1 Introduction

Nanoliquids are two phase mixtures of liquid (carrier liquids) and solid (nanoparticles). The nanoparticles concentrations are normally small (< 10% by volume), and their size generally does not surpass 100 nm. The carrier liquids are characteristic coolants often found in heat transport applications. Nanoliquids established by Choi and Eastman^[1] have attracted large consideration from the scientific community from 2000 and onwards since the heat transport improvement approaches. Such liquids have the capability to improve thermal effectiveness of conventional heat carrier liquids. Nanoliquids have ample utilizations like airplanes, power generator, melt-spinning, micro-reactors, nuclear reactor cooling, drying and cooling of papers, production of glass-fiber. Numerous factors regarding mechanism of nanoliquids concept which include Brownian movement, particle to particle coupling, micro-convection, thermal diffusion, conduction and thermophoretic through aggregates were addressed by numerous researchers recently. It is proven from these studies that nanoparticles Brownian movement is the major factor that significantly influences thermal performance of liquids. Several studies regarding consideration of nanoliquids have been reported (see Refs. [2]-[15]).

The concept regarding flow due to porous media is encountered widely in several areas like soil mechanics, petroleum engineering, filtration theory. Production of crude oil, grain storage, petroleum reservoirs, packed bed reactors, porous insulation, ground water pollution, resin transport modeling, and nuclear waste removal are examples of such systems. The available literature confirms that Darcy's concept^[16] which is effective when weaker porosity and smaller velocities situations are imposed is implemented by several scientists to model and formulate distinct problems. However, non-uniformity of the porosity distribution occurs in numerous practical utilizations leading to Darcy's theory failure in such situations. Thus, the non-Darcian concept^[17] should be employed for better modeling of the physical phenomenon. The non-Darcian characteristics in hydromagnetic stretchable viscous liquid flows were described by Seddeek^[18]. He noted the higher velocity and temperature distributions subject to larger injection and magnetic parameters. Singh et al.^[19] established analytic solutions for hydromagnetic unsteady convection flows subject to the non-Darcian and heat generation aspect using a perturbation approach. They noted an increasing trend in the temperature distribution for the heat generation factor, while the reverse situation is noticed for the heat absorption factor. Non-Darcy and thermal radiation impacts in dusty nanoliquid flows considering viscous dissipation, magnetic field and heat generation aspects were disclosed by Gireesha et al.^[20]. They developed numerical solutions of governing systems with a Runge-Kutta Fehlberg (RKF) scheme. Their presented analysis reports a reduction in the thickness of thermal layer when dust particles are suspended in clean fluids. Khan et al.^[21], Sadiq et al.^[22], and Hayat et al.^[23] elaborated non-Darcian characteristics in flow induced by the moving surface. They computed the nonlinear systems with the shooting and homotopy schemes. Their outcomes indicate that the velocity field diminishes when the Hartman number is enhanced.

There are numerous physical phenomena which comprise viscoelastic liquids in industry and engineering such as polymer solutions, molten plastics, food stuffs, exotic lubricants, synthetic propellants, colloidal and suspension solutions. These liquids have been modeled and formulated via diverse manners with their constitutive relations altering significantly in complexity. The Maxwell liquid is a kind of viscoelastic liquids which are scrutinized widely (see Refs. [24]-[29]). The characteristics of Maxwell liquids can be illustrated by an elastic spring^[30]. A polymer extrusion process can be described by the Maxwell fluid. It effectively elaborates the features of viscoelastic liquids.

Our focus in this research is to describe non-Darcian characteristics in thermally radiated nonlinear convective steady-state flows of Maxwell nanomaterials subject to heat generation. Moreover, this phenomenon of convective conditions and stratifications is incorporated simultaneously. A mixed convection term is considered nonlinear, which makes the modeled problem much more complicated and comprehensive. Mathematical expressions are simplified using the concept of boundary-layer. A homotopic scheme^[31-38] is used for the treatment of non-dimensional expressions. Salient features of emerging variables versus the nanoparticles concentration, the temperature, the velocity, and Nusselt and Sherwood numbers are addressed through graphs.

