1 Introduction

The combined consideration of heat and mass transport has potential importance in various engineering and industrial processes such as energy production, cooling of nuclear reactors, cooling of electronic devices, heat conduction in tissues, crop damage due to freezing, air conditioning, desalination, and food processing. The problems of heat and mass transport have been elaborated by use of the classical theories of Fourier^[1] and Fick^[2]. These are not adequate due to the fact that the relaxation time changing will affect both the thermal field and the concentration field. Cattaneo^[3] initially made a modification in the Fourier theory of heat diffusion by considering the effect of the extra thermal relaxation stress. The further advancement in the Cattaneo theory has been made by Christov^[4] by replacing the partial time differentiation with the Oldroyd upper convective derivative. Ciarletta and Straughan^[5] reported the unique solutions of the Cattaneo-Christov expression for energy. Straughan^[6] addressed the natural convective flows of incompressible viscous liquid via the Cattaneo-Christov theory of heat transfer. Li et al.^[7] studied the Cattaneo-Christov heat diffusion model for the hydromagnetic viscoelastic liquid over a vertically moving sheet. Abbasi et al.^[8] elaborated the importance of the Cattaneo-Christov heat diffusion theory for the two-dimensional steady flow of the Oldroyd-B liquid with a variable conductivity. Waqas et al.^[9] presented a mathematical analysis of the generalized Burger liquid flow. The features of heat transfer were explored through the Cattaneo-Christov heat diffusion model. Sui et al.^[10] recently extended the Cattaneo-Christov theory for the mass diffusion equation by considering the Maxwell nanoliquid flow under the velocity slip condition.

The complex liquids such as colloidal suspensions, polymeric diluents, and polymeric melts have common features of viscoelasticity. These liquids generally occur in the polymer processes such as injection molding, blow molding, and extrusion. The various aspects of fluid flows have been affected by the viscoelastic behaviors, e.g., hydraulic resistance, energy dissipation, flow instability, mixing performance, and transportation efficiency. Therefore, the modeling and simulation of viscoelastic liquids are important from both technical and scientifical points of view. The intra-and inter-molecular reactions of polymer chains generally cause the occurrence of viscoelasticity, and greatly depend on the polymer mechanism and molecule mass. In the literature, various models of viscoelastic liquids have been proposed according to their physical features. The most important category of viscoelastic liquids is known as the rate type fluids, which generally express high and low molecular weight liquid features. The present work focuses on the Maxwell fluid model, which predicts the extra stress relaxation features. The mathematical expression of the Maxwell liquid can be used to describe the behaviors of the dilute polymeric solutions, glycerine, crude oil, toluene, etc. Some interesting investigations on the Maxwell fluid can be seen in Refs. [11]-[20].

The present attempt considers the steady laminar flow of a three-dimensional Maxwell liquid towards a bidirectionally moving surface. The variable conductivity of the liquid is taken into account because it is now well-established that the conductivity of polymer melts changes from 255.37 K to 477.594 K^[21]. The involvement of the heat and mass transport is modelled through the utilization of the Cattaneo-Christov theory for double diffusion. The term of the first-order chemical reaction rate is encountered in the mass species expression. Such reactions generally appear in various chemical and metallurgical processes. The utilization of the boundary layer assumptions leads to the governing mathematical expressions of the velocities, temperature, and concentration. The constructed mathematical model is evaluated through the homotopic criteria^[22-32]. The homotopic scheme is valid for all the problems whether they have small or large physical parameters. It gives the easiest way to verify the convergence of the approximate solutions. This scheme provides a freedom to select the base functions and auxiliary parameters. The non-dimensional quantities are sketched and examined for various values of the physical constraints.

2 Problem developments

We assume that the laminar incompressible flow of the Maxwell liquid is over a moving surface. The governing expressions for the conservation laws of the mass and linear momentum for the three-dimensional flow of the Maxwell fluid after making use of the boundary layer approximation are

(1)

(2)

(3)

(4)

(5)

The conditions for the considered flow phenomenon are

(6)

(7)

In the above expressions, u, v, and w are the velocity components in the x-, y-, and z-directions, respectively, υ is the kinematic viscosity, δ₁ is the relaxation time, δ₂ is the thermal relaxation time of the heat diffusion, ρ is the fluid density, c_p is the specific heat, κ is the temperature-dependent thermal conductivity, δ₃ is the concentration relaxation time of the mass diffusion, D is the mass diffusion coefficient, k₁ is the chemical reaction parameter, a and b are the constants with time inverse dimensions, and T_w and C_w are the temperature and concentration at the wall, respectively.

The equations (2) -(7) can be reduced into the dimensionless forms by introducing the following new variables^[28]:

(8)

The relation for the thermal conductivity is κ =κ_∞(1 + ϵθ), where ϵ can be expressed as follows^[21]:

(9)

In the above equation, κ_w is the thermal conductivity at the wall. The non-dimensionless expressions of the momentum, energy, and mass species can be expressed as follows:

(10)

(11)

(12)

(13)

(14)

(15)

where De₁ =δ₁a is the Deborah number with respect to the relaxation time, De₂ =δ₂a is the Deborah number due to the relaxation time of heat diffusion, Pr =υ /α is the Prandtl number, De₃ =δ₃a is the Deborah number due to the relaxation time of mass diffusion, Sc=υ / D is the Schmidt number, γ =k₁ /a is the chemical reaction rate, and δ = b/a is the ratio parameter.

