# Melting Heat Transition in a Spinning Flow of Silver-Magnesium Oxide/Engine Oil Hybrid Nanofluid Using Parametric Estimation

Authors: Muhammad Bilal; Taza Gul; Abir Mouldi; Safyan Mukhtar; Wajdi Alghamdi; Souhail Mohamed Bouzgarrou; Nosheen Feroz
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

This study reports the three-dimensional (3D) flow of Ag-MgO hybrid nanofluid (HNF) over a spinning disc of flexible thickness in the presence of modified Fourier law. The HNF is contained of silver and magnetic nanoparticulate in the base fluid engine oil. The energy transition has been examined in the involvement of melting heat propagation. The highly nonlinear system of partial differential equations (PDEs) is processed by adopting the proper similarity conversions to attain the coupled ODE system. The obtained system of modeled equations is numerically solved by employing the Parametric Continuation Method (PCM). The nature of various constraints, as opposed to the velocities, energy, and mass transmission, is portrayed and described. In comparison to the simple nanofluid flow, the hybrid nanoliquid flow’s velocity and heat conduction are observed to have a significant influence. As a result, the functionality of the hybrid nanoliquid is significantly superior to that of the conventional nanofluid. The positive variation in power-law exponentn and Reynold number Re significantly enhances the fluid velocity. The effect of both melting coefficient and thermal relaxation term reduces fluid temperature.


