Numerical and Analytical Investigation for Darcy-Forchheimer Flow of a Williamson Fluid over a Riga Plate with Double Stratification and Cattaneo-Christov Dual Flux

The Darcy-Forchheimer ﬂ ow of a Williamson ﬂ uid over a Riga plate was analyzed in this paper. Energy and mass equations are modeled with Cattaneo-Christov theory and double strati ﬁ cations. The governing PDE models are altered into ODE models. These models are numerically solved by MATLAB bvp4c and analytically solved by the homotopy analysis method. The impact of governing ﬂ ow parameters on ﬂ uid velocity, ﬂ uid temperature, ﬂ uid concentration, skin-friction coe ﬃ cient, local Nusselt number, and local Sherwood number is scrutinized via graphs and tables. We acknowledged that the speed of the ﬂ uid becomes diminishes for more presence of porosity parameter. Also, we noted that the thermal and solutal boundary layer thicknesses are waning due to their corresponding strati ﬁ cation parameters. In addition, the maximum decreasing percentage of skin friction is obtained when the suction/injection parameter varies from 0.0 to 0.4 for Williamson and viscous ﬂ uids. The maximum increasing percentage of local Nusselt number occurs when the suction/injection parameter varies from 0.4 to 0.8 for Williamson and viscous ﬂ uids.


Introduction
Non-Newtonian fluids are extensively implemented in diverse industrial processes such as petroleum drilling, drawing of plastic films, fibre spinning, and food production. The Williamson fluid model is one of the simplest non-Newtonian models to replicate the viscoelastic shear-thinning attributes, see Williamson [1]. The flow of thermally radiative Williamson fluid on a stretching sheet with chemical reaction was disclosed by Krishnamurthy et al. [2]. They proved the fluid temperature falling off due to the presence of the Williamson parameter. Khan et al. [3] demonstrated the impact of slip flow of Williamson nanofluid in a porous medium. They exposed that the surface drag force suppresses due to rising the Williamson fluid parameter. The 2D unsteady radiative Williamson fluid flow on a permeable stretching surface was deliberated by Hayat et al. [4]. They noticed that the fluid speed becomes slow when the Williamson parameter is high. Nadeem et al. [5] examined the Williamson fluid flow past a stretching sheet, and they found that the skin friction coefficient decreases with enhancing the Williamson parameter. Make use of the Keller box procedure to solve the problem of MHD flow of Williamson fluid over a stretching sheet by Salahuddin et al. [6]. Their outcome shows that the Williamson fluid parameter leads to suppress the fluid velocity. Few significant analysis for this area is seen in Refs. [7,8].
Fluid flow over a porous medium is confronted in plentiful applications in industry. Few applications are wood drying, nuclear waste storage, food processing, oil purifying, drainage, and irrigation. Darcy's principle is applied to analyze the flow behavior under the condition of small velocity and low porosity. When the quantity of Reynolds number overcomes unity, the Darcy principle was not applicable. Forchheimer [9] defeated this limitation by inserting the square velocity term in the momentum equation. After that, this is known as the Forchheimer number, which is applicable for working higher Reynolds number. Numerical analysis for a Darcy-Forchheimer flow of viscous fluid over a plate was inspected by Mukhopadhyay et al. [10]. They noted that the permeability parameter leads to a decrease in the warmth of the fluid. Hayat et al. [11] demonstrate the 3D Williamson nanomaterial flow on a Darcy-Forchheimer porous medium. They concluded that the surface shear stress diminishes for growing the Forchheimer number. The Darcy-Forchheimer flow of a viscous fluid with heterogeneous-homogeneous chemical reactions was portrayed by Khan et al. [12]. Their results clearly show that the fluid speed becomes slowdown due to the availability of Darcy number. Haider et al. [13] scrutinize the Darcy-Forchheimer and slip flow of hybrid nanofluid on a rotating disk. They proved that the larger estimation of Forchheimer enhances the fluid temperature. Steady 3D Darcy-Forchheimer flow of carbon nanotubes on a rotating disk was revealed by Sadiq et al. [14]. Some important studies for these concepts are collected in Refs. [15][16][17][18].
The magnetic field plays a significant role in the development of fluid thermophysical traits. The demeanour of broadly used fluids like liquid metals, plasma, and electrolytes has a low conductor of electricity. Therefore, an external agent is required to boost up the heat transfer attributes through superior conductivity and thermophysical traits. A magnetic bar with permanently fixed magnets and alternate electronics, known as a Riga plate, can be acted as an external agent to improve fluid electricity. This plate was introduced by Gailitis and Lielausis [19]. Nanofluid flow over a Riga plate was deliberated by Ahmad et al. [20]. Nazeer et al. [21] inspected the chemically reacting Eyring-Powell nanofluid on a Riga plate. They proved that the fluid speed enhances when enhancing the modified Hartmann number. Chemically reacting Prandtl fluid on a Riga plate was addressed by Gireesha et al. [22]. Their results show that the velocity boundary layer thickens due to the more presence of the modified Hartmann number. Mehmood et al. [23] performed the impact of Soret and Dufour effects of a Casson fluid flow on a Riga plate with chemical reaction. Ayub et al. [24], Nayak et al. [25], and Rasool et al. [26] are few essential studies of fluid flow over a Riga plate.
Stratification is a natural process that combines two or more fluids with different densities, temperatures, and concentrations. The double stratification occurs due to both the heat and mass transfer differences. Cheng [27] examined the mass and heat transfer analysis of a power-law fluid in a stratified medium. He noticed that the heat transfer gradient declines for escalating the thermal stratification parameter. The radiative flow of a hyperbolic tangent fluid with chemical reaction and dual stratification's in a porous medium was elucidated by Khan et al. [28]. They found that the fluid concentration downturns for the high magnitude of the stratified thermal parameter. Rehman et al. [29] evaluated the problem of a chemically reacting Williamson fluid with dual stratification, and they have seen that the rate of heat transfer rate is declined for the presence of thermal stratification parameter. The impact of solutal and thermal stratification of a Williamson nanofluid was deliberated by Khan et al. [30]. It is noticed that the horizontal velocity suppresses when the higher magnitude of the thermal stratification parameter. Mallawi et al. [31] derived the series solution of thermally radiative non-Newtonian fluid flow with double stratifications. They have seen that the fluid concentration depresses for enhancing the solutal stratification parameter. Time-dependent MHD nanofluid flow with dual stratifications was performed by Hayat et al. [32]. They proved that surface shear stress enriches for higher values of thermal and solutal stratification parameters.
The aforementioned inspection announces that most of the researchers are involved in revealing the nature of the Darcy-Forchheimer flow with Cattaneo-Christov theory through prescribed wall temperature but not analyzed dual stratifications on a Riga plate. Therefore, our key argument is to fulfill this gap. So, our study elucidates the outcome of the Darcy-Forchheimer flow of a Williamson fluid in the presence of double stratifications, thermal radiation, and chemical reaction on a Riga plate. These types of outcomes will be definitely helpful for a thermal engineer to modeling the thermal systems. Here, the heat and mass transfer phenomena are illustrated by the Cattaneo-Christov dual flux model and the Riga plate is used to control the fluid flow.