3 Analysis and discussion

We adopt the homotopic procedure to compute the solutions of the governing dimensionless equations (10) -(13) along with the conditions (14) and (15). The homotopic expressions of the solutions involve the auxiliary parameter . This parameter has the key importance to compute the convergent solutions. The consideration of the proper values of is very essential when we obtain the solutions through the homotopic approach. To select the proper values of , we elaborate the -curves in Fig. 1. From this figure, we can see that the proper regions for the convergent solutions are

We also calculate the numerical data at various orders of the homotopy analysis method (HAM) deformations when

We compute the values of f"(0), g"(0), θ '(0), and φ '(0) in Table 1 by setting

The solutions of f"(0) and g"(0) converge from the 15th-order of approximations. It is also noticed from Table 1 that the values of θ'(0) and φ'(0) are larger when De₂ =0.3=De₃, i.e., for the case of the Cattaneo-Christov theory.

Figures 2 and 3 are elaborated here to evaluate the variations of the velocities f'(η) and g'(η), the temperature θ (η), and the concentration φ (η) for different values of the Deborah number De₁ when δ =0.6. A decay in the liquid velocities f'(η) and g'(η) appears when De₁ enhances (see Fig. 2). The maximum velocity profiles are attained when De₁ =0, i.e., the viscous liquid flow situation. The relaxation time δ₁ is involved in the definition of De₁. Here, δ₁ is larger for higher De₁, which resists the liquid flow, due to which the decreases in f'(η) and g'(η) are examined (see Fig. 2). From Fig. 3, we explore that θ (η) and φ (η) increase with an increase in De₁. Here, the resistance in the fluid flow occurs due to the larger relaxation time corresponding to higher θ (η), φ (η) and their related boundary layer thicknesses. It is also visualized that the curves of θ (η) are nearer the wall than those of φ (η).

The importance of the ratio parameter δ on the velocities f'(η) and g'(η), the temperature θ (η), and the concentration φ (η) by setting De₁ =0.5 is characterized in Figs. 4 and 5. Figure 4 represents that f'(η) reduces while g'(η) enhances when δ increases. Physically, the lateral sheet moves towards the y-direction when δ increases from zero. Due to this, f'(η) reduces, while g'(η) rises. Here, the two-dimensional flow situation can be generated by setting δ = 0. The temperature θ (η) and the concentration φ (η) are retarded when we use larger values of δ (see Fig. 5). It is also evaluated that the curves of φ (η) are achieved far away from the wall in comparison with θ (η).

The various curves of θ (η) corresponding to the Deborah number De₂ are shown in Fig. 6. From the figure, we can see that θ (η) and its associated thickness of the boundary layer are smaller for larger De₂. The thermal relaxation time δ₂ is involved in De₂, and increases when De₂ increases. This larger thermal relaxation time creates a reduction in the temperature θ (η). We also observe that the temperature in the case of the Fourier law of heat diffusion (δ₂ = 0) is higher than that in the case of the Cattaneo-Christov heat diffusion model (δ₂ > 0). The effect of various values of the Prandtl number Pr on θ (η) is elaborated in Fig. 7. Higher Pr corresponds to smaller temperature θ (η) and thinner boundary layer thickness. When the thermal conductivity parameter ϵ increases, θ (η) increases (see Fig. 8). It is a physical phenomenon that the liquids with larger conductivity possess higher temperature. When we increase the values of ϵ, the thermal conductivity of the liquid becomes larger, which leads to an increase in the temperature. The constant conductivity problem is obtained by setting ϵ = 0. The conductivity parameter ranges from 0 to 1.

Figure 9 shows the variations of the concentration φ (η) for different Deborah numbers De₃. The concentration φ (η) starts to reduce when the value of De₃ enhances. The maximum concentration is achieved when De₃ =0.0. Here, De₃ =0.0 implies the Fick model of mass diffusion. The concentration field is stronger for the case of the Fick model of mass diffusion than for the Cattaneo-Christov theory. Figure 10 elucidates that higher Schmidt number Sc corresponds to lower concentration φ (η). The mass diffusion coefficient is involved in Sc. Larger Sc implies weaker mass diffusion coefficient, due to which the lower concentration profiles are achieved (see Fig. 10). The features of the chemical reaction rate γ are examined in Fig. 11. We note that φ (η) is stronger in absence of γ, i.e., γ = 0.

4 Conclusions

This work reports the features of the Cattaneo-Christov heat and mass diffusion theory for the three-dimensional laminar flow of the Maxwell liquid induced by the bidirectional stretching of sheet. The effect of the first-order chemical reaction is incorporated in the mass species expression. The results show that f'(η) and g'(η) decrease while θ (η) and φ (η) increase when the Deborah number De₁ increases. This situation is arisen due to the involvement of the relaxation time in De₁. When De₂ =0, the temperature θ (η) is higher for the Fourier theory of heat diffusion than for the Cattaneo-Christov heat diffusion model. When De₂ > 0, the Cattaneo-Christov heat diffusion decreases when De₂ increases. The presence of the thermal relaxation time in De₂ leads to a reduction in the liquid temperature. It is examined that the temperature and its related thickness of the boundary layer increase when the thermal conductivity parameter increases. The minimum temperature curve appears when ϵ =0, i.e., for the case of constant thermal conductivity. The concentration φ (η) is smaller for the Cattaneo-Christov model of mass diffusion than for the Fick model of mass flux. We also notice that the concentration φ (η) decreases when the chemical reaction γ increases.