# A Study of Magnetic/Nonmagnetic Nanoparticles Fluid Flow under the Influence of Nonlinear Thermal Radiation

Authors: Ayesha Shaukat; Muhammad Mushtaq; Saadia Farid; Kanwal Jabeen; Rana Muhammad Akram Muntazir
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The present research work scrutinizes numerical heat transfer in convective boundary layer flow having characteristics of magnetic (Fe3O4) and nonmagnetic (Al2O3) nanoparticles synthesized into two different kinds of Newtonian (water) and non-Newtonian (sodium alginate) convectional base fluids of casson nanofluid which integrates the captivating effects of nonlinear thermal radiation and magnetic field embedded in a porous medium. The characterization of electrically transmitted viscous incompressible fluid is taken into account within the Casson fluid model. The mathematical formulation of governing partial differential equations (PDEs) with highly nonlinearity is renovated into ordinary differential equations (ODEs) by utilizing the suitable similarity transform that constitutes nondimensional pertinent parameters. The transformed ODEs are tackled numerically by implementing bvp4c in MATLAB. A graphical illustration for the purpose of better numerical computations of flow regime is deliberated for the specified parameters corresponding to different profiles (velocity and temperature). To elaborate the behavior of Nusselt and skin friction factor, a tabular demonstration against the distinct specific parameters is analyzed. It is perceived that the velocity gradient of Newtonian fluids is much higher comparatively to non-newtonian fluids. On the contrary, the thermal gradient of non-Newtonian fluid becomes more condensed than that of Newtonian fluids. Graphical demonstration disclosed that the heat transfer analysis in non-Newtonian (sodium alginate)-based fluid is tremendously influenced comparatively to Newtonian (water)-based fluid, and radiation interacts with the highly denser temperature profile of non-Newtonian fluid in contrast to that of Newtonian fluid. Through such comparative analysis of magnetic or nonmagnetic nanoparticles synthesized into distinct base fluids, a considerable enhancement in thermal and heat transfer analysis is quite significant in many expanding engineering and industrial phenomenons.


