Impact of melting heat transfer and thermal radiation on MHD hybrid nanofluid flow via a stretching cylinder in a porous medium

Introduction

Many applications of manufacturing like food production, underground water transportation, solar heated, thermal insulating properties and fuel production etc. are consequences of heat transfer past a fluid flow. The outcome of the product quality in these industries depends upon the velocity gradient and the rate of heat transfer through the surface. The application of nano technology in various fields has attracted many researchers to work on it. The concept of nano fluid was firstly given by Choi [1] in 1995 and it was investigated by him that nanofluid has a higher thermal conductivity with respect to the base fluid. The proficiency of heat transfer in nanofluid has a significant impression on the size of particle and percentage of volume fraction of particle dispersed in the base fluids. Dilute solutions like water, oils and ethylene glycol form nanofluids when particles (nano-meter sized) of copper, carbon or the other material having some thermal conductivity are suspended in it. The devices like refrigerators, nuclear reactor, cancer therapeutics require nanofluids for their better working. Nanofluids have a great importance in electronic and automobile industries. Several manufacturing exercises like biomedicine / pharmaceutical processes, fuel cells, pasteurization of food, microelectronics etc admits the applications of heat transfer. Later on Lee et al. [2] , Choi et al. [3] , Yoo et al. [4] and Hai et al. [5] observed that the addition of a small amount of nanoparticles in nanofluids increase the thermal conductivity of the fluid significantly. Crane [6] was the first who found the exact analytical solution of the problem dealing with the boundary layer flow through a stretching of elastic sheet moving linearly in its own plane. The heat transfer flows under the effect of a magnetic field in presence of a porous medium play important role cross-hatching on ablative surfaces, transportation cooling of rocket boosters and combustion chambers in which film vaporization occurs.

The process of filtration and purification in chemical engineering, the underground water resource's study in agriculture engineering and the study of motion of water, oil and natural gas via oil reservoirs are the applications of flow in a porous medium. Various approach have been constructed to control the functioning of the boundary layer but the principle of MHD affect the flow of the fluid as per the requirement by making changes in the boundary layer patterns. A wide relevant works is present related to the heat transfer in fluidflow through continuously stretching rods and sheets. The boundary layer flow with natural convection in a nanofluid embedded in a porous medium is studied by Gorla and Chamkha [7] . Keeping in mind the application of flow via a porous medium Raptis and Kafoussias [8] and Kim [9] worked on MHD heat and mass transfer flow carrying in presence of porous medium. Wang and Chiu-On Ng [10] studied the effect of slip in the fluid flow due to a stretching cylinder numerically and analytically and observed that the velocity lowers greatly due to slip.

Nomenclature

'

Derivative with respect to η ϕ nanoparticle volume fraction '' Second derivative with respect to η '''

Third derivative with respect to η

The problem dealing with the fluid flow in the outer of a stretching cylinder has been studied by Wang [10] in 1988 which enables hollow tube extrusion in the external of fluid flow through it. He maintained the lowering of thickness of the hollow tube with keeping outer diameter of the tube almost stagnant and found that by doing so the solution resembles with the exact results of the Navier-Stokes. Ishak et. Al. [11] studied steady laminar flow involving incompressible fluid bearing electrical conductivity in the outer of a stretched cylinder and obtained numerical results for MHD fluid flow and transfer of heat. Ahmad et al. [12] studied the influence of heat source/sink on the nanofluid flow passing through a stretching cylinder. He observed that there is a dominant impact on local Nusselt number as solid volume fraction is increased. The impression of surface heat flux on steady laminar flow due to a stretching cylinder place in an incompressible viscous fluid is examined by Norfifah Bachok and Anuar Ishak [13] . It is observed by them that with mounting values of curvature parameter, at the surface of the cylinder heat transfer rate and surface shear stress increase. S. Mukhopadhyay [14] investigated the impact of mixed convection on heat transfer through a permeable stretching cylinder and founded that the velocity and temperature have opposite trends as permeability parameter is increased.