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## Body

## 1. Introduction

In the analysis of HNF due to its substantial participation in engineering constraints and modern machinery, the examination of HNF flow over a turning disk with energy communication has taken significant interest [1, 2]. The well-recognized uses consist of electric control techniques, cocircling apparatus, aerodynamic systems, whirling machines, biochemical reactions, supercomputer management, and hydrothermal sectors [3]. Lv et al. [4] investigated the effects of magnetism and Hall potential on nanofluid flow across a revolving disc. Their target was to increase the level of heat dissipation for technological reasons. As per the conclusions, the modification of CNTs in water is substantially more favorable than that of other nanoparticles due to their C–C interaction. Li et al. [5] employed the bvp4c packages to perform a percentage approximation for Darcy HNF flow over a pierced rotation disc with heat slip. Khan et al. [6] investigated the chemical reaction that influences Maxwell fluid flow over a diagonally gyrating oscillating disc with the magnetic flux during unstable motion. It should be observed that the energy transference ratio raises drastically when the disc radiation and rotation factors increase. The unsteady slip flow with entropy production over a revolving disc under the action of a ferromagnetic material was studied by Shuaib et al. [7] and Bilal et al. [8]. The slip factor seems to be effective in regulating flow and heat characteristics. The Oldroyd-B fluid flow was investigated by Hafeez et al. [9] using a whirling disc. As the relaxation time factor is increased, the flow rate is spotted to diminish. The fluid potential spectrum is also lowered as the thermal relaxation phase develops. HNF flow through a swaying disc with sessile microbes and chemical reactions was discovered by Waqas et al. [10]. A concordance with previous research and a full simple geometric presentation for key variables aid the presented rebuttal. Tassaddiq et al. [11] created an HNF flow over an indefinite impermeable rotating disc. The role of magnetic flux was used to accurately analyze the positive rotation of nanofluid flow. Their primary project purpose was to raise public awareness about energy usage in scientific and technological contexts. The addition of ferric oxide Fe3O4 nanoparticles enhances the heat transition rate considerably [12].Nanofluids are a new type of solution that operates efficiently in heat exchanger when compared to traditional fluids. When the thermal sensitivity is high enough, nanocomposite can be used in a wide range of thermal processes, including freezing [13–17]. Nanofluid flow is used in a variety of applications, including heat converters, geothermal energy, heat pumps, metallurgy, climate control, the automobile sector, turbines, microelectronics, nuclear condenser networks, ships, medicine, and circuit condensation [18–21]. The vast demand for thermal energy in the era of development of science and technology cannot be met with widely utilized fluids. When similar base liquids were produced with the addition of tiny-sized particles, however, a substantial improvement in thermal properties was seen [22]. In the present analysis, we have utilized the MgO (magnesium oxide) and Ag (silver) nanomaterials in the base fluid. MgO is a chemical made up of Mg2+ and O2- ions at 700–1500°C [23]. For metallurgical and electronic operations, MgO is more practical [24]. Similarly, Ag nanoparticulates’ might be exploited to control bacterial movement in an array of products, involving dental work, injuries and wound therapy, surgery, and biomedical apparatus [25, 26]. Ahmadian et al. [27] investigate a 3D simulation of an unstable Ag-MgO HNF flow with heat conduction induced by a curvy spinning disc going up and downwards. With the dispersion of Ag-MgO nanocrystals, the HNF is created. The issue was solved using the PCM technique. The usage of Ag-MgO is thought to be more effective in overcoming poor energy transfer. Among metal and metal oxide, silver and magnesium oxide nanoparticles have been widely recorded to have broad-spectrum antibiotic assets [28]. Silver nanoparticles are the most widely utilized inorganic nanoparticles, having several applications in biomaterial detection and antibacterial activities [29]. Anuar et al. [30] used Ag and MgO nanocrystals in water to evaluate the energy distribution of a hybrid nanoliquid through an extending sheet with suction and buoyant force effects. The findings show that improving the quantity of Ag nanoparticles in HNF lowers the energy transference. Gangadhar et al. [31] arithmetically addressed the heat transport properties of a hybrid nanofluid mixture combining Au and MgO nanoparticulate. Hiba et al. [32] evaluated the thermal performance of HNF including magnesium oxide and Silver and across a highly permeable hollow microplate under magnetic impact. Recently, several researchers have been reported on the study of hybrid nanofluid flow [33–36].PCM tackles a lot of challenging nonlinear boundary value problems that other numerical techniques cannot solve. Convergence is subject to the relaxation variables and initial strategy for many problems that are generally addressed by traditional computational approaches [37–40]. The PCM’s goal is to determine that the proposed methodology can be used to solve complex nonlinear problems related to industry [41]. Shuaib et al. [42] emphasized the 3D oscillating fluid and energy conductivity across the surface of an irregular elastic revolving disc. The fluid flow has been examined in the context of an external magnetism flux. The phenomena of an ionic fluid flow throughout a spinning disc were discovered by Shuaib et al. [43]. The Poisson’s and Planck models were used to computing the molecular interactions. Dombovari et al. [44] investigated the robustness of nonlinear hydrological systems using a parametric continuation technique. They also looked into static bifurcation, which arises while addressing complex initial value systems with distinctive roots, and devised a method for efficiently determining the points of bifurcation. Ref. [45, 46] may be used to solve the stated challenge in the future.The assessment was aimed at reporting the 3D flow of Ag- and MgO-based HNF over a spinning disk of flexible thickness. The HNF is synthesized with the composition of silver and magnetic nanomaterials in the engine oil. The energy transition is examined with the involvement of melting heat propagation. To evaluate the behaviors of the fluid flow, Tiwari and Das’s model is employed. The nonlinear system of PDEs is processed through the proper similarity conversions to attain the coupled ODE system. The obtained system of modeled equations is numerically solved employing the Parametric Continuation Method (PCM). In the next section, the formulation, solution methodology, and results and discussion have been discussed in detail.