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## Body

## 1. Introduction

The study of non-Newtonian fluids enhanced the substantial attention of researchers due to its extensive utilization in mechanical processing, food industry, paper production, biological engineering, polymer industry, and numerous other concerned industries during the last few decades. In fact, in the mathematical and physical modeling of non-Newtonian fluids, there exist many models which encompass the Walter-B model, the power law model, the Williamson model, and the viscoelastic model that bring forward the analysis of such fluids. However, these models do not fulfill all the rheological properties of non-Newtonian fluids. The Casson model is one of the most preferred models initiated in 1959 [1], which incorporates shear thinning behavior and yield stress. The classification of Casson fluid in the category of non-Newtonian fluids is related with the relationship between the shear stress-strain and its rheological characteristics. It switches its behavior towards a Newtonian fluid above a critical stress and acts like an elastic solid at low shear strain. The following liquids, i.e., soup, tomato, sauce, and human blood, correspond with the characteristics of Casson fluid [2].Within the exceeding developments in industrial and engineering sectors such as transportation, production and supply of energy, and electronics, the requirements for the evolvement of heat transfer and higher thermal conductivity have been increased. To overcome the needs for enhancing the thermal conductivity, small-sized solid particles were suspended into regular fluids and a disruption in the fluids was observed as consequences. Furthermore, nanosized particles were used into convectional-based fluids, which cause hike in the thermal conductivity. Such inclusion of microsized particles flourished the concept of nanofluid [3–5]. The nativity of nanofluid is imputed to exhaustive idea of adding nanosized particles (having capacity of 10–100 nm) into base fluids (glycol, ethylene, and oil) to achieve the high thermal conductivity (which is the prominent attribute of nanofluids). In heat transfer fields, nanomaterials (magnetic, nonmetallic, and nanotubes) own a bright future of heat transfer fluids. By dispersing a small amount of nanoparticles, thermal conductivity can be enhanced to a remarkable level. However, it can be managed by changing the nanoparticle size, as it improves with the declination in nanomaterials. Initially, the research in the field of nanomaterials was based on the enhancement of the conductivity of fluids in the numerous engineering field transport engineering (automotive and aerospace), power plant cooling, drilling thermal storage, etc. Later on, in the era of nanotechnology, inventions were significantly enlarged in the fields of inorganic-based nanoparticles, bio-convection, microorganisms, stem cells, etc. [6, 7].Generally, the formation of nanoparticles consists of titanium dioxide, copper, graphite, aluminium oxide, and silicon monoxide, and these nanoparticles cause elevation of the thermal conductivity than the base fluids. In [8], the dispersion of Cu nanosized particles into ordinary base fluid (ethylene glycol) tremendously enhanced 40% thermal conductivity of fluid. Later on, it was reported that the thermal conductivity of fluids increased more than 20% by the suspension of just 1−5% nanoparticles. So, the analysis of thermal conductivity was experimentally by Boungiorno’s model [9]. Eventually, another model was proposed by the same researcher, and it was perceived that the effects of Brownian diffusion and thermophorsis also have the capability to incline the thermal conductivity [10]. The succession of research on nanofluid can be further found in [11–15]. The synthesis of magnetic nanoparticles broads up a new era of research which is called ferrofluid [16]. These nanoparticles are helpful in the usage of numerous base fluids having roots of applications in many engineerings and industrial fields. Such prescribed particles urge to tackle with the rapidly emerging needs for numerous industries (viomedical engineering and energy needs in engineering). To overcome the challenges of waste water treatment processes, magnetic nanoparticles are tested to eliminate the toxins from such processes. These are also beneficial for targeting the specified portion of the human body to demolish the cancer or tumor cells. Some pioneering research about the ferrofluids can be established in [17–25].The innovations in the field of science are the need of time. This need urges the researchers attention towards flow in a porous medium. Porous medium has numerous directions of exploration in different fields of engineering. Flow in subterranean porous layers is the main cause of energy transformation, i.e., power plants. So, these phenomenons are beneficial in energy engineering. These are also helpful during energy exploration (petroleum refinery engineering), biomedical engineering, and civil engineering. Different models were proposed for handling of such a phenomenon in which amendments are being made with the passage of time. As customary, the porous media dealt by Darcy and Darcy-modified flow models and homogeneous and isotropic porous media. Modern and acceptable models are structures with coarse porosity and heterogeneous media. The urge for the further new models results in the directions of aerosol transport and porous filters [26, 27].The phenomenon of boundary layer flow persuades by stretching surface used in many fabricating processes expeditiously. The problem of stretching surface is highly recommended due to its exceptionally range of applications. It is beneficial in numerous industrial and engineering processes such as extrusion of plastic sheets, hot rolling, melt-kneading, glass fibril, manufacturing of plastic and caoutchouc sheets, and refrigeration of wide metallic plate in a bath, which can be brine. By continuous extrusion of the polymer, filaments and polymer sheets are manufactured in industry from a die to a windup roller and kept away at a finite distance. The thin polymer sheet constitutes a continuously moving surface with a nonuniform velocity through an ambient fluid [28–30].A combination of Newtonian- and non-Newtonian-based fluids along with magnetic and nonmagnetic nanosized particles was firstly explored by A. Hakeem in 2017. Investigations revealed that the effect of magnetic field on magnetic nanoparticles is more influential than on the nonmagnetic nanoparticles. It was also claimed that the specific quantities (Nusselt number and skin friction coefficient) are higher in the phenomenon of non-Newtonian-based nanofluids in comparison of Newtonian-based nanofluids. They extended their investigation towards the two different sorts of Newtonian (water)- and non-Newtonian (sodium alginate)-based fluids of two-dimensional convective flow and heat transfer along a flat plate. The dispersion of nonmagnetic nanoparticle and magnetiteFe3O4 nanoparticles in convectional base fluids was taken into account and concluded effects of the slip conditions and thermal radiation. To compensate with the highly emerging needs for nanoparticles, the improvisation of magnetic/nonmagnetic nanoparticles is analyzed by observing the performance of different base fluids on heat and mass transfer [31, 32]. The variation in thermophysical properties of nanoparticles fluid flow (ρ,Cp,k,σ) for two different base fluids (water and sodium alginate) is briefly examined. Due to the prescribed properties of magnetic and nonmagnetic nanoparticles into Newtonian and non-Newtonian base fluids are chosen for the present study. The current research will be beneficial and sustainable in numerous industries and will overcome the emerging needs of science.


