1 Introduction

The exploration of steady flow and heat transfer over a rotating disk has developed one of the magnificent areas in academia and industry owing to its widespread technical applications, such as geothermal industry, computer storage, rotating machinery, chemical process, turbo-machinery, and lubrication. The innovative work on Newtonian fluid flow over an infinite rotating disk was initiated by von Kármán^[1], in which the Navier-Stokes equations were transformed into a set of ordinary differential equations by applying the classical similarity transformations. His groundwork was the base for many researchers who devoted their studies towards the flow and heat transfer problems on a rotating disk by taking several industrial and technology applications into account. Bachok et al.^[2] considered the problem of incompressible viscous nanofluid over a rotating disk. The finite difference scheme was used for solution of transformed boundary layer equations. The effects of magnetic field in the flow of viscous fluid over a radially stretchable rotating disk were examined by Turkyilmazoglu^[3]. Rashidi et al.^[4] studied the entropy generation in magnetohydrodynamic (MHD) slip flow due to a rotating porous disk. Khan et al.^[5] analyzed the non-Newtonian Powell-Eyring fluid flow over a rotating disk under the influence of transverse magnetic field. The outcomes revealed that the shear stress augments in the radial direction at disk vicinity and vanishes far away from the disk as the Reynolds number enhances. Griffiths^[6] investigated the boundary-layer flow for a number of generalised Newtonian fluid due to a rotating disk. Guha and Sengupta^[7] analyzed von Kármán's swirling flow on a rotating disk in Bingham fluids. Ming et al.^[8] studied the power law fluid by a rotating disk with non-Fourier heat transfer. The conclusion was that the uplifting values for the Prandtl number and power-law index thin the thermal boundary layer thickness. Hayat et al.^[9] discussed heat in two stretchable rotating disks with nanofluid.

The rheology of non-Newtonian fluid mechanics has been an inspiring and fascinating subject area for researchers because of its growing technical and industrial importance such as in polymer, petroleum, chemical, and food processing industries. The fluid flow behavior frequently encountered in industries, for instance, biological fluid, polymeric liquid, and motor oils, and other mixtures can be modeled in the form of non-Newtonian fluid model. The non-Newtonian fluids include a special type of fluids called viscoelastic fluids in which the shear stress is a memory function of the deformation rate. In these kinds of fluids, when the applied shear stress is removed, the deformation rate gradually decreases. This phenomenon is called the stress relaxation. The upper-convected Maxwell model is the frequently used viscoelastic model. During last few years, the researchers have focused on the flow and heat transfer features of Maxwell fluid due to its effectiveness for polymers allowing lower molecular weight. Tan and Xu^[10] investigated the Maxwell fluid model to examine the viscoelastic property of the fluid with no-slip condition. The discrete inverse transform method was adopted to find the exact solution. Jamil and Fetecau^[11] computed helical flows considering the Maxwell liquid. Here, the flow was caused by the applied shear stress at plate. Han et al.^[12] studied the Cattaneo-Christov model for heat transfer mechanism in the Maxwell fluid flow. Mustafa^[13] addressed analytically the Maxwell fluid flow due to a rotating frame by utilizing the non-Fourier heat flux theory. Sui et al.^[14] addressed heat and mass transfer mechanisms of upper-convected Maxwell nanofluid over a stretched surface with Cattaneo-Christov double-diffusion and slip velocity. Afifya and Elgazery^[15] focused on the influence of a chemical reaction in the MHD boundary layer flow of a Maxwell liquid over a stretched surface with nanoparticles. Liu and Guo^[16] discussed the coupling model for time dependent MHD flow of generalized Maxwell fluid with radiation thermal transform. Cao et al.^[17] studied the Maxwell fluid with heat source/sink in presence of nanoparticles over a stretching porous surface. Hsiao^[18] reported a numerical study on a combined electrical MHD heat transfer thermal extrusion system in the radiative flow of Maxwell fluid with dissipation effects. Shahid et al.^[19] utilized the Cattaneo-Christov heat flux model in the MHD flow of Maxwell fluid together with thermal radiation and chemical reaction effects. Jusoh et al.^[20] numerically investigated the flow and heat transfer of MHD three-dimensional Maxwell nanofluid over a permeable stretching/shrinking surface with convective boundary conditions. Shahzad et al.^[21] studied the three-dimensional chemical reactive Maxwell liquid by using the Cattaneo-Christov heat and mass flux model.

