rate considering the case of pure fluid as well as the case of nanofluid with the work of Rizwan et al. [36], and this shows a good corroboration. The graphical illustration shows the significant behavior of these parameters associated with the flow phenomena. Furthermore, the tabular simulated results indicate the rate coefficients, i.e., shear rate and Nusselt number. However, throughout the computation, the following values of the parameters are considered as fixed whereas the variation of particular parameters are presented in the corresponding figures, and these are ϕ=0.2, b=0.1, M=1, Da=1, fw=0.5, and δ=1.Table 1
Thermos-physical properties of base fluid and nanoparticles.
ρkg/m3cpJ/kg KkW/mKPure water997.141790.613Copper (Cu)8933385401Aluminum oxide (Al2O3)397076540SWCNTs26004256600Table 2
Validation of shear rate.
Nanofluidsϕf″0Rizwan et al. [36]f″0PresentCu-H2O0.01.481134191.4810210.11.711055041.7110030.21.751387281.751128Al2O3-H2O0.01.481134191.4810210.11.434384551.4316250.21.330967581.330727SWCNT-H2O0.01.481134191.4810210.11.450882351.4501220.21.355718791.355526The role of particle concentration due to its appearance through the thermo-physical properties is a vital part of this investigation. Figure2 describes the significance of particle concentration on the velocity for the Cu, Al2O3, and SWCNT-water-based nanofluids. Several characteristics of the suction/injection on each profile are displayed. Here, the parameter fw>0 represents the role of suction whereas fw<0 indicates the injection and fw=0 characterizes the behavior when the flow through impermeable region. The decelerating nature of the profiles shows the increasing width of the bounding surface thickness for the increasing particle concentration. The range of the concentration is treated within ϕ=0.0,0.2. The impermeability region for the pure fluid is similar to the results obtained by Mishra et al. [19], and it can be obtained by considering fw=0 and ϕ=0. Furthermore, the increasing suction enriches the profiles, but the thickness of the bounding surface decreases; however, injection reveals opposite impact on the profile. It reveals the density of the Cu particles and diminishes the profile width in comparison to the particles of Al2O3 and SWCNTs. Figure 3 illustrates the behavior of the unsteadiness parameter in association with the suction/injection on the nanofluid velocity profiles. Here, b≠0 indicates the unsteady case on the velocity of the three different water-based nanofluid. An augmentation in the profiles is rendered for the increasing unsteadiness that causes a deceleration in the bounding surface thickness. The profiles of SWCNT nanofluid are lesser than the other nanoparticles of Al2O3 and Cu, respectively. Also, for each of the profiles, interestingly, the thickness decelerates more in case of injection in comparison to impermeable and the case of suction successively. Figure 4 portrays the role of magnetic parameter on the nanofluid velocity profiles with the interaction suction/injection. The magnetic field expresses the influence of moving electric charges along with electric current and magnetic materials. In modern technology, there are various applications of magnetic field such as in both the electric motors and generators; the use of rotating magnetic field is important. The profile augments lead to decelerate the thickness of the velocity-bounding surface for the augmented magnetic parameter. This is due to the resistance offered by the resistive force produced with the interaction of magnetic parameter, i.e., the Lorentz force. It is seen that SWCNTs have greater retardation than that of Al2O3 and Cu-water nanofluid. However, suction also favors to decelerate the profile significantly than that of injection. Furthermore, Figure 5 examines the significance of the permeability parameter on the nanofluid velocity distribution. Similar to the magnetic parameter, resistive force offered by porosity also causes a similar behavior on the velocity profiles for each of the nanofluids. The influence of the suction/injection has the same tendency on the profiles as described in the earlier description. The control of suction/injection due to the permeable surface is shown in Figure 6 for the velocity distribution of nanofluids. Generally, the pressure differential occurs by the elimination of air from the space. Therefore, the limited pressure is exerted by the external air. The pressure in one part of the system is reduced in comparison to another; there will be force exerts from the fluid of higher pressure region to lower. However, with escalating suction, the pressure increases, and this leads to decelerate the surface thickness, whereas impact is reversed for the case of injection. The case of impermeability is a particular case which validates with the earlier result. The significant characteristics of the controlling parameters on the fluid temperature is observed and presented. The role of these parameters enhances the thermos-physical properties significantly. Therefore, the current study discloses the properties of particle concentration, magnetic and porosity parameters, suction/injection, and unsteadiness parameter. Figure 7 displays the role of particle concentration on the nanofluid temperature with an interaction of suction/injection. The three-layer variation explains the distribution of different parameters on the Cu, Al2O3, and SWCNT-water-based nanofluids, respectively. Furthermore, the fluid temperature boosts its maximum trend in case of Cu-water nanofluid since it is well known that Cu is a good conductor of heat. Furthermore, with the increase in volume fraction, the profile decelerates significantly. Moreover, suction produces more energy to boost the profile rather than the impermeability of the surface and the case of injection. Figure 8 demonstrates the role of unsteadiness parameter that has important characteristics on the nanofluid temperature. Again, increasing unsteadiness, the fluid temperature decelerates in an order of preference such as Cu, Al2O3, and SWCNT-water nanofluid. Therefore, it suggests that effectiveness of the Cu nanoparticle is higher than that of other nanoparticles presented in this study. This gives a suggestive measure for the increasing thermal properties of the Cu-water nanofluid since the proposed thermal conductivity of the nanofluid enhances due to increase in particle concentration. Figure 9 exhibits the effects of the heat source on the nanofluid temperature distribution for different suctions/injections. The inclusion of additional heat suppresses the fluid temperature. In a comparative analysis, it is marked that the Cu-water nanofluid exhibits its maximum strength than other nanofluids. However, no significant change is marked for the variation of suction/injection. Figure 10 depicts the behavior of the thermal radiation in conjunction to the other contributing parameters on the temperature distributions of the nanofluids. Thermal radiation is due to the release of the electromagnetic waves from the fluid particles that is nothing but the renovation of thermal energy into the electromagnetic energy. With an increase in thermal radiation, the profile rises up and therefore the fluid temperature boosts up. This is because most of the solids and fluids are considered to be the surface phenomena and the interior molecules help to emit the radiations.Figure 2
Variation ofϕ on velocity profile.Figure 3
Variation ofb on velocity profile.Figure 4
Variation ofM on velocity profile.Figure 5
Variation ofDa on velocity profile.Figure 6
Variation offw on velocity profile.Figure 7
Variation ofϕ on temperature profile.Figure 8
Variation ofb on temperature profile.Figure 9
Variation ofδ on temperature profile.Figure 10
Variation ofNr on temperature profile.Finally, the simulated results of the rate coefficients for several contributing parameters are obtained and presented in Table3. The nanoparticle concentration enhances the shear rate coefficients whereas heat transfer rate decreases in magnitude. From the tabular results, it is quite clear to see that the rate coefficients are much higher in case of Cu-water nanofluid in comparison to other nanofluids. Furthermore, the resistive forces such as magnetic and porosity of the medium favors to enhance the shear rate and opposite trend is rendered for the heat transfer rate. An increase in suction enriches the rate coefficients significantly.Table 3
Numerical computations of physical quantities used in engineering applications.
ϕCu-waterAl2O3-waterSWCNTs-waterMDabfwPrδNrCfNuxCfNuxCfNux0.110.10.50.56.210.52.6073−4.38912.2782−4.38742.1858−4.22660.153.0849−3.76912.5915−3.77282.4523−3.57000.23.6065−3.21902.9466−3.23122.7597−3.00290.122.9684−4.41952.6735−4.42392.5920−4.263733.2863−4.44433.0137−4.45282.9387−4.292710.22.6450−4.39242.3200−4.39142.2290−4.23080.32.6820−4.39562.3610−4.39532.2713−4.23480.10.12.8167−4.15212.4481−4.14392.3432−3.99320.32.7139−4.27262.3645−4.26782.2657−4.11190.5−0.51.5958−2.14901.5570−2.16021.5442−2.150802.0563−3.11861.8913−3.12141.8433−3.05200.512.6073−1.56442.2782−1.55872.1858−1.510722.6073−2.26702.2782−2.26022.1858−2.18726.222.6073−5.08172.2782−5.09712.1858−4.939632.6073−5.68482.2782−5.71292.1858−5.555310.12.6073−4.04412.2782−4.04082.1858−3.89260.32.6073−4.22042.2782−4.21802.1858−4.0634


