Liangliang Chen,Madeeha Tahir,Sumeira Yasmin,Taseer Muhammad,Muhammad Imranand Fenghua Liu

1Chongqing Water Resources and Electric Engineering Collage,Chongqing,402160,China

2Department of Mathematics,Government College Women University,Faisalabad,38000,Pakistan

3Department of Mathematics,Government College University Faisalabad,Layyah Campus,Layyah,31200,Pakistan

4Department of Mathematics,College of Sciences,King Khalid University,Abha,61413,Saudi Arabia

5Department of Mathematics,Government College University,Faisalabad,38000,Pakistan

ABSTRACT The energy produced by the melting stretching disks surface has a wide range of commercial applications,including semi-conductor material preparation,magma solidification,permafrost melting,and frozen land refreezing,among others.In view of this,in the current communication we analyzed magnetohydrodynamic flow of Maxwell nanofluid between two parallel rotating disks.Nanofluids are important due to their astonishing properties in heat conduction flows and in the enhancement of electronic and manufacturing devices.Furthermore,the distinct tinysized particles Al2O3 and TiO2 in the Maxwell water-based fluid for enhancing the heat transfer rate are analyzed.The heat equation is developed in the occurrence of thermal radiation.The influences of melting impacts are incorporated.The mathematical model is developed in the form of partial differential expressions then converted to ordinary differential equations by employing tool of similarity variables.Finite element method(FEM)is chosen for solving the nonlinear governing ordinary differential equations(ODEs)with necessary conditions.The consequence of flow parameters against the velocity profiles and heat transport field is considered.The noted novelty of this communication is to discuss the thermal transfer of Maxwell nanofluid model through double stretching disks with thermal radiation and melting phenomenon.Further, Al2O3/water and TiO2/water are considered in the modeling.

KEYWORDS Maxwell nanofluid;melting phenomenon;thermal radiation;revolving stretching disks;finite element method(FEM)

Nomenclature

1 Introduction

Advanced technologies realize the critical value of a unique type of energy transport fluids known as nanoliquid,because of the growing requirements of heat.The most significant aspect of the heat transfer system is the heat efficiency of the base fluid.Since non-metallic materials have poorer thermal efficiency than metallic substances.High thermal efficiency nanoparticles dispersed in regular fluids,which are typically made up of metals and oxides,greatly improve the heat proficiency of the host liquid.Therefore,metals are more useful to improving the thermal transfer rate.Choi and Eastman in(1995)coined the term “nanofluid” to describe the regular fluid that contained very tiny(1–100 nm)nanomaterials.Hayat et al.[1]reviewed the analysis of activation in Ree-Eyring nanofluid flow inside double disks.Qayyum et al.[2]scrutinized the entropy production inspirations on Williamson nanoliquid flow insides double rotating disks.Muhammad et al.[3]observed the slip impacts with activation energy across a three-dimensional sheet.Rafiq et al.[4]examined the numerical computations effects of nanofluid containing six different particles namelyAg,Cu,CuO,Fe3O4,TiO2andAl2O3.Hassan et al.[5]debatedCu?Ag/water hybrid nanofluid flow through a cone.Sheikholeslami et al.[6]illustrate the impact of electro-hydrodynamic flow ofFe3O4?ethylene glycol nanofluid over an enclosure.The effects of magnetic fields on the hybrid nanomaterials are scrutinized by Shah et al.[7].Arefmanesh et al.[8]discussed the influence of mixed convection in nanofluid through wavy wall cavity.Ashraf et al.[9]introduced the thermal transfer improvement in blood-base nanofluid across wavy tube.The theoretical analysis of three-dimensional Newtonian nanofluids with permeability porous medium is also examined by Ullah et al.[10].Lu et al.[11]examined the squeezing flow of nanoliquid by adopting MDP.

The development of hybrid nanofluids,which are essentially an aqueous mixture of two or more forms of nanostructures in mixture or composite shape,is the next advancement in nanofluids technology.Hybrid nanofluids are being developed to solve the drawbacks of single suspension and to take benefit of nanoparticle synergy.Hybrid nanomaterials demonstrated that nanofluids have enhanced energy transfer and thermal conductivity,resulting in cost savings in commercial applications.There is little experimental,theoretical,and computational research on hybrid nanofluids.Li et al.[12]simulated the fractional study of hybrid nanoliquid flow across a spinning disk.Gul et al.[13]highlights the conical gap between cone and disk under hybrid nanofluid flow by adopting Homotopy analysis method.Waqas et al.[14]introduced the hybrid nanofluid flow over radiative disk.Shafee et al.[15]introduced the Entropy generation effects along NEPCM charging mechanics utilizing hybrid nanomaterials.Armaghani et al.[16]discussed the role of magnetized hybrid nanofluid flow through L-shape cavity.Shoaib et al.[17]explored the joule heating effects on magneto hybrid nanofluid over radiative rotating disk.The significance of magnetic dipole in hybrid nanofluid across stretchable surface is illustrated by Gul et al.[18].Several evaluations have been discussed and presented in the literature,see for illustration[19,20].