## 2. Mathematical Formulation

In this study, we considered the steady and incompressible flow of Ag-MgO hybrid nanoliquid over a gyrating disk of variable thicknessz=a1+r∗−m, moving with fixed angular velocity Ω about the z-axis. Here, u, v, and w are the velocity component along r,θ,z direction, respectively. T∞ is the free stream temperature, and Tm is the temperature of the melting surface. Figure 1 reveals the flow mechanism over a spinning disk. The modeled equations can be rebound as [47–49]
(1)∂u∂r+ur+∂w∂z=0,(2)u∂u∂r+v∂u∂z=νhnf∂2u∂z2+ν2r−σhnfB02u,(3)u∂v∂r+w∂v∂z=νhnf∂2v∂z2−uvr−σhnfB02v,(4)u∂T∂r+w∂T∂z=khnfρCphnf∂2T∂z2−λu2∂2T∂r2−w2∂2T∂z2+u∂u∂r∂T∂r+w∂u∂z∂T∂r,+2uw∂2T∂r∂z+∂T∂zu∂w∂r+w∂w∂z,(5)u∂C∂r+v∂C∂z=Dhnf∂2C∂z2−kC−C0.Figure 1
Spinning disk geometry.Here,u,v,w exhibit the velocity element, and νhnf, khnf and ρCphnf reveal the kinematic viscosity, thermal conductivity, and volumetric heat capacity, respectively.The boundary conditions are(6)u=0,w=0,v=rΩ,T=T∞,C=C∞atz=0,khnf∂T∂zz=a1−r∗−m=ρhnfλ∗+CsTm−T0wr,z,u⟶0,v⟶0,T⟶T∞,C⟶C∞whenz⟶∞.Table1 shows the thermophysical properties of nanofluids and hybrid nanofluids in terms of viscosity, density, heat capacity and thermal conductivity.Table 1
The thermophysical properties of hybrid nanofluid and model [47].
ϕnf=ϕMgOorϕAgϕhnf=ϕMgO+ϕAgViscosityμμf/1−ϕAuμf/1−ϕAgμf/1−ϕhnf2.5Densityρ1−ϕAuρf+ϕAuρAu1−ϕAgρf+ϕAgρAg1−ϕhnfρf+ϕMgOρMgO+ϕAgρAgHeat capacityρCp1−ϕAuρCpf+ϕAuρCpAu1−ϕAgρCpf+ϕAgρCpAg1−ϕhnfρCpf+ϕMgOρCpMgO+ϕAgρCpAgThermal conductivitykkAu+2kf−2ϕAukf−kAu/kAu+2kf+2ϕAukf−kAukfkAg+2kf−2ϕAgkf−kAg/kAg+2kf+2ϕAgkf−kAgkfkAg+2knf−2ϕAgknf−kAg/kAg+2knf+2ϕAgknf−kAg×kMgO+2kf−2ϕMgOkf−kMgO/kMgO+2kf+2ϕMgOkf−kMgOkfDiffusivityαknf/ρCpnfkhnf/ρCphnfIncorporating the following transformation in Equations (1)–(5) and (6)
(7)u=ΩrFη,v=ΩrGη,w=−R0Ω1+r∗−mR02Ωρf/μf1/n+1Jη,θη=T−TmT∞−Tm,ϕη=C−CmC∞−Cm,η=zR01+r∗−mR02Ωρfμf1/n+1,we get(8)J′η+2Fη+ηmεF′η=0,(9)μhnfμfRe1−n/1+n1+r∗2mF′′η+ρhnfρf−F2η−JηF′η+G2η−ηεmFηF′η=0,(10)μhnfμfRe1−n/1+n1+r∗2mG″η+ρhnfρf−2GηFη−JηG′η−ηmεFηG′η=0,(11)khnfkfRe1−n/1+n1+r∗2mPrθ″η−γρCphnfρCpfmm−1ηε2F2ηθ′η+η2m2ε2Fηθ′′η+J2ηθ′′η+ηmεF2ηθ′η+η2m2ε2FηF′ηθ′η+ηmεθ′ηJηF′η+ηmεθ′ηJ′ηFη+JηJ′ηθη−ηmεθ′ηFη−Jηθ′η=0,(12)φ″ηα21+r∗−2m−1Scd1φη+1ScmaεηR0+1Fηφη−1Sc1+r∗−maGηφη=0.The transform conditions are(13)Fα=0,Gα=1,khnfkfMeRe1−n/1+n1+r∗2mθ′α+ρhnfρfPrJα=0,θα=1,φα=1,F∞=0,θ∞=0,φ∞=0,G∞=0.Here,α is the disk thickness coefficient, Re is the Reynold number, Me is the melting constant, ε is the constant coefficient, r∗ is the dimensionless radius parameter, γ is the thermal relaxation parameter, and Pr is the Prandtl number defined as [47]
(14)α=aR0R02Ωρfμf1/n+1,Re=R02Ωρfμf,Me=T∞−TmCpCsTm−T0+λ∗,ε=rR0+r,r∗=rR0,γ=Ωλ,Pr=μfCpnfkf,where F, G, and J denote the radial, tangential, and axial velocities, and ϕ,θ show the dimensionless concentration and temperature. The deformations are expressed as
(15)fξ=fη−α=Fη,jξ=jη−α=Jη,gξ=gη−α=Gη,θξ=θη−α=θη,φξ=φη−α=φη.Using Equation (15), Equations (7)–(12) take the form
(16)J′ξ+ξmεf′ξ+αmεf′ξ+2fξ=0,(17)μhnfμfRe1−n/1+n1+r∗2mf″ξ+ρhnfρf−f2ξ−jξf′ξ−ξmεfξf′ξ+g2ξ−αmεfξf′ξ=0,(18)μhnfμfRe1−n/1+n1+r∗2mg″ξ+ρhnfρf−jξg′ξ−ξmεfξg′ξ−2gξfξ−αmεfξg′ξ=0,(19)khnfkfRe1−n/1+n1+r∗2mPrθ″ξ−ξ+αmεfξθ′ξ−jξθ′ξ−γρCphnfρCpfmm−1ξ+αε2f2ξθ′ξ+ξ+α2fξθ′′ξ+j2ξθ″ξ+α+ξεmf2ξθ′ξ+ξ+α2m2ε2f2ξf′ξθξ+ξ+αmεjξf′ξθ′ξ+ξ+αmεj′ξfξθ′ξ+jξj′ξθξ,(20)φ″ξα21+r∗−2m−1Scd1φξ+1ScmaεξR0+1fξφξ+1Scma2εR0+1fξφξ−1Sc1+r∗−magξφξ=0,(21)f0=0,g0=1,θ0=1,φ0=1,khnfkfMeRe1−n/1+n1+r∗2mθ′0+ρhnfρfPrj0=0,f∞=0,g∞=0,θ∞=0,φ∞=0.The skin friction is stated as(22)Cfr=τwrρfR0Ω2,Cfθ=τwθρfR0Ω2.Shear forces are(23)τwr=μhnf∂u∂zz=1+r∗−m,τwθ=μhnf∂v∂zz=1+r∗−m.The nondimensional form is(24)Ren/n+1Cfr=μhnfμf1+r∗mr∗f′0,Ren/n+1Cfθ=μhnfμf1+r∗mr∗g′0.