When Recently, hybrid nanofluid have gained its remarkable attraction in various applications used for energy storage in photo electric devices, optical fibre coating, drug reduction, nuclear reactor cooling, cooling of electronic devices, coolant in machines, welding, lubrication, heat exchanger in solar collector and many more. Hybrid nanofluids are liquids in which two nano particles of different materials are dispersed into base fluid. Hybrid nanofluid causes significant enhancement in the features of heat transfer. Silver-water and titanium dioxide-water nanofluid flow through a stretching sheet was studied by Acharya et al. [15] and observed that Nusselt number (magnitude) falls with rising positive values of volume fraction Silver-water nanofluid. Sureh et al. [16] firstly found hybrid nanofluid practically and noted that increase in volume fraction of nanoparticles results hike in viscosity as well as thermal conductivity of hybrid nanofluid. Devi and Devi [17] numerically studied the impact of hybrid nanofluid ( )

flow past a stretching surface placed in porous medium. Javad et al. [18] analyzed the influence of thermal radiation on MHD hybrid nanofluid flow through a stretching surface. Madhesh and Kalaiselvam [19] conducted experimental analysis on hybrid nanofluid and explored the impacts of forced convection on heat transfer coefficient, thermal characteristics and thermal conductivity. Shahirah Abu Bakar et al. [20] studied the impact of thermal radiation in a porous medium on hybrid nanofluid

/ Ag TiO H O past a stretching sheet. Melting heat transfer is associated with challenges in phase change occurring in the process of manufacturing taking place in glass treatment, laser ablation, electromagnetic crucible systems, metallic processing etc. Gorlaa et al. [21] analyzed the nanofluid flow through a melting surface in a porous medium and found that within the boundary layer temperature of the fluid falls as the melting parameter increases. Naganthran et al. [22] work involved the influence of melting heat transfer on Carreau thin hybrid nanofluid film flow through a sheet in an unsteady acceleration. They noted that without affecting the film thickness (liquid) melting heat transfer lessens the heat transfer rate. It was examined by Hayat et al [23] in the study of melting magneto convective nanofluid flow across a stretching sheet under viscous dissipation and Joule heating that heat transfer rate is squeezed with melting parameter. It was explored by Soomro et al [24] under the study of flow MHD Sisko nanofluid via a linear stretching sheet that mounting value of melting parameter results in lowering the shear stress rate. A study of impact of thermal radiation and fluid-particle suspension along with melting heat transfer on hyperbolic tangent fluid through a stretching sheet was explored by Ganesh Kumar et al [25] . Hayat et al. [26] analyzed the comparative study of the flow of hybrid nanofluid (MWCNTs + Ag + Kerosene oil) through a stretched cylinder with nanofluid (MWCNTs + Kerosene oil) under melting heat transfer. Waqas et al. [27] investigated the impact of melting heat transfer on hybrid nanofluid flow along a stretched surface in porous medium. Kaya and Ahmad [28] explores how changes in thermal conductivity and viscosity affect the flow of magneto hydrodynamic (MHD) heat transfer over a stretching or shrinking sheet. They found that an increase in the melting parameter brings results to an enhanced heat transfer rate which indicates that heat flows from the regions of lower to higher melting rates. Additionally, the fluid temperature is observed to be higher during the process of shrinking in comparison to the stretching process. Farroq et al. [29] explores the behaviour of heat transfer of (AA7075-AA7072/Methanol) hybrid nanofluid flow by incorporating the effects of non-uniform heat sources with the heat flux model of Cattaneo-Christovl. The outcomes found by them hold significant potential for improving in the field of thermal management occurs in aerospace applications, commonly used in aircraft engines. Additionally, their results could lead to significant improvement in efficiency of automotive cooling, enhance heating dissipation in electronic systems and encourage the development of technologies in renewable energy in concentrated solar power plants. Ramprasad et al. [30] examined magneto hydrodynamic (MHD) fluid flow within diverging and converging channel geometries, highlighting the role of melting heat transfer. The analysis integrates a hybrid approach, integrating numerical methods particularly the finite element Galerkin method along with artificial neural network (ANN) modelling to deal with study of the fluid dynamics and thermal behaviour throughout the different channel configurations.

Mathematical formulation

The laminar boundary layer flow of the hybrid nanofluid containing nanoparticles of CuO and 2 3 Fe O in the base fluid 2 H O across the permeable stretching cylinder of radius a and with the constant surface mass transfer w U is considered.