Magnetohydrodynamics(MHD)is a class of physics that describes the magnetic properties of electrically interacting liquids.It usually influences thermal transport and manifests itself as Joule heating and Lorentz strength.MHD includes things like refrigerator cooling,saltwater,plasma,tumor therapy,X-ray radiation,and electrolytes.Many excellent works describing various aspects of MHD have been released.Reddy et al.[21]explored the MHDCu?Ag/watermono nanoliquid flow.The significant features of flow and thermal transfer inCu/waternanofluid inside double plates are scrutinized by Wakif et al.[22].The effects of entropy generation on magnetized hybrid nanoliquid flowing configured by a sheet are investigated by Mumraiz et al.[23].

The concept of porous media is used in various fields of technology and applied science,including mechanics,filtration,petroleum engineering,construction engineering,hydrogeology,and biophysics.MHDCuO?waternanofluid flow within the porous cavity was explored by Sheikholeslami[24].Waqas et al.[25]analyzed the viscous nanofluid flowing across a porous stretchable revolving disk embedded in porous medium.The features of second order slip on nanofluid contain nano-sized nanomaterials past curved channels under porous permeability are explored by Riaz et al.[26].

The thermal radiation behavior in thermal transfer has applications in a variety of thermal engineering fields,including hybrid power solar systems,rocket propulsion,nuclear reactors,rockets,aircraft,and communication technology.It is worth noting that linear radiation is adequate to produce thermal equipment and is widely used in several technological processes.Hayat et al.[27]analyzed the impacts of thermal radiation onCarbon/waternanofluid.The behavior of thermal radiation in hydrothermal nature of nanofluid is scrutinized by Sheikholeslami et al.[28].Li et al.[29]numerically investigate the radiative modified second grade nanofluid aspects.Naqvi et al.[30]studied the heat radiation effects on Casson nanofluid in the occurrence of magnetic field.

Melting heat has piqued the interest of scientists and researchers owing to its various uses in novel industrial processes.In recent times,scientists have concentrated their efforts on creating more long-term,reliable,and affordable energy storing technologies.Unintended thermal efficiency,solar energy and power are all examples of such technologies.Roberts[31]the first to depict the nature of melting thermal transfer in a hot air stream of ice surface.The behavior of melting impact with bioconvection in generalized second grade nanofluid configured by stretching surface is illustrated by Waqas et al.[32].Hayat et al.[33]introduced the melting thermal transport in stagnation point flow of nanofluid with zero mass flux condition across a stretching surface.Ullah et al.[34]discussed the numerical solution ofSWCNTs?MWCNTs/Engine oil hybrid nanofluid flow with melting mechanism.The behavior of melting heat on magneto nanofluid flow past stretching sheet is analyzed by Sharma et al.[35].

As reviewed above,the non-Newtonian nature of nanofluid has vital applications.The key determination of this article is to scrutinize the MHD flow and melting heat transfer for a nanofluid model through a two parallel rotating disks.Novelty of the communication is firstly to scrutinize the axisymmetric flow of Maxwell-water based nanofluid with thermal radiation.Secondly the numerical solution of current problem is solved numerically by applied Finite element method.The numerical solutions are obtained and the role of physical factors on the subjective dimensionless profiles is scrutinized and reflected by graphs.

2 Mathematical Modeling

2.1 Mathematical and Physical Flow Description

Considerable the effect of steady incompressible,axisymmetric electromagnetic flow with heat transformation by using different particles based on Maxwell nanofluid between both revolving stretching disks at the distancehfrom upper to lower disks.The upper and lower disks have the distinct angular velocities denoted by(ω1&ω2)and both disks have distinct stretching rates(b1&b2)along the radial direction and axial direction correspondingly.The magnetic field strengthB0alongz?axisis presented and the cylindrical coordinate system(r?,θ?,z?)is considered in Fig.1.

Figure 1:Physical configuration of flow issue

2.2 Dimensional Governing Boundary Flow Expressions

The governing flow equations are given bellow[36,37]:

With,

The melting condition is addressed as

Here the electrical conductivity represented byσnf,the thermal diffusivity simplified asαnf,the density for nanofluid isρnf,T0be the melting temperature,the nanofluid heat capacitance symbolized as(ρcp)nf,the dynamic viscosity denoted byμnf,the thermal conductivity for nanofluid demonstrated byknfthe kinematic viscosity of base nanofluid characterized byνf.

wherenfelucidated the thermophysical properties of nanofluid,solid particles symbolized ass,base fluid can be written asfand the solid volume fraction denoted byφf(shuō)or fluid nanoparticles.“For the said flow the equation of continuity is satisfied for detail see Appendix”.

