1 Introduction

Non-Newtonian fluid models are beneficial in industrial and manufacturing processes, such as hot rolling, paper production, drilling muds, plastic polymers, metal spinning, and cooling of metallic plates in cooling baths. The non-Newtonian Jeffrey liquid is one of the rate type fluid. This liquid model is accomplished by describing the features of retardation and relaxation time. For this reason, the Jeffrey fluid model has gained much attention from the researchers. Kothandapani and Srinivas^[1] analyzed the magnetohydrodynamic (MHD) Jeffrey fluid flow under the peristaltic transport in an irregular channel. Shehzad et al.^[2] investigated the flow of Jeffrey fluid at the stagnation point with Dufour and Soret effects. An analysis of unsteady flow and heat transport mechanism in the Jeffrey fluid over a stretching sheet was made by Hayat et al.^[3]. Ahmad and Ishak^[4] discussed the convective Jeffrey liquid flow at a stagnation point over a linearly elongated vertical surface. Ahmed et al.^[5] presented exact solutions for the MHD Jeffrey liquid flow through a convectively heated stretched surface. The stagnation-point flow of a Jeffrey liquid due to a convectively heated stretched disk under the effect of viscous dissipation and Joule heating was examined by Hayat et al.^[6]. They reported the convergent series solutions to the problem. Khan et al.^[7] presented analytical solutions for Jeffrey liquid flow subject to cross-diffusion characteristics due to a stretching of a permeable cylinder in presence of Newtonian flux condition.

There is a huge development in modern nanotechnology due to its broad involvement in the physiological and industrial processes. The mixture of non-metallic or metallic nanoparticles and conventional heat transfer fluids is referred as a nanofluid. The nanoparticles improve the thermal characteristics of the base fluids. Such practical features of nanofluids have special importance in several industrial and technological developments such as vehicle cooling, transformer and electronic device cooling, biomedicine, heat exchanger, and nuclear reactor cooling. Convective heat transfer with nanofluids can be modelled using the single phase or two-phase Buongiorno nanofluid approach. The first assumes that the fluid phase and particles are in thermal equilibrium and move with the same velocity. It only includes the influence of nanoparticle volume fraction aspect. There are several other physical aspects (mainly Brownian motion and thermophoresis) which are the reason for an enhancement in thermal conductivity. The employed model would incorporate these two physical aspects. Also, this approach is simpler and requires less computational time. Choi^[8] first introduced the term nanoliquid. The mathematical expressions of nanofluid through thermophoretic and Brownian movement aspects were developed by Buongiorno^[9]. The Brownian motion acts against concentration. In addition, thermophoresis moves the particles from hot regions to cold ones. The nanoliquid flow by an elongated surface was initially investigated by Khan and Pop^[10]. Hayat et al.^[11] addressed the three-dimensional MHD flow of viscous nanofluid over an impermeable nonlinearly stretched surface with the convective condition. The MHD Maxwell nanoliquid flow in a saturating non-Darcy porous medium was investigated by Muhammad et al.^[12]. They reported the analytical solutions using the homotopic analysis method. Sheikholeslami et al.^[13] studied the MHD flow of nanofluid inside a permeable cavity with hot elliptic obstacle through the lattice Boltzmann method. Later, many researchers have been engaged in recent developments on nanofluids (see Refs. [14]-[24]).

The flowing and mixing of liquid are generated by density variation due to difference in temperature within the liquid, and it is called convection or natural convection. Near the hot surface, the density of the liquid is smaller than that of the cold fluid. The buoyant force appears due to gravity that lifts the heated liquid upward. The problems of convective flow of working fluids have numerous engineering applications, such as solar collectors, food preservation, energy storage, nuclear reactor technology, and cryogenic devices. Moreover, the nonlinear convection produces due to nonlinear density temperature variations in the buoyancy force term, and it has a significant impact on the flow characteristics. Keeping this fact in mind, Vajravelu et al.^[25], Kameswaran et al.^[26], Shaw et al.^[27-28], Ramreddy and Pradeepa^[29], Bandaru et al.^[30], and Mahanthesh et al.^[31] investigated the influence of nonlinear convection in flow and heat transfer under different physical aspects.

