1 Introduction

The topic of convective heat transfer in nanofluids has achieved much attention. Ethylene glycol, water and oil are examples of poor heat conducting fluids. To achieve enhanced thermal conductivity of such fluids, particles are suspended in base fluids to form nanofluids. Choi^[1] was the first one to observe thermal conductivity enhancement in Cu-water nanofluids. Because of the enhanced heat transfer capability, these fluids are used in many applications like space cooling, microelectronics cooling, transformer oil cooling, improving the effectiveness of hybrid-powered engines and few more. The phenomenon of effective thermal conductivity fluctuations with temperature in nanofluids was observed by Masuda et al.^[2]. Shirvan et al.^[3] studied nanofluid flows filled in the wavy square cavity. The numerical investigation of effectiveness of double piped heat exchanger filled with nanofluids was presented by Shirvan et al.^[4]. The impact of nanofluids on the entropy generation and productivity in the single solar still using the volume of fluid model was addressed by Rashidi et al.^[5]. The mechanism of heat transfer in nanofluids filled in the porous medium with the stretching wall was discussed by Sheikholeslami et al.^[6]. The influence of activation energy and chemical reactions in the Couette-Poiseuille nanofluid flow was analyzed by Zeeshan et al.^[7]. Hayat et al.^[8] analyzed the convective nanofluid flow with characteristics of the variable thickness. The analysis of magneto-fluid has very important applications in metallurgy, chemistry, engineering, physics, in manufacturing polymer, droplet filters pump, electrostatic filters, cooling of reactors, power generators and in designing heat exchangers. Fluids with an electrically conducting property induced by the applied magnetic field, have many applications including boundary layer control, bearings and magnetohydrodynamics (MHD) generators. The applied magnetic field rearranges the particles in the fluid causing a change in heat transfer characteristics of that fluid. Zhang et al.^[9] investigated MHD nanofluid flows in the porous medium with the impacts of chemical reactions and the variable surface flux. The thermal radiation impact on the MHD flow of nanofluids filled between rotating plates was observed by Sheikholeslami et al.^[10]. Srinivasacharya and Upendar^[11] examined micropolar fluid flows with free convection and double stratification. Hayat et al.^[12] analyzed the viscous nanofluid flow induced using rotating disk with the partial slip impact. Heat transfer effects in stream wise transverse flows about the porous barrier were observed by Rashidi et al.^[13]. The MHD flow by a rotating disk with the slip effect and an entropy generation has been investigated by Arikoglu et al.^[14]. Shehzad et al.^[15] studied on three-dimensional MHD flows induced by the radiative surface using Oldroyd-B fluids. Abdel-Wahed and Emam^[16] and Abdel-Wahed and Akl^[17] studied the MHD nanofluid flow caused by rotating disk in addition with the Hall current effect and viscous dissipation.

Lately, the focus has shifted towards the study of fluid flow by rotating disk because of its various applications in industry and engineering including medical appliances, food processing machine, aerodynamical engineering, machines for electric power generating systems, air cleaning and gas turbines. Flows induced by an infinite rotating disk were initially studied by Karman^[18]. In his work, he used appropriate transformations to convert the Navier-Stokes equations into ordinary differential equations. Stewartson^[19] observed the fluid flow between two rotating disks. Turkyilmazoglu^[20] used the rotating disk and examined the mechanism of heat transfer in the nanofluid flow. Jiji and Ganatos^[21] analyzed the laminar flow generated by the two parallel disks. Hayat et al.^[22] examined flows in magnetite-nanoparticles by stretchable rotating disks with partial slip effects.

The association between heterogeneous reaction occurring on the catalyst surface and homogeneous reaction that has been carried out in fluids is very complex. Some general applications of such reactions include dispersion and fog formation, hydrometallurgical industry and polymers, ceramics production, food processing and so on. In many systems, chemical reactions are comprised of both homogeneous-heterogeneous reactions like biochemical systems, catalysis and combustion. These reactions proceed very slowly or not at all except in the presence of a catalyst. Chaudhary and Merkin^[23] discussed the homogeneous and heterogeneous reactions effects in the stagnation-point fluid flow. They considered different diffusivities of the autocatalyst and the reactants. Hayat et al.^[24] considered the Oldroyd-B fluid flow with effects of homogeneous-heterogeneous reactions. Rashidi et al.^[25] reported the effect of chemical reaction in the mixed convection boundary layer flow. Bachok et al.^[26] examined stagnation-point flows in addition with homogeneous-heterogeneous reactions.

