# Effect of Diffusion-Thermo on MHD Flow of a Jeffrey Fluid Past an Exponentially Accelerated Vertical Plate with Chemical Reaction and Heat Generation

Authors: Ahmad Shafique; Zaib Un Nisa; Muhammad Imran Asjad; Mudassar Nazar; Fahd Jarad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In many flow phenomenons of fluid with medium molecular weight, the energy flux is effected due to the inhomogeneity of concentration of mass. This contribution of the concentration to the energy flux is considered as diffusion-thermo effect or Dufour effect. In this research article the diffusion-thermo effect is addressed for the magnetohydrodynamics (MHD) flow of Jeffrey’s fractional fluid past an exponentially accelerated vertical plate with generalized thermal and mass transports through a porous medium. For the generalization of the thermal and mass fluxes the constant proportional Caputo (CPC) fractional derivative is utilized. The governing of this generalized flow are reduced to non-dimensional forms and then solved semi analytically by Laplace transform. In additions the physical aspects of flow and material parameters especially the effect ofDu and fractional parameters are discussed by sketching the graphs. From the graphical illustration, it is concluded that in the presence of Dufour effect flow speeds up. Moreover, a comparison between fractionalized and ordinary velocity fields is also drawn and it is also observed that fractional model with constant proportional derivative is of the more decaying nature as compare to the model contracted with classical Caputo and Caputo fractional derivatives.


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## Body

## 1. Introduction

The non-Newtonian fluids posses the diverse nature from Newtonian fluids due to their complex rheological properties. Now a days the study of non-Newtonian becomes a popular research area due to its scientific and technological applications in the processing industry, and biological sciences, like making of plastic sheets, lubricant’s performance and motion of biological fluid. There are several non-Newtonian fluid models have been presented to demonstrate the distinction between Newtonian and non-Newtonian fluids but the Jeffrey fluid model is more efficient to demonstrate attribute of stress relaxation for memory time scale (the relaxation of time). Mohd-Zin et al. [1] studied the porosity effect on unsteady MHD free convection flow of Jeffrey fluid past an oscillating vertical plate with ramped wall temperature. Bajwa et al. [2] solved a problem of transient flow of Jeffrey fluid semi-analytically over permeable wall. Asgir et al. [3] discussed the heat transfer analysis of channel flow of MHD Jeffrey fluid with porosity. The model on Jeffrey fluid be simplest and most popular. Some of the work on Jeffrey fluid are of Das [4] and Qasim [5]. Ali et al. [6] discussed the magnetohydrodynamic fluctuating free convection flow of incompressible electrically conducting viscoelastic fluid in a porous medium in the presence of a pressure gradient. Shah et al. [7] worked on the new semi analytical technique for the solution of fractional Order Navier Stokes equation.The fractional derivative is the generalization of ordinary derivative by taking the non-integer order of differentiation. Due its generalized property the fractional derivative becomes a potent tool to describe the heat and mass transfer phenomenons and has attained the attention by researchers. Chandra [8] present the MHD flow for Jeffrey fluid past an inclined porous plate by applying