The heat transfer process is a hot topic of modern technology, and it arises when there is a difference in temperature between the physical systems. It has plenty of applications in modern industry and technology, for instance, power generation, atomic reactor cooling, and energy production. The theory of heat conduction for heat transfer attributes was estimated by Fourier^[32], which is highly useful for characterization of heat transport process in various circumstances. Cattaneo^[33] improved the Fourier law^[32] by insertion of thermal relaxation time. Christov^[34] improved the Cattaneo model^[33] by introducing the Oldroyd's upper-convected derivative. The convective flow of Newtonian fluid by considering the Cattaneo-Christov heat flux model was investigated by Straughan^[35]. The spinning flow of Maxwell fluid was studied by Mustafa^[36] through the Cattaneo-Christov theory of heat transport. Khan et al.^[37] analyzed the viscoelastic fluid flow by employing Cattaneo-Christov heat flux aspects. Waqas et al.^[38] explored the aspect of Cattaneo-Christov model on the flow of generalized Burgers fluid. The comparison of two classes of viscoelastic fluids through Cattaneo-Christov diffusion expressions was examined by Hayat et al.^[39]. Shehzad et al.^[40] investigated the three-dimensional hydrodynamic flow of Maxwell liquid through the Cattaneo-Christov heat and mass flux model.

Here, the main goal is to investigate the nonlinear convection flow of Jeffrey liquid across an elongated sheet with heat and mass fluxes via Cattaneo-Christov double diffusion expressions. Further, the influence of thermophoretic and Brownian movement, nonlinear radiation, and convective boundary conditions is accounted. So far, the above-stated model has not been considered yet. The numerical solutions via the Runge-Kutta-Fehlberg (RKF) method along with the shooting system are constructed. For a particular case, the obtained numerical results are compared with the previously reported results of Hayat et al.^[3] and good agreement is achieved.

2 Mathematical formulation

The nonlinear convective and radiated flow of Jeffrey nanoliquid due to the stretched sheet is considered. U_w (=aX) is the stretching velocity. The coordinate system is assumed that, the sheet is along with the X-axis, while Y measures normally outward to it (see Fig. 1). The sheet is supposed to be heated by a hot fluid with the concentration C_f and the temperature T_f which are related with the mass/heat transfer coefficient h₂/h₁. T_∞ and C_∞ are the ambient temperature and the concentration, correspondingly.

The relevant equations are^{[3, 30, 36]}

(1)

(2)

(3)

(4)

with the boundary conditions^{[11, 23]}

(5)

whereas

U and V are velocities, and ν, μ, ρ, λ₁, λ₂, g, β₀, β₁, β₂, β₃, T, and C symbolize the kinematic viscosity, the dynamic viscosity, the density, the ratio of relaxation to retardation time, the retardation time, the acceleration due to gravity, the linear volumetric thermal expansion coefficient, the nonlinear volumetric thermal expansion coefficient, the linear volumetric solute expansion coefficient, the nonlinear volumetric solute expansion coefficient, the temperature, and the concentration, respectively. α=k/(ρc)_f is the thermal diffusivity, k is the thermal conductivity, (ρc)_f is the heat capacity of liquid, (ρc)_p is the nanoparticle effective heat capacity, D_B is the Brownian diffusion coefficient, D_T is the thermophoretic diffusion coefficient, λ_E and λ_C are the relaxation time of heat and mass fluxes, respectively, and σ^* and k^* are the Stefan-Boltzman constant and the mean absorption coefficient, respectively. Substituting the following similarity variables

(6)

into (1)-(5) yields

(7)

(8)

(9)

with

(10)

where λ is the local mixed convection parameter, Gr_x is the local Grashof number, Re_x is the local Reynolds number, α₁ and α₂ are the nonlinear thermal and solutal convection parameters, respectively, N is the buoyancy ratio parameter, β is the Deborah number, Le is the Lewis number, Pr is the Prandtl number, N_B is the Brownian movement parameter, R is the parameter of radiation, N_T is the parameter of thermophoretic, Bi₁ is the thermal Biot number, δ₁ is the thermal relaxation parameter, δ₂ is the concentration relaxation parameter, Bi₂ is the concentration Biot number, and θ_w is the temperature ratio parameter. These parameters are defined as follows:

(11)

The skin friction coefficient, the local Nusselt number, and the local Sherwood number are

(12)

where τ_w, q_w, and q_m are

(13)

The dimensionless forms of (12) are

(14)

3 Method of solution and validation

The nonlinear ordinary differential equations (7)-(9) subject to the boundary conditions (10) are solved by the shooting technique along with the RKF method. The system of (7)-(9) is reduced to a first-order differential equation system by establishing the following new variables:

Accordingly, we have

with

To determine the unknowns using the shooting method, first, guess the values for m₁, m₂, m₃, and m₄. In a while, the subsequent initial value problem is treated numerically by the RKF scheme. We set the convergence criteria and step size as 10^-6 and 10^-3, correspondingly. The values of f''(0) are compared with the previously published homotopy analysis method (HAM) results of Hayat et al.^[3] for the specific case. The comparison results are found in outstanding agreement and presented in Table 1.

4 Physical interpretation

This section emphasizes the significance of physical parameters on the liquid velocity f'(η), the temperature θ(η), and the concentration φ(η). The behaviors of the skin friction C_f, the local Nusselt number Nu_x, and the local Sherwood number Sh_w for various physical constraints are visualized through the numeric data presented in tables. Figure 2 illustrates the impact of the Deborah number on the velocity distribution f'(η). It is visualized that f'(η) increases with an increase in β. In fact, the Deborah number is dependent on the retardation time. Therefore, an increase in β leads to an increase in the retardation time, which reduces the viscous force. Consequently, the liquid velocity field increases.

The impact of λ on f'(η) is illustrated in Fig. 3. It is noted that the velocity distribution shows an increasing behavior corresponding to higher values of the mixed convection parameter. This is because the convection parameter and the buoyancy force are directly proportional.

The influence of λ₁ on the velocity is demonstrated in Fig. 4. We visualize that f'(η) decreases for higher values of λ₁. Physically, λ₁ denotes the ratio of relaxation and retardation time.

f'(η) is higher for large values of α₁ for both absence and presence of the Cattaneo-Christov theory of heat and mass diffusion (see Fig. 5). Physically, the nonlinear convection (α₁) increases the temperature gradient at the wall and enhances the buoyancy force due to convection which forces the liquid to move away from the sheet. Consequently, f'(η) is increased.

The impact of N on f'(η) is depicted in Fig. 6. Here, as the buoyancy ratio parameter increases, the momentum boundary layer becomes thicker. Figure 7 depicts the variations of Bi₁ on θ(η). It is shown that the layer of thermal boundary is superior for larger Biot numbers. The heat transfer coefficient h₁ increases at the boundary with an increase in Bi. Consequently, heat energy is restored in liquid. The thermal boundary layer grows thicker.

The variation in θ(η) for thermal relaxation is explored in Fig. 8. The larger thermal relaxation δ₁ (=aλ_E) increases the relaxation time of heat flux λ_E. As a result, an increment in δ₁ shows a reduction in the temperature profile. Figures 9 and 10 depict the influence of α₁ and λ on the temperature profile, respectively. Here, θ(η) decays for larger values of both α₁ and λ.

Moreover, the temperature field is weaker in the presence of Cattaneo-Christov heat and mass flux compared with their absence. The effect of λ₁ on θ(η) is demonstrated in Fig. 11. It is observed that the temperature of fluid is enhanced with higher values of λ₁.

The variations of θ(η) due to the effects of N_B are shown in Fig. 12. Here, θ(η) enhances for higher N_B. Physically, the random motion of nanoparticles increases by increasing N_B. As a result, the temperature profile increases. The temperature field is lower for higher values of the buoyancy ratio parameter (see Fig. 13).

Figures 14 and 15 are presented to explore the effects of R and θ_w on θ(η). In fact, the radiation process generates more heat into the fluid system, due to which θ(η) enhances. It is far clear that an enhancement within the temperature ratio parameter corresponds to a higher wall temperature in comparison with the ambient fluid.

