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

Authors: S. Eswaramoorthi; Nazek Alessa; M. Sangeethavaanee; Ngawang Namgyel
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

The Darcy-Forchheimer flow of a Williamson fluid over a Riga plate was analyzed in this paper. Energy and mass equations are modeled with Cattaneo-Christov theory and double stratifications. 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 flow parameters on fluid velocity, fluid temperature, fluid concentration, skin-friction coefficient, local Nusselt number, and local Sherwood number is scrutinized via graphs and tables. We acknowledged that the speed of the fluid 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 stratification 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 fluids. 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 fluids.


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## Body

## 1. 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. [15–18].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.


## 2. 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 byTw and Cw 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:
(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=ν∂2u∂y2+2Γν∂u∂y∂2u∂y2−νk2u−Cbxk2u2+πJ0M08ρexp−πa1y,(3)u∂T∂x+v∂T∂y+λTΩT=α∂2T∂y2+1ρCp16σ∗T∞33k∗∂2T∂y2+QρCpT−T∞,(4)u∂C∂x+v∂C∂y+λCΩC=DB∂2C∂y2−k1C−C∞,where
(5)ΩT=u∂u∂x∂T∂x+v∂v∂y∂T∂y+u2∂2T∂x2+v2∂2T∂y2+2uv∂2T∂x∂y+u∂v∂x∂T∂y+v∂u∂y∂T∂x,ΩC=u∂u∂x∂C∂x+v∂v∂y∂C∂y+u2∂2C∂x2+v2∂2C∂y2+2uv∂2C∂x∂y+u∂v∂x∂C∂y+v∂u∂y∂C∂x,where u,v is the velocity in x and y directions, ν is the kinematic viscosity, Γ is the time constant, J0 is the current density, ρ is the density of the fluid, M0 is the magnetization of the magnet, a1 is the width of the magnet and the electrodes, Cb is the drag coefficient, k2 is the permeability of porous medium, T is the fluid temperature, λT is the relaxation time of heat flux, α is the thermal diffusivity, Cp 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, λC is the relaxation time of mass flux, DB is the mass diffusivity, and k1 is the chemical reaction parameter.Figure 1
Sketch of Riga plate (a) and physical configuration of the problem (b).
(a)(b)The boundary conditions are(6)u=Uwx,v=−Vwx,T=Tw=T0+bx,C=Cw=C0+cx,aty=0,u⟶0,∂u∂y⟶0,T⟶T∞=T0+dx,C⟶C∞=C0+exaty⟶∞.Now, we consider the following dimensionless variables:(7)η=yaν,u=xaf′,v=−νaf,θη=T−T∞Tw−T0,ϕη=C−C∞Cw−C0.By using (7), we can modify equations (2)–(4) as follows:
(8)f′′′−f′2+ff′′+Wef′′f′′′−λf′−Frf′2+Hae−βη=0,(9)1Pr1+43Rθ′′+Hgθ+fθ′−f′θ−S1f′−Γ1f′2θ+S1f′2−ff′θ′−ff′′θ−S1ff′′+f2θ′′=0,(10)1Scϕ′′−f′ϕ−S2f′+fϕ′−Crϕ−Γ2f′2ϕ+S2f′2−ff′ϕ′−ff′′ϕ−S2ff′′+f2ϕ′′=0.We=Γx2a3/ν is the Weissenberg number, Ha=πJ0M0/8ρa2x is the modified Hartmann number, β=π/a1a/ν1/2 is the dimensionless parameter, λ=ν/k2a is the local porosity parameter, Fr=Cb/k21/2 is the Forchheimer number, Pr=ν/a is the Prandtl number, R=4σ∗T∞3/kk∗ is the radiation parameter, S1=d/b is the thermal stratification parameter, Hg=Q/ρCpa is the heat absorption/generation parameter, Γ1=λTa is the heat relaxation time parameter, Sc=ν/DB is the Schmidt number, Cr=k1/a is the chemical reaction parameter, S2=e/c is the solutal stratification parameter, Γ2=λCa is the mass relaxation time parameter, and fw=−Vw/aν is the suction/injection parameter.The corresponding boundary conditions are(11)f0=fw,f′0=1,θ0=1−S1,ϕ0=1−S2,(12)f′∞=0,θ∞=0,ϕ∞=0.The skin friction coefficient, local Nusselt number, and local Sherwood number are expressed as(13)Cf=2τwρUw2;Nu=xqwkTw−T∞;Sh=xjwDBCw−C∞;here, the wall shear stress, heat, and mass flux are as follows:
(14)τw=μ∂u∂y1+Γ12∂u∂y;qw=−k∂T∂y+qr;jw=−DB∂C∂y.The dimensionless form of the above parameters are expressed as(15)12CfRe=f′′0+We2f′′02;NuRe=−1+43Rθ′0;ShRe=−ϕ′0.