Now by employing the Rosseland evaluation of radiative heat fluxqris considered as

Here consider that the heat transfers in the flow field are such that the expressionT4may be designed as thermal dependent linearly functions.This is capable by expendingT4in Taylor sequence about the ambient heatT∞as bellow described.

In above Eq.(8)eliminate the terms of multi order and outer the first degree in(T?T∞)then we get:

Now putting Eq.(9)into Eq.(7)then we have,

2.3 Similarity Transformations

The stream functionψcan be demarcated as follows[38]:

The transformations are as follows:

2.4 Dimension-Less Equations

After applying the similarity variables,we obtain the non-dimensional expressions are as follows:

With

2.5 Non-Dimensional Parameters

The dimension-less sundry parameters are given bellow:

Here the Eq.(13)differentiated w.r.t.toζand eliminateεthen we get shorter expression is as follows:

Through Eqs.(13)and(17)the parameterεwhich shows the pressure can be expressed as:

The above Eq.(15)integrated with respect toζand determined the pressure equation,by taking the limits starting(0to ζ).

2.6 Physical Quantities

The Physical industrial inters in which the coefficients of skin frictions(C1&C2)and the coefficients of Nusselt numbers(Nux1&Nux2)for both discs are described.

Here we take stress(τzr and τzθ)elucidated the shear stresses for the lower disc along radial as well as tangential direction and the heat fluxqwcorrespondingly.

The skin frictions coefficients in non-dimensional forms are

And the coefficients of Nusselt numbers in non-dimensional forms are

Here the total shear stressτwaddressed as:

3 Numerical Process:Finite Element Method(FEM)

Here,the nonlinear dimensionless ODE’s(14)–(16)and(19)with specific boundary conditions(17)are tackled by utilizing the finite element method(FEM)or finite element analysis(FEA).These ODEs are highly nonlinear,so analytical techniques are not useful to solve these expressions.Therefore,the(FEM)is used to compute the solution of highly nonlinear system.The FEM is more effective than other technique.

3.1 Finite Element Technique(FET)

We get the exact solution through the finite element analysis the coupled nonlinearly ODE’s structure(14)–(16)and(19)with specified boundary conditions(17)firstly we let

With

3.2 Variational Formulation

The variational method associated with the expressions(28)–(32)by typical linear component(ζe,ζe+1)is explain as

Here(ω1,ω2,ω3and ω4)described the arbitrary test functions and may be observed as the values in(f,s,g and θ)etc.

3.3 Finite-Element Description

Substitute finite element estimation of the form declared below;the finite element method may be attained from the abovementioned expressions:

By

ω1=ω2=ω3=ω4=ψi(i=1,2,3).

For typical components(ζe,ζe+1),the above-represented form functionψican be defined as

The structure of finite-element method for nonlinear governing equations is mentioned as:

Here[Knm]and[rn](m,n=1,2,3,4)are established as

3.4 Validation of Results

We used finite element method for tackling the nonlinear set of ODEs.Because fewer nodes are required in finite element,less memory is required to perform the full program,resulting in improved accuracy and,as a result,a shorter calculation time than FEM.Physical features like density,heat capacity,thermal conductivity and electrical conductivity of some particles and base liquid are given in Table 1.Characteristics for some physical properties of nanofluids are given in Table 2.For validation of code we compare current outcomes with the numerical outcomes found by Lance et al.[40],Turkyilmazoglu[41]and Kumar et al.[42]for the rotation parameterB3by fixedB1=B2=M=φ=0 andRe=1.0 as summarized in Table 3.We observed good agreement between our results and published literature.Here minimal percentage errors are also display in Table 3.