## 5. Conclusion

The radiative heat transport phenomenon on the two-dimensional flow of water-based nanofluids over an elastic surface is carried out in the current investigation. Here, the electrically conducting nanofluid past a porous surface embedding with porous matrix is presented. The effect of heat source is also included to examine the heat transfer properties. Numerical approach is employed for the solution of the flow phenomena designed by the proposed model. Furthermore, the important characteristics of the physical parameters are laid down here:(i)
Comparative analysis shows a pathway for the further investigation of the current problem under study for the behavior of several nanoparticles in the water-based fluid with the interaction of various characterizing parameters.(ii)
Particle concentration decelerates the velocity distributions causing a special effect to enhance the bounding surface thickness whereas the thermal bounding surface behaves in the reverse order, and it clarifies that Cu nanoparticle has a greater role in both the profile in comparison to Al2O3 and SWCNT nanoparticles.(iii)
The unsteadiness overshoots the velocity profiles for which the thickness of the bounding surface thickness retards; moreover, similar trend is marked for the temperature distribution. However, steady state conditions preserve maximum magnitude for both the profiles.(iv)
An augmentation in suction enriches the profiles of velocity in comparison to injection, whereas heat source diminishes the fluid temperature significantly.(v)
The shear rate coefficient rises with increase in particle concentration, whereas heat transfer rate shows its opposite impact.



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