Chemical reactions are identified as homogeneous or heterogeneous process relying on whether they arise in bulk of fluid (homogeneous) or happen on some catalytic surfaces (heterogeneous). Homogeneous and heterogeneous reactions occur in various chemical reacting systems including catalysis, biochemical systems, and combustion. The relationship between homogeneous and heterogeneous reactions associated with formation and consumption of reactant species at different rates both within the fluid and on the catalytic surfaces is usually very complex. The homogeneous-heterogeneous reactions involving the boundary layer flow were studied by Chaudhary and Merkin^[22]. In another paper, Merkin^[23] used the idea of heterogeneous and homogeneous processes in flow of viscous liquid over a stretched surface. He examined the homogeneous reaction through cubic autocatalysis and considered first-order process for heterogeneous reaction. Khan and Pop^[24] assessed the flow behavior in a two-dimensional stagnation point due to a permeable surface with homogeneous-heterogeneous reactions. The nonlinear equations were tackled by employing the implicit finite difference scheme, and showed that the flow characteristics were affected by the mass transfer parameter. Abbas et al.^[25] studied the influence of homogeneous-heterogeneous reactions in the viscous fluid flow over a stretching/shrinking sheet. Khan et al.^[26] reported numerically the three-dimensional Sisko fluid flow with heterogeneous-homogeneous reactions and the Cattaneo-Christov heat flux model. Nadeem et al.^[27] reported the characteristic of homogeneous-heterogeneous reactions and magnetic dipole effects on ferrofluid motion due to a stretching cylinder. analyzed the homogeneous-heterogeneous reactions in the flow of Maxwell liquid. Rauf et al.^[28] studied the Powel-Eyring fluid for chemically reactive flow with double diffusive Cattaneo-Christov heat and mass flux theories. Recently, Hashim et al.^[29] made investigation on dual solutions by taking the homogeneous-heterogeneous reactions in the flow of a non-Newtonian Carreau fluid.

The emission of electromagnetic radiation in all directions by a heated surface is called thermal radiation. The effect of thermal radiation is important in high temperature process and space technology. In polymer processing industry, the heat transfer process is mainly controlled by the application of thermal radiations. The thermal radiations play a significant role in the surface as a consequence of convective heat transfer coefficient. Hayat et al.^[30] addressed the mixed convection radiative stagnation point flow in the Maxwell fluid. Effects of thermophoresis along with Joule heating and convective conditions were also discussed by Hayat et al.^[31]. Nadeem and Haq^[32] studied the influence of thermal radiations in the MHD flow of nanofluid past a convectively heated stretching surface. Atlas et al.^[33] investigated the squeezing nanofluid flow in a channel with thermal radiation effects. Some recent studies on linear and nonlinear thermal radiations within the aspects of heat transfer can be seen in Refs.[34]-[36].

Literature survey reveals that considerably less attention has been devoted to the corresponding non-Newtonian rotating disk problem. To be more specific, no single attempt is devoted to study the non-Newtonian Maxwell fluid over a rotating disk. Thus, the main objective here is to analyze the flow pattern in the Maxwell fluid with heat and mass transfer characteristics together with homogeneous-heterogeneous reactions. The heat transfer enhancement is carried out under the influence of nonlinear thermal radiation and convective boundary conditions. The governing nonlinear ordinary differential equations, namely, mass, momentum, energy, and concentration equations, are solved with the Runge-Kutta Felberg fourth-order and fifth-order (RKF45) in the MAPLE software.

2 Physical model and governing equations

A three-dimensional Maxwell fluid flow by a rotating stretchable disk subject to the uniform magnetic field B₀ applied in the axial direction is considered. The induced magnetic field is ignored by considering small magnetic Reynolds numbers. Moreover, the heat transfer features are characterized through the nonlinear radiative heat flux. Here, we assume a cylindrical coordinate system to understand the physical model, as shown in Fig. 1.

