1 Introduction

Nowadays, the study of nanofluid is a popular area of research. A nanofluid is a liquid consisting of suspension of solid nanoparticles into the base fluid. Such fluids (water, oil, and ethylene glycol) possess the higher effective thermal conductivity in comparison to the pure base fluids. Nanoparticles with a size of (10 nm, 100 nm) are formed using carbon nanotubes, various metals like Ag, Cu, Fe, Si, and Au, oxides, carbides, and nitrides of these metals. Nanofluids have wide applications in engineering, heating and cooling devices, medicines like nanodrugs, optical sensors, and electromechanical systems. Choi^[1] was the first one who observed the enhanced thermal conductivity of fluids by the use of nanoparticles. Hwang et al.^[2] utilized Al₂O₃ nanoparticles in water-based nanofluids to study the characteristics of convective heat transfer of fluids possessing laminar flows. Mustafa et al.^[3] studied Bödewadt flows and heat transfer of nanofluids. Bhatti and Rashidi^[4] examined Williamson nanofluid flows using a porous stretching sheet with thermo-diffusion and thermal radiation. Hatami et al.^[5] numerically analyzed flows of nanofluids by transferring nanoparticles through porous walls with the reduction and enhancement of the gaps within the walls. Hayat et al.^[6] studied flows of silver and copper water nanofluids with mixed convection and nonlinear thermal radiation. Akbar et al.^[7] analyzed heat transfer effects in peristaltic flows of the porous tube using copper oxide water base nanofluids. Ellahi et al.^[8] discussed shape effects of nanoparticles with entropy generation. Akbarzadeh et al.^[9] examined a sensitivity analysis on thermal and pumping power for the channel flow of nanofluids. A magneto nanofluid is the study of nanofluid in the presence of magnetic field. Some applications of magnetohydrodynamic (MHD) involve magnetic drugs targeting, power generators, MHD sensors, plasma confinement, cooling of nuclear reactors and accelerators. Magneto nanofluids include plasmas, liquid metals and salt water or electrolytes composed of soft ferro-magnetic or paramagnetic particles that are dispersed in a carrier fluid. Magneto nanofluids are significantly used in optical grating, optical switching, nonlinear optical materials, and optical modulators (see Refs. [10]-[15]).

Fluid flows by two disks with rotation have gained much importance amongst the researchers. It is due to its useful applications in gas turbines, aircrafts, engines, computer disk drives, car brake system, rotational air cleaner, extractors, atomizers, evaporators, medical equipments and food processing technologies. Turkyilmazoglu^[16] examined effects of suction in a rotating disk using the asymptotic approach. Hatami et al.^[17] used the least square method to study the heat transfer characteristics for laminar flows of nanofluids between contracting and rotating disks. Yan and Soong^[18] examined the heat transfer analysis in mixed convection flows with transpiration in walls using two permeable rotating disks. You et al.^[19] developed the numerical analysis of elastic-plastic rotating disks with the variable thickness and density. Soong^[20] theoretically studied axisymmetric mixed convective flows between rotating coaxial disks. Hayat et al.^[21] addressed ferrofluid flows with chemical reaction by a rotating disk.

Darcy flows are appropriate in a porous medium with extremely less permeability or with the use of a very low Reynolds number. In comparison to Darcy flows, the situation of non-Darcy flow takes place in the presence of a medium with the high velocity of a fluid and the large Reynolds number. Non-Darcy flow occurs near high flux wells during oil and gas production, water pumping and liquid waste injection. Baytas et al.^[22] studied non-Darcy flows using double diffusive natural convection in an enclosure filled with a step type porous layer. Gireesha et al.^[23] found a numerical solution for a non-Darcy flow in a porous medium using hydromagnetic fluid and heat transfer effects over a stretching surface. Lebeau and Konrad^[24] discussed effects of non-Darcy flow and thermal radiation in the convective embankment modeling. Mabood et al.^[25] investigated non-Darcian convective flows of micropolar fluids saturating a porous medium in the presence of MHD and non-uniform heat source or sink. Rosca et al.^[26] studied mixed convection non-Darcy flows of nanofluids in a porous medium.