## 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 first-order differential equations with the boundary conditions also modified in the same manner. For this, let us take
(16)f=y1,f′=y2,f′′=y3,θ=y4,θ′=y5,ϕ=y6,ϕ′=y7.The system of equations are(17)y′1=y2,y′2=y3,y′3=y22−y1y3−Hae−βη+λy2+Fry221+Wey3,y′4=y5,y′5=−y1y5+y2y4+S1y2−Hgy4+Γ1y22y4+S1y22−y1y2y5−y1y3y4−S1y1y31/Pr1+4/3R−Γ1y12,y′6=y7,y′7=−y1y7+y2y6+S2y2+Cry6+Γ2y22y6+S2y22−y1y2y7−y1y3y6−S2y1y31/Sc−Γ2y12.With the boundary conditions(18)y10=fw,y20=1,y2∞=0,y40=1−S1,y4∞=0,y60=1−S2,y6∞=0.The above set of equations are numerically solved by MATLAB built-in function bvp4c.


### 3.2. 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 f0η=fw+1−1/eη, θ0η=1−S1/eη, and ϕ0η=1−S2/eη, and linear operators are Lf=D3f−Df, Lθ=D2θ−θ, and Lϕ=D2ϕ−ϕ, where D is the differential operator and the linear property is LfC1+C2eη+C31/eη=0=LθC4eη+C51/eη=LϕC6eη+C71/eη, where Ckk=1−7 are constants.After implementing theith order HAM technique, we found the following:
(19)fiη=fi∗η+C1+C2eη+C31eη,θiη=θi∗η+C4eη+C51eη,ϕiη=ϕi∗η+C6eη+C71eη.Here, fi∗η,θi∗η, and ϕi∗η are the particular solutions.These HAM techniques have the parameters (hf,hθ, and hϕ), and these are responsible for the convergence of solutions, see Refs. [34–37]. Figure 2 portrays the range value of hf,hθ, and hϕ are −1.3≤hf≤−0.4, −1.6≤hθ≤−0.25, and −1.4≤hϕ≤−0.35. We assign hf=hθ=hϕ=−0.9 for better convergency.Figure 2
h− curve with 18th order of approximation.


## 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 first-order differential equations with the boundary conditions also modified in the same manner. For this, let us take
(16)f=y1,f′=y2,f′′=y3,θ=y4,θ′=y5,ϕ=y6,ϕ′=y7.The system of equations are(17)y′1=y2,y′2=y3,y′3=y22−y1y3−Hae−βη+λy2+Fry221+Wey3,y′4=y5,y′5=−y1y5+y2y4+S1y2−Hgy4+Γ1y22y4+S1y22−y1y2y5−y1y3y4−S1y1y31/Pr1+4/3R−Γ1y12,y′6=y7,y′7=−y1y7+y2y6+S2y2+Cry6+Γ2y22y6+S2y22−y1y2y7−y1y3y6−S2y1y31/Sc−Γ2y12.With the boundary conditions(18)y10=fw,y20=1,y2∞=0,y40=1−S1,y4∞=0,y60=1−S2,y6∞=0.The above set of equations are numerically solved by MATLAB built-in function bvp4c.


