Muhammad Sohail , Umair Ali , Fatema Tuz Zohra , Wael Al-Kouz , Yu-Ming Chu and Phatiphat Thounthong

Utilization of updated version of heat flux model for the radiative flow of a non-Newtonian material under Joule heating: OHAM application

De Gruyter | Published online: March 10, 2021

Abstract

This study reports the thermal analysis and species transport to manifest non-Newtonian materials flowing over linear stretch sheets. The heat transfer phenomenon is presented by the Cattaneo–Christov definition of heat flux. Mass transportation is modeled using traditional Fick’s second law. In addition, the contribution of Joule heating and radiation to thermal transmission is also considered. Thermo-diffusion and diffusion-thermo are significant contributions involved in thermal transmission and species. The physical depiction of the scenario under consideration is modeled through the boundary layer approach. Similar analysis has been made to convert the PDE model system into the respective ODE. Then, the transformed physical expressions are calculated for momentum, thermal, and species transport within the boundary layer. The reported study is a novel contribution due to the combined comportment of thermal relaxation time, radiation, Joule heating, and thermo-diffusion, which are not yet explored. Several engineering systems are based on their applications and utilization.

1 Introduction

The study of the boundary layer flow over a stretch sheet has recently received a great deal of attention because of its important applications and great potential for use as a technological tool in many engineering applications such as paper production, polymer sheet, insulation materials, food processing, wire coating, and fiber. Sakiadis [1] first solved the fundamental flow problem in the boundary layer theory for Newtonian and non-Newtonian fluids. In addition, many relationships have been found for this boundary layer problem class for different scenarios. Sohail et al. [2] studied the mechanism of bio-convection in three-dimensional Oldroyd-B nanofluid obeying thermal conductivity. In Ref. [3], Hayat et al. analyzed the inspiration of Soret and Dufour effects for MHD flow of Casson liquid. Flow is produced because of the stretching of the surface. Yokus [4] discussed the numerical and exact solutions to the model of the FitzHugh–Nagumo equation. He computed the numerical solution via finite difference methods (FDMs). He presented the stability analysis via the von Neumann stability analysis. He also presented the error norms in his exploration. Numerically, the influence of generalized Buongiorno’s model for the flow of thermohydrodynamic model was studied by Wakif et al. [5]. Mackolil and Mahanthesh [6] examined the statistical analysis for the Casson model by considering mass and heat flux conditions. They used Laplace transform to record the parametric comportment. Theoretical examination was reported by Megahed [7] with the temperature model of thermal conductivity in the Casson model. The flow is produced because of the stretching of an exponential stretching sheet. Thermal profile is presented with viscous dissipation. Mahanthesh et al. [8] examined the flow past over an elongating surface under the inspiration of numerous physical considerations. They used the boundary layer approach to diminish the complex problem into a simpler one and then solved it by using the shooting approach. They established the comparative study in their report. Mohyud-Din et al. [9] examined the mechanism of heat transfer for the squeezed flow of yield exhibiting a non-Newtonian fluid model between two parallel disks. They considered the involvement of viscous dissipation. Moaaz et al. [10] presented new results for the solutions of third-order nonlinear neutral differential equations. Their results complement and improve some previous results in the literature. Usman et al. [11] premeditated the thermal transport in ferrofluid with magnetic effects. They reported that skin friction and heat transfer rate upshift by mounting the values of magnetic parameter. Yokus and Bulut [12] examined the effectiveness of the solution of Cahn–Allen equation using FDM. They also computed the exact solution for comparison. Mathematica 11.0 package is used to perform the computational procedure. Two- and three-dimensional graphs are prepared to judge the comparison. Moaaz et al. [13] used a refinement of the Riccati transformations and comparison principles. They established new sufficient conditions for the oscillation of solutions of second-order neutral differential equations with distributed deviating arguments along with some illustrated examples. The exact solution for thermal transport in water-based nanofluids was studied by Abrar et al. [14]. The impact of viscous dissipation is considered in the temperature equation. They prepared a flow line graph to show the velocity field behavior. Numerically, the Carreau fluid flow by heat transport using a general model for heat flux over a stretch sheet was studied by Rashid et al. [15]. They demonstrated that the value of the magnetic parameter installation reduced the flow and booted the thermal distribution. Mahanthesh et al. [16] investigated the enhancement in thermal transmission by considering the contribution of nanoparticles. Flow is assumed to past over a stretchable disk. Bazighifan and Postolache [17] used multiple techniques for studying asymptotic properties of a class of differential equations with variable coefficients. They were concerned with the oscillatory properties of fourth-order differential equations with variable coefficients. They applied the following methods, namely, the comparison method, the Riccati technique, and the integral averaging technique along with two given examples. Megahed [18] studied the thermal transport in the Sisko fluid by considering heat generation past over a nonlinear stretching sheet. Also, he considered viscous dissipation. Numerically, the system of modeled equations approximated. Mebarek-Oudina et al. [19] numerically studied the flow in porous upright annulus cylindrical medium, which is magnetized. Flow features have been shown graphically using finite volume scheme. Important parametric studies have been established. Several other important contributions are presented in ref. [34–49].