## 2. Mathematical Formulation of the Flow Model

Forced convectional, steady, laminar, and incompressible flow of Newtonian and non-Newtonian base fluids dispersed into magnetic and nonmagnetic nanoparticles of Casson nanofluid is contemplated in the existence of porous medium. The suspended magnetic and nonmagnetic nanoparticles into convectional base fluids are being considered with the assumption of thermal equilibrium. The flow starts off at the dominating edge of the sheet which is inspected as the origin in the frame of reference. The coordinates (x, y) are measured along the stretching sheet in horizontal and vertical direction, respectively, as illustrated in Figure 1. The rheology of incompressible casson nanofluid state can be written down as [33](1)τ=τ0+μσ∗,(2)orτmn=2μB+pλ2πemn,π>πc,2μB+pλ2πcemn,π<πc,where π=emnemn indicates that the factor of the deformation rate that is connected to itself by the usual multiplication and the critical value of such product based of non-Newtonian fluid is represented by πc. emn is the m,nth factor of the deformation rate; μB and pλ symbolize the plastic dynamic viscosity and yield stress of Casson fluid, respectively.Figure 1
Flow model configuration.The free stream or far field velocity of nanofluid is supposed to beU∞. Due to small magnetic Reynolds number, the effects of induced magnetic field is negligible and a uniform intensity of magnetic field B0 is considered. Under these considerations, the boundary layer estimations yield the dimensional form as follows [34, 35]:(3)∂u∂x+∂v∂y=0,(4)ρnfu∂u∂x+v∂u∂y=μnf1+1β∂2u∂y2−μnf1ku1+1β−σnfB2u,(5)u∂T∂x+v∂T∂y=αnf∂2T∂y2−1ρCpnf∂qr∂y,where components of velocity in horizontal and vertical directions are u and v, T is temperature of fluid, and β is the Casson parameter.The quantities μnf,νnf=μnf/ρnf,ρnf,αnf=knf/ρnfCpnf,andknf symbolize dynamic viscosity, kinematic viscosity, density, effective thermal diffusivity, and effective thermal conductivity of nanofluid, respectively, which are mentioned below:μnf=μf/1−Φ2.5σnf=αnfρCpnfσ=σs/σfρnf=1−ϕρf+ϕρsknf/kf=ks+2kf−2ϕkf−ks/ks+2kf+ϕkf−ksσnf=σfϕ1ϕ1=1+3σ−1ϕ/σ+2−σ−1ϕFurthermore,ρCpnf=1−ΦρCpf+ϕρCps is the capability of specific heat and electrical conductivity of nanosized fluid is σnf=αnfρCpnf, where ϕ,s,andf refer to the volume fraction, properties of solids, and fluid of nanoparticles of the fluid flow, respectively.The radiative heating fluxqr is analyzed and expressed using Rosseland approximation in the energy equation [36], i.e.,(6)qr=−4σ∗3k∗∂T4∂y=−16σ∗3k∗T3∂T∂y,where Stefan–Boltzmann constant is σ∗ and k∗ is the mean absorption coefficient. The variation of temperature is assumed to be high enough to continue nonlinear expansion of T4 in Taylor series in the negligence of higher order terms. From equation (6), the energy equation (5) takes the form(7)u∂T∂x+v∂T∂y=αnfd2Tdy2+16σ∗T3ρCpnf3k∗d2Tdy2.The current mathematical model is associated with the following boundary conditions:(8)u=U∞=ax,v=0,T=Tw,aty=0,u⟶0,T⟶T∞,aty⟶∞.