Laplace transform method. Jameel et al. [9] obtain the analytic solutions for the incompressible unsteady flow of fractionalized MHD Jeffrey fluid over an accelerating porous plate with linear slip effect by using Caputo fractional derivative. Abro et al. [10] obtain the solution of Jeffrey fluid flow acquired by non-singular kernel (Caputo-Fabrizio) using integral transform technique. During the last decade, different generalized fractional derivatives have appeared in the literature that are derivatives of Caputo, Caputo-Fabrizio, Atangana-Baleanu [11, 12]. Jawad et al. [13] observed the behavior of Caputo time fractional model based on generalized Fourier’s and Fick’s laws for Jeffrey Nano fluid. Siddique et al. [14] studied the unsteady double convection flow of a magnetohydrodynamics (MHD) differential-type fluid flow in the presence of Dufour effect, Newtonian heating, and heat source over an infinite vertical plate with fractional mass diffusion and thermal transports.Sandeep et al. [15] discussed the behavior of momentum and heat transfer on Jeffrey, Maxwell and Oldroyd-B nanofluids over a stretching surface in the presence of transverse magnetic field, non-uniform heat source/sink. A magnetohydrodynamic Jeffrey fluid flow with thermal and mass transfer on an infinitely rotating upright cone investigated by Saleem et al. [16]. Sulochana et al. [17] solved a problem of a MHD radiative Casson fluid flow numerically over a wedge to analyze the heat and molecular transfer. A bio convective nanoliquid flow with the effect of second order slip and chemical reaction between the parallel plates studied by Acharya et al. [18]. The radiative couple stress fluid and chemically reactive nanofluid over a stretched cylinder with magnetic effect analyzed by Acharya et al. [19, 20]. Riaz et al. [21] obtained the solution of peristaltic flow of Prandtl fluid by using homotopy perturbation method. The role of hybrid nanoparticles in thermal performance of peristaltic flow of Eyring-Powell fluid model have discussed by Riaz et al. [22].Different types of fractional operators has discussed by Yuri-Luchko [23]. Baleanu et al. [24] presented a new fractional operator combining proportional Caputo and solve different kind of example with CPC derivative. Chu et al. [25] worked on fractional model of Second grade fluid induced by generalized thermal and molecular fluxes with constant proportional Caputo. Siddique et al. [26] analyzed the blood liquor model fractionalized with constant proportional Caputo fractional derivative and Dolat et al. [27] investigate a fractional model of MHD viscous fluid with heat transfer by using the constant proportional Caputo fractional derivative.The purpose of this article is to utilized the Constant proportional Caputo fractional derivative for the investigation of diffusion-thermo effect on MHD flow of Jeffrey fluid past an exponentially accelerated vertical plate in the presence of chemical reaction, and heat generation through a porous medium with generalized heat and mass fluxes. Constant proportional Caputo fractional derivative is considered for the generalization of constitutive equations for heat and mass fluxes. Initially, the proposed governing equations are reduced to non-dimensional form then solved semi analytically via Laplace transform. The physical aspects of flow regarding involved parameters are also discussed by sketching some graphs for velocity field of Jeffrey fluid. Moreover the obtained result for velocity with CPC will compare with ordinary result by sketching the graphs.