It is evident from the aforementioned contributions that so far no attempt has been made to investigate the thermal transmission in the Casson liquid with the involvement of thermal relaxation time, radiation, and Joule heating. This exploration fills this huge gap in the available open literature. In addition, transportation of mass is also explored. This contribution can be used as a base for the further research in this direction, and it has numerous applications in thermal systems. Moreover, we presented the comparative investigation of our work with those of the published findings. The performed finding has been organized as follows: literature survey is listed in Section 1; modeling is presented in Section 2; important quantities of practical use are explained in Section 3; methodology with convergence, validity of results, and parametric analysis are presented in Sections 4 and 5; and the key findings of the performed research are presented in Section 6.

2 Mathematical drafting via boundary layer theory (MDVBLT)

An incompressible MHD flow of the Casson model over a stretching heated sheet is considered in this examination as presented in Figure 1. Flow is produced because of the stretching of the sheet, which is shown in Figure 1. Here, velocity of the wall is assumed to be u w ( x ) = c x , c > 0 . For c < 0 , case of shrinking is established. Temperature and concentration at y = 0 are denoted by T w and C w , respectively. In the transverse direction to the surface, a constant magnetic field B 0 is applied. The rheological equation for the flow under consideration [3,6,9,23,24,25] is presented as follows:

(1) τ i j = 2 μ B + p y 2 π e i j , π π c .

Figure 1 
               Depiction of considered physical happening.

Figure 1

Depiction of considered physical happening.

Under usual boundary layer approximations, the flow, heat transfer, and mass transfer phenomena are governed by the following equations [3]:

(2) u 1 x + u 2 y = 0 ,
(3) u 1 u 1 x + u 2 u 2 y = v a 1 + 1 β 2 u 1 y 2 σ ρ B 0 2 u 1 ,
(4) u 1 T x + u 2 T y + λ T u 1 u 1 x T x + u 2 u 2 y T y + u 1 u 2 x T y + u 2 u 1 y T x + 2 u 1 u 2 2 T x y + ( u 1 ) 2 2 T x 2 + ( u 2 ) 2 2 T y 2 = α T 2 T y 2 + D m k T C 0 c p 2 C y 2 + ( σ ) 2 ( u 1 ) 2 B 0 2 ρ c p σ + 16 σ 3 ρ c p k T 3 2 T y 2 ,
(5) u 1 C x + u 2 C y = D m 2 C y 2 + D m k T T m 2 T y 2 .

The boundary conditions [3] for the considered physical situation are as follows:

(6) u 1 = u w = c x , u 2 = 0 , T = T w = T + a x , C = C + b x at y = 0 , u 1 0 , T T , C C as y

Assuming the following similarity renovations [3],

(7) ξ = y c v a , u 1 = c x f ( ξ ) , u 2 = c v a f ( ξ ) , θ ( ξ ) = T T T w T , ϕ ( ξ ) = C C C w C .

