Darcy-Forchheimer flow with nonlinear mixed convection

An analysis of the mixed convective flow of viscous fluids induced by a nonlinear inclined stretching surface is addressed. Heat and mass transfer phenomena are analyzed with additional effects of heat generation/absorption and activation energy, respectively. The nonlinear Darcy-Forchheimer relation is deliberated. The dimensionless problem is obtained through appropriate transformations. Convergent series solutions are obtained by utilizing an optimal homotopic analysis method (OHAM). Graphs depicting the consequence of influential variables on physical quantities are presented. Enhancement in the velocity is observed through the local mixed convection parameter while an opposite trend of the concentration field is noted for the chemical reaction rate parameter.


Introduction
The fluid flow through a porous space and objects of different shapes embedded in the porous space has recently attracted the attention of researchers. Its applications can be found in diversified disciplines such as nuclear engineering, bioengineering, mechanical engineering, geothermal physics, civil engineering, and applied mathematics. Geothermal energy utilization, solidification of casting, blood flow in lungs or in arteries, buried electrical cables, pollutants dispersion in aquifers, porous heat pipes, chemical catalytic connectors are some processes which involve the fluid flow through a porous space. Darcy's law is extensively used to interpret the flow filling the porous space. Darcy's law becomes invalid for effects of high velocity and turbulence in the porous space. The second-order polynomial function was introduced in the momentum equation by Forchheimer [1] to account for the effects of inertia on apparent permeability. Muskat [2] named it as the Forchheimer factor. Several studies considered the flow through a porous space by using the Darcy-Forchheimer relation in different of density than the linear variation.
In view of the aforementioned studies, it is analyzed that the combined impact of nonlinear mixed convection and activation energy past a nonlinear inclined stretching surface has not been studied yet. Fluids filling the porous space with the Darcy-Forchheimer expression are also considered. Heat transfer aspects are considered in presence of heat generation/absorption. Such investigation is useful in high-temperature polymeric mixtures, aerosol technique, and solar collector which operates at moderate to high temperatures. The nonlinear system of equations is obtained through suitable transformations. Analytical solutions are computed by the optimal homotopy analysis method (OHAM) [39][40][41][42][43][44][45][46][47][48][49] . Aspects of emerging parameters are physically illustrated. Graphs are portrayed for the effects of emerging parameters on physical quantities.

Model development
We consider the two-dimensional, incompressible, mixed convective flow of viscous fluids induced by a nonlinear stretching sheet. The surface is inclined at an acute angle ξ. Viscous fluids filling the porous media are specified by the Darcy-Forchheimer relation. Impacts of activation energy and heat generation/absorption are also considered. The Cartesian coordinate framework is adopted such that the surface is lined up in the x-and y-directions normal to the surface. Surface deforms continuously in the x-direction with a nonlinear velocity U w = ax n (a > 0, n 0). Using boundary-layer and Boussinesq approximation, the resulting problems are obtained as follows [16,38] : Here, u and v depict the velocity components along the x-and y-directions, F = C * b K * 1/2 is the nonlinear inertia coefficient of the porous space, g is the gravity, C * b represents the drag coefficient, β 0 , β 1 , β 2 , and β 3 are the first-order and second-order expansions of thermal and solutal coefficients, respectively, k r is the chemical reaction constant, k * is the Boltzmann constant, and E * is the activation energy.
Select [20] Then, Eq. (1) is identically justified, and Eqs. (2)-(5) yield 1 Sc In the above expressions, λ depicts the local porosity parameter, R i is the local mixed convection parameter, F r is the Forchheimer number, α 1 is the nonlinear density-temperature parameter, α 2 is the nonlinear density-concentration parameter, N * is the Buoyancy ratio parameter, δ is the local heat generation/absorption coefficient, P r is the Prandtl number, Λ * is the temperature difference variable, E is the activation energy parameter, Λ is the local reaction rate parameter, and Sc represents the Schmidt number. Here, one has The skin friction coefficient and local heat and mass transfer rates are in which Re x = ax n+1 ν is the local Reynolds number.