## 3. Numerical Solution

The basic steps of PCM are as follows:Step 1: simplifying Equations (16)–(20) to 1st order with the boundary conditions
(25)χ1=fξ,χ2=f′ξ,χ2′=B∗2χ12+χ2χ5+mξ+αεχ1−χ32A∗1+r∗2mRe1−n/1+n,χ3=gξ,χ4=g′ξ,χ4′=B∗2χ1χ3+χ4χ5+mξ+αεχ1−χ5χ2A∗1+r∗2mRe1−n/1+n,χ5=jξ,χ5′=−−2χ1+mξ+αεχ22,χ6=θξ,χ7=θ′ξ,χ7′=D∗γPr∗χ5χ6χ5′+χ7mm−1ξ+αεχ12+mεχ5χ1+mξ+αεχ12m2ξ+α2ε2χ1χ2+mξ+αεχ5χ2+mξ+αεχ1χ5′−mεχ1χ5−mξ+αεχ1χ7+χ5χ7C∗C1∗1+r∗2mRe1−n/1+nχ52+m2ξ+α2ε2χ12,χ9′=1/Scd1χ8+χ9−1/ScmaεξR0+1χ1−1/Scma2εR0+1χ1+1/Sc1+r∗−maχ3α21+r∗−2m,χ10=0,χ30=1,C∗C1∗1+r∗2mRe1−n/1+nMeχ7+D∗Pr∗χ5,χ6=1,χ1∞,χ3∞,χ6∞,χ8∞.Step 2: introducing parameterp(26)χ1=fξ,χ2=f′ξ,χ2′=B∗2χ12+χ2−1pχ5+mξ+αεχ1−χ32A∗1+r∗2mRe1−n/1+n,χ3=gξ,χ4=g′ξ,χ4′=B∗2χ1χ3+χ4−1pχ5+mξ+αεχ1−χ5χ2A∗1+r∗2mRe1−n/1+n,χ5=jξ,χ5′=−−2χ1+χ5−1p−χ5+mξ+αεχ22,χ6=θξ,χ7=θ′ξ,χ7′=D∗γPr∗χ5χ6χ5′+χ7−1pmm−1ξ+αεχ12+mεχ5χ1+ξ+αmεχ12+m2ξ+α2ε2χ1χ2+ξ+αmεχ5χ2+mξ+αεχ1χ5′−mεχ1χ5−mξ+αεχ1χ7+χ5χ7C∗C1∗1+r∗2mRe1−n/1+nχ52+m2ξ+α2ε2χ12,χ9′=1/Scd1χ8+χ9−1p1/Sc−maεξR0+1χ1−1/Scma2εR0+1χ1+1/Sc1+r∗−maχ3α21+r∗−2m.Step 3: apply the Cauchy principle and discretized Equation (26)
(27)Ui+1−UiΔη=AUi+1,Wi+1−WiΔη=AWi+1.Finally, we get the iterative form as(28)Ui+1=I−ΔηA−1Ui,Wi+1=I−ΔηA−1Wi+ΔηR.