Table 1:Physical features[39]of particles and base liquid

Table 2:Characteristics of nanofluid

Table 3:Validation of outcomes of distinguished amounts of B3 for f′′(0) by selecting Re=1.0 when B1=B2=M=φ=0

4 Results and Discussion

Fig.2 indicates the radial velocity profilef′for varying variations in magnetic parameterMforAl2O3/waterandTiO2/waternanofluid flow between double radiative nanofluids.It is clearly illustrated that radial velocity fieldf′is reduced via magnetic parameterM.Magnetic parameter develops a resistance force known Lorentz force that is a delaying force in flow of fluid.The radial velocity componentf′of Maxwell nanoliquid with change in Reynolds numberReis depicted in Fig.3.From this picture it is learnt that the Reynolds numberRedepresses the radial component of velocityf′.The physical reason for this is because when the lower disk rotates;the inertial impacts grow,slowing down the flow rate.Fig.4 reveals the vital role of the radial velocity of fluidf′for the various variations of the nanoparticles volume fractionφ.The increment in the velocity componentf′is analyzed for higher nanoparticles volume frictionφ.The converse relationship between the volume fraction and the dynamic viscosity of nanofluid causes the increase in velocity as a physical phenomenon.As a result,the viscosity of ordinary fluid is reduced in response to an increase in nanoparticle volume fraction,and fluid flow is increased[43].The variation in radial velocityf′of water-based Maxwell single nanofluid with stretching parameterB2for upper disk is clarified in Fig.5.It is mentioned that radial velocityf′of nanoliquid is depresses with stretching parameter for upper diskB2.Fig.6 illustrate the effect of fluid parameterβagainst radial flowf′ofAl2O3/waterand 44 nanofluid for distinguish values of fluid parameterβ.Here we observed that larger fluid parameterβ,decays the velocity of fluid.The radial velocityf′forAl2O3/waterandTiO2/waternanofluid against melting parameterMais elucidated in Fig.7.Improve in melting parameterMareduces the radial velocity fieldf′of single nanoliquid inside parallel two disks.The significant decrement in radial velocity profilef′depicts in Fig.8,due to growing variations of porosity parameterK.Fig.9 is plotted to show the behavior of stretching parameter for upper diskB2against tangential velocitygofAl2O3/waterandTiO2/waternanofluid flow.It is illustrated that with an increment in variation of stretching parameterB2,the tangential flowgof nanofluid increases.Fig.10 signifies the significant features of melting parameterMa vs.tangential velocitygofAl2O3/waterandTiO2/waternanofluid flow over two parallel disks.It is examined that tangential velocity of fluidgis decreases via larger variations in the melting parameterMa.

Figure 2:Importance of M via f′

Figure 3:Importance of Re via f′

Figure 4:Importance of φ via f′

Figure 5:Importance of B2 via f′

Figure 6:Importance of β via f′

Figure 7:Importance of Ma via f′

Figure 8:Importance of K via f′

Figure 9:Importance of B2 via g

Figure 10:Importance of Ma via g

The effect of thermal radiation parameterRdagainst thermal field of nanomaterialsθis disclosed in Fig.11.It is analyzed that heat transferθis increased with greater thermal radiation parameterRd.Physically the absorption coefficient reduces for greater thermal radiation parameter and therefore the thermal field enhances.Fig.12 examines the estimation in energy fieldθwith different amounts of melting parameterMaforAl2O3/waterandTiO2/waternanofluid flow over two parallel disks.We noted that temperature distributionθis decays via larger melting parameterMa.Fig.13 is designed to scrutinize the behavior of Prandtl numberPragainst heat profileθ.It is mentioned that by growing the Prandtl numberPr,the solutal field of species is declined.Since the Prandtl number indicates how quickly thermal diffusion occurs in compared to momentum diffusion,increasing the Prandtl number causes the thermal diffusivity to decrease and the temperature to drop.Fig.14 shows that temperature distributionθis decreased with solid volume fractionφofAl2O3/waterandTiO2/waternanofluid flow.

Figure 11:Importance of Rd via θ

Figure 12:Importance of Ma via θ

Figure 13:Importance of Pr via θ

Figure 14:Importance of φ via θ

5 Conclusions

The current study novelty is recent progress in melting heat transfer transportation of MaxwellAl2O3?TiO2/waternanofluid flow through double parallel rotating disks.The conclusive remarks of the present work are as follows:

?An increase in radial velocity of single Maxwell water-based nanofluid is analyzedvs.an improvement in nanoparticles volume friction.

?Tlhe radial velocity of nanofluid is reducing function of larger magnitudes of the melting parameter.

?The fluid flow declines with fluid parameter.

?The significant value of the tangential velocity is noticed via stretching parameter for upper disk.

?Temperature distribution raises for thermal radiation parameter.

?The thermal field of the fluid reduces as Prandtl number[44]increases.

?Larger solid volume friction decreases the temperature distribution.

Acknowledgement:This work was sponsored in part by National Natural Science Foundation of China(No.51869031);and Science and Technology Research Program of Chongqing Municipal Education Commission(Grant No.KJQN201903801)and Huzhou Key Laboratory of Green Building Technology.

Funding Statement:This work is financially supported by the Government College University,Faisalabad and Higher Education Commission,Pakistan.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the current analysis.

Appendix