2 Modeling

Here, simultaneous features of double stratification and convective conditions in stretchable flows of Maxwell nanomaterials are formulated. Nanofluid properties are analyzed through thermophoresis and Brownian movement aspects. Heat generation, nonlinear mixed convection, and thermal radiation aspects are retained. A non-Darcian concept is used to elaborate a porous medium effect. The fluid is assumed in compression. Viscous dissipation is not accounted. Keeping such assumptions in view, we have the following governing expressions^[38]:

(1)

(2)

(3)

(4)

(5)

(6)

Here, signifies the kinematic viscosity, μ is the dynamic viscosity, ρ_f is the liquid density, λ₁ is the relaxation time, K is the porous medium permeability, g is the gravitational acceleration, represents the variable inertia coefficient of the porous medium, Λ₁ and Λ₃ are the linear thermal and concentration expansion coefficients, respectively, C_b is the drag coefficient, Λ₂ and Λ₄ are the nonlinear thermal and solutal expansion coefficients, respectively, k^* is the coefficient of mean absorption, is the thermal diffusivity, Q₀ is the the coefficient of heat absorption/generation, is the heat capacity ratio, T and C are the liquid temperature and nanoparticle concentration, respectively, D_T and D_B are the thermophoresis and Brownian diffusion coefficients, respectively, (ρc)_f is the liquid heat capacity, (ρc)_p is the nanoparticles effective heat capacity, u_w (x) is the stretching velocity, h_f and h_g are the heat and mass transfer coefficients, respectively, k is the thermal conductivity, T_∞ and C_∞ are the ambient liquid temperature and concentration, respectively, c, d₁, and d₂ are the dimensional constants, T₀ and C₀ are the reference temperature and concentration, respectively, T_f =T₀ + a₁x, C_f = C₀ + a₂x are the heated liquid temperature and concentration, respectively, and u and v are the components of velocities in the x- and y-directions, respectively.

The following equations are introduced:

(7)

Equation (1) is automatically justified, whereas Eqs. (2)-(6)

(8)

(9)

(10)

(11)

(12)

Here, differentiation with respect to η is signified by ('), β is the Deborah number, λ is the porosity parameter, δ is the mixed convection variable, is the local inertia coefficient, is the drag coefficient per unit length, Gr_x is the thermal buoyancy variable, N is the ratio of concentration to thermal buoyancy, Gr_x^* is the concentration buoyancy variable, β_t and β_c are the nonlinear thermal and concentration convection variables, respectively, Nt is the thermophoresis variable, S₁ and S₂ are the thermal and concentration stratification variables, respectively, Pr is the Prandtl number, Nb is the Brownian motion variable, R is the radiation variable, S is the heat generation variable, Sc is the Schmidt number, and γ₁ and γ₂ are the thermal concentration Biot numbers, respectively. These variables are described as follows:

(13)

The local Nusselt and Sherwood numbers, i.e., and , in the dimensional forms can be expressed as

(14)

(15)

We have dimensionless forms

(16)

(17)

3 Computational procedure

The optimal homotopic scheme is implemented to compute the governing nonlinear differential systems. The minimization concept provided by Ref. [31] is used to acquire optimal values of the auxiliary variables ħ_f, ħ_θ, and ħ_ϕ. We found h_f =-1.256~18, h_θ =-1.123~97, h_ϕ =-0.800~07, and the total averaged squared residual error ε_m^t =1.67 × 10^-3. The total residual errors versus f, θ, and ϕ are revealed in Fig. 2. Moreover, Table 1 illustrates the individual average squared residual errors using the optimal values of convergence control variables at m=2. Clearly, the averaged squared residual errors decrease with higher order approximations.