## 2. Mathematical Model

The flow of Jeffrey fluid is studied in the presence of MHD by considering the diffusion-thermo effect through a porous media. The rectangular coordinates system is oriented in such a way that thex∗-axis is pointing along the plate in the vertical direction and y2∗-axis is pointing normal to the plate as shown in Figure 1. At the time t2∗=0+ the system with the plate and fluid is suppose to be at rest, at the temperature T∞∗ with concentration C∞∗. However, for the time t2∗>0, the plate suppose to be moves with exponential velocity Ueat2∗ in its own plane. The plate’s temperature as well as concentration of the fluid raised to Tw∗ and Cw∗ respectively. Magnetic field effect is also considered normally with a constant strength β0. Energy flux due to concentration gradient is also considered. In view of above assumptions, the Jeffrey fluid flow model with Boussinesq’s approximation appears in the following form [3, 13, 28](1)∂u1y2∗,t2∗∂t2∗=ν1+λ11+λ2∂∂t2∗∂2u1y2∗,t2∗∂y2∗2−σβ02ρu1y2∗,t2∗−μϕρK1u1y2∗,t2∗+gβT1∗T1∗−T∞∗+gβC1∗C1∗−C∞∗.Thermal Eq. is(2)ρCp∂T1∗y2∗,t2∗∂t2∗=−∂q1∗y2∗,t2∗∂y2∗+Q1TT1∗−T∞∗−ρKTDmDCS∂J1∗∂y2∗.Fourier’s Law states that [29](3)q1∗y2∗,t2∗=−K2∂T1∗y2∗,t2∗∂y2∗.Diffusion Eq. is(4)∂C1∗y2∗,t2∗∂t2∗=−∂J1∗y2∗,t2∗∂y2∗−K3C1∗−C∞∗.Fick’s Law states that,J1∗y2∗,t2∗ is written as [30](5)J1∗y2∗,t2∗=−D∂C1∗y2∗,t2∗∂y2∗.The initial as well as boundary conditions of the flow problem are [20](6)u1y2∗,t2∗=0,T1∗y2∗,t2∗=T∞∗,C1∗y2∗,t2∗=C∞∗,y2∗>0,t2∗=0,(7)u10,t2∗=Uea1t2∗,T1∗0,t2∗=Tw∗,C1∗0,t2∗=Cw∗,a1>0,t2∗>0,(8)u1y2∗,t2∗⟶0,T1∗y2∗,t2∗⟶0,C1∗y2∗,t2∗⟶0,y2∗⟶∞,t2∗>0.The dimensionless form of the flow parameters are(9)v=u1U,t=U2t2∗ν,Q=Q1νU2ρCp,y=Uy2∗ν,λ=U2λ2ν,J1=J1∗J,q1=q1∗q,Du=ρKTDmJUDCSCpTw−T∞,Gm=gνβC1∗Cw∗−C∞∗U3,Gr=gνβT1∗Tw∗−T∞∗U3,M=β02σνρU2,R=K3νU2,K=μϕνρU2K1,C=C1∗−C∞∗Cw∗−C∞∗,T=T1∗−T∞∗Tw∗−T∞∗,a=a1νU2.Using non-dimensional variables of equation (9) into equations (1-8), we have(10)∂vy,t∂t=1+λ1−11+λ∂∂t∂2vy,t∂y2+GrTy,t+GmCy,t−M+1Kvy,t,∂Ty,t∂t=−n2∂q1y,t∂y+QTy,t−Du∂J1∂y−m1∂T∂y=q1,∂Cy,t∂t=−n3∂J1y,t∂y−RCy,t−n1∂C∂y=J1,vy,t=0,Ty,t=0,Cy,t=0,y>0,t=0,v0,t=eat,T0,t=C0,t=1,a>0,t>0,vy,t=Ty,t=Cy,t⟶0,y⟶∞,n2=qTw−T∞−1UρCp,n3=Cw−C∞UJ−1,n1=DUCw−C∞−1Jν,m1=K2UTw−T∞−1qν,where Pr,λ, K, Sc, R, and M represent the Prandtl number, Jeffrey parameter, non-dimensional permeability, Schmidt number, chemical reaction, and magnetic parameter respectively.


## 3. Generalization of Model

Eq. (10) can be written as(11)∂vy,t∂t=11+λ1+λ1+λ1∂∂t∂2vy,t∂y2−M+1/Ky,t+GrTy,t+GmCy,t,Fractional form of Fourier’s law [25, 31] is used to generalize Eq. (3) in dimensionless form(12)q1=−m1CPCDtγ∂Ty,t∂y,1≥γ>0.Fractional form of Fick’s Law [25] is used to generalize Eq. (5) in dimensionless form(13)J1=−n1CPCDtα∂Cy,t∂y,1≥α>0.Using equation (12) into (2), and (3) into (3), we get(14)∂Ty,t∂t=n2∂∂ym1CPCDtγ∂Ty,t∂y+QTy,t+Du∂∂yn1CPCDtα∂C∂y,(15)∂Cy,t∂t=n3∂∂yn1CPCDtα∂Cy,t∂y−RCy,t.Equations (14) and (15) can be written as(16)∂Ty,t∂t=1PrCPCDtγ∂2Ty,t∂y2+QTy,t+Dun1CPCDtα∂2C∂y2,(17)∂Cy,t∂t=1ScCPCDtα∂2Cy,t∂y2−RCy,t.where DtαCPCfy,t indicates the CPC fractional derivative of fy,t [24] as(18)DtαCPCfy,t=1Γ1−α∫0tK1αfy,τ+K0αf′y,τt−τ−αdτ.


## 4. Solution of Problem

The formulated initial and boundary value value problem can be solved by means of the Laplace transform method.


### 4.1. Concentration Field

(17) be solved via Laplace transform for concentration species as(19)sC¯y,s=Sc−1k1αs+k0αsαy,s−RC¯y,s,with(20)C¯0,s=1s,C¯y,s⟶0,y⟶∞.Put (20) into (19), and result is(21)C¯y,s=1se−yR+sSc−1K1α/s+K0αsα.This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (21).