## 3.2. 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 f0η=fw+1−1/eη, θ0η=1−S1/eη, and ϕ0η=1−S2/eη, and linear operators are Lf=D3f−Df, Lθ=D2θ−θ, and Lϕ=D2ϕ−ϕ, where D is the differential operator and the linear property is LfC1+C2eη+C31/eη=0=LθC4eη+C51/eη=LϕC6eη+C71/eη, where Ckk=1−7 are constants.After implementing theith order HAM technique, we found the following:
(19)fiη=fi∗η+C1+C2eη+C31eη,θiη=θi∗η+C4eη+C51eη,ϕiη=ϕi∗η+C6eη+C71eη.Here, fi∗η,θi∗η, and ϕi∗η are the particular solutions.These HAM techniques have the parameters (hf,hθ, and hϕ), and these are responsible for the convergence of solutions, see Refs. [34–37]. Figure 2 portrays the range value of hf,hθ, and hϕ are −1.3≤hf≤−0.4, −1.6≤hθ≤−0.25, and −1.4≤hϕ≤−0.35. We assign hf=hθ=hϕ=−0.9 for better convergency.Figure 2
h− curve with 18th order of approximation.


## 4. 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 numberWe, local porosity parameter λ, Forchheimer number Fr, modified Hartmann number Ha, thermal radiation parameter R, thermal stratification parameter S1, heat generation/absorption parameter Hg, heat relaxation time parameter Γ1, chemical reaction parameter Cr, solute stratification parameter S2, mass relaxation time parameter Γ2, and the suction/injection parameter fw. The numerically obtained values are compared with the results fetched by the analytical approach by HAM.Table1 provides the HAM order and CPU timings. From this table, we concluded that 13th order is sufficient for all profiles. 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, S1, and Γ1 over the heat flux. The heat transfer gradient decimates when developing the Hg and S1 values, and it grows when growing the R and Γ1 values. Table 4 helps to figure out the shift of mass flux for the various values of Cr,S2, and Γ2. The mass transfer rate escalates for the enriching values of Cr and Γ2, and it suppresses for increasing S2 values. Also, we proved that our numerical and analytical results are almost same.Table 1
Order of order approximations and CPU timing.
Order−f′′0−θ′0−ϕ′0CPU timings (sec.)11.270001.210541.430540.5851341031.226871.458814.73101.342901.226921.4586033.06131.342941.226921.4586047.64151.342941.226921.45860139.76201.342941.226921.45860646.61251.342941.226921.458602376.22301.342941.226921.458607157.92Table 2
Numerical and analytical computation of local skin friction coefficient, local Nusselt number, and local Sherwood number for various values ofWe,λ,Fr,fw, and Ha.
WeλFrfwHaSkin friction coefficientLocal Nusselt numberLocal Sherwood numberNumericalAnalyticalNumericalAnalyticalNumericalAnalytical0.00.20.40.30.3-1.21627-1.216242.063382.063441.468401.468410.1-1.19072-1.190702.054712.054771.463831.463850.2-1.16263-1.162592.044812.044871.458581.458600.3-1.13089-1.130832.033142.033191.452331.452350.4-1.09319-1.091642.018482.018591.444381.444420.5-0.99995-0.999981.997201.997821.432571.432940.20.00.40.30.3-1.07827-1.078182.074082.074221.472981.473030.1-1.12135-1.121302.059042.059131.465581.465610.2-1.16263-1.162592.044812.044871.458581.458600.3-1.20228-1.202252.031322.031361.451941.451950.4-1.24046-1.240422.018512.018531.445631.445640.5-1.27728-1.277252.006302.006341.439611.439640.20.20.00.30.3-1.05656-1.056522.068052.068111.470741.470760.3-1.13706-1.137022.050382.050431.461491.461510.5-1.18763-1.187582.039402.039451.455751.455760.8-1.25946-1.259422.023972.024061.447681.447731.0-1.30495-1.305442.014302.014981.442621.443220.20.20.4-1.00.3-0.68939-0.689361.229351.229400.913490.91350-0.5-0.83580-0.835791.473331.473391.076731.076750.0-1.02546-1.025431.796331.796391.292211.292230.5-1.26411-1.264082.241482.241531.591031.591051.0-1.55195-1.551822.888262.889232.033432.032420.20.20.40.30.0-1.29871-1.298691.992541.992551.433321.433330.5-1.07378-1.073742.076122.076191.473991.474011.0-0.85646-0.856412.145562.145681.508821.508861.5-0.64420-0.644112.205872.205861.539701.539892.0-0.43568-0.434312.259652.260841.567711.57081Table 3