4 Analysis of results

The contributions of emerging variables against f', θ, ϕ, , and are described in Figs. 3-19. The effect of β on f' is addressed in Fig. 3. Here, f' and the associated thickness of momentum layer show a decaying trend for larger β. Physically, β comprises the phenomenon of relaxation time. Such time is taken by the material to acquire an equilibrium state when the stress is employed. Materials with lower β exhibit liquid characteristics whereas larger β corresponds to solid characteristics. In fact, larger β increases the liquid viscosity which yields reduction of f'. Figure 4 interprets f' variation for different values of F_r. Clearly, larger F_r corresponds to a decay in f' and the related thickness layer. In fact, larger values of F_r lead to a resistance in the liquid flow.

The impact of S₁ on θ is interpreted in Fig. 5. As expected, the thermal layer increases with S₁. There is the decay in the temperature difference between the surface and the surrounding when S₁ is increased. Figure 6 elaborates θ outcome against R. Presence of the radiation aspect provides additional heat to liquids due to enhancement of θ. The coefficient of mean absorption k^* diminishes by increasing R which is accountable for enhancement of θ. Features of γ₁ on θ are explored in Fig. 7. The thermal Biot number γ₁ signifies the strength of convective heating. Consequently, larger γ₁ implies strong convective heating on the surface which increases θ. The situation γ₁ =0 illustrates the isoflux wall situation, whereas the condition of isothermal wall is jumped when γ₁ → ∞. Figure 8 exhibits behavior of Pr on θ. Here, θ and the corresponding thickness layer are reduced for larger Pr. As liquid diffusivity occurs in Pr definition which becomes weaker when Pr is increased. Such weaker diffusivity corresponds to reduction of θ. Consequences of Nt and Nb on θ and ϕ are described in Figs. 9-12. We note an improvement in θ subject to larger Nt and Nb, whereas ϕ shows decaying behavior for Nb and it rises when Nt is enhanced. Nanoparticles close to the hot boundary are being pushed towards the cold region at the ambient owing to the thermophoretic force contribution. Thus, one expects thicker thermal/nanoparticles concentration layers for the thermophoretic phenomenon. Moreover, material elements random movement improves for larger Nb due to which additional heat is delivered. Consequently, θ increases, and ϕ decays.

The effect of Sc on ϕ is visualized in Fig. 13. As expected, ϕ diminishes when Sc is augmented. Physically, the mass diffusivity decreases when Sc is increased. Thus, ϕ and the related nanoparticles concentration layer decay. Figure 14 explains ϕ variations subject to S₂. Larger S₂ decreases the volumetric fraction between surface and reference nanoparticles due to which ϕ diminishes. Characteristics of the mass Biot number γ₂ on ϕ are reported through Fig. 15. We note that ϕ and the corresponding nanoparticles concentration layer are decaying functions of γ₂. It is due to the reason that the mass Biot number γ₂ and coefficient of mass transfer h_g are directly proportional. Therefore, an increment in γ₂ yields the higher concentration distribution.

Figures 16 and 17 are designed to interpret effects of R, γ₁, Nt, and Nb on . Clearly, increases when R is enhanced. However, the reverse behavior is found for Nb. Figure 18 illustrates characteristics for Sc and γ₂. As expected, rises for larger Sc.

5 Conclusions

The novel idea of combined convective conditions and double stratification for rate-type (Maxwell) liquid flow is introduced. The salient aspects such as the nonlinear convection, the Darcy-Forchheimer relation, the thermal radiation, and the heat generation are also retained. The present analysis yields the following outcomes:

(ⅰ) The Deborah number β and the inertia coefficient F_r yield the lower velocity f'.

(ⅱ) Biot numbers γ₁ and γ₂ improve the temperature θ and nanoparticles concentration ϕ, whereas the stratified variables S₁ and S₂ decay the temperature θ and the nanoparticles concentration ϕ.

(ⅲ) Larger Brownian Nb and thermophoretic Nt variables have the same impact on the temperature θ whereas these variables have a reverse impact on the nanoparticles concentration ϕ.

(ⅳ) The nanoparticle concentration ϕ is reduced when the Schmidt number Sc is increased.

(ⅴ) The Nusselt number increases with the radiation factor R.

(ⅵ) The present study yields the viscous nanomaterial situation for β =0.