## 4. Results and Discussion

The discussion segment analyzed the comportment of velocity, energy, and mass-circulation as compared to the deviation of numerous physical parameters for hybrid nanoliquid consisting of Ag and magnetic nanoparticles. The default values used while solving the set of 1st order ODEs through PCM code are ϕ1=ϕ2=0.01,Re=0.5,n=1.0,Me=0.2,γ=0.1,Sc=0.2,d1=0.01 and τ=0.4.


### 4.1. Velocity Profile

Figures2(a)–2(e) expose the nature of radial velocity fη and tangential velocity gη profiles versus volume friction ϕ1, volume friction ϕ2, Reynold number Re, power-law exponent n, and Reynold number Re. Figures 2(a) and 2(b) particularize that the velocity field rises with the growing values of volume friction of both silver and magnetic nanoparticulates. Physically, the specific heat capacity of engine oil is much higher than silver and magnesium compounds that is why the increasing quantity of such nanomaterials reduces the average heat capacity of HNF and causes the elevation of fluid velocity. Figures 2(c) and 2(e) display the dominance of both radial fη and tangential velocity gη profiles against Reynold number Re. The upshot of Reynold’s number increases the rotation of the disk, which accelerates the fluid particles and exercises their kinetic energy, which causes the improvement in the velocity field. Similar behavior of radial velocity has been observed versus the increment of power-law exponent n in Figure 2(d). The positive variation in the power-law exponent significantly enhances the fluid velocity in the radial direction.Figure 2
The nature of radial velocityfη and tangential velocity gη profiles versus (a) volume friction ϕ1, (b) volume friction ϕ2, (c) Reynold number Re, (d) power-law exponent n, and (e) Reynold number Re.
(a)(b)(c)(d)(e)


### 4.2. Energy Distribution Profile

Figures3(a)–3(e) illuminate the nature of energy transition θη versus volume friction ϕ1, volume friction ϕ2, Reynold number Re, melting coefficient, and thermal relaxation parameter γ, respectively. As discussed in Figure 2, the specific heat capacity of engine oil is much higher than silver and magnesium compounds that is why the increasing quantity of such nanomaterials reduces the average heat capacity of hybrid nanofluid and causes a rise in internal heat, which encourage both velocity and energy transmission rate. Figure 3(c) reports the Reynold number upshot on the energy profile. The number of rotations enhances with the variation of Reynold number that is why due to internal kinetic energy fluid temperature θη also enhances. The effect of both melting coefficient and thermal relaxation term γ reduces energy contour as shown in Figures 3(d) and 3(e). The specific heat capacity of fluid improves with the flourishing values of melting coefficient, as a result, energy transference enhances.Figure 3
The energy outlinesθη versus (a) volume friction ϕ1, (b) volume friction ϕ2, (c) Reynold number Re, (d) melting coefficient Me, and (e) thermal relaxation parameter γ, respectively.
(a)(b)(c)(d)(e)