## 3. Similarity Transformation and Nondimensionalization

For the mathematical solutions of the problem, similarity transforms are being introduced as follows [37]:(9)η=yxRex,ψ=νfRex,f=U∞xνffη,T=T∞+T0bxθη,where the local Reynolds number Rex depend on the free stream velocity (U∞) and kinematic viscosity νf of the fluid, and the similarity variable η is brought into use. The selection of stream function ψx,y assures to satisfy equation (3):(10)u=∂ψ∂y,v=−∂ψ∂x.The nondimensional transformed flow model of governing equations (3)–(5) obtained by utilizing the similarity transformation in (9) is as follows:(11)1+1βf′′′+1−ϕ2.51−ϕ+ϕρsρf×12ff′′−1−ϕ2.5ϕ1Mf′−1+1βK1f′=0,(12)1Prknfkf+Rd1+θw−1θ311−ϕ+ϕρCps/ρCpfθ′′+12fθ′−2f′θ=0.The following pertinent parameters are being used during the transmission of the conventional flow model of PDEs into the frame of reference of ODEs:(13)Pr=μcpkf,M=σρfB021U∞,θw=TfT∞,Rd=16T∞33σ∗k∗kf,subjected to the following transformed boundary conditions at η⟶0andη⟶∞:(14)f′0=1,f0=0,θ0=1,f′∞⟶0,θ∞⟶0.


## 4. Numerical Method

The nondimensional coupled nonlinear ODEs in the coordination of eminent boundary conditions are handled numerically by the implementation of thebvp4c scheme in the utilization of MATLAB and renovated into initial value problem. Such renovation of the boundary value problem is allocated some appropriate restricted values as (η⟶∞) for a far field boundary conditions say (η∞=5). We introduce the new set of dependent specified variables as follows:(15)f=z1,f′=z2,z2′=z3,θ=z4,z4′=z5,z3′=1−ϕ2.5ϕ−1−ϕρs/ρf×1/2z1z3+1−ϕ2.5ϕ1Mz2+K11+1/βz21+1/β,(16)z5′=Pr×2z2z4−z1z5×1−ϕ+ϕρCps/ρCpf×1/2knf/kf+Rdθw−1θ3,with the associate boundary conditions:(17)z20=1,z10=0,z40=1,z2∞⟶0,z4∞⟶0.The MATLABbvp4c tool primarily encompasses by default the finite difference code. Generally, this scheme is the fourth-order collocation method. The residual of continuous solution is in charge of the error control and mesh resolution in the grid continuation and tolerance in this method is specified as 10−7.


## 5. Physical Quantities: Surface Drag Coefficient and Thermal Gradient

In the present problem, the shear stress and rate of heat transfer are indicated by engineering quantities of physical interest skin friction coefficientCf, and the Nusselt number Nux is defined as(18)Cf=τwxρfUx2,Nux=xkfqwTf−T∞.The surface drag coefficient at the wall along thex direction is specified as(19)τwx=μnf1+1β∂u∂y|y=0,and the surface heat flux which is denoted by qw is(20)qw=−knf∂T∂y|y=0−16σ∗3k∗T3∂T∂y|y=0,where local Reynolds number is fixed as(21)Rex=ax2νf.After utilizing the nondimensional relation, equation (18) takes the form(22)⟹CfRex1/2=1+1βf″011−ϕ2.5,(23)⟹NuxRex−1/2=−θ′0knfkf1+Rdθw3.