Numerical and analytical computation of local Nusselt number for various values ofR,Hg,S1, and Γ1.
RHgS1Γ1Local Nusselt numberNumericalAnalytical0.0-0.50.20.11.723511.723550.31.928211.928260.52.044812.044870.72.150022.150091.02.291562.291640.5-0.50.20.12.044812.04487-0.21.885311.885440.01.764941.940910.21.625861.626950.51.215301.271390.5-0.50.00.12.334792.334810.22.044812.044870.41.754831.754920.61.464841.464980.81.174861.175031.00.884880.885080.5-0.50.20.01.905911.905970.12.044812.044870.22.188182.188240.32.336092.336150.42.488592.48865Table 4
Numerical and analytical computation of local Sherwood number for various values ofCr,S2, and Γ2.
CrS2Γ2Local Sherwood numberNumericalAnalytical-0.50.20.10.903170.89525-0.21.063131.063910.01.144981.145171.51.315651.315681.01.458581.458601.00.00.11.694691.694690.21.458581.458600.41.222481.222500.60.986370.986400.80.750260.750301.00.514160.514211.00.20.01.372971.372980.11.458581.458600.21.547161.547180.31.638811.638840.41.723991.73363Figures3(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 illustrated in 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 S1 (a) and Γ1 (b) for heat generation/consumption fluid. We revealed that the fluid warmness becomes subsides for hike values of S1 and Γ1. In addition, the thermal boundary layer thickness is large in the heat generation case compared to the heat consumption case. The significance of S2 and Γ2 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 S2 and Γ2.Figure 3
The velocity profile for disparate values offw (a) and Ha (b).
(a)(b)Figure 4
The temperature profile for disparate values ofHa (a) and R (b).
(a)(b)Figure 5
The temperature profile for disparate values ofS1 (a) and Γ1 (b).
(a)(b)Figure 6
The concentration profile for disparate values ofS2 (a) and Γ2 (b).
(a)(b)The variations of SFC for distinct combination offw and Ha (a) for DF flow (lower plate) and NDF flow (upper plate) and (b) for DF flow of Williamson fluid (upper plate) and viscous fluid (lower plate) are portrayed in Figures 7(a) and 7(b). We found that the plate shear stress depresses for enhancing values of fw, and it raises for rising the values of Hartmann number. Figures 8(a) and 8(b) disclose the changes of LNN for disparate combination of fw and Ha (a) for DF flow (lower plate) and NDF flow (upper plate) and (b) for DF flow of Williamson fluid (lower plate) and viscous fluid (upper plate). We noted that heat transfer gradient escalates for increasing the values of fw and Ha. The alternations of LNN for various combination of (a) fw and Γ1 with Hg=−0.5 (upper plate) and Hg=0.5 (lower plate) and (b) fw and S1 with Hg=−0.5 (upper plate) and Hg=0.5 (lower plate) are plotted in Figures 9(a) and 9(b). We noticed that the heat transfer gradient upgrades for the available of fw, and it weakens for high quantity of S1 and Γ1. In addition, the larger heat transfer gradient occurs in heat consumption case compared to heat generation case. Figures 10(a) and 10(b) express the LSN variation with respect to the distinct combination of fw and Ha (a) for DF flow (upper plate) and NDF flow (lower plate) and (b) for DF flow of Williamson fluid (upper plate) and viscous fluid (lower plate). We proved that the mass transfer gradient slashes due to larger values of fw and Ha. The deviation of LSN for different combination of (a) fw and Γ2 with generative chemical reaction (upper plate) and destructive chemical reaction (lower plate) and (b) fw and S2 with generative chemical reaction (upper plate) and destructive chemical reaction (lower plate) is shown in Figures 11(a) and 11(b). We concluded that the LSN declines for upgrading the values of fw, Γ2, and S2.Figure 7
