1 Introduction

Fluid flow analysis has been a key issue for many decades in experimental and theoretical fluid mechanics. Such analysis includes determining the flow properties in separation systems, calculating moments and forces on aircraft, and investigating atmospheric dynamics, cloud motion, and oceanographic streams. Rotating flow dynamics has critical importance for product applications, physical processes, and modeling of diverse products such as vacuum cleaners, turbomachinery, jet engines, atmospheric flows, centrifuges, geotropic turbulence, and chaos. Fluid flow investigations with a rotating disk have gained tremendous interest among scientists and engineers for the purpose of modernizing industrial processes such as studying acidic fluid kinetic reactions in the petroleum industry and investigating reactions between solid surfaces and fluids, swirl flow around conical diffuser, cooling of disks through impinging jet, shrouded disk rotation, spin coating, spinning disk reactors, and centrifugal pumps. The topic of fluid motion driven by a rotating disk is a classical problem first discussed by von Karman^[1], in which well-known similarity transformations were introduced. Asymptotic solutions were acquired for an incompressible fluid through a rotating disk by Cochran^[2]. Stuart^[3] discussed the impact of uniform suction on the viscous fluid via a rotating disk. Benton^[4] extended Ref. [1] to the case of impulsively-started flow from rest, and extended Ref. [2] to obtain steady-state accuracy of the higher order. This provides an opportunity to discuss the fascinating scenario of fluid motion owing to a rotating disk with different physical and theoretical aspects. Attia and Hassan^[5] explored the Hall current characteristics of fluid motion caused by a rotating disk. Turkyilmazoglu^[6-8] discussed various flow features generated by a rotating disk under different conditions. Xun et al.^[9] elaborated numerically the Ostwald-de Waele liquid attributes of the rotating disk with variable thickness. Yin et al.^[10] elaborated the nanofluid features by considering the geometry of a rotating disk. Mustafa^[11] adopted the nanofluid Buongiorno model to characterize the flow phenomenon considering the rotating disk. Aziz et al.^[12] developed a numerical scheme to discuss the nanofluid flow over a rotating disk with slip effects. Munawar et al.^[13] conducted numerical investigations of viscous fluid flow over an oscillatory disk.

Heat transfer in fluid flow problems is a topic of vast interest owing to its role in numerous industrial and technical processes in the petroleum industry, marine engineering, cooling of metal plates/sheets, distillation columns, fiber spinning, plastic films, aerodynamic extrusion, and nuclear processes. In the field of continuum physics, the model of Fourier's law of heat conduction is the most prominent one. The main shortcoming in the Fourier's law is the generation of parabolic equation for the temperature field, in which any initial interference is detected instantly throughout the whole medium. This effect creates a heat conduction paradox. Various attempts have been made to overcome this paradoxical limitation. One of the best solutions is the Maxwell-Cattaneo law^[14]. The main term in this theory is thermal inertia, in which the thermal relaxation characteristic time denotes the time delay required to initiate steady-state heat conduction in the presence of a temperature gradient. Furthermore, the partial derivative of time in the constitutive relation of Maxwell-Cattaneo law contributes to the development of a damped hyperbolic equation with which heat disturbance propagation attains a finite speed. Li et al.^[15] obtained the numerical results for the viscoelastic fluid flow via a stretching surface with the Cattaneo-Christov law. Shehzad et al.^[16] explored flow features by considering the Cattaneo-Christov theory on the third-grade fluid flow and assuming the surface of exponential. Heat transfer attributes using the Cattaneo-Christov model for the Maxwell fluid were reported by Abbasi and Shehzad^[17]. Shehzad et al.^[18] considered the hydromagnetic flow of Maxwell liquid owing to a moving surface according to the Cattaneo-Christov theory. Meraj et al.^[19] discussed the flow features of Jeffrey fluid with the Cattaneo-Christov heat flux model subject to a vertical sheet. Rauf et al.^[20] tackled the Powell-Eyring fluid flow problem in a numerical way by considering the Cattaneo-Christov heat-mass flux theories. Kumar et al.^[21] considered magnetohydrodynamic (MHD) flow through cone and wedge utilizing the Cattaneo-Christov model.