After consuming the aforementioned alteration, prevailing equations condense to the following form:

(8) 1 + 1 β f + f f ( f ) 2 ( Ha ) 2 f = 0 ,
(9) 1 + 4 3 R d θ + Pr θ + Pr M e ϕ + Pr E c ( Ha ) 2 ( f ) 2 + Pr λ a [ f f θ f 2 θ ] = 0 ,
(10) ϕ + Pr L e f θ Pr L e f ϕ + S r L e θ = 0 ,

The dimensionless boundary conditions [3] that help the flow over an overextended surface are as follows:

(11) f ( 0 ) = 0 , f ( 0 ) = 1 , f ( ) = 0 , θ ( 0 ) = 1 , θ ( ) = 0 , ϕ ( 0 ) = 1 , ϕ ( ) = 0 ,

In the aforementioned equations, Ha , Pr , λ a , E c , and L e are the Hartman, Prandtl, thermal relaxation parameter, local Eckert, and Lewis numbers, respectively, expressed by

(12) ( Ha ) 2 = σ B 0 2 ρ c , Pr = v a α T , λ a = λ T c , L e = α T M e , E c = c 2 x 2 c p ( T w T ) , R d = 4 σ T 3 k k ρ c p ,
where M e and S r are Dufour and Soret Numbers, respectively, and are defined by Hayat et al. [ 3].
(13) M e = D m k T C w C C s c p T w T , S r = D m k T T w T C s c p C w C .

3 Physical quantities of interest

To discuss the fluid flow problems’ skin friction ( C F ) , heat transfer coefficient ( N u ) and mass transfer coefficient S u [3] are the significant terminologies that are demarcated as follows:

C F = 2 τ wall ρ ( u 1 ) 2 = 2 μ B + p y 2 π u 1 y + u 2 x y = 0 ρ ( u 1 ) 2 , N u = x ( 1 + 16 σ 3 ρ c p k T 3 ) T y y = 0 k T w T , Sh = x C y y = 0 C w C .

The dimensionless expressions of these quantities are presented as follows:

(14) R e 1 2 C f = 1 + 1 β f ( 0 ) ,
(15) R e 1 2 N u = 1 + 4 3 R d θ ( 0 ) ,
(16) R e 1 2 Sh = ϕ ( 0 ) ,
where R e = u w ( x ) x v a is the local Reynolds number.

4 Optimal homotopy analysis method

Several approaches are used to handle the nonlinear system of differential equations. Since then, the equation resulting from the estimated flow provided by the sheet extension is very nonlinear, and the exact solution to these equations cannot be found. For the approximate solution, optimal homotopy analysis method (OHAM) proposes that the scheme has the potential to calculate the solutions of the nonlinear-coupled problems that arise in various phenomena in mathematical physics. The scheme is free of discretization, linearization, or perturbation and decrease the computational cost. It is more reliable and consistent for the nonlinear problems arising in the mathematical physics. Moreover, the scheme is also easy to implement, effective, and simple to handle.

4.1 STEP-I

Selection of linear operator.

4.2 STEP-II

Selection of initial guess.

The convergence of the proposed scheme depends on the correct resolution of the initial guesses. For the estimated modeling problem, the initial guesses are their respective linear operators:

(17) f 0 ( ξ ) = 1 1 e ξ , θ 0 ( ξ ) = 1 e ξ , ϕ 0 ( ξ ) = 1 e ξ ,
(18) L 1 = ( D 3 D ) f , L 2 = ( D 2 1 ) θ , L 3 = ( D 2 1 ) ϕ ,
and these linear operators conform the following features:
(19) L 1 s 1 + s 2 e ξ + s 3 e ξ = 0 ,
(20) L 2 [ s 7 e ξ + s 8 e ξ ] = 0 ,
(21) L 3 [ s 9 e ξ + s 10 e ξ ] = 0 ,
where s n ( n = 1 12 ) are the constants that are computed by engaging the mentioned constraints. Averaged squared residuals as suggested in ref. [ 2, 25, 26, 27, 28, 29, 30, 31, 32, 33] for velocity, thermal, and solutal fields with controlling parameters f , θ , and ϕ , respectively, are described as follows:
(22) Q r f ˜ = 1 B + 1 J = 0 B G f I = 0 r f ˆ ( ξ ) ξ = J δ ξ 2 ,
(23) Q r θ ˜ = 1 B + 1 J = 0 B G θ I = 0 r f ˆ ( ξ ) , I = 0 r θ ˆ ( ξ ) , I = 0 r ϕ ˆ ( ξ ) ξ = J δ ξ 2 ,
(24) Q r ϕ ˜ = 1 B + 1 J = 0 B G ϕ I = 0 r f ˆ ( ξ ) , I = 0 r θ ˆ ( ξ ) , I = 0 r ϕ ˆ ( ξ ) ξ = J δ ξ 2 .

