1 Introduction

The importance of non-Newtonian materials has increased tremendously in the present world of modern science and technology. Such materials have influential usage in our routine life. Examples of non-Newtonian materials include certain oils, colloidal solutions, cosmetic products like shampoo and soap, sugar solution, apple sauce, mud, polymeric liquid, paint, and exotic lubricant. All such non-Newtonian materials have diverse characteristics and complicated physical nature. Till now, there is not a single non-Newtonian model which can scrutinize all the characteristics of non-Newtonian liquids. In the past, several non-Newtonian fluid models have been developed according to the physical nature of non-Newtonian materials. The researchers of recent world have paid great attention to explore the nature of rate type non-Newtonian liquids. Such fluids possess the properties of stress relaxation and retardation which cannot be described by differential type non-Newtonian fluid models. The simplest constitutive expression of rate type models is known as the Maxwell fluid, which is restricted only to exploit the nature of stress relaxation. In the present research work, we consider the Jeffrey fluid model, which is capable of exploring the phenomena of both stress relaxation and retardation. Some potential works on the Jeffrey fluid can be found in Refs. [1]-[10].

The analysis of heat transfer mechanism is very essential to obtain the best quality end product in engineering and industrial manufacturization. The heat flux theory developed by Fourier^[11] is well-known to describe the heat transport mechanism under various conditions and circumstances. However, it contradicts with the causality principle. The propagation speed of heat disturbance can be made finite by introducing thermal relaxation time in Fourier's formula^[12]. The material invariant can be preserved by utilizing Oldroyd's upper convective derivative in the Maxwell-Cattaneo model^[13]. Tibullo and Zampoli^[14] reported the uniqueness of the solutions obtained for the case of incompressible fluids under the Cattaneo-Christov heat diffusion. Khan et al.^[15] elaborated the numerical solutions of the viscoelastic fluid flow generated by an exponential stretching sheet, and discussed the heat transfer mechanism by using the Cattaneo-Christov heat diffusion equation. Shehzad et al.^[16] used the Cattaneo-Christov heat flux theory to explore the behavior of the third grade liquid by an exponentially stretching sheet with heat transfer. Li et al.^[17] reported the hydromagnetic Maxwell fluid flow with convective heat transport through the heat flux model of Cattaneo-Christov. Abbasi et al.^[18] discussed an optimal homotopic analysis of the Oldroyd-B fluid flow with the Cattaneo-Christov heat flux theory. Sui et al.^[19] considered the Cattaneo-Christov heat flux theory for the analysis of heat and mass transfer in the Maxwell liquid flow under the slip condition. Waqas et al.^[20] addressed the heat transfer phenomenon in the laminar flow of Burgers' liquid by utilizing the Cattaneo-Christov model.

The convective heat transport phenomenon in porous media has potential role in the applications related to geophysical and chemical engineering systems. Examples of these systems include crude oil production, petroleum reservoirs, grain storage, packed bed reactors, ground water pollution, porous insulation, nuclear waste disposal, resin transfer modeling, etc. The existing literature clearly indicates that much more attention has been devoted to the problems modelled and formulated with Darcy's theory^[21]. Darcy's theory of porous media is valid for smaller velocities and weaker porosity conditions. In various practical applications, the non-uniformity of porosity distribution can take place, leading to the failure of Darcy's theory under such circumstances. The better modelling of physical phenomena can be made by including the non-Darcian effects in the convective transport analysis in porous media^[22]. Seddeek^[23] explored the non-Darcian effects in a variable viscosity liquid saturated through porous media. Seddeek^[24] addressed the Darcy-Forchheimer flow of a viscous liquid by considering the simultaneous impact of thermophoretic and viscous dissipation. Singh et al.^[25] reported the convective flow under the heat generation/absorption saturated in the Darcy-Brinkman-Forchheimer porous medium. Gireesha et al.^[26] numerically described the non-Darcy flow of the viscous nanofluid with applied magnetic field.