Solution convergence
The solution expressions consist of f , θ , and φ which play an essential role in convergent series solutions. The concept of minimization is utilized for obtaining optimal data of f , θ , and φ . The average squared residual errors as recommended by Liao [39] are given as where ε t m depicts the total squared residual error, δζ = 0.5, and k = 20. The optimal data of convergence control variables for n = 0.5 and n = 1.   Table 1 Individual averaged squared residual errors when n = 0.5 [6]

Discussion
This section intends to inspect the contribution of the local porosity parameter λ, the local Forchheimer number F r , the local mixed convection parameter R i , the nonlinear density-temperature parameter α 1 , the nonlinear density-concentration parameter α 2 , the inclination angle ξ, the Buoyancy ratio parameter N * , the local heat generation/absorption coefficient δ, the Prandtl number P r, the activation energy parameter E, the temperature difference parameter Λ * , the local reaction rate parameter Λ, and the Schmidt number Sc on the velocity f (ζ), thermal θ(ζ) and concentration φ(ζ) fields. The computations have been done for distinct values of 0 λ 1.2, 0 F r 1.4, 0 R i 0.9, 0 α 1 12, 0 α 2 12, 0 ξ π 2 , 0 N * 2.0, −0.3 δ 0.3, 0.7 P r 1.4, 0 E 3, 1 Λ * 4, 0 m 3, 0 Λ 1.5, and 0.5 Sc 1.5 [6,38] . The curves of f (ζ) for λ estimations are deliberated in Fig. 3. Here, f (ζ) lowers for higher λ. Physically, the presence of the porous space creates resistance in the smooth movement of fluid particles, which consequently declines the velocity field. Figure 4 portrays the significant impact of F r on f (ζ). Reduction in f (ζ) is noticed for larger F r . Figure 5 shows the salient features of R i on f (ζ). Higher estimation of R i predicts a strong buoyancy force within the fluid flow which intensifies the velocity for both n = 0.5 and n = 1.5. The variation of f (ζ) via α 1 and α 2 is pointed out in Figs. 6 and 7, respectively. Clearly, f (ζ) enhances for increasing values of α 1 and α 2 . They signify the relative impact of thermal and solutal buoyancy forces on viscous hydrodynamic forces, respectively. Thus, the velocity increases due to the enhancement of buoyancy forces [28] . Figure 8 is analyzed for the role of N * on f (ζ). An enhancement in f (ζ) is observed for larger N * for both n = 0.5 and n = 1.5. N * > 1 corresponds to the situation when solutal buoyancy forces exceed thermal buoyancy forces, N * < 1 when solutal buoyancy forces are less than thermal buoyancy forces, and N * = 1 when both the buoyancy forces are of the same magnitudes. Figure 9 presents the consequences of ξ on f (ζ). It describes that f (ζ) reduces for higher ξ. Attributes of δ against θ(ζ) are declared in Fig. 10. It is investigated that an enhancement in δ yields stronger θ(ζ) and a larger related layer thickness for both n = 0.5 and n = 1.5. It is due to the increase in buoyancy forces for heat generation which influences the flow rate. This enhancement in the flow rate causes stronger θ(ζ). θ(ζ) against P r is sketched in Fig. 11. P r possesses a converse relation with the thermal diffusivity. The less thermal diffusivity is noted for higher P r which reduces the fluid temperature. Aspects of E on φ(ζ) are displayed in Fig. 12. An increment in E gives rise to stronger φ(ζ) and a larger associated layer thickness. Physically, higher E represents the decrease in the modified Arrhenius function which pushes the generative chemical reaction. The impact of Λ * on φ(ζ) is plotted in Fig. 13. Larger Λ * indicates the decrease in φ(ζ) and the related layer thickness for both n = 0.5 and n = 1.5. From Fig. 14, it is recognized that larger m produces weaker φ(ζ) and a smaller associated layer thickness. Figure 15 is sketched to examine the variation in φ(ζ) for Λ. An increase in Λ leads to the destructive chemical reaction that dissolves the liquid species more effectively which causes weaker φ(ζ). The role of Sc on φ(ζ) is pointed out in Fig. 16. Clearly, both φ(ζ) and the corresponding layer thickness are reduced for higher Sc. Figures 17 and 18 characterize the consequences of α 1 , α 2 , and ξ on the skin friction coefficient C fx Re 1/2 x . It is analyzed that C fx Re  Tables 3 and 4 are arranged for the justification of the current results which are compared with those by Vajravelu and Sastri [15] . It is analyzed that the current results are in good agreement with those in Ref. [15].

Conclusions
The nonlinear mixed convective flow by a nonlinear inclined stretching surface with the activation energy and the Darcy-Forchheimer porous space is modeled. Heat generation/ absorption is also considered. The key findings of the present analysis are outlined as follows.
(i) The velocity is an increasing function of R i .
(ii) The improvement in the velocity is observed through α 1 and α 2 .
(iii) The opposite trend of the velocity is noticed for N * and ξ.
(iv) The stronger temperature is noted for larger δ. License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.