The variations of SFC for distinct combination offw and Ha (a) for DF flow (lower plate) and NDF flow (upper plate) and (b) for DF flow of Williamson fluid (upper plate) and viscous fluid (lower plate).
(a)(b)Figure 8
The variations of LNN for distinct combination offw and Ha (a) for DF flow (lower plate) and NDF flow (upper plate) and (b) for DF flow of Williamson fluid (lower plate) and viscous fluid (upper plate).
(a)(b)Figure 9
The variations of LNN for distinct combination of (a)fw and Γ1 with Hg=−0.5 (upper plate) and Hg=0.5 (lower plate) and (b) fw and S1 with Hg=−0.5 (upper plate) and Hg=0.5 (lower plate).
(a)(b)Figure 10
The variations of LSN for distinct combination offw and Ha (a) for DF flow (upper plate) and NDF flow (lower plate) and (b) for DF flow of Williamson fluid (upper plate) and viscous fluid (lower plate).
(a)(b)Figure 11
The variations of LSN for distinct combination of (a)fw and Γ2 with generative chemical reaction (upper plate) and destructive chemical reaction (lower plate) and (b) fw and S2 with generative chemical reaction (upper plate) and destructive chemical reaction (lower plate).
(a)(b)The decrement percentage of SFC for various values offw on Williamson fluid and viscous fluid is plotted in Figures 12(a) and 12(b) and observed that the maximum decreasing percentage of surface shear stress is obtained when fw varies from 0.0 to 0.4 for both fluids. Figures 13(a) and 13(b) give the increment percentage of LNN for various values of fw on Williamson fluid and viscous fluid. The maximum increment percentage has occurred when fw vary from 0.4 to 0.8 for both fluids. The decrement percentage of LSN for various values of fw on Williamson fluid and viscous fluid is illustrated in Figures 14(a) and 14(b), and we have seen that the maximum decrement percentage occurred when fw varies from 0.4 to 0.8 for both fluids. Figures 15(a) and 15(b) display the decrement/increment percentage of LNN for various values of S1 and LSN for various values of S2. We concluded that the heat transfer gradient enhances in heat generation case and maximum increment percentage is obtained when S1 varies from 0 to 0.2 and it suppresses in heat consumption case and maximum decrement percentage is obtained when S1 varies from 0.8 to 1. The mass transfer gradient enhances in destructive chemical reaction case and maximum increment percentage is obtained when S2 varies from 0.8 to 1, and it suppresses in the generative case and maximum decrement percentage is obtained when S2 varies from 0 to 0.2.Figure 12
Decrement percentage of SFC for various values offw on (a) Williamson fluid and (b) viscous fluid.
(a)(b)Figure 13
Increment percentage of LNN for various values offw on (a) Williamson fluid and (b) viscous fluid.
(a)(b)Figure 14
Decrement percentage of LSN for various values offw on (a) Williamson fluid and (b) viscous fluid.
(a)(b)Figure 15
Decrement/increment percentage of (a) LNN for various values ofS1 and (b) LSN for various values of S2.
(a)(b)


## 5. Conclusion

The current study figures out the Darcy-Forchheimer flow of the Williamson fluid over a Riga plate with Cattaneo-Christov double diffusion and double stratification. The solutions are fetched numerically by a bvp4c solver in MATLAB and analytically by HAM. The outcomes are employed as follows:(i)
The fluid velocity drops when the suction/injection parameter enriches(ii)
The larger values of the thermal radiation parameter boost up the temperature(iii)
The high temperature is noticed in the heat generation case, and the low temperature is produced in the heat absorption case(iv)
The generative chemical reaction case upturns the fluid concentration while the destructive chemical reaction case declines(v)
The high wall shear stress is produced when enlarging the modified Hartmann number



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