Flow models considering a porous medium have numerous technical applications. The theory of porous media generally consists of a differential equation that describes the macroscopic motion of fluid. Darcy^[22] examined transport phenomena by experiments on the unidirectional flow, and obtained a linear relationship between the velocity and the pressure gradient through a uniform medium. Several technological processes depend on porous media theories such as hydrology, oil exploration, solar collectors, porous insulation, packed beds, chromatography, heterogeneous catalysis, control of shear stress at the seabed bottom, and oscillatory flow through seabed ripples. The Darcy theory has wide applications in the field of biomedicine and the development of biological clogging and flow through biological tissues^[23]. Sintering of granular material^[24], where a significant amount of pore structures exist, has numerous applications in manufacturing, paper, ceramic products, and textiles. Attia et al.^[25] developed a numerical solution for unsteady, non-Newtonian fluid subject to a rotating disk in a porous medium. Ali et al.^[26] explored the flow characteristics of viscoelastic fluid over a oscillatory surface by considering the impact of porous medium. Reddy et al.^[27] examined the flow behavior over a rotating disk through a porous medium. Hasnain and Abbas^[28] described the flow in immiscible fluids through a porous medium in the annular region of concentric cylinders. Numerical computations were performed for the three-dimensional flow through a bi-directional surface in a porous medium by Siddiq et al.^[29]. Sheikholeslami^[30] applied the control-volume finite-element method (CVFEM) to solve the natural convection flow of a magnetic nanofluid model subject to a porous medium. Sheikholeslami and Shamlooei^[31] discussed the influence of the variable-type magnetic field on the flow of nanofluid in a porous medium by implementing the CVFEM. Khan and Ali^[32] adopted the Jeffrey fluid model to discuss the flow over a stretching surface immersed in a porous medium.

The MHD flow plays a significant role in engineering processes and industry. The MHD flow has notable applications in the purification process of crude oil, electromagnetic pumps, metal coating, drug delivery, biological transportation, sterilized devices, optical grating, and nuclear plants. Sheikholeslami et al.^[33] discussed the influence of Lorentz forces on heat transfer during solidification in an energy storage system. Sheikholeslami^[34] conducted a numerical analysis of the effects of a magnetic field on nanofluid convection heat transfer in a porous lid using the lattice Boltzmann method. Sheikholeslami^[35] numerically investigated the salient flow features of MHD nanofluid via a permeable medium. Analysis of the entropy and exergy of nanofluid under Lorentz forces was conducted by Sheikholeslami^[36].

Our main focus here is to present the numerical results for the MHD flow problem of viscous fluid subject to an oscillatory rotating disk drenched/immersed in a porous medium. The impact of the Maxwell-Cattaneo model on Fourier's law of heat conduction is also considered for the present flow model. The numerical analysis of the Maxwell-Cattaneo law for an oscillatory disk is still not available in the literature. The remainder of this paper is organized as follows. Mathematical modeling is elucidated in Section 2. The numerical solution is developed in Section 3. Section 4 discusses the results of the present work, and key findings are summarized in Section 5.

2 Mathematical modeling

An unsteady, three-dimensional, incompressible, hydromagnetic viscous fluid flow is reviewed. The fluid motion is caused by periodic oscillations and a constant rotation of the disk occupying the porous space. The disk oscillates with the amplitude ε and rotates with the angular speed γ. The location of the disk is in the plane z=0. A cylindrical coordinate system is a suitable choice for the present problem. The velocity vector considered is

and the velocity components (u, v, w) are chosen along the increasing direction of (r, θ, z). The derivatives along tangential coordinates are excluded owing to the assumption of axisymmetric flow. The electric field is also ignored. A magnetic field with a uniform potency β₀ is applied in the axial direction. However, the induced magnetic field is assumed to be zero due to a low Reynolds number.

Heat transfer characteristics are examined by implementing the Maxwell-Cattaneo model. The wall temperature is denoted by T_w, while the ambient temperature is denoted by T_∞ such that T_w>T_∞. The fluid flow model considering the above assumptions is expressed as^[13-14]

(1)

(2)

(3)

(4)

(5)

The boundary conditions for t>0 are

(6)

with the following initial conditions:

(7)

Here, ρ, μ, and υ are the fluid density, viscosity, and kinematic viscosity, respectively, δ_e is the electrical conductivity of the liquid, β₀ is the magnetic flux density, ϕ₁ and k₁ are the porosity and permeability of the porous medium, respectively, is the coefficient of heat diffusion, τ₀ is the heat flux relaxation time, and ω is the angular frequency.