Figures 16 and 17 show the variations of θ(η) and φ(η) for N_T, respectively. Here, θ(η) and φ(η) increase significantly for larger N_T.

The variation in the concentration profile for the concentration relaxation parameter (δ₂) is illustrated in Fig. 18. By increasing δ₂ (=aλ_C), the concentration and its boundary layer thickness decrease. Physically, the relaxation time of mass flux λ_C raises with an increase in δ₂.

Figures 19 and 20 establish the effects of Biot numbers Bi₁ and Bi₂ on the nanoparticle concentration profile, respectively. The nanoparticle concentration profile increases with the Biot numbers. Thus, the convective conditions can be used as key aspects to control the development of thermal and concentration boundary layers on the surface. φ(η) decreases for large values of α₂ (see Fig. 21).

The variation of φ(η) for N is displayed in Fig. 22. Here, φ(η) is reduced with larger values of the buoyancy ratio parameter. Figure 23 depicts the influence of Le on φ(η). As expected, the concentration field diminishes for inflation of the Lewis number parameter. As higher values of the Lewis number Le (=ν/D_B) decay the strength of random moment of nanoparticles, φ(η) decreases.

Tables 2-4 present numerical data of the skin friction coefficient (C_fRe_x^1/2), the Nusselt number (Nu_xRe_x^-1/2), and the Sherwood number (Sh_wRe_x^-1/2) when Bi₁=Bi₂=λ=λ₁=0.5, N=β=0.4, N_B=N_T=R=0.3, α₁=α₂=0.3, Pr=0.71, Le=0.5, θ_w=1.2, and δ₁=δ₂=0.2. The varying parameter values are specified in the tables. From Table 2, we observe that larger values of Bi₁ decay Sh_wRe_x^-1/2, while they enhance C_fRe_x^1/2. C_fRe_x^1/2 and Nu_xRe_x^-1/2 are significantly varied for Bi₁ from 0.5 to 20. After that, small variations can be noticed.

From Table 3, it is observed that nonlinear thermal and concentration convections display better friction factors. Also, the heat and mass transfer rate is higher. It is also noticeable that the friction factor is a decreasing function of λ₁ whereas it is an increasing function of λ and N. λ and N have propensity to increase the Nusselt number, while an opposite trend is perceived for λ₁. This outcome is exactly similar for the Sherwood number (see Table 3). Interestingly, it is also noted that the effect of β is not reliable on the friction factor, Nusselt number, and Sherwood number profiles. All three distributions increase initially by increasing the value of β and then decrease (see Table 3). Finally, from Table 4, it is clear that the rate of heat transfer coefficient enlarges for larger values of R, θ_w, Pr, and δ₁. However, we can see an opposite result with an increase in N_B, N_T, δ₂, and Le.

5 Concluding remarks

A nonlinear convection flow of Jeffrey nanofluid is examined through double diffusions. The main features are

(ⅰ) The Deborah number is constructive for the velocity profile.

(ⅱ) The parameter of thermophoresis leads to higher curves of temperature and concentration.

(ⅲ) Consequence of the buoyancy ratio parameter on the velocity and temperature fields is quite opposite.

(ⅵ) The nonlinear convection parameters α₁ and α₂ give rise to the local Nusselt and Sherwood numbers and also better friction factor.

(ⅴ) The rate of heat transport coefficient increases for R, θ_w, Pr, and δ₁ but decreases for N_B, N_T, δ₂, and Sc.

(ⅳ) The velocity and the heat and mass transport rate are lower in presence of δ₁ and δ₂ compared with their absence.

Acknowledgements One of the authors P. B. SAMPATH KUMAR is thankful to University Grant Commission (UGC), New Delhi, for their financial support under National Fellowship for Higher Education (NFHE) of ST students to pursue M. Phil/PhD Degree (F117.1/201516/NFST201517STKAR2228/(SAIII/Website) Dated: 06-April-2016). Also, the author B. MAHANTHESH is thankful to the Management of Christ University, Bengaluru, India, for the support through Major Research Project to accomplish this research work.