The theme of the present contribution is to characterize the role of the Cattaneo-Christov heat diffusion formula in the mixed convective flow of the Jeffrey fluid over a moving sheet with the consideration of the non-Darcy porous medium. The thermal conductivity of the liquid is not constant. It varies linearly with the temperature. The boundary layer assumptions are used to govern the mathematical expressions of the arising physical phenomenon. The governing mathematical model is treated by the homotopic algorithm^[27-35]. There are various analytical techniques available in the literature. Researchers have employed analytical techniques for different fluid flow problems, e.g., the Adomian decomposition method (ADM)^[36-37], the differential transform method (DTM)^[38-39], the homotopy perturbation technique^[40-41], and the optimal homotopy analysis technique^[42]. The variations of the arising parameters on the velocity and temperature are elaborated and examined in detail.

2 Problem development

We consider an incompressible laminar flow of the Jeffrey liquid due to a vertical sheet. The steady flow of an incompressible liquid fills the porous space. The sheet is stretched along the x-direction in such a manner that the origin remains constant. All the physical characteristics of the liquid are constant except the thermal conductivity k. The effects of the magnetic field and viscous dissipation are neglected. The considered fluid is non-Newtonian, and maintains in the thermal equilibrium at each face. The porosity effects are taken into account by using the Darcy-Forchheimer term. The continuity and momentum expressions under mixed convection can be described as follows^{[5, 23]}:

(1)

(2)

The conditions for the present flow model are

(3)

where u and v denote the velocity components along the x- and y-directions, respectively, λ₁ is the ratio of the relaxation time to the retardation time, λ₂ is the retardation time, υ =μ /ρ is the kinematic viscosity, μ is the dynamic viscosity, ρ is the density, g is the gravitational acceleration, β is the thermal expansion coefficient, k is the permeability parameter, C_F is the Forchheimer coefficient, and u_w is the stretching velocity. The porous medium effects for viscosity are incorporated here only.

The energy expression through the Cattaneo-Christov heat flux model is^{[14, 17]}

(4)

where q shows the heat flux, λ^* is the relaxation time of the heat flux, k₁ is the thermal conductivity, T is the temperature, and V is the velocity. Fourier's law can be retrieved when λ^* =0. The energy equation for the laminar incompressible fluid can be expressed as follows^[17]:

(5)

The governing equation of temperature with variable thermal conductivity is

(6)

The conditions on the temperature are

(7)

where c_p is the specific heat, T_w is the temperature at the wall, and T_∞ is the ambient fluid temperature. Here, we consider the constant wall temperature condition instead of the variable heat flux condition.

The following variables are used to reduce Eqs. (2) , (3) , (6) , and (7) in dimensionless forms:

(8)

The expression for the variable thermal conductivity^[43] is

where

(9)

in which k_w denotes the thermal conductivity of liquid at the wall, k_∞ is the liquid free stream conductivity, and prime represents the differentiation with respect to η .

The incompressibility condition is trivially satisfied while the momentum and energy expressions are

(10)

(11)

(12)

(13)

where De₁ =λ₂c is the Deborah number with respect to the relaxation time and the retardation time, λ ₃ =υ /kc is the porosity parameter, is the inertia parameter, λ =Gr_x/Re²_x is the mixed convection parameter where Re_x =u_w(x)x/υ is the local Reynolds number and Gr_x = gβ(T_w − T_∞)x³/ν² is the local Grashoff number, De₂ =λ^*c is the Deborah number with respect to the relaxation time of the heat flux, and Pr=υ /α is the Prandtl number.

3 Analysis and discussion

We develop the homotopic algorithm to obtain the solutions of the velocity and temperature. The homotopic solutions generally contain the auxiliary parameters known as and . The solution convergence is strongly based on the suitable selection of these auxiliary parameters. To find the proper values of and , we have drawn the $\hbar $-curves in Fig. 1. The figure clearly indicates that the suitable convergence regions for the present problems are

Table 1 is given to investigate the values of f"(0) and θ'(0) at different orders of calculation in the case of the Cattaneo-Christov and Fourier's law heat diffusion models when

The convergent values of f"(0) and θ'(0) are achieved at the 28th-order of deformations for the Cattaneo-Christov model while the 26th-order data give the convergent solutions for Fourier's law model. The convergent values of f"(0) are smaller in the case of Fourier's law of heat diffusion than those obtained in the Cattaneo-Christov heat flux case.