The radiation effect has provided a lot of scopes in the fields of physics and industries including glass production, nuclear reactors, polymer processing and also in space technology like in rocket, power plant and missiles. Hayat et al.^[27] discussed the radiative micropolar nanofluid flow accounting Brownian moment and thermophoresis. Fluid flows in Jeffrey nanofluids with the thermal radiation effects and Hall and ion slip were analyzed by Hayat et al.^[28]. Impacts of nonlinear thermal radiation and partial slip on the MHD flow of nanofluids were inspected by Hayat et al.^[29]. Mukhopadhyay^[30] reported flows by the stretching sheet in addition with thermal stratification and the heat transfer effects. Melting heat transfer has much significance in making of semi-conducting materials, permafrost melting, solidifying magma, defrosting in frozen ground, laser manufacturing (selective sintering, welding, and drilling), metal casting and thermal energy storage etc. The laminar flow over the flat plat with characteristics of melting heat transfer was examined by Epstein and Cho^[31]. Melting heat transfer in fluid flows over the flat plate in the presence of porous medium, forced and natural convective modes was discussed by Kazmierczak et al.^[32-33].

The present article examines the impact of melting heat transfer in Cu-water and Ag-water nanofluids flows. The analysis has been carried out in the presence of homogeneous and heterogeneous reactions. Significant aspects of viscous dissipation and thermal radiation are taken into consideration. Since thermal conductivity has a key role in the coefficient of heat transfer, various methods are used to enhance thermal conductivity of poor conducting fluids. Their conductivity is enhanced by suspending nanoparticles in the base fluid. The nanofluids are much effective in industrial cooling, heat exchangers, latest cooling systems, evaporators based thermal electronic systems and nano electromechanical devices applications. Moreover, the magnetic field becomes a part of various applications in physics and engineering. The direction and applied intensity of the magnetic field strongly affect the flow behavior. This magnetic field disorientates the added particles and then rearranges them in fluids resulting in alterations of heat transfer properties of flows. The fluid flow induced by the rotating disk has gained much importance in geophysical and engineering applications including flows in liquid metals pumping, centrifugal machinery, spin coating and turbo-machinery. Furthermore, the heat transfer and modeling for the rotating disk are undoubtedly useful in an electronic cooling system and a micro-electro mechanical system. Thermal radiation is the principal mechanism of heat transfer that is generated by charged particles thermal motion. Its applications in physics engineering and industry include solar power technology, electrical power generation, polymer processing, astrophysical flows and nuclear reactors. Melting plays a significant role in advanced technological developments. It has broad ranges of applications like making of semi-conducting materials, permafrost melting, solidifying magma, defrosting in frozen ground, laser manufacturing (selective sintering, welding, and drilling), metal casting and thermal energy storage. Highly accurate solutions are implemented using a homotopy analysis method (HAM)^[34-40]. Impacts of pertinent parameters are emphasized by graphical results.

2 Formulation

We consider incompressible flows of Cu-water and Ag-water nanofluids. Flows are induced by a stretchable rotating disk at z=0. The disk is rotating about z-axis with the constant angular velocity Ω, while the stretching velocity is considered to be U_w(r)=C₀r, where C₀ is a constant. The magnetic field of strength B₀ is applied parallel to the z-axis. Impacts of induced electric and magnetic fields are considered negligible compared with the applied magnetic field. Viscous dissipation and thermal radiation effects are also considered. Let T_m be the melting surface temperature while T_∞ be the temperature of ambient fluids (see Fig. 1).