The disk surface temperature is because of convective heating process characterized by the coefficient of heat transport h_f and temperature of the hot fluid T_f. The homogeneous-heterogeneous reactions in boundary layer flow and catalyst surface are taken, respectively, as inspired by Chaudhary and Merkin^[22]. The homogeneous reaction for cubic autocatalysis is

(1)

For isothermal first-order reaction of the form, we have

(2)

where A and B denote chemical species, a and b are their concentrations, and k_c and k_s are the rate constants. We further presume that the process of these reactions is isothermal. In the absence of viscous dissipation and Joule heating, the governing equations for the Maxwell three-dimensional flow with the magnetic field are

(3)

(4)

(5)

(6)

(7)

where S is the extra stress tensor for the non-Newtonian Maxwell fluid which is defined as

(8)

in which A₁ represents the first Rivlin Erickson tensor written as

λ₁ is the relaxation time parameter, and is the upper convected derivative. The following expressions hold for a second rank tensor S:

(9)

where L= grad V.

In Eq. (4),

denotes the velocity vector with components in radial-, azimuthal-, and axial-directions. Further,

is the electric current density with σ^* being the electrical conductivity of the base fluid,

is the magnetic field strength, p is the liquid pressure, k is the thermal conductivity of fluid, and ρ_f is the density of liquid. Considering the expressions (8) and (9) and applying the boundary layer approximations, the governing continuity, momentum, energy, and concentration equations become

(10)

(11)

(12)

(13)

(14)

(15)

where D_A and D_B are the diffusion coefficients, α is the thermal diffusivity, and T is the fluid temperature. Incorporating the Rosseland approximation, we have the following simplest form of radiative heat flux:

(16)

where σ^* is the Stephan-Boltzmann constant, and k^* is the mean absorption coefficient. Assume that the temperature differences within the flow are sufficiently small. Therefore, expressing T⁴ in the Taylor series about T_∞ and omitting the terms for higher order yield

(17)

In view of Eq. (17), the energy equation (13) takes the following form:

(18)

with the boundary conditions

(19)

where c is the stretching rate, Ω is the swirl rate, and a₀ is the positive dimensional constant.

2.1 Non-dimensionalization and parameterization

Introduce the following similarity variables:

(20)

where the dimensionless radial-, azimuthal-, and axial-velocity components are F(η), G(η), and H(η), respectively, the dimensionless temperature is θ(η), the dimensionless concentration of homogeneous bulk fluid is g(η), and the dimensionless concentration of heterogeneous catalyst at the surface is h(η). By using the above similarity transformations, Eqs. (10)-(12), (14), (15), (18), and (19) reduce to the following dimensionless forms:

(21)

(22)

(23)

(24)

(25)

(26)

(27)

where denotes the rotation strength parameter measuring the ratio of swirl to stretch rates, and ω =0 implies pure stretching without rotation. Further, M is the magnetic field (hydrodynamic situation can be retrieved by replacing M=0), β₁ is the Deborah number, R_d is the radiation parameter, θ_w is the temperature ratio parameter, Pr is the Prandtl number, γ is the Biot number, k₁ is the measure of strength of homogenous reaction, Sc is the Schmidt number, and k₂ is the measure of strength of heterogeneous reaction. These dimensionless quantities are, respectively, defined as

(28)

We assume that the diffusion coefficients of chemical species A and B are of the same magnitude. This hypothesis leads to explore the study, where the diffusion coefficients D_A and D_B are equal, i.e., δ =1 (see Ref. [23]). Actually, this assumption gives the relation

(29)

Thus, Eqs. (25) and (26) turn into

(30)

with the conditions

(31)

It is worth mentioning here that the present problem of Maxwell model reduces to the case of viscous fluid when the parameter β ₁=0.

2.2 Heat transfer performance

The convective heat transfer performance over the disk surface can be estimated by the local Nusselt number N_ur. Physically, the Fourier law is utilized to define the Nusselt number N_ur. The mathematical expression of this quantity is given as

(32)

with

(33)

In the non-dimensional form, we can write

(34)

where

is the local Reynolds number.