### 4.3. Mass Transfer Profile

Figures4(a)–4(d) spot the nature of mass transition φη versus volume friction ϕ1, volume friction ϕ2, Schmidt number Sc, and chemical reaction d1, respectively. Mass transmission enhances with the rising quantity of nanoparticles, because, as we have discussed earlier, the rising values of volume friction parameters ϕ1 and ϕ2 significantly elevated the heat and fluid velocity that is why the mass transition also enhances their effects as shown in Figures 4(a) and 4(b). The upshot of the Schmidt number boosts the fluid kinetic viscosity, which results in the reduction of concentration profile φη as illustrated in Figure 4(c). The energy transport rate is reduced by the chemical reaction variable, while the mass transport rate is increased. An increase in the intensity ofd1indicates that the species concentration interaction is less with the thermal boundary layer and more with the momentum Figure 4(d).Figure 4
The nature of mass transitionφη versus (a) volume friction ϕ1, (b) volume friction ϕ2, (c) Schmidt number Sc, (d) chemical reaction d1, respectively.
(a)(b)(c)(d)Figure5 reports the comparative assessment of PCM technique with Matlab code bvp4c. From Figures 5(a) and 5(b), it can be clearly observed that both techniques show best settlement and PCM procedure is a reliable method. Table 2 describes the experimental values of Ag, MgO, and engine oil. Table 3 displays the comparative analysis of the current work with the existing literature, in which the present work revealed the best settlement with them. Table 4 presents the numerical results for skin friction. It has been observed that the drag force enhances along both radial and tangential direction with the variation of parameter m and τ while reduces the effect of Reynold number. Table 5 communicates the numerical outcomes for Nusselt number versus melting coefficient, Reynold number, and both nanofluid and hybrid nanofluid. As comparative to the simple nanofluid, hybrid nanofluid energy transition rate is faster against volume friction coefficient ϕ1,ϕ2, respectively. Table 6 displays the Sherwood number versus volume friction parameter ϕ1,ϕ2, Schmidt number, and chemical reaction d1 constant, respectively. The mass transference rate diminishes with the rising effect of Schmidt number while enhances against the increasing quantity of chemical reaction parameter and volume friction constants.Figure 5
The comparison between PCM and Matlab built-in package bvp4c.
(a)(b)Table 2
The experimental values of Ag, MO, and engine oil [47].
ρkg/m3CpJ/kg·KkW/mKEngine oil88419100.114Magnesium oxide356095545Silver10,500235429Table 3
Comparative analysis with the existing literature, whenn=1 and ϕhnf=0..
Order of approximationsf′0−g′0Present work0.4993210.500761Zhanget al. [47]0.4972010.509623Xun et al. [50]0.4102210.515911Minget al. [51]0.4102000.515901Table 4
Statistical outcomes for skin frictionf′0,−g′0.
mτReϕ2f′0−g′00.10.41.00.030.04745350.07023280.50.41.00.030.05541230.07399321.00.41.00.030.06648520.08158431.50.41.00.030.07911070.11302611.00.11.00.030.06645880.08171751.00.31.00.030.06647600.07163061.00.71.00.030.06649680.08153621.00.41.00.030.06651510.08143361.00.40.10.030.06294180.11614211.00.40.50.030.06854270.08572231.00.41.50.030.06569350.07925181.00.42.50.000.06441280.07640151.00.41.00.010.07265200.11364611.00.41.00.020.07057450.08930141.00.41.00.030.06860360.08529581.00.41.00.040.06673250.0815843Table 5
Arithmetic results for Nusselt numberknf/kfθ′0,khnf/kfθ′0.
ParametersPCMPCMbvp4cMeReϕ1,ϕ2knf/kfθ′0khnf/kfθ′0khnf/kfθ′00.10.04745350.04845310.04843400.50.03541230.03661220.03660311.00.03648520.03698530.03695421.50.02911070.02714070.02712680.20.05645880.05545550.05544630.30.05747600.05759610.05758700.40.05789620.05899650.05898340.010.06734200.06834600.06833520.020.06832410.06932710.06931600.030.06903240.07131420.07130210.040.07234190.07433190.0743237Table 6
Numerical outcomes for Sherwood number.
Scd1ϕ1,ϕ2Dnf/Dfφ′0Dhnf/Dfφ′00.20.06323280.06423220.40.0629320.06393360.60.06158430.06148450.80.59302610.59102400.10.07143360.07343310.20.07614270.07714260.30.08172230.08672240.40.08225160.08825130.010.06276120.06776350.020.06387320.07287540.030.66874500.77745040.040.70267180.7906714