## 6. Results and Discussion

The current work specifically explores the behavior of the magnetic/nonmagnetic nanoparticles (magnetite and aluminium oxide) suspensions into the Newtonian and non-Newtonian base fluids (water and sodium alginate) together with the consideration of heat transfer. Steady, two-dimensional forced convective boundary layer flow of Casson nanofluid due to stretching sheet in the presence of magnetic field effect and thermal radiation suppressed in a porous media is analyzed. Numerical computations are being obtained using MATLAB for many sundry parameters effects on different profiles with the help of bvp4c tool, and the consequences are illustrated tremendously numerically and graphically. Table1 depicts the thermophysical properties of distinct base fluids along with magnetic and nonmagnetic nanoparticles. CfRex1/2 and NuxRex−1/2 are listed in Tables 2 and 3 for Newtonian and non-Newtonian base fluids at distinct nondimensional pertinent parameters. It is eminent that variation in the different sundry parameters ϕandM,Kp,β causes to upsurge and lessen the velocity gradient for both magnetic and nonmagnetic nanoparticles in Newtonian and non-Newtonian base fluids, respectively. While, a decreasing trend is followed up for the thermal gradient. It is to be noted that CfRex1/2 of Newtonian fluids is much higher comparatively to non-Newtonian fluids. The thermal gradient of non-Newtonian fluid becomes more condensed than that of Newtonian fluids.Table 1
Thermophysical properties of distinct base fluids and magnetic and nonmagnetic nanoparticles.
Properties of nanofluidρkgm−3CpJkg−1K−1kWm−1K−1σΩm−1Water (Newtonian base fluid)997.141790.6135.5×10−6Sodium alginate (non-Newtonian base fluid)98941750.63762.6×10−4Nanoparticle (Al2O3)39707654035×106Nanoparticle (Fe3O4)51806709.72.5×104Table 2
Variation in velocity gradientCfRex1/2 for Newtonian and non-Newtonian base fluids at distinct nondimensional pertinent parameters.
ϕMKpβWaterSodium alginateAl2O3Fe3O4Al2O3Fe3O40.05−1.187141−1.216334−1.187925−1.2173330.1−1.182780−1.233517−1.184154−1.2352370.2−1.139906−1.217345−1.142027−1.2199400.3−1.060873−1.149444−1.063316−1.1523920.5−1.160537−1.199047−1.161573−1.2003621.5−1.672274−1.697887−1.672956−1.6987702.5−2.064541−2.084847−2.065081−2.0855493.5−2.394215−2.411515−2.394674−2.4121130.5−1.579767−1.607058−1.580495−1.6079981.5−2.518318−2.534707−2.518753−2.5352742.5−3.193786−3.206538−3.194124−3.2069793.5−3.749776−3.760568−3.750062−3.7609420.5−0.601767−0.637716−0.602745−0.6389291.5−0.777435−0.828521−0.778832−0.8302362.5−0.842285−0.898578−0.843826−0.9004663.5−0.876324−0.935300−0.877939−0.937278Table 3
Variation in thermal gradient−NuxRex−1/2 for Newtonian and non-Newtonian base fluids at distinct nondimensional pertinent parameters.
ϕMKpβWaterSodium alginateAl2O3Fe3O4Al2O3Fe3O40.054.0904384.1035664.1931104.2076110.13.8967203.9213433.9944154.0216190.23.5496453.5950053.6382433.6882690.33.2405003.3067723.3208323.3936190.52.7829262.8227922.8540642.8976841.52.6873662.7304712.7586112.8054902.52.6130642.6578832.6843052.7329143.52.5504812.5964742.6216532.6714520.52.7047992.7474292.7760322.8224331.52.5269672.5733572.5980972.6482992.52.4001892.4484152.4709232.5230033.52.2982922.3476792.3684932.4217730.52.9078162.9496342.9787193.0242571.52.8753452.9142442.9462972.9889272.52.8629782.9008182.9339552.9755313.52.8564062.8936912.9273982.968421The manifestation of the effect of magnetic field (M) of nanoparticles Al2O3 and Fe3O4 into suspensions for both Newtonian (water) and non-Newtonian (sodium alginate) conventional base fluids on velocity fη and temperature θη profiles is given in Figures 2(a), 2(b) and 3(a), 3(b).Figure 2
Variation in magnetic field onf′η for (a) Al2O3 and (b) Fe3O4.
(a)(b)Figure 3
Variation in magnetic field onθη for (a) Al2O3 and (b) Fe3O4.
(a)(b)In Figures2(a) and 2(b), the momentum boundary layer thickness depreciates by enhancing the values of magnetic field for both magnetic and nonmagnetic nanoparticles due to presence of magnetic flux in the perpendicular direction of electrical transmission of nanoparticles fluid flow which cause exertion a resistive drag force called Lorentz force, that opposes the direction of flow and reduce the velocity of fluid. While such retarding force contributes to orchestrate the nanoparticles due to augmentation in kinetic energy, and the temperature profile faces a dominant surge for the increasing values of M. Figures 4 and 5 accurately exhibit the effects of Casson parameter β on different boundary layer profiles. It necessitates that the velocity boundary layer thickness is influenced by the augmented β and gets to diminish for both geometries of nanoparticles. This trend followed up by the fact that β crucially has an ability to remold the properties of fluid such as plasticity and yield stress. The yield stress and β are inversely correlated, and enlarging β reduces the yield stress. Hence, the velocity profile reduces. On the contrary, in Figure 5(a), considerable inclination in the fluid patterns for the increasing values of parameter is beneficial enough in enhancing the thickness of the thermal boundary layer. This is due to the fact that the viscosity of fluid escalates with an increase in the values of β, and such escalation in viscosity leads the temperature to rise.Figure 4
Variation in the Casson parameter onf′η for (a) Al2O3 and (b) Fe3O4.