In Fig. 2, we elaborate the curves of the velocity f'(η ) and the temperature θ (η ) for multiple values of the Deborah number De₁. From this figure, we observe that the velocity f'(η ) and its related boundary layer thickness are enhanced whereas the temperature θ (η ) and the thermal boundary layer decay. The retardation time directly appears in the definition of De₁ . The higher retardation time occurs for larger De₁, leading to higher flow velocity f'(η ) and lower temperature θ (η ). The features of the ratio of the relaxation time to the retardation time λ₁ on the velocity f'(η ) and the temperature θ (η ) are addressed in Fig. 3. Here, the temperature decreases when we increase the values of λ₁ while increases when we increase the values of the velocity f'(η ).

Figure 4 shows the variations of the velocity f'(η ) and the temperature θ (η ) due to the mixed convective parameter λ when

The buoyancy force takes place in λ, which is responsible for the variations of f'(η ) and θ (η). The buoyancy force is stronger for larger λ and weaker for smaller λ . Here, increasing λ corresponds to stronger buoyancy force that gives rise to the velocity. The temperature θ (η ) and its related thermal boundary layer are reduced for larger λ .Figure 5 presents the impact of the porosity parameter λ₃ on the velocity f'(η ) and the temperature θ (η ) when λ ₁ =0.3=λ , De₁ =0.2, Fr=0.1, ε =0.4, Pr =1.2, and De₂ =0.3. It is noted that f'(η ) decays while θ (η ) enhances when λ₃ increases. The permeability of a porous medium is inversely related to λ₃ due to which higher porosity parameter corresponds to smaller permeability, leading to lower velocity and thinner momentum boundary layer thickness. Figure 6 shows the curves of the velocity f'(η ) and the temperature θ (η ) for various values of Fr when λ ₁ =0.3=λ, De₁ =0.2=λ ₃, ε =0.4, Pr =1.2, and De₂ =0.3. From Fig. 6, we find that, when the values of the Darcy-Forchheimer parameter Fr increase, the temperature and thermal boundary layer increase, while the velocity f'(η ) and its related boundary layer become weaker. The results in the case of Darcy's flow can be retrieved by setting Fr=0.

Figure 7 depicts the changes in f'(η ) and θ (η ) for multiple values of the Deborah number De₂ when λ ₁ =0.3=λ, De₁ =0.2=λ ₃, Fr=0.1, ε =0.4, and Pr =1.2. Figure 8 shows the changes in f'(η ) and θ (η ) for various values of ε when λ ₁ =0.3=λ , De₁ =0.2=λ ₃, Fr=0.1, Pr =1.2, and De₂ =0.3. Here, an increase in De₂ corresponds to decreases in the velocity and temperature. The Deborah number De₂ appears in the energy equation. It is due to the consideration of the heat flux relaxation time, which enhances for larger Deborah number. Such larger heat flux relaxation time shows lower velocity and temperature fields. It also shows that the change in the temperature θ (η ) due to De₂ is very prominent in comparison with the variation in the velocity. The Cattaneo-Christov heat flux model can be reduced to Fourier's law heat flux model when De₂ =0. It is evident from Fig. 8 that f'(η ) and θ (η ) are increased when we enhance the values of the thermal conductivity parameter ε . It is a very common argument that higher thermal conductivity liquid possesses larger temperature. Due to this argument, the changes in θ (η ) are more pronounced than the changes in f'(η).

4 Conclusions

The influence of the Cattaneo-Christov heat flux theory for the laminar flow of the Jeffrey fluid in the Darcy-Forchheimer porous space is addressed. The considered fluid has variable thermal conductivity. The results show that the velocity is enhanced while the temperature decreases for larger values of the Deborah number De₁ . The increasing values of the mixed convection parameter λ result in higher velocity and thicker boundary layer thickness. The velocity and its associated boundary layer thickness are reduced when the Darcy-Forchheimer parameter Fr increases. When Fr increases, the temperature and thermal boundary layer thickness increase. The effects of the Deborah number De₂ are more pronounced on the temperature field in comparison with the velocity profile. When De₂ increases, the velocity and the temperature decrease. The heat flux relaxation time parameter occurs in De₂, corresponding to smaller temperature and velocity fields. Moreover, larger ε shows higher velocity and temperature, and higher ε implies stronger thermal conductivity, due to which the momentum and the thermal boundary thicknesses are thicker.