In the view of ref. [2,25,26,27,28,29,30,31,32,33], we have

Q r t ˜ = Q m f ˜ + Q r θ ˜ + Q r ϕ ˜ ,
where Q r t ˜ attitudes for total squared residuals errors, δ ξ = 0.5 and B = 20 . A situation has been reconnoitred, where Pr = 1.0 , Ha = 0.5 , E c = 0.6 , λ a = 0.2 , M e = 0.2 , S r = 0.5 , and β = 0.7 . It is perceived that at the fourth order of approximation, the values of optimal convergent control parameters are f = 0.3971394 , θ = 1.029516 , ϕ = 1.01234 , and the total squared residuals error is 0.0002590136.

5 Results and discussion

5.1 Convergence analysis

The convergence of the desired solution is presented in Table 1, which declares that by increasing the order of approximations, errors guaranteeing convergence of the proposed scheme diminution. From Table 1, it is evident that higher order approximations reduce the errors. Reduction in errors guarantees the convergence of OHAM solutions. Moreover, this table presented the authenticity of the applied scheme. Similarly, the last column of this table shows the time for the mentioned iterations from 2 to 50. Computationally, the applied algorithm converges, which is depicted in Table 1. Table 2 is organized to note the encouragement of the magnetic parameter on dimensionless stress. Noise stress is the increasing function of the magnetic parameters (Table 2). Moreover, this analysis presents the validity of the anticipated scheme. Also, it presents that our obtained results are in close agreement with the investigations reported in Refs. [2022]. The rate of heat and mass transport against innumerable developing parameters is presented in Tables 3 and 4, respectively. These tables clearly indicate that the solutions found are in excellent arrangement with the work designated in the open literature. Moreover, these tables present the strength of the applied scheme.

Table 1

Convergence scrutiny of performed methodology

B ε B f ε B θ ε B ϕ CPU time (s)
2 0.000120126 0.0000189128 0.0000265693 0.7657
4 6.04977 × 10 6 1.596 × 10 6 0.0000109456 2.5698
6 7.24237 × 10 7 2.9847 × 10 7 2.31806 × 10 6 6.1096
10 1.26725 × 10 7 1.0608 × 10 7 1.88947 × 10 7 37.5163
12 1.58932 × 10 9 2.66867 × 10 8 9.0298 × 10 8 74.2982
16 2.25355 × 10 11 3.29468 × 10 9 1.24480 × 10 8 193.1398
18 2.62284 × 10 12 2.41065 × 10 9 6.39224 × 10 9 288.5361
20 3.11266 × 10 13 9.9970 × 10 10 3.91447 × 10 9 435.9568
22 3.95205 × 10 14 3.39264 × 10 10 2.00081 × 10 9 678.1165
24 5.74914 × 10 15 2.43546 × 10 10 8.98722 × 10 10 1044.5182
26 1.01937 × 10 15 1.95977 × 10 10 4.91498 × 10 10 1460.6771
28 2.21954 × 10 16 9.89696 × 10 11 3.27548 × 10 10 1943.9862
30 5.64755 × 10 17 3.56767 × 10 11 1.94651 × 10 10 2529.3342
34 1.92023 × 10 18 1.28945 × 10 11 3.78563 × 10 11 3012.3457
38 1.21621 × 10 18 3.27894 × 10 12 2.95701 × 10 11 3310.1673
42 2.29043 × 10 19 1.29082 × 10 12 4.29083 × 10 12 4671.2681
50 3.19846 × 10 20 2.98431 × 10 15 2.98432 × 10 13 5217.6258
Table 2

Comparative investigation of dimensionless stress when β =

Ha [20] [21] [22] Present
0.0 1.9991 1.9991 1.9991 1.998023
0.1 2.0101 2.0101 2.0101 2.074134
0.5 2.1102 2.1102 2.1102 2.087394
1.0 2.3902 2.3902 2.3902 2.241731
Table 3

Comparative investigation of rate of heat transfer against different emerging parameters when R d =