3 Numerical solution procedure

To compute the numerical solution of proposed problem (21)-(24), (27), (30), and (31), the RKF45 integration scheme is implemented. In order to get the required accuracy during the computation of the results of the problem, two step sizes are used. A comparison is made at each mesh point related to a large step size. If the required accuracy is not obtained, the step size h should be kept smaller as compared with the previous size. The process shall be continued till up to the desired accuracy. Once the results of two iterations are matched, such an approximation shall be accepted. However, if the results are more accurate than the required accuracy, the step size shall be increased accordingly. Thus,

(35)

(36)

(37)

(38)

(39)

(40)

An approximate solution to the initial value problem (IVP) is furnished by a Runge-Kutta method of fourth-order,

(41)

where the values of functions W₁, W₃, W₄, and W₅ are used. It is noted here that W₂ is not utilized in the expression (41). The Runge-Kutta method of fifth-order is used for the better value of solution,

(42)

The optimal step size sh can be determined by multiplying the scalar s times the current step size h. The scalar s is

(43)

where ε is the specified error control tolerance.

4 Physical interpretation

This section focuses on physical explanation of involved parameters in flow, heat, and mass transfer distributions. The above stated numerical computation is carried out for the magneto Maxwell fluid due to a stretchable rotating disk. The heat transfer characteristics are investigated with nonlinear thermal radiation effects. Homogeneous-heterogeneous reactions are considered on the fluid volume concentration. The velocity components F(η), G(η), and H(η), the temperature field θ(η), and the nanoparticle concentration field g(η) are examined graphically through the parametric study. The impact of parameters such as rotation 0.8≤ω ≤6.0 and magnetic field 0.0≤M≤3.0 are examined on the velocity components F(η), G(η), H(η), and the temperature field θ(η). Effects of radiation parameter 0.0≤ R_d≤0.6 and Biot number 0.5≤γ ≤2.0 on the temperature distribution θ(η) are inspected. The effects of the homogeneous reaction strength 0.2≤k₁≤0.8, the heterogeneous (surface) reaction strength parameter 0.1≤ k₂≤0.7, the magnetic field 0.0≤M≤3.0, and the Schmidt number 1.0≤Sc≤2.5 on the volume concentration g(η) are studied.

Figures 2(a)-2(d) are displayed to investigate the role of rotation parameter ω on the velocity components and the temperature field. Figures 2(a) and 2(b) show that the radial and azimuthal velocity components represented by F(η) and G(η), respectively, increase with the value of the rotation parameter ω. The rotation parameter ω is the ratio of the swirl rate Ω to the stretch rate c. Thus, it gives us measure of the swirl rate to the stretch rate. When the rotation ω starts flourishing, this means that the swirl rate becomes larger compared with the stretch rate. The case ω =0 implies stretching without rotation. For the special choice of magnetic parameter (M=1.0) and Deborah number (β₁=0.1), radial and azimuthal velocity components represented by F(η) and G(η), respectively, grow up. The main reason behind it is that with the increase in the rotation parameter, the centrifugal force pumps the fluid particles in the radial direction. It is further observed that the change in radial and azimuthal velocities due to rotation diminishes gradually when moving away from the disk. Physically, this trend is expected because the influence of centrifugal force is limited at the disk vicinity. The usual enhancement in the angular component of velocity with flourishing rotation near the disk surface can be visualized in Fig. 2(b). Figure 2(c) depicts a decreasing trend in the axial velocity component H(η) with stronger rotation rates. This happens because with the accelerated rotation rate, the fluid particles are forced to move in the radial direction, and this is compensated by the particles which are drawn towards the disk surface in the negative axial direction. It appears from Fig. 2(d) that the fluid temperature decreases as the rotation parameter becomes strengthened without Joule heating and dissipation effects.

The effects of magnetic field M on the velocity and temperature curves at a prescribed rotation parameter ω and Deborah number β₁ are sketched in Figs. 3(a)-3(d). It is obvious that as the magnetic field M turns into powerful, the velocity curves reduce considerably in all directions. Contrary to the velocity profiles, the temperature field θ(η) rises, which is because of the fact of growth in the skin-friction which offers great resistance to fluid particles, and as a result, heat is generated in the fluid by the occurrence of magnetic field in the vertical direction. Physically, a resistive force called the Lorentz force is developed due to presence of the applied magnetic field in the electrically conducting fluid. The feature of this force is to retard the flow over the disk at the expense of enhancing its temperature. This is portrayed by the decline in the radial, tangential, and axial velocity components and uplift in the temperature field as M develops (see Figs. 3(a)-3(d)).