### 4.2. Sherwood Number

The local coefficient of the rate of mass transfer is define in term of Sherwood number, and define by the following relation(22)Sh=∂C∂y|y=0=−L−1∂C¯∂y|y=0=L−11sR+sSc−1K1α/s+K0αsα.


### 4.3. Temperature Field

(16) be solved via Laplace transform for temperature profile as(23)sT¯y,s=Pr−1k1γs+k0γsγ∂2T¯y,s∂y2+QT¯y,s+n1Duk1αs+k0αsα∂2C¯y,s∂y2,with(24)T¯0,s=1s,T¯y,s⟶0,y⟶∞.Put (24) into (23), and result is(25)T¯y,s=1se−ys−QPr−1K1γ/s+K0γsγ+n1DuR+sPr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγ×e−ys−QsPr−1K1γ/s+K0γsγ−e−yR+sSc−1K1α/s+K0αsα.This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (25).


### 4.4. Nusselt Number

The local coefficient of the rate of heat transfer is define in term of Nusselt number as(26)Nu=∂T∂y|y=0=−L−1∂T¯∂y|y=0=L−11ss−QPr−1K1γ/s+K0γsγ+n1DuR+sPr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγ×s−QsPr−1K1γ/s+K0γsγ−R+sSc−1K1α/s+K0αsα.


### 4.5. Velocity field

(11) be solved via Laplace transform for velocity field as(27)sv¯y,t=1+λ1−11+λs∂2v¯y,t∂y2−M+1/Kv¯y,t−GrT¯y,t−GmC¯y,t,with(28)v¯0,s=s−a−1,v¯y,s⟶0,y⟶∞.Put (28) into (27), we get(29)v¯y,s=1s−ae−ys+M+1/K1+λ11+λs+Gr/s+n1DuGrR+s/Pr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγPrs−Q1+λs/K1γ/s+K0γsγ1+λ1−s+M+1/Ke−ys+M+1/K1+λ11+λs−e−ys−QPr−1K1γ/s+K0γsγ+Gm/s−n1DuGrR+s/Pr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγScR+s1+λs/K1α/s+K0αsα1+λ1−s+M+1/Ke−ys+M+1/K1+λ11+λs−e−yR+sSc−1K1α/s+K0αsα.This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (29).


### 4.6. Skin Friction

The expression of skin friction is define by using (29) as(30)τ=∂u∂y|y=0=−L−1∂u¯∂y|y=0=L−11s−as+M+1/K1+λ11+λs+Gr/s+n1DuGrR+s/Pr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγPrs−Q1+λs/K1γ/s+K0γsγ1+λ1−s+M+1/K×s+M+1/K1+λ11+λs−s−QPr−1K1γ/s+K0γsγ+Gm/s−n1DuGrR+s/Pr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγScR+s1+λs/K1α/s+K0αsα1+λ1−s+M+1/K×s+M+1/K1+λ11+λs−R+sSc−1K1α/s+K0αsα.


## 4.1. Concentration Field

(17) be solved via Laplace transform for concentration species as(19)sC¯y,s=Sc−1k1αs+k0αsαy,s−RC¯y,s,with(20)C¯0,s=1s,C¯y,s⟶0,y⟶∞.Put (20) into (19), and result is(21)C¯y,s=1se−yR+sSc−1K1α/s+K0αsα.This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (21).


## 4.2. Sherwood Number

The local coefficient of the rate of mass transfer is define in term of Sherwood number, and define by the following relation(22)Sh=∂C∂y|y=0=−L−1∂C¯∂y|y=0=L−11sR+sSc−1K1α/s+K0αsα.


## 4.3. Temperature Field

(16) be solved via Laplace transform for temperature profile as(23)sT¯y,s=Pr−1k1γs+k0γsγ∂2T¯y,s∂y2+QT¯y,s+n1Duk1αs+k0αsα∂2C¯y,s∂y2,with(24)T¯0,s=1s,T¯y,s⟶0,y⟶∞.Put (24) into (23), and result is(25)T¯y,s=1se−ys−QPr−1K1γ/s+K0γsγ+n1DuR+sPr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγ×e−ys−QsPr−1K1γ/s+K0γsγ−e−yR+sSc−1K1α/s+K0αsα.This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (25).