It is assumed that there exist thermal equilibrium and no slip between the base fluid water and the nanoparticles. The coordinate system is chosen in such a way that z and r axes represent the axis and radial direction of the cylinder respectively as shown in Fig. 1. B is employed to the cylinder. For neglecting the impact of magnetic field induced, Magnetic Reynolds number is considered to be very small as compared to the applied uniform magnetic field. Moreover, the motion of the hybrid nanofluid is considered in a situation in which no effect of viscous dissipation, Ohmic heating and Hall Effect exists. The thermo physical properties of the hybrid nanoparticles and the base fluid are displayed in table [1]. The differential equations describing the boundary layer motion of the fluid under above specified assumptions in dimensional form are expressed as follows:

Subject to the boundary conditions:

Here the hybrid nanofluid velocity components along the direction r and z axes are u and w respectively.

where f µ is the dynamic viscosity of the base fluid, φ is the non-dimensional concentration, f ρ

where dash denotes differentiation with respect to η and m deals with the melting heat transfer parameter. The similarity transformations given by eq (10) reduce the governing equations (2-4) to following ordinary differential equations as:

Where the Reynolds number Re , the magnetic parameter M , porosity parameter Λ , Heat source parameter Q and thermal radiation parameter are given by

Further, the constant , , , , A B C D E are taken as ( )

Using the similarity transformations defined in the problem, the boundary conditions given by eq. ( 5) and ( 6) become:

The pressure can now be determined from eq. ( 13) in the following form:

The skin friction coefficient and the Nusselt number are defined as follows:

Furthermore, the skin friction w τ and the heat transfer w q from the surface of the cylinder are given as:

where k denotes the thermal conductivity. Using the similarity transformations given by ( 10) in ( 14) and ( 15) we get:

Results and Discussion

The effects of various parameters in the current problem on the fluid flow velocity, temperature, skin friction and Nusselt number have been discussed in order to deliver a good insight of the problem addressed in this study by setting numerical values to the parameters involved in the problem. For most of the results unless otherwise stated, the values of the parameters are chosen as

and 1 9 ≤ Λ ≤ has been performed to observe the influence on the flow of hybrid nanofluid through the cylinder. The velocity and temperature profiles for various values of porosity parameter Re with constant magnetic field and Prandtl number are displayed in figure [2] and figure [3] respectively. It can be noted from the figures that the impression of increase in Reynolds number Re causes decrease in both the velocity and temperature of the hybrid nanofluid flow. Figure [4] and [5] Fe O nano particle volume fraction 1 φ and 2 φ on velocity and temperature profile is realized in figures [6] and [7] . It is inferred from the figures [6] and [7] that increasing values of 1 The impression of different values of melting parameter on the velocity distribution and temperature profile is displayed in figures [8] and [9] . The consequences of boosting the values of melting parameter appearing in these figures show that within the boundary layer there is a decrease in both the velocity and temperature distribution. The consequences of heat source sink parameter and thermal radiation on temperature profile of the fluid flow is layout in figures [10] and [11] respectively. These figures realize that the temperature of the fluid declines away from the surface of the cylinder and becomes zero far away from the cylinder for a particular value of Q and Rd . It can be perceived also from the figures [10] and [11] that there is an upsurge in the temperature of fluid for growing values of Q and Rd . The figure [12] set forth the variation of Reynolds number Re and porosity parameter Λ on ( )

'' 1 f (skin factor or surface drag force). Surface drag force get elevated monotonically with applying increments Re and Λ . The surface heat transfer rate (or Nusselts number) ( ) ' 1 θ is displayed in figure [13] for different Re and Λ . It is observed from this figure that Nusselts number shows uplift when Reynolds number values are mounted and a contradictory pattern is noted with rising values of porosity parameter. Table [3] presents the comparison of current results with the outcomes found by the researchers in past studies and it can be concluded from this comparison that the current results resembles with them. Wang [10] Ishak et. Al. [11] Present Results ( )

for different values of Re and Λ .

Conclusions

Melting heat transfer impression on hybrid nanofluid flow consisting and nanoparticles through a stretching cylinder placing in porous medium is considered in this paper. The model structure defining the motion of fluid by partial differential equations along with suitable boundary conditions is transformed to ordinary differential equations by imposing similarity transformations. The transformed ODEs including boundary conditions are numerically solved by using software MATLAB and shooting technique. Graphical representation of various physical quantities has been drawn to get detailed information about the problem entertained. For controlled parameters with realistic values, it is noted that there is a decline in velocity of the fluid with growing values of various parameters such as Reynolds number, porosity parameter and melting heat parameter. It can be seen from table [2] and various graphs drawn between temperature and varying parameters used in the problem that the temperature is magnified with heat source parameter, radiation parameter, porosity parameter, and volume fraction of nanoparticles. It is observed that for high temperature involvement nanoparticle concentration of both the nanomaterial has a significant role. From this point of view it may be banked on that this investigation may be favorable in refrigeration systems, cooling systems in air conditioning, automobile devices, electronic devices and nuclear device.

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