We assume the model for homogeneous and heterogeneous reactions as suggested by Chaudhary and Merkin^[23]. For cubic autocatalysis, homogeneous reaction is as follows:

(1)

and the rate is k_cab², while on the catalyst surface, heterogeneous reaction is

(2)

and the rate is k_sa, where the chemical species A₁ and A₂ have concentrations a and b, respectively, while k_c and k_s are rate constants. Moreover, these reactions are assumed to be isothermal. Then, continuity, momentum, energy, and concentration equations are given as^{[35, 37]}

(3)

(4)

(5)

(6)

(7)

(8)

(9)

with boundary conditions

(10)

and

(11)

where u, v, and w are the velocity components, λ is the latent heat of fluid, σ^* is the Stefan-Boltzmann constant, c_p is the specific heat, k^* is the mean absorption coefficient, ρ is the density, T is the temperature, p is the pressure, c_s is the solid surface heat capacity, C₀ is the rate constant, and a₀ is the positive dimensional constant. In addition, (ρc_p)_nf is the effective heat capacitance of nanofluids, μ_nf is the dynamic nanofluid viscosity, α_nf is the thermal diffusivity, k_nf is the effective thermal conductivity of nanofluids, ρ_nf is the effective nanofluid density, D_A and D_B are diffusion coefficients, and σ_nf is the electrical conductivity of nanofluids, which are defined as

(12)

(13)

(14)

(15)

(16)

(17)

where the subscript nf is used for the nanofluid, f is for the base fluid, and s is for the nanoparticle, c is the heat capacity, and ψ is the solid volume fraction of nanoparticles. The boundary condition (11) states that heat carried to the melting surface is equal to the heat needed to raise the temperature of surface from T₀ to T_m (i.e., the melting temperature) plus heat of melting.

Now consider von Karman transformations,

(18)

By using the transformations above, Eqs. (3)-(9) reduce to the following non-dimensional equations:

(19)

(20)

(21)

(22)

(23)

(24)

and the boundary conditions (10) and (11) become

(25)

in which M_e is the dimensionless melting parameter given by

(26)

where δ is the ratio of diffusion coefficients, M is the Hartmann number, Ec is the Eckert number, R_d is the radiation parameter, Sc is the Schmidt number, Re is the local Reynolds number, k₁ is the homogeneous reaction parameter, Pr is the Prandtl number, S_t is the dimensionless parameter corresponding to the stretching rate, and k₂ is the heterogeneous reaction parameter. These dimensionless parameters are given by

(27)

The melting parameter is a combination of the Stefan numbers c_f(T_∞ -T_m)/λ and c_s(T_m-T₀)/λ for the liquid and solid phases, respectively. Furthermore,

(28)

The diffusion coefficients D_A and D_B are assumed to be comparable and hence the ratio of diffusion coefficients reduces to 1, i.e., δ =1, and thus

(29)

Then, Eqs. (23) and (24) become

(30)

subject to the boundary conditions

(31)

The local skin friction coefficient C_f and the Nusselt number Nu_r are given by

(32)

where is the surface tangential stress, is the surface radial stress, and is the surface heat flux. Therefore, the local skin friction coefficient C_f and the Nusselt number Nu_r can be written as

(33)

3 Solution procedure 3.1 Zeroth-order deformation problems

Let H₀, f₀, g₀, θ₀, and Φ₀ be initial approximations as follows:

(34)

with the linear operators , and as

(35)

The linear operators satisfy

(36)

with the constants c_i (i=1, 2, ..., 9).

The zeroth-order deformation problems are presented as

(37)

(38)

(39)

(40)

(41)

(42)

in which ħ_H, ħ_f, ħ_g, ħ_θ , and ħ_Φ are nonzero auxiliary parameters, while the nonlinear operators , and are given by

(43)

(44)

(45)

(46)

(47)

where q is the embedding parameter.

3.2 mth-order deformation problems

The problems at this order are expressed as follows:

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

The general solutions H_m, f_m, g_m, θ_m, and Φ_m in terms of particular solutions H_m^* , f_m^* , g_m^* , θ_m^* , and Φ_m^* are

(60)

(61)