Thermal radiation is a way to control the rate of heat transfer in numerous processes involving the low convective heat transfer coefficient. Thermal radiation is applied to many combustion chambers, energy transfer furnaces, rockets, missiles, engines, power plants and nuclear explosions. Turkyilmazoglu^[27] examined time dependent MHD flows with the variable viscosity and thermal radiation. Sheikholeslami et al.^[28] discussed MHD flows of nanofluids and heat transfer over a radiative surface. Zeeshan et al.^[29] observed stretched flows of ferrofluids in the presence of the thermal radiation and magnetic field. Rashidi et al.^[30] analyzed Buoyancy and thermal radiation effects in MHD flows of nanofluids over a stretching sheet. Hayat et al.^[31] examined MHD three-dimensional flows of nanofluids with velocity slip and nonlinear thermal radiation. Hayat et al.^[32] also analyzed convective flows of carbon nanotubes between rotating stretchable disks with thermal radiation effects. The present problem involves the study of nanofluids between two rotating stretchable disks. A concept of the Darcy flow is employed. Characteristics of thermal radiation, Joule heating and viscous dissipation are also examined. Convergent series solutions are obtained using the homotopy analysis method (HAM)^[33-41]. Impacts of various involved parameters on the velocity, temperature, skin friction coefficient, and Nusselt number are examined and analyzed.

2 Model development

We consider axisymmetric flows of nanofluids between two disks. One disk is located at z=0 while the other one is positioned at z=h. Constant temperatures T₂ and T₂ are maintained at these disks. Here, Ω₁ and Ω₂ are the angular velocities of both disks. Rates of stretching for both the disks are a₂ and a₂. The nanofluid comprises of nanoparticles of copper and silver with water as a base fluid. An incompressible fluid fills a porous medium. The porous medium is characterized by the Darcy-Frochheimer relation. The magnetic field with the constant magnitude B₀ is applied along the radial and tangential directions. The Joule heating effect is presented. The induced magnetic field is neglected under the assumption of the low magnetic Reynolds number (see Fig. 1).

With cylindrical coordinates, the governing equations for the present flow are

(1)

(2)

(3)

(4)

(5)

The boundary conditions related to the present problem are

(6)

In the above equations, u, v, and w represent the components of velocity in r-, θ-, and z-directions, respectively, T₁ is the temperature of the lower disk while T₂ is the temperature of the upper disk, σ^* is the Stefan Boltzmann constant, k^* is the mean absorption coefficient, µ_nf, ρ_nf, (ρc_p)_nf, σ_nf, and k_nf are the effective dynamic viscosity, density, heat capacity, electrical conductivity, and thermal conductivity of the nanofluid, and F and K are the Forchheimer coefficient and the permeability of the porous medium, respectively. These definitions are given as follows^[42-45]:

(7)

(8)

(9)

(10)

(11)

Here, ϕ represents the solid volume fraction of the nanoparticles, and the subscripts s, f, and nf are used for nano-solid-particles, base fluid, and thermo-physical properties of the nanofluid.

Suitable transformations are given as follows:

(12)

The equation of continuity is identically satisfied, and Eqs. (2)-(6) are converted into the set of equations given as follows:

(13)

(14)

(15)

(16)

(17)

Non-dimensional parameters appearing in these equations are

(18)

where Re, Ha, λ, F^*, Pr, Ω, R, and Ec are the Reynolds number, Hartman number, porosity parameter, local inertia-coefficient, Prandtl number, rotation parameter, radiation parameter and Eckert number while A₂ and A₂ are the scaled stretching parameters.

Shear stresses τ_rz and τ_zθ in radial and tangential directions at the lower disk are

(19)

The total shear stress is written as follows:

(20)

The skin friction coefficients C₂ and C₂ are

(21)

(22)

where is the local Reynolds number. The Nusselt numbers at both disks are

(23)

and the wall heat flux q_w and the radiative heat flux q_r at lower and upper disks are

(24)

(25)

3 Solutions procedure

Linear operators L_f, L_g, and L_θ and initial guesses f₀(η), g₀(η), and θ₀(η) are

(26)

(27)

with properties

(28)

where c_i (i = 1, 2, …, 7) are the constants.

3.1 Zeroth-order deformation problems

Let q ∈ [0, 1] be the embedding parameter and , and be the non-zero auxiliary parameters. Then, zeroth-order deformation problems are

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

3.2 mth-order deformation problems

The mth-order deformation equations are

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

The general solutions (f_m, g_m, θ_m) in terms of particular solutions (f_m^*, g_m^*, θ_m^*) are

(46)

(47)

(48)

where c_i (i = 1, 2, …, 7) are the constants.

4 Analysis of results 4.1 Convergence of derived series solutions

The HAM involves auxiliary parameters , and . The convergence region of the series solution is controlled and adjusted by using these auxiliary parameters. Thus, in order to obtain proper ranges for these parameters, the curves at 11th-order of approximation are plotted in Figs. 2-4. Appropriate ranges for both nanofluids are , and . Further, the convergence of the series solution for both component of velocity is obtained up to five decimal places, and convergence of temperature is achieved up to four decimal places when . Tables 1 and 2 present properties of the base fluid and nanoparticles and convergence of HAM solutions, respectively.