Figure 4(a) elaborates the curves of thermal field θ(η) for various values of radiative parameter R_d. The thermal field θ(η) and their associated thickness of boundary layer rise for enhancement in the values of R_d. As expected, the existence of radiative parameter implies to absorb more heat by the fluid that corresponds to higher temperature. The features of local Biot number γ with various values on the fluid temperature are displayed in Fig. 4(b). Physically, a gradual increment in the Biot number γ results in the larger convection at the disk surface which elevates the fluid temperature. Furthermore, the thermal boundary layer thickness is significantly pronounced by the stronger Biot number γ.

Figure 5(a) reveals the actions of parameter k₁, denoting the homogeneous reaction strength, on the fluid concentration g(η). The concentration profile reduces and the consequent concentration boundary layer thickness diminishes as the homogeneous reaction k₁ strengthens. Figure 5(b) shows the variation of concentration profile for numerous values of heterogeneous (surface) reaction strength parameter k₂, the magnitude of concentration profile g(η) reduces with uplifting the strength of heterogeneous reaction k₂. This happens from the reason that augmentation in either k₁ or k₂ suggests the consumption of reactants in the flow field which corresponds to a reduction in the fluid volume concentration g(η).

The ratio between viscosity and mass diffusivity is called the Schmidt number Sc. Figure 6(a) elaborates the plots of concentration profile with different values of the Schmidt number Sc. It is observed that a larger Schmidt number leads to an enhancement in the concentration profile. As expected, an increase in the Schmidt number corresponds to the decrease in the mass diffusivity, and as a result, the fluid concentration increases. The impact of the magnetic parameter M is observed in Fig. 6(b). The increasing values for the magnetic field reduce the concentration profile due to the resistive force, namely, the Lorentz force, which is produced due to application of magnetic force.

Tables 1 and 2 are organized for the authentication of present numerical computations. For this, we have calculated the numerical values for the radial F'(0), angular skin friction -G'(0), vertical velocity -H(∞), and the Nusselt number -θ'(0) in limiting cases for different values of the rotation parameter ω with the magnetic field (M=0, M=2). The attained outcomes match in an outstanding way with those of Turkyimazoglu^[3] which confirms the accuracy of the applied numerical scheme. The numerical computations of the local Nusselt number N_ur is included in Table 3. It is observed that the heat rate represented by the expression -(1+R_d(1+(θ_w-1) θ(0)) ³) θ'(0) enhances with the increase in the physical parameters such as the Prandtl number 1.0≤Pr≤7.0, the thermal radiation 0.1≤R_d≤0.4, the Biot number 0.5≤γ ≤2.0, and the temperature ratio 1.2≤θ_w≤1.8. A decreasing trend is observed for the magnetic field 0.0≤M≤ 6.0.

5 Concluding remarks

The impact of nonlinear thermal radiation and homogeneous-heterogeneous reactions on the Maxwell magneto-fluid flow over an infinite stretchable rotating disk with convective boundary conditions is studied in this work. The velocity, temperature, and concentration fields are studied numerically and depicted graphically in detail. This investigation can be summarized in the following main points.

(ⅰ) The influence of the centrifugal force is observed strongly in the vicinity of the disk with boosting disk rotation which in turn increases the radial and azimuthal velocity components and results in a decrease in the axial velocity component.

(ⅱ) The presence of magnetic field is to reduce all three velocity components in their respective directions which results in an increase in the fluid temperature.

(ⅲ) The thermal boundary layer thickness is improved significantly with the radiation parameter and Biot number.

(ⅳ) The concentration boundary layer thickness is found to be a decreasing function of both the homogeneous and heterogeneous parameters.

(ⅴ) The heat transfer rate at the disk surface is increased by the Prandtl number, the radiation parameter, the temperature ratio parameter, and the Biot number.

(ⅵ) A significant growth is observed in the concentration profile by increasing the Schmidt number.