4 Convergence of the series solutions

The HAM is used to determine the series solutions for linear and non-linear problems. This method involves some auxiliary parameters. By choosing appropriate values for these parameters, the desired convergence of homotopic solutions can be achieved. The ħ-curve is sketched at the 10th-order of approximation to determine the suitable ranges of these parameters for both Cu-water and Ag-water nanofluids (see Figs. 2-5). The admissible ranges for ħ_H, ħ_f, ħ_g, ħ_θ , and ħ_Φ are -1.5≤ ħ_H≤ -0.9, -1.5 ≤ ħ_f≤ -1, -1.1≤ ħ_g≤ -0.7, -1.5 ≤ ħ_θ ≤ -1.1, and -2.3≤ ħ_Φ ≤ -0.4 for the Cu-water nanofluid, and -1.5≤ ħ_H≤ -0.9, -1.4≤ ħ_f≤ -1, -1.2≤ ħ_g≤ -0.7, -1.3 ≤ ħ_θ ≤ -1.1, and -2.2≤ ħ_Φ ≤ -0.7 for the Ag-water nanofluid. Furthermore, series solutions converge in a whole region of η (0≤ η ≤ ∞) when ħ_H=-1.1, ħ_f=-1.2, ħ_g=0.9, ħ_θ =-1.2, and ħ_Φ =-0.8. Tables 1 and 2 illustrate the convergence of velocities, temperature, and concentration. Presented values clarify that convergence is achieved up to the 23rd-order of approximation. Thermo-physical properties of water, Ag, and Cu are given in Table 3. It is worth mentioning that the obtained numerical results are in good agreement with the previously published results (see Table 4).

5 Analysis of results

Effects of pertinent parameters on the velocity, temperature, and concentration distributions are observed graphically in this section. The radial velocity distribution f(η) is observed for all considered values of the parameter M in Fig. 6. The magnetic field causes the resistive force to act between the particles resulting in the smaller velocity. Hence, increasing M decreases the velocity. However, M has an opposite effect on the temperature θ(η) (see Fig. 7). This resistive force generates heat that results in the increase in temperature.

Variations in the velocity profile for various values of the melting parameter M_e can be observed in Fig. 8. Since melting leads to the enhanced molecular motion, the velocity along with the boundary layer thickness enhances.

As the volume fraction of nanoparticle ψ is increased, the velocity profile f(η) decreases while the temperature increases (see Figs. 9-10). From the physical point of view, thermal conductivity enhances by adding more nanoparticles into the base fluid, but by such addition, particles get closely packed resulting in the smaller velocity. Figure 11 depicts that the temperature profile θ(η) increases for the larger value of the radiation parameter R_d. Here, an absorption coefficient decreases with the increase in R_d which causes enhancement in the radiative heat transfer rate. Variations of Eckert number Ec are observed in Fig. 12. When Ec is increased, heat is produced in fluids due to friction force, and hence the temperature increases.

Figure 13 presents the behavior of the homogeneous reaction parameter k₂ on the concentration profile. Since k₂ is inversely related to diffusion, larger k₂ causes less diffusion resulting in the increase in concentration. Effects of Schmidt number Sc are depicted in Fig. 14. Since Sc is the ratio of viscous diffusion to the molecular diffusion rate, for the increasing values of Sc, the fluid concentration Φ(η) increases.

Figures 15 and 16 present the skin friction coefficient C_f(Re)^1/2 and the local Nusselt number Nu_r(Re)^-1/2 for increasing values of the nanoparticles volume fraction ψ. Also, the skin friction coefficient C_f(Re)^1/2 is an increasing function of M. It is observed that the skin friction coefficient is smaller for Cu-water nanofluids compared with Ag-water nanofluids. It is found that increasing values of Re result in an increase of the Nusselt number Nu_r(Re)^-1/2.

Figure 17 demonstrates the impact of k₁ on surface concentration for increasing values of k₂. In fact, k₁ is associated with the consumption of reactants. An enhancement in k₁ reduces the surface concentration.

6 Conclusions

The chemical reactions and melting heat transfer in Cu-water and Ag-water nanofluids are analyzed. The following results are observed.

(ⅰ) The nanofluid volume fraction and Hartman number serve to decrease the radial velocity.

(ⅱ) The fluid temperature increases for larger values of the Eckert number, Hartman number, radiation parameter, and nanofluid volume fraction.

(ⅲ) Homogeneous and heterogeneous reaction parameters have opposite effects on the concentration distribution.

(ⅳ) The surface drag force for Cu-water nanofluids is smaller than that for Ag-water nanofluids comparatively.

(ⅴ) The magnitude of surface concentration enhances for decreasing values of the Schmidt number.