The well established self-similar transformations are^[13]

(8)

The self-similar transformations result in the following system:

(9)

(10)

(11)

with the authorized boundary conditions:

(12)

where the prime (') is used for differentiation with respect to the dimensionless variable z, the subscript t denotes differentiation with respect to the dimensionless time t, is the unsteady parameter, is the magnetic parameter, is the porosity parameter, is the Prandtl number, is the swirl parameter, and λ₁=aτ₀ is the thermal relaxation time parameter.

The shear stresses (radial shear stress and frictional torque) are defined as

(13)

where

denotes the local Reynolds number.

Substituting (8) into (4), the pressure distribution is

(14)

3 Numerical solution

The obtained system of equations is of third order in F and second order in G and θ. Firstly, the order reduction technique is applied to convert (9) into second order. The following substitution is made:

(15)

(16)

(17)

(18)

The required form of boundary conditions (12) is

(19)

Next, the domain truncation method is applied to convert the semi-infinite [0, ∞) domain into a finite domain [0, N]. The boundary conditions implemented at infinity are located at the domain endpoint N. The (R+1) uniformly-distributed points are taken to discretize the spatial variable η such that the step-size is considered to be

The finalized system is solved numerically using the finite difference method. The first- and second-order backward difference formulae are approximated with respect to time,

(20)

First- and second-order central difference approximations are implemented with respect to η,

(21)

Here, i and j are the spatial and time levels, respectively.

The equation system (16)-(19) in view of (20)-(21) is written as

(22)

(23)

(24)

with applicable boundary conditions,

(25)

An iteration process is applied using the successive over-relaxation method to obtain the solution. For the integration of (15), the Adam Moulton and Milne methods are executed. Various time scales t_k=kδt(k=0, 1, 2, …) are used in the calculation process.

4 Discussion

Numerical computations are performed for the unsteady, hydromagnetic flow problem of a viscous fluid using the Maxwell-Cattaneo model. A rotating disk, which also oscillates, causes the fluid motion when immersed in a porous medium. Dimensionless quantities of the nonlinear system are discussed via graphical representations and tables. These quantities include the magnetic parameter M, the constant amplitude ε, the porosity parameter P, the swirl parameter Ω, the Prandtl number Pr, and the thermal relaxation time parameter λ₁.

Figures 1 and 2 show the impact of the magnetic parameter on the radial and axial velocities F(0.25, t) and F(0.25, t), respectively, versus the time t when η=0.25. The increasing magnetic parameter values result in declining velocity curves in both Figs. 1 and 2. The application of a magnetic field generates a Lorentz force known as a resistive force that creates resistance in the flow field and thus decreases the velocity in the flow field. Furthermore, the hydrodynamic case of M = 0.0 is considered (see Figs. 1 and 2).

Figures 3 and 4 present different flow velocities at a constant amplitude versus t. In the first scenario, when ε=0.0, the amplitude of oscillations decreases and the special case is attained for the linearly rotating disk. In addition, the non-zero reducing values of ε develop oscillatory amplitude profiles. The amplitude interval in the radial velocity curve is larger than that in the axial velocity curve.

Figures 5 and 6 describe the behavior of velocity curves with the increasing porosity parameter. The radial and axial velocity oscillatory profiles versus the dimensionless time t show a tendency to decline. It is further observed in Fig. 6 that the amplitude of velocity curves is closer compared with the profiles in Fig. 5 for different porosity values. The first case shows that the flow does not saturate the porous space. The flow oscillatory behaviors of F'(0.25, t) and F(0.25, t) for the swirl parameter case are exhibited in Figs. 7 and 8, respectively. The profiles present an increasing trend against time for enlarging swirl parameter values. As the stretching rate is inversely proportional to the swirl parameter, increasing the swirl parameter causes a decrease in the stretching rate. At the starting level, the amplitude of oscillatory curves is small and then strengthens over time because of the brisk thrust of fluid particles. We also observe that the profiles are similar to those in Ref. [13].