## 4.4. Nusselt Number

The local coefficient of the rate of heat transfer is define in term of Nusselt number as(26)Nu=∂T∂y|y=0=−L−1∂T¯∂y|y=0=L−11ss−QPr−1K1γ/s+K0γsγ+n1DuR+sPr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγ×s−QsPr−1K1γ/s+K0γsγ−R+sSc−1K1α/s+K0αsα.


## 4.5. Velocity field

(11) be solved via Laplace transform for velocity field as(27)sv¯y,t=1+λ1−11+λs∂2v¯y,t∂y2−M+1/Kv¯y,t−GrT¯y,t−GmC¯y,t,with(28)v¯0,s=s−a−1,v¯y,s⟶0,y⟶∞.Put (28) into (27), we get(29)v¯y,s=1s−ae−ys+M+1/K1+λ11+λs+Gr/s+n1DuGrR+s/Pr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγPrs−Q1+λs/K1γ/s+K0γsγ1+λ1−s+M+1/Ke−ys+M+1/K1+λ11+λs−e−ys−QPr−1K1γ/s+K0γsγ+Gm/s−n1DuGrR+s/Pr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγScR+s1+λs/K1α/s+K0αsα1+λ1−s+M+1/Ke−ys+M+1/K1+λ11+λs−e−yR+sSc−1K1α/s+K0αsα.This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (29).


## 4.6. Skin Friction

The expression of skin friction is define by using (29) as(30)τ=∂u∂y|y=0=−L−1∂u¯∂y|y=0=L−11s−as+M+1/K1+λ11+λs+Gr/s+n1DuGrR+s/Pr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγPrs−Q1+λs/K1γ/s+K0γsγ1+λ1−s+M+1/K×s+M+1/K1+λ11+λs−s−QPr−1K1γ/s+K0γsγ+Gm/s−n1DuGrR+s/Pr−1R+s/K1α/s+K0αsα−Sc−1s−Q/K1γ/s+K0γsγk1γ+sk0γsγScR+s1+λs/K1α/s+K0αsα1+λ1−s+M+1/K×s+M+1/K1+λ11+λs−R+sSc−1K1α/s+K0αsα.