Mathematical Formulation
Let us consider the 2D Darcy-Forchheimer flow of a Williamson fluid on a Riga plate. Here, the surface temperature and the concentration are denoted by T w and C w which are always larger than the free stream temperature T ∞ and the free stream concentration C ∞ , respectively, see Figures 1(a) and 1(b). The thermal radiation and first-order chemical reaction are taking into account. Flow situation is manifested with double stratifications. The fluid phase is heat consumption/generation. In addition, the heat and mass transfer phenomenon is inspected through Cattaneo-Christov dual models. The governing equations are modeled as follows: Advances in Mathematical Physics where where u, v is the velocity in x and y directions, ν is the kinematic viscosity, Γ is the time constant, J 0 is the current density, ρ is the density of the fluid, M 0 is the magnetization of the magnet, a 1 is the width of the magnet and the electrodes, C b is the drag coefficient, k 2 is the permeability of porous medium, T is the fluid temperature, λ T is the relaxation time of heat flux, α is the thermal diffusivity, C p is the specific heat, σ * is the Stefan-Boltzmann constant, k * is the mean absorption coefficient, Q is the heat generation/absorption coefficient, C is the fluid concentration, λ C is the relaxation time of mass flux, D B is the mass diffusivity, and k 1 is the chemical reaction parameter.
The boundary conditions are Now, we consider the following dimensionless variables: By using (7), we can modify equations (2)-(4) as follows: Sc We = Γx ffiffiffiffiffiffiffiffiffiffiffi 2a 3 /ν p is the Weissenberg number, Ha = πJ 0 M 0 /8ρa 2 x is the modified Hartmann number, β = π/ða 1 ða/νÞ 1/2 Þ is the dimensionless parameter, λ = ν/k 2 a is the local porosity parameter, 3 Advances in Mathematical Physics the radiation parameter, S 1 = d/b is the thermal stratification parameter, Hg = Q/ρC p a is the heat absorption/generation parameter, Γ 1 = λ T a is the heat relaxation time parameter, Sc = ν/D B is the Schmidt number, Cr = k 1 /a is the chemical reaction parameter, S 2 = e/c is the solutal stratification parameter, Γ 2 = λ C a is the mass relaxation time parameter, and fw = −V w / ffiffiffiffiffi aν p is the suction/injection parameter. The corresponding boundary conditions are The skin friction coefficient, local Nusselt number, and local Sherwood number are expressed as here, the wall shear stress, heat, and mass flux are as follows: The dimensionless form of the above parameters are expressed as 3. Solutions 3.1. Numerical Solution. In this section, the bvp4c solver has been used for gaining the solution. In order to solve the problem, equations (8)-(10) are commuted into a system of firstorder differential equations with the boundary conditions also modified in the same manner. For this, let us take   Advances in Mathematical Physics The system of equations are , y ′ 6 = y 7 , y′ 7 = −y 1 y 7 + y 2 y 6 + S 2 y 2 + Cry 6 + Γ 2 y 2 2 y 6 + S 2 y 2 2 − y 1 y 2 y 7 − y 1 y 3 y 6 − S 2 y 1 y 3

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With the boundary conditions The above set of equations are numerically solved by MATLAB built-in function bvp4c.

