Effect of Magnetic Field with Parabolic Motion on Fractional Second Grade Fluid

Abstract

This paper is an analysis of the flow of magnetohydrodynamics (MHD) second grade fluid (SGF) under the influence of chemical reaction, heat generation/absorption, ramped temperature and concentration and thermodiffusion. The fluid was made to flow through a porous medium. It has been proven in many already-published articles that heat and mass transfer do not always follow the classical mechanics process that is known as memoryless process. Therefore, the model using classical differentiation based on the rate of change cannot really replicate such a dynamical process very accurately; thus, a different concept of differentiation is needed to capture such a process. Very recently, new classes of differential operators were introduced and have been recognized to be efficient in capturing processes following the power law, the decay law and the crossover behaviors. For the study of heat and mass transfer, we applied the newly introduced differential operators to model such flow. The equations for heat, mass and momentum are established in the terms of Caputo (C), Caputo–Fabrizio (CF) and Atangana–Baleanu in Caputo sense (ABC) fractional derivatives. The Laplace transform, inversion algorithm and convolution theorem were used to derive the exact and semi-analytical solutions for all cases. The obtained analytical solutions were plotted for different values of existing parameters. It is concluded that the fluid velocity shows increasing behavior for $\kappa$, $G_r$ and $G_m$, while velocity decreases for $P_r$ and M. For $K_r$, both velocity and concentration curves show decreasing behavior. Fluid flow accelerates under the influence of $S_r$ and R. Temperature and concentration profiles increase for $S_r$ and R. Moreover, the ABC fractional operator presents a larger memory effect than C and CF fractional operators.

1. Introduction

Over the past thirty years, fractional derivatives have fascinated multiple investigators as compared to classical derivatives. Moreover, fractional derivatives are more credible in mathematical modeling of real-world problems [1,2,3,4].

In classical calculus, derivatives and integrals are uniquely computed. A similar situation exists in the case of fractional integrals. For example, Samko et al. [5], Podlubny [6], Oldham and Spanier [7], and Miller and Ross [8] used a similar definition to compute the fractional integrals. However, the circumstances are complicated in fractional order derivatives (FODs) because several different competing definitions exist in the literature. For instance, a few of those approaches include the Riemann-Liouville, the Caputo, the Hadamard, the Marchaud, the Granwald–Letinkov, the Erdelyi–Kober, the Riesz–Feller, the Caputo–Fabrizio and the Atangana–Baleanu approach. These definitions coincide only with some particular cases. Among all these approaches to define fractional differentiation and fractional integration, the approach of Riemann–Liouville is significant. However, the approach of Riemann–Liouville does not properly address the physics of some fractional derivative initial-boundary value problems. Furthermore, this definition can exhibit the derivative of a constant function other than zero. To overcome this problem, Caputo proposed an alternate definition of FOD in 1967 [9], and it was used in fluid dynamics to explain the theory of viscoelasticity [10]. Recently, Caputo and Fabrizio [11] provided a modern definition of a non-integer order derivative including exponential function and the Atangana and Baleanu [12], based on Mittatag–Leffler function, which is a generalization of the exponential function.

The FOD inherits a nonlocal nature, so it is an excellent tool to obtain a better understanding of the hereditary properties of different processes and materials. Possibly, the first utilization of non-integer calculus in physical problems was noticed due to the work of Abel [13] while finding the solution of integer order equation, known as tautochrone problem. In this problem, the curve of an object (frictionless wire), lying in a vertical plane, was determined by using the operator $D_0^{\frac{1}{2}}$ and assuming the dependence of the time position not on the starting point. Bagley [14] presented the first PhD thesis on the applications of FC in viscoelasticity models. Recently, the applications of FC have been observed in psychology to determine the time variation of the emotions of mankind [15,16]. The applications of FOD problems can be seen in dynamics and control systems [17], marine sciences and wave dynamics [18,19,20,21], diffusion processes [22,23,24], solid mechanics [25,26,27], medical sciences [28,29,30,31] and many more [32,33,34,35].

Convective flow is a self-sustained flow with a temperature gradient. Convective flow and magnetic effect combined play a very important role in real value problems. The SGF with warmth-transferring porous medium was discussed by Tan and Masuoka [36]. Mixed MHD convection on an upright plate with permeable space was analyzed by Aldose et al. [37]. Rashidi et al. [38] also investigated the difference between two normal kinds of liquid stream between clear liquid and permeable medium. The authors of [39] investigated the precarious MHD stream of turning SGF past a swaying plate. Khan et al. [40] discussed the precise answers for quickened flow behavior of a pivoting SGF in a permeable space. Bilal et al. [41] and Ali et al. [42] analyzed the SGF with a swaying plate under different conditions. The literature shows more interest developed in the numerical and approximate solutions on the convection flow of SGF [43,44,45,46,47,48,49]. Exact solutions for viscoelastic SGF fluid have been investigated by researchers [50,51,52]. In 2010, the SGF with Laplace transform methodology over a wavering plate was explored by Nazar et al. [53]. Ali et al. [54] explored the MHD fluid with permeable space. The procedure of heat transfer is expressed by using momentum, energy and continuity equations with stress and heat flux. The heat transfer phenomenon consists of the fins of the heat exchanger, the tabulator inside the tube or plates and the convective physical transport phenomenon [55]. MHD free convection radiative stream with different conditions has been considered by researchers [56,57,58,59]. Ali et al. [42] considered SGF with porous surface and determined the exact solutions. Some recent and useful work has also been done for SGF with fractional differential operators [60,61,62,63,64,65]. Heat and mass transfer phenomena in nanofluids with a porous medium have been investigated by [66,67,68]. Some significant work in the field of nanoparticles and carbon nanotubes via fractional derivatives have been done by researchers [69,70,71]. Recently, Rehman et al. [72] discussed heat and mass transfer of MHD unsteady SGF in the presence of ramped conditions. Song et al. [73] used the definition of ABC in order to study SGF with exponential heating and Darcy’s law. Moreover, Riaz et al. explored MHD SGF with ramped conditions via special functions [74].