## 5. Results and Discussion

In this article the diffusion-thermo effect is investigated for the magnetohydrodynamics (MHD) flow of Jeffrey’s fractional fluid past an exponentially accelerated vertical plate with generalized thermal and mass transports through a porous medium. The semi-analytical results for the concentration, temperature, and velocity fields are obtained. Moreover to see the physical effect of involved parameters, the concentration, temperature, and velocity fields are postured by some graphs.Figure 1
Physical model.Velocity profile is sketched in the Figure2(a) to twilight the influence magnetic field over the fluid velocity and it is observed that fluid slows down for greater values of M. In the presence of magnetic field a retarding force is created which oppose the fluid motion as depicted in Figure 2(a). The effect of porosity parameter K on fluid flow is defected in Figure 2(b) and it is noted that the fluid motion enhance with increasing value of K. The larger value of K refers to the less Darcy resistance to the flow that is why flow speeds up with increasing value of K. The influence of mass Gm on fluid velocity vy,t is illustrate in Figures 3(a) and 3(b). It is noted that fluid motion increases as values of Gm increasing. Physically higher values of Gm refers the higher concentration gradients and greater the buoyancy force and more current in flow domain so velocity increases. Figure 4(a) represent the result of Gr on fluid velocity vy,t. The fluid motion rises up with maximizing the values of Gr, and it represents the more impact of thermal buoyancy force as compare to the viscous effect. Therefore maximizing the values of Gr refers the higher temperature gradient due to which velocity field rises. The impact of Pr for velocity field is presented in Figure 4(b).Figure 2
(a) Velocity profilevy,t for parameter M at R = 3.2, Q = 4, Gr = 14, Gm = 8, Du = 0.4, Sc = 2.5, Pr = 6, and K = 2. (b) Velocity profile vy,t for parameter K at R = 3.2, Q = 4, Gr = 14, Gm = 8, M = 4, Sc = 2.5, Pr = 6, and Du = 0.4.
(a)(b)Figure 3
(a) Velocity profilevy,t for parameter Gm at R = 3.2, Q = 4, Gr = 14, Du = 0.4, M = 4, Sc = 2.5, Pr = 6, and K = 2. (b) Velocity profile vy,t for parameter Gm at R = 3.2, Q = 4, Gr = 14, Du = 0.4, M = 4, Sc = 2.5, Pr = 6, and K = 2.
(a)(b)Figure 4
(a) Velocity profilevy,t for parameter Gr at R = 3.2, Q = 4, Gm = 8, M = 4, Sc = 2.5, Pr = 6, and K = 2. (b) Velocity profile vy,t for parameter Pr at R = 3.2, Q = 4, Gr = 14, Gm = 8, M = 4, Sc = 2.5, Du = 0.4, and K = 2.
(a)(b)The diffusion-thermo or Dufour effect (Du) over the velocity field are discussed in the Figures5(a) and 5(b). An enhancing flow pattern is observed due to the increasing variations of the Du. The concentration gradient give a contribute to the temperature gradient which creates an additional energy flux in the flow domain. Therefore an increasing value of the Du refer to the more energy flux which generates the more flow current that is why the flow speeds up with increasing value of Du. Moreover the negative values refer to the opposite direction of the additional energy flux therefore more negativity of Du generates the more opposite flow current in the flow domain.Figure 5
(a) Velocity profilevy,t for parameter Du at R = 3.2, Q = 4, Gr = 14, Gm = 8, M = 4, Sc = 2.5, Pr = 6, and K = 2. (b) Velocity profile vy,t for parameter Du at R = 3.2, Q = 4, Gr = 14, Gm = 8, M = 4, Sc = 2.5, Pr = 6, and K = 2.
(a)(b)Figures6(a) and 6(b) are drawn to see impact of Jeffrey parameters λ and λ1 over velocity field vy,t respectively. As parameter λ refer to the viscosity of the fluid therefore fluid velocity falls due to the increasing value of λ and the parameter λ1 refers to the elasticity of the fluid material so fluid speed up with increasing value λ1. The influence of heat generation on vy,t is reported in Figure 7(a). An increasing value of R decreases the fluid velocity as shown in Figure 7(b). The impact of Sc on vy,t is illustrate in Figure 8(a). Here, maximizing the values of Sc slow down the fluid motion due to decay of molecular diffusion. It is analyzed that motion of fluid increases with raising values of α=γ as depicted in Figure 8(b). The graphical behavior of Du, Pr, Q, and t on Ty,t are shown in Figures 9(a), 9(b), 10(a) as well as 10(b) respectively. Figures illustrate that temperature reduces by exceeding the values of Du, and Pr, and temperature increases by exceeding the values of Q and t. The impacts of Schmidt number Sc as well as chemical reaction R on Cy,t are present in