β Ha Pr S r M e L e 1 + 4 3 R d θ ( 0 ) [3] 1 + 4 3 R d θ ( 0 ) Present
0.8 0.5 0.7 0.4 0.5 1.0 0.65027 0.6502502
1.4 0.62182 0.6218174
3.0 0.59241 0.5924382
2.0 0.0 0.63157 0.6315692
0.6 0.59648 0.5964527
1.2 0.51923 0.5192471
0.5 0.5 0.47751 0.4775010
1.0 0.77142 0.7714026
1.5 0.99905 1.9990825
0.7 0.0 0.55231 0.5523421
0.5 0.64117 0.6411602
1.0 0.73679 0.7367721
0.4 0.0 0.81654 0.8165324
0.7 0.50425 0.5042731
1.5 0.32168 0.3216602
0.5 0.0 0.69482 0.6948132
1.3 0.61294 0.6129627
2.0 0.51821 0.5182052
Table 4

Analysis of mass transfer coefficient against involved parameters

β Ha Pr S r M e L e ϕ ( 0 ) [3] ϕ ( 0 ) Present
0.8 0.5 0.7 0.4 0.5 1.0 0.71096 0.7109047
1.4 0.68252 0.6825432
3.0 0.63305 0.6330348
2.0 0.0 0.69227 0.6922494
0.6 0.65714 0.6571684
1.2 0.57903 0.5790586
0.5 0.5 0.52859 0.5285826
1.0 0.84393 0.8439237
1.5 1.08874 1.0887453
0.7 0.0 0.81655 0.8165478
0.5 0.57530 0.5753187
1.0 0.33297 0.3329586
0.4 0.0 0.60513 0.6050998
0.7 0.69848 0.6984964
1.5 0.75579 0.7557698
0.5 0.0 0.50957 0.5095532
1.3 0.78412 0.7841021
2.0 1.07266 1.0726532

5.2 Parametric analysis and physical justification

The consequence of collective consideration of the employment of thermal relaxation, radiation, joule heating, and Soret and Dufour effects for the Casson fluid (that makes the model novel) has been explained elaborately in this section through graphs. For this determination, Figures 2–14 are plotted. Figure 2 is organized to scrutinize the comportment of β on f ( ξ ) . Here, it is shown that the velocity weakens against β . Physically, β is directly associated with plastic dynamics viscosity. As β increases, plastic dynamic viscosity increases, which retards the flow. Moreover, against the growing values of β , yield stress decreases. Consequently, fluid velocity depreciates. Figure 3 is portrayed to inspect the bearing of magnetic parameter Ha on fluid velocity. An escalation in magnetic parameter corresponds to increase in the Lorentz force. We know that the Lorentz force is resistive in nature, and it controls the sudden oscillation in fluid velocity due to which the velocity field decreases. Import of the magnetic parameter on fluid temperature and concentration is plotted in Figures 4 and 5, respectively. Direct bearing of magnetic parameter Ha is shown with the help of these plots. Physically, an increase in Ha corresponds to an increase in the Lorentz force due to which more heat is produced. As a result, fluid temperature θ ( ξ ) and concentration grow. The Hartman number is proportional to the magnetic field, that is, the increase in the Hartman number enhances the Lorentz force. The Lorentz force resists the fluid motion. Due to the resistance and collision of the fluid particles, some energies are converted into heat [25] and the concentration of the particles in the fluid flow also increases. Thus, for higher Hartman ( Ha ) number, the velocity profile decreases (Figure 3), but the temperature profile and concentration profile increase (Figures 4 and 5). As the temperature and concentration profile increase, the heat and mass transfer rates decrease, as presented in Tables 3 and 4. Figure 6 is planned to judge the involvement of β on fluid temperature. Due to the increase in β , frictional force increases. As a result, heated particles transfer extra energy to the neighboring particles. Consequently, fluid temperature upsurges significantly. Influence of fluid temperature against the Prandtl number is plotted in Figure 7. Physically, the Prandtl number gives the information about momentum and thermal boundary layers because the Prandtl number is inversely related to thermal diffusion. For higher values of the Prandtl number, momentum diffusion is higher than the thermal diffusion. As a result, the temperature field decreases. The phenomena are helpful to overcome the energy losses, which is necessary for the thermal stability. Behavior of radiation parameter ( R d ) on the fluid temperature is shown in Figure 8. It is presented that an increment in ( R d ) cause to enhance the temperature field. Due to an increase in radiation parameter, moving fluid particles transmit energy. Consequently, thermal profile upsurges. The influence of the local Eckert number on the fluid temperature and the velocity field is plotted in Figures 9 and 10, respectively. Mounting values of E c enhance the velocity and temperature fields. Compartment of the Dufour number M e on fluid temperature and concentration is plotted in Figures 11 and 12, respectively; in these figures, opposite impact of M e on temperature and concentration property is noted. Involvement of Soret number ( S r ) on dimensionless concentration is shown in Figure 13. Direct relation of S r and ϕ ( ξ ) is revealed through this figure. Figure 14 presents that mounting values of λ a correspond that heated particles transport the thermal energy from one point to another by taking the extra time and hence this fluid temperature decreases, which is shown in Figure 14.