HAM Solution.
The obtained ODE's (8)-(10) with conditions (11) are analytically solved by applying the HAM scheme. Because this method is powerful tool for solving nonlinear problems, see Sarwar and Rashidi [33]. Let the initial approximations are chosen as f 0 ðηÞ = fw + 1 − 1/e η , θ 0 ðηÞ = ð1 − S 1 Þ/e η , and ϕ 0 ðηÞ = ð1 − S 2 Þ/e η , and linear oper- where D is the differential operator and the linear property is After implementing the i th order HAM technique, we found the following: Here, f * i ðηÞ, θ * i ðηÞ, and ϕ * i ðηÞ are the particular solutions. These HAM techniques have the parameters (h f , h θ , and h ϕ ), and these are responsible for the convergence of solutions, see Refs. [34][35][36][37].

Results and Discussion
Here, we revealed the results by graphs and tables which describes the shift in velocity, temperature, concentration, skin friction coefficient, local Nusselt number, and local Sherwood number concerning the disparate values of the parameters, such as Weissenberg number ðWeÞ, local porosity parameter ðλÞ, Forchheimer number ðFrÞ, modified Hartmann number ðHaÞ, thermal radiation parameter ðRÞ, thermal stratification parameter ðS 1 Þ, heat generation/absorption parameter ðHgÞ, heat relaxation time parameter ðΓ 1 Þ, chemical reaction parameter ðCrÞ, solute stratification parameter ðS 2 Þ, mass relaxation time parameter ðΓ 2 Þ, and the suction/injection parameter ðfwÞ. The numerically obtained values are compared with the results fetched by the analytical approach by HAM.   Table 2 delineates the changes of skin friction coefficient, local Nusselt number, and local Sherwood number for the distinct values of We, λ, Fr, Ha, and fw. We noted that the surface shear stress upsurges when heightening the We and Ha values and it declines for enhancing the λ, Fr, and fw values. The local Nusselt and Sherwood numbers reduce for raising the We, λ, and Fr, and it rises for increasing the Ha and fw. Table 3 describes the influence of R, Hg, S 1 , and Γ 1 over the heat flux. The heat transfer gradient decimates when developing the Hg and S 1 values, and it grows when growing the R and Γ 1 values. Table 4 helps to figure out the shift of mass flux for the various values of Cr, S 2 , and Γ 2 . The mass transfer rate escalates for the enriching values of Cr and Γ 2 , and it suppresses for increasing S 2 values. Also, we proved that our numerical and analytical results are almost same. Figures 3(a) and 3(b) establish the impact of fw (a) and Ha (b) on velocity profile for DFRP and NDFRP. We uncovered that the fluid speed aggravates due to more presence of Hartmann number and quite the opposite behavior is obtained for the fw parameter. The MBLT is high in NDFRP compared to DFRP for both parameters. The outcomes for disparate values of Ha and R on temperature profile are   Figures 4(a) and 4(b) for DFRP and NDFRP (a) and heat consumption/generation (b). We ascertained that the fluid temperature dwindles because of the high quantity of Hartmann number. However, it is enhanced for raising the radiation parameter. Figures 5(a) and 5(b) explains the changes of fluid temperature for distinct values of S 1 (a) and Γ 1 (b) for heat generation/consumption fluid. We revealed that the fluid warmness becomes subsides for hike values of S 1 and Γ 1 . In addition, the thermal boundary layer thickness is large in the heat generation case compared to the heat consumption case. The significance of S 2 and Γ 2 on concentration profile on destructive chemical reaction and the generative chemical reaction is plotted in Figures 6(a) and 6(b). We ascertained that the fluid concentration lessens for large values of S 2 and Γ 2 .

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for various values of S 1 and LSN for various values of S 2 . We concluded that the heat transfer gradient enhances in heat generation case and maximum increment percentage is obtained when S 1 varies from 0 to 0.2 and it suppresses in heat consumption case and maximum decrement percentage is obtained when S 1 varies from 0.8 to 1. The mass transfer gradient enhances in destructive chemical reaction case and maximum increment percentage is obtained when S 2 varies from 0.8 to 1, and it suppresses in the generative case and maximum decrement percentage is obtained when S 2 varies from 0 to 0.2.

Conclusion
The