4.2 Results and discussion

This section explains the graphical results of various involved parameters versus the radial and tangential components of velocity and temperature of fluid. Effects of these parameters on the surface drag force and rate of heat transfer are also observed, and the solid and dashed lines in the following figures represent the Ag-water and Cu-water nanofluids, respectively.

4.2.1 Radial velocity profile

Figures 5 and 6 display the effects of Reynolds number Re and nanoparticles volume fraction ϕ on the radial velocity for both nanofluids. In both figures, the radial velocity is reversely affected. The large Re causes increases in viscous effects because fluid flows are enhanced. In Figs. 7 and 8, similar effects of radial velocity are found for increasing values of stretching parameters A₂ and A₂ for both nanofluids. The effect of porosity parameter λ is displayed in Fig. 9 in which λ decreases the radial velocity. The effect of F^* is exhibited in Fig. 10 which shows a decrease in the radial velocity. Figure 11 shows that, with an enhancement of Ha, there is a decrease in the radial velocity near disks for both nanofluids. It is because around disks the Lorentz force offers resistance to the fluid flow.

4.2.2 Tangential velocity profile

Figure 12 shows the effects of Re on the tangential velocity for both nanofluids. The tangential velocity is a decreasing function of Re. Figure 13 depicts effects of ϕ on the tangential velocity. The tangential velocity is a decreasing function of ϕ. Figure 14 describes the increasing behavior of tangential velocity for both nanofluids for larger Ω. Effects of Ha are discussed for both nanofluids in Fig. 15. Increasing the value of Ha results in a decrease in the tangential velocity, since larger Ha increases the Lorentz force that leads to decrease the magnitude of the tangential velocity.

4.2.3 Dimensionless temperature profile

Figure 16 reveals the effect of Re on the temperature for both nanofluids. The increase in Re decreases the fluid viscosity which reduces the temperature of fluid. Figure 17 shows influence of ϕ on the temperature. Since an increase in the volume fraction enhances the thermal conductivity, both thermal boundary layer and temperature are enhanced. Figure 18 shows increasing effects of Ha on the temperature for both nanofluids. In fact, increasing Ha enhances the strength of applied magnetic field. It provides more heat in order to increase the temperature of the fluid. The temperature is an increasing function of Ec as shown in Fig. 19. An increase in Ec enhances the kinetic energy that results in the enhancement of fluid temperature. Figure 20 shows that the impact of R is to decrease the value of mean absorption coefficient. An increase in the radiative heat transfer leads to the temperature enhancement of the fluid. Figure 21 manifests the increasing behavior of temperature for larger λ. Figure 22 shows that an increase in F^* decelerates the flow of a fluid. Thus, the loss of energy in form of heat is experienced by the fluid which results in an enhancement of temperature.

4.2.4 Skin friction coefficient and Nusselt number

Table 3 depicts the values of skin friction coefficient at the lower and upper disks for both Ag-water and Cu-water nanofluids. Numerical values in this table show that the skin friction coefficient for both disks is reversely affected by increasing Re, Ha, λ, F^*, Ω, and ϕ while similar effects of skin friction coefficient for both disks can be seen for A₂ and A₂ for both nanofluids. Table 4 shows impacts of Nusselt number at both disks for Ag-water and Cu-water nanofluids. From this table, it can be seen that the Nusselt number at both disks is affected in a similar manner for large R and ϕ while it is oppositely affected at both disks depending upon Ha, λ, F^*, and Ec.

5 Concluding remarks

Steady flows of Ag-water and Cu-water nanofluids are analyzed between two rotating stretchable disks using a non-Darcy porous medium. Main points are listed below.

(ⅰ) The radial velocity shows similar behavior with increasing values of porous and non-Darcy parameters.

(ⅱ) Increasing the value of Reynolds number causes the decrease in the tangential velocity.

(ⅲ) The tangential velocity enhances with the increase in the rotation parameter.

(ⅳ) Similar effects of porous and non-Darcy parameters are observed on the temperature profile.

(ⅴ) The temperature is enhanced for both Ag-water and Cu-water nanofluids when ϕ is increased.

(ⅵ) The skin friction coefficient shows the same result at both disks when values of A₂ and A₂ for both nanofluids are increased.

(ⅶ) The higher rate of heat transfer at the upper disk is observed for the Cu-water nanofluid compared with the Ag-water nanofluid.