## 4.1. Velocity Profile

Figures2(a)–2(e) expose the nature of radial velocity fη and tangential velocity gη profiles versus volume friction ϕ1, volume friction ϕ2, Reynold number Re, power-law exponent n, and Reynold number Re. Figures 2(a) and 2(b) particularize that the velocity field rises with the growing values of volume friction of both silver and magnetic nanoparticulates. Physically, the specific heat capacity of engine oil is much higher than silver and magnesium compounds that is why the increasing quantity of such nanomaterials reduces the average heat capacity of HNF and causes the elevation of fluid velocity. Figures 2(c) and 2(e) display the dominance of both radial fη and tangential velocity gη profiles against Reynold number Re. The upshot of Reynold’s number increases the rotation of the disk, which accelerates the fluid particles and exercises their kinetic energy, which causes the improvement in the velocity field. Similar behavior of radial velocity has been observed versus the increment of power-law exponent n in Figure 2(d). The positive variation in the power-law exponent significantly enhances the fluid velocity in the radial direction.Figure 2
The nature of radial velocityfη and tangential velocity gη profiles versus (a) volume friction ϕ1, (b) volume friction ϕ2, (c) Reynold number Re, (d) power-law exponent n, and (e) Reynold number Re.
(a)(b)(c)(d)(e)


## 4.2. Energy Distribution Profile

Figures3(a)–3(e) illuminate the nature of energy transition θη versus volume friction ϕ1, volume friction ϕ2, Reynold number Re, melting coefficient, and thermal relaxation parameter γ, respectively. As discussed in Figure 2, the specific heat capacity of engine oil is much higher than silver and magnesium compounds that is why the increasing quantity of such nanomaterials reduces the average heat capacity of hybrid nanofluid and causes a rise in internal heat, which encourage both velocity and energy transmission rate. Figure 3(c) reports the Reynold number upshot on the energy profile. The number of rotations enhances with the variation of Reynold number that is why due to internal kinetic energy fluid temperature θη also enhances. The effect of both melting coefficient and thermal relaxation term γ reduces energy contour as shown in Figures 3(d) and 3(e). The specific heat capacity of fluid improves with the flourishing values of melting coefficient, as a result, energy transference enhances.Figure 3
The energy outlinesθη versus (a) volume friction ϕ1, (b) volume friction ϕ2, (c) Reynold number Re, (d) melting coefficient Me, and (e) thermal relaxation parameter γ, respectively.
(a)(b)(c)(d)(e)