The effect of the Prandtl number at t=π/2 on the temperature field is revealed in Fig. 9. The fixed time t=π/2 shows no oscillation in the temperature profiles. It is observed that the temperature curves become steeper as the thermal boundary layer thickness decreases. This occurs because the Prandtl number and the thermal diffusivity have an inverse relation. The thermal diffusivity is weaker owing to higher Prandtl numbers. Hence, a decreasing phenomenon in the profiles is detected (see Fig. 9). The profile in Fig. 9 is similar to the results of Turkyilmazoglu^[7]. Figure 10 shows how the thermal relaxation time alters the temperature curves at t=π/2. Larger λ₁ produces a reduction in the temperature field with the linked thermal boundary layer thickness. By increasing λ₁, it takes longer for fluid molecules to transfer energy to neighboring molecules. Therefore, the temperature curves demonstrate a decreasing nature/behavior.

The three-dimensional fluid flow phenomena of axial velocity, radial velocity, and temperature versus η and t are shown in Figs. 11-13. Figures 11 and 12 show the oscillatory fluid flow behaviors. A two-dimensional flow phenomenon of the radial velocity versus η is shown in Fig. 14, while Fig. 15 shows a two-dimensional flow representation of η versus t. Figure 16 elucidates two-dimensional contours. It is predicted that if η<1, the behaviors of contours are very close to each other. The time-series velocity fields at various distances η=0.15, 0.35, 0.55, and 0.75 from the disk surface are shown in Fig. 17. Owing to the oscillatory motion of the disk, the amplitude of oscillations increases near the disk surface. However, with the increasing distance from the disk, the amplitude of oscillations decreases (see Fig. 17).

Figures 18 and 19 are designed to study the impact of unsteady and thermal relaxation time parameters on velocity and temperature fields, respectively. It is observed that out-of-phase situations occur in the flow because of varying unsteady and thermal relaxation time parameters. The phase-log increases owing to the enhancement of St and λ₁. The phase-log in Fig. 19 increases from 0° to 180° for λ₁ =0.0, 0.1, 0.3, and 0.5. The oscillatory curves depict a declining behavior by increasing St and λ₁. A zoomed small portion of temperature curves is also shown in Fig. 19 for better understanding of the phase-shift phenomena.

Table 1 presents the radial shear stress τ_rz and the frictional torque τ_zθ. The effects of the magnetic parameter, the constant amplitude, the porosity parameter, and the swirl parameter are described at different time scales t= π/6, π/4, π/2, and 3π/4. It is perceived that both shear stresses (τ_rz and τ_zθ) strengthen at various time scales by enhancing M, ε, and P. The improved swirl parameter reduces the shear stress in the radial direction and increases the frictional torque at several time levels, as seen in Table 1. It is also perceived that for ε=0.0, when there is no oscillation and hence representing the case of linearly rotating disk, the radial shear stress has the same values at different time levels. A similar result is obtained for the frictional torque case. Table 2 summarizes the impact of the Prandtl number and the thermal relaxation time on the heat transfer rate at distinct time scales. At t= π/6, π/4, π/2, and 3π/4, the augmented values of Pr and λ₁ increase the heat transfer rate.

5 Conclusions

Numerical computations are carried out for the hydromagnetic swirling flow of viscous fluid subject to a rotating disk. The oscillatory disk is contained in a porous medium. The Maxwell-Cattaneo theory is applied to an energy equation. The obtained system is then solved using the successive over-relaxation method after utilizing self-similar transformations. The key learning points are as follows:

(ⅰ) Both the radial and axial velocity curves demonstrate a declining behavior with the increasing magnetic and porosity parameters. The axial velocity curves are closer to each other compared with the radial velocity curves.

(ⅱ) The amplitude of oscillatory velocity curves decreases by decreasing the values of the constant amplitude. Moreover, the amplitude decreases with zero value of the constant amplitude.

(ⅲ) The Prandtl number and the thermal relaxation time parameter have a decreasing tendency in the temperature field.

(ⅳ) At the increased distance from the disk surface, the amplitudes of the velocity oscillatory curves diminish.

(ⅴ) The radial shear stresses decrease and the frictional torque increases because of the increased swirl parameter.

(vi) The heat transfer rate increases by increasing the Prandtl number and the thermal relaxation time parameter.