(a)(b)Figure 5
Variation in the Casson parameter onθη for (a) Al2O3 and (b) Fe3O4.
(a)(b)Evident results which integrate with porosity parameterKp are portrayed in Figures 6 and 7. Due to the existence of porous media, the fluid velocity belittles and turns out as the downfall in the thickness of the momentum boundary layer. Physically, the boundary layer has to tackle with the soaking of a immense amount of fluid in the existence of the porosity factor, and it is perceived that the velocity boundary layer decelerates in the whole flow field. Interestingly, the increasing values of Kp prompt to relatively high temperature of the fluid due to the fact that porous media exerts a force which have a tendency to enhance the resistance in fluid. Such resistive force tends to slow down the motion of fluid and as a result the thermal resistance rises in prominence that can be easily visualized in Figure 7. In comparison between these two geometries of Al2O3 and Fe3O4 nanoparticles, it is observed that the resistance of magnetite nanoparticles (Fe3O4) is higher than that of the nonmagnetic nanoparticles in connection with the flow field. The effects of variation in velocity and temperature profiles against the volume fraction ϕ are precisely delineated in Figures 8 and 9. The fluid velocity and temperature bear a trend of being enlarged within the increment of ϕ. Such behavior follows up due to the increase in the density of nanoparticles which retards the fluid viscosity and consequently the velocity profile got enriched. The thermal conductivity of water-based magnetite and sodium alginate-based aluminium oxide nanoparticles tend to rise because ϕ causes detection of energy from the boundary layer which results in the acceleration in the thermal state of fluid. Figure 10 shows the behavior of temperature profile along with the mounting values of thermal radiation (R) for Al2O3 and Fe3O4. For the different increasing values of pertinent parameter (R), fluid temperature is seen to be upsurged. Thermal diffusivity achieves an enhancement because such effects of thermal radiation brings a prominence expand in temperature. Moreover, the effects of radiation prompt more heat energy towards the boundary layer, and advancement in the heat can be easily observed in the entire flow region which initiates the thermal nanoparticles profiles to proceed.Figure 6
Variation in the porosity parameter onf′η for (a) Al2O3 and (b) Fe3O4.
(a)(b)Figure 7
Variation in the porosity parameter onθη for (a) Al2O3 and (b) Fe3O4.
(a)(b)Figure 8
Variation in volume fraction onf′η for (a) Al2O3 and (b) Fe3O4.
(a)(b)Figure 9
Variation in volume fraction onθη for (a) Al2O3 and (b) Fe3O4.
(a)(b)Figure 10
Variation in the radiation parameter onf′η for (a) Al2O3 and (b) Fe3O4.
(a)(b)Finally, the ramifications of temperature profile with the variation of temperature ratio parameterθw for water-based magnetic and sodium alginate-based nonmagnetic nanosized particles are established in Figure 11. Eventually, the temperature nanoparticles distribution explores an enrichment with the increasing values of θw. Such behavior fits with the physical agreement that the mounted values of θw reflects the elevation in surface temperature in contrast with ambient temperature; thus, enhancement in the state of temperature is observed. It is of high consideration that magnetic water-based nanoparticles experienced high-temperature profile than those of nonmagnetic sodium alginate-based nanoparticles.Figure 11
Variation in the temperature ratio parameter onθη for (a) Al2O3 and (b) Fe3O4.
(a)(b)The skin friction coefficient, which is an exciting physical engineering quantity in evaluating the viscous stress acting on the stretching sheet and Nusselt number, is analyzed graphically in Figures12 and 13. Figures 12(a), 12(b), 13(a), and 13(b) show the effect of M variations on skin friction coefficient and, in the presence of ϕ for two geometries, face a diminution in surface drag force. It is perceived that the non-Newtonian fluid is much denser than Newtonian fluid. While, comparison between magnetic and nonmagnetic nanoparticles in the existence of magnetic field reveals that both the nanoparticles show more or less the same behavior because induced magnetic field is negligible as compared to applied magnetic field.Figure 12
Variation in the magnetic field parameter on the skin friction coefficient for (a)Al2O3 and (b) Fe3O4 against ϕ.
(a)(b)Figure 13
Variation in the magnetic field parameter on the Nusselt number forAl2O3 and Fe3O4 against ϕ for Al2O3 and Fe3O4 against ϕ.
(a)(b)Figures14(a) and 14(b) show the fluctuation in the Nusselt number with varying volume fraction of nanoparticles against temperature ratio. The magnification in the nanoparticles’ volume fraction enriches the Nusselt number for both Newtonian and non-Newtonian convectional base fluids. This is due to the fact that an enhancement in volume fraction causes to increase the thermal conductivity of the fluids. The presence and absence of magnetic field diminishes and increases the thermal boundary layer thickness. It is observed that the Nusselt numbers for magnetic nanoparticles are minimum and maximum for nonmagnetic nanoparticles.Figure 14
Variation in the volume fraction parameter on the Nusselt number for (a)Al2O3 and (b) Fe3O4 against temperature ratio.
(a)(b)