## 4.3. Mass Transfer Profile

Figures4(a)–4(d) spot the nature of mass transition φη versus volume friction ϕ1, volume friction ϕ2, Schmidt number Sc, and chemical reaction d1, respectively. Mass transmission enhances with the rising quantity of nanoparticles, because, as we have discussed earlier, the rising values of volume friction parameters ϕ1 and ϕ2 significantly elevated the heat and fluid velocity that is why the mass transition also enhances their effects as shown in Figures 4(a) and 4(b). The upshot of the Schmidt number boosts the fluid kinetic viscosity, which results in the reduction of concentration profile φη as illustrated in Figure 4(c). The energy transport rate is reduced by the chemical reaction variable, while the mass transport rate is increased. An increase in the intensity ofd1indicates that the species concentration interaction is less with the thermal boundary layer and more with the momentum Figure 4(d).Figure 4
The nature of mass transitionφη versus (a) volume friction ϕ1, (b) volume friction ϕ2, (c) Schmidt number Sc, (d) chemical reaction d1, respectively.
(a)(b)(c)(d)Figure5 reports the comparative assessment of PCM technique with Matlab code bvp4c. From Figures 5(a) and 5(b), it can be clearly observed that both techniques show best settlement and PCM procedure is a reliable method. Table 2 describes the experimental values of Ag, MgO, and engine oil. Table 3 displays the comparative analysis of the current work with the existing literature, in which the present work revealed the best settlement with them. Table 4 presents the numerical results for skin friction. It has been observed that the drag force enhances along both radial and tangential direction with the variation of parameter m and τ while reduces the effect of Reynold number. Table 5 communicates the numerical outcomes for Nusselt number versus melting coefficient, Reynold number, and both nanofluid and hybrid nanofluid. As comparative to the simple nanofluid, hybrid nanofluid energy transition rate is faster against volume friction coefficient ϕ1,ϕ2, respectively. Table 6 displays the Sherwood number versus volume friction parameter ϕ1,ϕ2, Schmidt number, and chemical reaction d1 constant, respectively. The mass transference rate diminishes with the rising effect of Schmidt number while enhances against the increasing quantity of chemical reaction parameter and volume friction constants.Figure 5
The comparison between PCM and Matlab built-in package bvp4c.
(a)(b)Table 2
The experimental values of Ag, MO, and engine oil [47].
ρkg/m3CpJ/kg·KkW/mKEngine oil88419100.114Magnesium oxide356095545Silver10,500235429Table 3
Comparative analysis with the existing literature, whenn=1 and ϕhnf=0..
Order of approximationsf′0−g′0Present work0.4993210.500761Zhanget al. [47]0.4972010.509623Xun et al. [50]0.4102210.515911Minget al. [51]0.4102000.515901Table 4
Statistical outcomes for skin frictionf′0,−g′0.
mτReϕ2f′0−g′00.10.41.00.030.04745350.07023280.50.41.00.030.05541230.07399321.00.41.00.030.06648520.08158431.50.41.00.030.07911070.11302611.00.11.00.030.06645880.08171751.00.31.00.030.06647600.07163061.00.71.00.030.06649680.08153621.00.41.00.030.06651510.08143361.00.40.10.030.06294180.11614211.00.40.50.030.06854270.08572231.00.41.50.030.06569350.07925181.00.42.50.000.06441280.07640151.00.41.00.010.07265200.11364611.00.41.00.020.07057450.08930141.00.41.00.030.06860360.08529581.00.41.00.040.06673250.0815843Table 5
Arithmetic results for Nusselt numberknf/kfθ′0,khnf/kfθ′0.
ParametersPCMPCMbvp4cMeReϕ1,ϕ2knf/kfθ′0khnf/kfθ′0khnf/kfθ′00.10.04745350.04845310.04843400.50.03541230.03661220.03660311.00.03648520.03698530.03695421.50.02911070.02714070.02712680.20.05645880.05545550.05544630.30.05747600.05759610.05758700.40.05789620.05899650.05898340.010.06734200.06834600.06833520.020.06832410.06932710.06931600.030.06903240.07131420.07130210.040.07234190.07433190.0743237Table 6
Numerical outcomes for Sherwood number.
Scd1ϕ1,ϕ2Dnf/Dfφ′0Dhnf/Dfφ′00.20.06323280.06423220.40.0629320.06393360.60.06158430.06148450.80.59302610.59102400.10.07143360.07343310.20.07614270.07714260.30.08172230.08672240.40.08225160.08825130.010.06276120.06776350.020.06387320.07287540.030.66874500.77745040.040.70267180.7906714


## 5. Conclusion

The 3D flow of Ag and MgO HNF past over a gyrating disk of varying thickness has been reported in the present estimation. The hybrid nanoliquid is synthesized by using silver, magnetic nanoparticulate, and engine oil. The energy transition consequences are examined in the involvement of melting heat propagation. The highly nonlinear system of PDEs is processed through the proper similarity conversions to attain the coupled ODE system. The obtained system of modeled equations is numerically solved through the PCM technique. The key points are rebound as follows:(i)
The radialfη and tangential gη velocities and energy propagation enhance with the rising values of volume friction of both silver ϕ1=Ag and magnetic nanoparticulates ϕ2=MgO(ii)
The upshot of Reynold numberRe improves the velocity and energy transition of fluid flow, due to an increase in the number of disk’s rotation(iii)
The positive variation in power-law exponentn significantly enhances the fluid velocity in the radial direction(iv)
The increasing quantity of nanomaterials reduces the average heat capacity of hybrid nanofluid and causes a rise in internal heat, which encourages both velocity and heat transition rate(v)
The effect of both melting coefficient and thermal relaxation termγ reduces fluid temperature



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