Figure 2 
                  Variation of 
                        
                           
                           
                              β
                           
                           \beta 
                        
                      on 
                        
                           
                           
                              
                                 f
                                 ′
                              
                              (
                              ξ
                              )
                           
                           f^{\prime} (\xi )
                        
                     .

Figure 2

Variation of β on f ( ξ ) .

Figure 3 
                  Influence of 
                        
                           
                           
                              Ha
                           
                           \text{Ha}
                        
                      on 
                        
                           
                           
                              
                                 f
                                 ′
                              
                              (
                              ξ
                              )
                           
                           f^{\prime} (\xi )
                        
                     .

Figure 3

Influence of Ha on f ( ξ ) .

Figure 4 
                  Variation of 
                        
                           
                           
                              Ha
                           
                           \text{Ha}
                        
                      on 
                        
                           
                           
                              θ
                              (
                              ξ
                              )
                           
                           \theta (\xi )
                        
                     .

Figure 4

Variation of Ha on θ ( ξ ) .

Figure 5 
                  Influence of 
                        
                           
                           
                              Ha
                           
                           \text{Ha}
                        
                      on 
                        
                           
                           
                              ϕ
                              (
                              ξ
                              )
                           
                           \phi (\xi )
                        
                     .

Figure 5

Influence of Ha on ϕ ( ξ ) .

Figure 6 
                  Variation of 
                        
                           
                           
                              β
                           
                           \beta 
                        
                      on 
                        
                           
                           
                              θ
                              (
                              ξ
                              )
                           
                           \theta (\xi )
                        
                     .

Figure 6

Variation of β on θ ( ξ ) .

Figure 7 
                  Comportment of 
                        
                           
                           
                              Pr
                           
                           \text{Pr}
                        
                      on 
                        
                           
                           
                              θ
                              (
                              ξ
                              )
                           
                           \theta (\xi )
                        
                     .

Figure 7

Comportment of Pr on θ ( ξ ) .

Figure 8 
                  Variation of 
                        
                           
                           
                              
                                 
                                    R
                                 
                                 
                                    d
                                 
                              
                           
                           {R}_{\text{d}}
                        
                      on 
                        
                           
                           
                              θ
                              (
                              ξ
                              )
                           
                           \theta (\xi )
                        
                     .

Figure 8

Variation of R d on θ ( ξ ) .

Figure 9 
                  Variation of 
                        
                           
                           
                              
                                 
                                    E
                                 
                                 
                                    c
                                 
                              
                           
                           {E}_{\text{c}}
                        
                      on 
                        
                           
                           
                              θ
                              (
                              ξ
                              )
                           
                           \theta (\xi )
                        
                     .

Figure 9

Variation of E c on θ ( ξ ) .

Figure 10 
                  Variation of 
                        
                           
                           
                              
                                 
                                    E
                                 
                                 
                                    c
                                 
                              
                           
                           {E}_{\text{c}}
                        
                      on 
                        
                           
                           
                              
                                 f
                                 ′
                              
                              (
                              ξ
                              )
                           
                           f^{\prime} (\xi )
                        
                     .

Figure 10

Variation of E c on f ( ξ ) .

Figure 11 
                  Comportment of 
                        
                           
                           
                              
                                 
                                    M
                                 
                                 
                                    e
                                 
                              
                           
                           {M}_{\text{e}}
                        
                      on 
                        
                           
                           
                              θ
                              (
                              ξ
                              )
                           
                           \theta (\xi )
                        
                     .

Figure 11

Comportment of M e on θ ( ξ ) .