## 7. Conclusion

This research work reports a comparative analysis of the phenomenon of Newtonian (water)/non-Newtonian (sodium alginate) base fluids with the magnetic (magnetite)/nonmagnetic (aluminium oxide) nanoparticles. The steady boundary layer flow of Casson nanofluid over a stretching sheet fixed in porosity media in association with the transverse magnetic field and thermal radiation is tremendously analyzed. The governing highly ordered nonlinear coupled PDEs are converted into nondimensional ODEs with the help of appropriate similarity transformation. For the purposes of better numerical computations, the ODEs are being handled by numerical technique,bvp4c. The novelty of pertinent parameters is delineated for nondimensional velocity and temperature profiles. The additional remarks of the present study encompass the results as follows:(i)
The velocity profile shares a diminution with the intensification in the values of transverse magnetic field (M), Casson parameter (β), and porosity parameter (Kp). The Newtonian base fluid is much thicker than the non-Newtonian fluid for both the magnetic and nonmagnetic nanosized particles.(ii)
The augmentation in the values of sundry parameters’ magnetic field, Casson parameter, porosity parameter, volume fraction of nanofluid, radiation parameter, and temperature ratio parameter yields hike in the thermal boundary layer. In temperature profile, the thickness of non-Newtonian fluid becomes more condensed than that of Newtonian fluid. The fluid temperature upsurges in the presence of Lorentz forces.(iii)
The increment in the radiation parameter possesses much contact with the temperature profile of non-Newtonian fluid (Casson fluid) as compared to Newtonian fluid.(iv)
The embedded porous media and magnetic parameter have the propensity to surge the heat transfer in the entire flow field.(v)
Heat transfer is greatly influenced by the thermal conductivity of nanoparticles, and nonmagnetic nanoparticles entail high thermal conductivity in contrast with the magnetic.(vi)
It is perceived that the velocity gradient of nonmagnetic nanoparticles’ Newtonian fluids is slightly higher comparatively to that of magnetic for both types of base fluids. On the contrary, the thermal profile shows reverse trend just opposite to velocity gradient.(vii)
Magnetic nanosized particles expedites viscosity (resistance) in comparison with nonmagnetic.(viii)
It was also perceived that Newtonian fluid (water) shares the least ability of heat and mass transfer comparatively to non-Newtonian (sodium alginate).



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