Figures 11(a) as well as 11(b) respectively. Figures 12(a) and 12(b) represent the comparison of velocity and temperature distribution of present work with Naseem [20] respectively. If we take fractional parameters α=γ⟶1,=λ=λ1=M=Gm=Du=Q=0, 1/K⟶0,1/γ⟶0 and ft=eat2∗ in Naseem [28], the velocity fields are identical that shows the validity of this present work. Moreover the comparison of present work with other fractional derivatives Caputo and Caputo Fabrizio used in Nazar et al. [34] and Nadeem et al. [35] respectively in the absence of K, Q, Gm, R, Du are shown in Figure 12(c). If we put α=1 the fluid curves are identical as shown in Figure 12(d). From the figures, it is concluded that Jeffrey fluid with CPC fractional derivative is the best choice to enhance the fluid motion. Figures 13(a) and 13(b) represent the authenticity of inversion algorithms for temperature and concentration distributions. The velocity distributions overlap which shows the authenticity of inversion algorithms as presented in Figure 14. Skin friction, rate of heat and mass transfer can be enhanced by increasing the values of fractional parameter and presented in table 1.Figure 6
(a) Velocity profilevy,t for heat generation parameter λ at R = 3.2, Q = 4, Gr = 14, Gm = 8, M = 4, Sc = 2.5, Pr = 6, and K = 2. (b) Velocity profile vy,t for chemical reaction parameter λ1 at R = 3.2, Q = 4, Gr = 14, Gm = 8, M = 4, Sc = 2.5, Pr = 6, and K = 2.
(a)(b)Figure 7
(a) Velocity profilevy,t for heat generation parameter Q at R = 3.2, Du = 0.4, Gr = 14, Gm = 8, M = 4, Sc = 2.5, Pr = 6, and K = 2. (b) Velocity profile vy,t for chemical reaction parameter R at Q = 4, Du = 0.4, Gr = 14, Gm = 8, M = 4, Sc = 2.5, Pr = 6, and K = 2.
(a)(b)Figure 8
Velocity profilevy,t for parameter Sc at R = 3.2, Q = 4, Gr = 14, Gm = 8, M = 4, Du = 0.4, Pr = 6, and K = 2. (b) Velocity profile vy,t for fractional parameters at Q = 4, Du = 0.4, Gr = 14, Gm = 8, M = 4, R = 3.2, Du = 0.4, Pr = 6, and K = 2.
(a)(b)Figure 9
(a) Temperature profileTy,t for parameter Du. (b) Temperature profile Ty,t for Pr.
(a)(b)Figure 10
Temperature profileTy,t for parameter Q. (b) Temperature profile Ty,t for t.
(a)(b)Figure 11
(a) Concentration profileCy,t for parameter Sc. (b) Concentration profile Cy,t for parameter R.
(a)(b)Figure 12
(a) Velocity distribution for comparison of our work with Naseem [28] asα=γ→0,λ=λ=M=Gm=Du=Q=0,1/K→0,1/γ→0. (b) Temperature for comparison of our work with Naseem [28] as α = γ → 1, Gm = Du = Q = 0. (c) Comparisons between different fractional derivatives α = 0.5 (d) Comparisons between different fractional derivatives α → 1.
(a)(b)(c)(d)Figure 13
(a) Concentration obtained by Stehfest’s and Tzou’s algorithms. (b) Temperature obtained by Stehfest’s and Tzou’s algorithms.
(a)(b)Figure 14
Velocity obtained by Stehfest’s and Tzou’s algorithms.• With decreasing fractional parameter values, the velocity distribution slows down.• Thermal buoyancy forces accelerate the fluid velocity.• Fluid velocity reduces as Schmidt number Sc, chemical reactionR, and magnetic parameter M rises.• The fluid velocity increased for larger values of Jeffrey parameter as well as fractional parameter.• The temperature distribution increases by the smaller values of Prandtl number Pr.•The concentration distribution decreases for greater values of Schmidt number Sc.• The concentration distribution decreases by exceeding the values of chemical reactionR.Table 1
Effect ofα on skin friction, Nusselt number, and Sherwood number.
Fractional Parameterskin frictionNusselt numberSherwood number0.11.458 939 711.363 328 251.293 234 410.21.469 733 541.376 590 531.306 809 850.31.480 812 221.389 919 451.320 272 420.41.492 212 411.403 398 901.333 699 190.51.503 987 041.417 120 901.347 160 760.61.516 206 971.431 188 591.360 720 510.71.528 970 611.445 719 091.374 431 120.81.542 413 611.460 842 281.388 325 480.91.556 732 821.476 693 471.402 406 850.991.572 220 171.493 394 871.416 634 77


## 6. Conclusions

The Magnetohydrodynamics flow model of Jeffrey’s fractional fluid model has been considered with the effect of magnetic field in the porous regime. Heat generation, and radiation are also consider with effect of chemical reaction in the flow domain. This flow model is solved analytically and results for velocity, temperature, and concentration fields are constructed by using Laplace transform. Further different graphs of optimising fields are plotted to highlight the influence parameters. The key outcomes of this flow model are:



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