Figure 12 
                  Bearing of 
                        
                           
                           
                              
                                 
                                    M
                                 
                                 
                                    e
                                 
                              
                           
                           {M}_{\text{e}}
                        
                      on 
                        
                           
                           
                              ϕ
                              (
                              ξ
                              )
                           
                           \phi (\xi )
                        
                     .

Figure 12

Bearing of M e on ϕ ( ξ ) .

Figure 13 
                  Influence of 
                        
                           
                           
                              
                                 
                                    S
                                 
                                 
                                    r
                                 
                              
                           
                           {S}_{\text{r}}
                        
                      on 
                        
                           
                           
                              ϕ
                              (
                              ξ
                              )
                           
                           \phi (\xi )
                        
                     .

Figure 13

Influence of S r on ϕ ( ξ ) .

Figure 14 
                  Behavior of 
                        
                           
                           
                              
                                 
                                    λ
                                 
                                 
                                    a
                                 
                              
                           
                           {\lambda }_{\text{a}}
                        
                      on 
                        
                           
                           
                              θ
                              (
                              ξ
                              )
                           
                           \theta (\xi )
                        
                     .

Figure 14

Behavior of λ a on θ ( ξ ) .

6 Concluding remarks

This comparative study is presented in the tabular form for the rate of thermal and species transport. A special case of the presented study is compared with the open literature. Excellent agreement in results is noted, which confirms the authenticity of the applied procedure. Moreover, the reverse bearing of the magnetic parameter and the Prandtl number is noticed for the thermal transportation mechanism. Following are the key findings noted from the performed investigation:

  • Velocity profile decelerates with the escalation of the magnetic parameter and accelerates with the escalation of the local Eckert parameter.

  • Higher estimation in the Prandtl number decreases the fluid temperature.

  • Temperature profile increases with the increment of the radiation parameter and demolishes with the increment of the Prandtl number.

  • Concentration field boosts with the increase of the Soret number and declines with the increase of the local Eckert number.

  • Mounting values of the local Eckert number and the radiation parameter enhance the thermal profile.

  • The fluid temperature weakens against the Prandtl number and the thermal relaxation parameter.

  • Enhancement in magnetic parameters has reverse bearing on velocity and temperature fields.

  • Higher values of the magnetic parameter reduce the fluid velocity.

  • Temperature field grows against higher estimation of the magnetic parameter.

  • Comparative analysis found to be an excellent settlement with those of the previous reported explorations.

Nomenclature

c

stretching constant

C s

concentration susceptibility

u 1 , u 2

velocity components

x , y

space variables

ξ

dimensionless independent variable

T w

wall temperature

ϕ ( ξ )

fluid dimensionless concentration

β

Casson fluid parameter

p y

yield stress

σ

electrical conductivity

v a

kinetic viscosity

L e

Lewis number

M e

Dufour number

T m

fluid mean temperature

ρ

fluid density

C F

skin friction

C

ambient concentration

Sh

mass transfer coefficient

R e

Reynolds number

1 + 4 3 R d θ ' ( 0 )

heat transfer rate

CBL

concentration boundary layer

MBL

momentum boundary

f 0 ( ξ )

initial guess for velocity

ϕ 0 ( ξ )

initial guess for concentration

c p

specific heat

θ ( ξ )

dimensionless fluid temperature

C w

wall concentration

B 0

magnetic field strength

Ha

magnetic parameter

q r

radiative heat flux

e i j

deformation rate

R d

radiation parameter

Pr

Prandtl number

E c

Eckert number

λ a

dimensionless fluid thermal relaxation time

k

mean absorption coefficient

S r

Soret number

T

ambient temperature

N u

Nusselt number

K T

thermal diffusion ratio

D m

mass diffusivity

σ

Stefan Bultmann constant

π c

critical value of this product based on the non-Newtonian model

ϕ ( 0 )

mass transfer rate

TBL

thermal boundary layer

f ( ξ )

dimensionless velocity

θ 0 ( ξ )

initial guess for temperature

L 1 , L 2 , L 3

linear operators

    Conflict of interest: Authors state no conflict of interest.

    Funding information: This research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).

    Data availability statement: The data used to support this study are included within the Manuscript.

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Received: 2020-11-03
Revised: 2021-02-03
Accepted: 2021-02-10
Published Online: 2021-03-10

© 2021 Muhammad Sohail et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.