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DOI: 10.32604/cmes.2022.017539


Flow and Melting Thermal Transfer Enhancement Analysis of Alumina, Titanium Oxide-Based Maxwell Nanofluid Flow Inside Double Rotating Disks with Finite-Element Simulation

Liangliang Chen1, Madeeha Tahir2,*, Sumeira Yasmin3, Taseer Muhammad4, Muhammad Imran5,* and Fenghua Liu1

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
*Corresponding Authors: Madeeha Tahir. Email: dr.madeeha.2017@gmail.com; Muhammad Imran Email: drmimranranchaudhry@gmail.com
Received: 19 May 2021; Accepted: 17 September 2021

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 tiny-sized 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)

β0Magnetic field strength
NuxNusselt number
θDimensionless temperature
B2Stretching parameter at upper disk
PrPrandtl number
T1Ambient temperature attained
qrRadiative heat flux
TFluid temperature
RdRadiation parameter
CfSkin friction coefficient
ReReynolds number
ρfBase fluid density
ρpNanoparticle mass density
f(ζ)dimensionless stream function
(ρcp)nfNanofluid heat capacitance
gGravitational acceleration
ρnfNanofluid density
μViscosity of fluid
ψStream function
KCoefficient of mean absorption
σElectric conductivity
σStephan-Boltzmann constant
ksThermal conductivity of nanoparticles
εPressure parameter
MMagnetic parameter
τwShear stress
αThermal diffusivity of base fluid
βMaxwell parameter
ζSimilarity variable
ω1,ω2Angular velocities
knfNanofluid thermal conductivity
b1,b2Different stretching rates

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 namely Ag, Cu, CuO, Fe3O4,TiO2 and Al2O3. Hassan et al. [5] debated CuAg/water hybrid nanofluid flow through a cone. Sheikholeslami et al. [6] illustrate the impact of electro-hydrodynamic flow of Fe3O4ethylene 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 MHD CuAg/water mono nanoliquid flow. The significant features of flow and thermal transfer in Cu/water nanofluid 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. MHD CuOwater nanofluid 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 on Carbon/water nanofluid. 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 of SWCNTsMWCNTs/Engineoil 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 distance h from 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 strength B0 alongzaxis is 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]:








The melting condition is addressed as


Here the electrical conductivity represented by σnf, the thermal diffusivity simplified as αnf, the density for nanofluid is ρnf, T0 be the melting temperature, the nanofluid heat capacitance symbolized as (ρcp)nf, the dynamic viscosity denoted by μnf, the thermal conductivity for nanofluid demonstrated by knf the kinematic viscosity of base nanofluid characterized by νf.

where nf elucidated the thermophysical properties of nanofluid, solid particles symbolized as s, base fluid can be written as f and the solid volume fraction denoted by ϕ for 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 flux qr is considered as


Here consider that the heat transfers in the flow field are such that the expression T4 may be designed as thermal dependent linearly functions. This is capable by expending T4 in Taylor sequence about the ambient heat T as bellow described.


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


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


2.3 Similarity Transformations

The stream function ψ can 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:







2.5 Non-Dimensional Parameters

The dimension-less sundry parameters are given bellow:


Here the Eq. (13) differentiated w.r.t. to ζ and eliminate ε then we get shorter expression is as follows:


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


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


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 (τzrandτzθ) elucidated the shear stresses for the lower disc along radial as well as tangential direction and the heat flux qw correspondingly.

The skin frictions coefficients in non-dimensional forms are



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


Here the total shear stress τw addressed 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








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,ω3andω4) described the arbitrary test functions and may be observed as the values in (f,s,gandθ) 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:




For typical components(ζe,ζe+1), the above-represented form function ψi can 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 parameter B3 by fixed B1=B2=M=ϕ=0 and Re=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.



4  Results and Discussion

Fig. 2 indicates the radial velocity profile f for varying variations in magnetic parameter M for Al2O3/water and TiO2/water nanofluid flow between double radiative nanofluids. It is clearly illustrated that radial velocity field f is reduced via magnetic parameter M. Magnetic parameter develops a resistance force known Lorentz force that is a delaying force in flow of fluid. The radial velocity component f of Maxwell nanoliquid with change in Reynolds number Re is depicted in Fig. 3. From this picture it is learnt that the Reynolds number Re depresses the radial component of velocity f. 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 fluid f for the various variations of the nanoparticles volume fraction ϕ. The increment in the velocity component f is analyzed for higher nanoparticles volume friction ϕ. 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 velocity f of water-based Maxwell single nanofluid with stretching parameter B2 for upper disk is clarified in Fig. 5. It is mentioned that radial velocity f~of nanoliquid is depresses with stretching parameter for upper disk B2. Fig. 6 illustrate the effect of fluid parameter β against radial flow f of Al2O3/water and 44 nanofluid for distinguish values of fluid parameter β. Here we observed that larger fluid parameter β, decays the velocity of fluid. The radial velocity f for Al2O3/water and TiO2/water nanofluid against melting parameter Ma is elucidated in Fig. 7. Improve in melting parameter Ma reduces the radial velocity field f of single nanoliquid inside parallel two disks. The significant decrement in radial velocity profile f depicts in Fig. 8, due to growing variations of porosity parameter K. Fig. 9 is plotted to show the behavior of stretching parameter for upper disk B2 against tangential velocity g of Al2O3/water and TiO2/water nanofluid flow. It is illustrated that with an increment in variation of stretching parameter B2, the tangential flow g of nanofluid increases. Fig. 10 signifies the significant features of melting parameter Ma vs. tangential velocity g of Al2O3/water and TiO2/water nanofluid flow over two parallel disks. It is examined that tangential velocity of fluid g is decreases via larger variations in the melting parameter Ma.


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 parameter Rd against thermal field of nanomaterials θ is disclosed in Fig. 11. It is analyzed that heat transfer θ is increased with greater thermal radiation parameter Rd. Physically the absorption coefficient reduces for greater thermal radiation parameter and therefore the thermal field enhances. Fig. 12 examines the estimation in energy field θ with different amounts of melting parameter Ma for Al2O3/water and TiO2/water nanofluid flow over two parallel disks. We noted that temperature distribution θ is decays via larger melting parameter Ma. Fig. 13 is designed to scrutinize the behavior of Prandtl number Pr against heat profile θ. It is mentioned that by growing the Prandtl number Pr, 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 distribution θ is decreased with solid volume fraction ϕ of Al2O3/water and TiO2/water nanofluid 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 Maxwell Al2O3TiO2/water nanofluid 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 analyzed vs. 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.


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We have


Putting all required values in Eq. (A1) we, have


Hence equation of continuity trivially satisfied.

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