In the present work, we propose the mathematical modeling of fractional SGF with the help of Laplace transform and fractional operators. Moreover, solutions are acquired for momentum, heat and mass profiles. An inversion algorithm is used for graphical interpretation of fractional models.

2. Mathematical Modeling

We begin with SGF on an upright and unbounded plate, with the impact of a magnetic field having strength $B_0$. The plate is perpendicular to $\tilde{\eta }$-axis and parallel to $\tilde{x}$-axis. At the start, the plate and fluid are not moving. ${\tilde{T}}_{\infty }$ is fixed temperature and ${\tilde{C}}_{\infty }$ is concentration at the surface. At the time $\tilde{\tau }>0$, the temperature of the plate is either raised or lowered to ${\tilde{T}}_{\infty }+({\tilde{T}}_w-{\tilde{T}}_{\infty })\frac{\tilde{\tau }}{{\tau }_0}$, when $\tilde{\tau }\le {\tau }_0$, and thereafter, for $\tilde{\tau }>t_0$, a constant temperature ${\tilde{\tau }}_w$ is maintained and the level of mass transfer at the surface of the wall is either raised or lowered to ${\tilde{C}}_{\infty }+({\tilde{C}}_w-{\tilde{C}}_{\infty })\frac{\tilde{t}}{{\tau }_0}$, when $\tilde{\tau }\le {\tau }_0$, and thereafter, for $\tilde{\tau }>{\tau }_0$ is maintained at the constant surface concentration ${\tilde{C}}_w$, respectively. The physical model of the problem can be given as follows in Figure 1 [75]. The detailed method to solve the problem is shown in Figure 2.

Governing equations for momentum, heat and mass are presented by [75]:

$$\begin{array}{c}\hfill \frac{\partial \tilde{u}}{\partial \tilde{\tau }}=\left(\upsilon +\frac{\gamma }{\rho }\frac{\partial }{\partial \tilde{\tau }}\right)\frac{{\partial }^2\tilde{u}}{\partial {\tilde{\eta }}^2}+g{\beta }_T\tilde{T}-g{\beta }_T{\tilde{T}}_{\infty }+g\widetilde{{\beta }_{\tau }}\tilde{C}-g\widetilde{{\beta }_C}{\tilde{C}}_{\infty }-\frac{\sigma B_o^2}{\rho }\tilde{u}-\frac{\mathrm{\Phi }}{\widetilde{k_1}}\left(\upsilon +\frac{\gamma }{\rho }\frac{\partial }{\partial \tilde{\tau }}\right)\tilde{u},\end{array}$$

(1)

$$\begin{array}{c}\hfill \rho C_p\left(\frac{\partial \tilde{T}}{\partial \tilde{\tau }}\right)=k\left(\frac{{\partial }^2\tilde{T}}{\partial {\tilde{\eta }}^2}\right)-\frac{\partial \widetilde{q_r}}{\partial \tilde{\eta }}+Q_0\tilde{T}-Q_0{\tilde{T}}_{\infty },\end{array}$$

(2)

$$\begin{array}{c}\hfill \left(\frac{\partial \tilde{C}}{\partial \tilde{\eta }}\right)=D_M\left(\frac{{\partial }^2\tilde{C}}{\partial {\tilde{\eta }}^2}\right)+D_T\left(\frac{{\partial }^2\tilde{T}}{\partial {\tilde{\eta }}^2}\right)-\widetilde{k_2}\tilde{C}-\widetilde{k_2}{\tilde{C}}_{\infty },\end{array}$$

(3)

and the imposed initial and boundary conditions are [75]:

$$\begin{array}{c}\hfill \tilde{\tau }\le 0,\tilde{u}\left(\tilde{\eta },0\right)=0,\tilde{T}\left(\tilde{\eta },0\right)={\tilde{T}}_{\infty },\tilde{C}\left(\tilde{\eta },0\right)={\tilde{C}}_{\infty },\eta \ge 0,\end{array}$$

(4)

$$\begin{array}{c}\hfill \tilde{\tau }>0,\tilde{u}\left(\tilde{\eta },\tilde{\tau }\right)=u_o{\tilde{\tau }}^2,\tilde{T}(0,\tilde{\tau })=\left\{\begin{array}{cc}{\tilde{T}}_{\infty }+({\tilde{T}}_w-{\tilde{T}}_{\infty })\frac{\tilde{\tau }}{{\tau }_0},\hfill & 0<\tilde{\tau }\le {\tau }_0;\hfill \\ {\tilde{T}}_w,\hfill & \tilde{\tau }>{\tau }_0,\hfill \end{array}\right.\end{array}$$

(5)

$$\begin{array}{c}\hfill \tilde{C}(0,\tilde{\tau })=\left\{\begin{array}{cc}{\tilde{C}}_{\infty }+({\tilde{C}}_w-{\tilde{C}}_{\infty })\frac{\tilde{\tau }}{{\tau }_0},\hfill & 0<\tilde{\tau }\le {\tau }_0;\hfill \\ {\tilde{C}}_w,\hfill & \tilde{\tau }>{\tau }_0,\hfill \end{array}\right.\tilde{\eta }=0,\end{array}$$

(6)

$$\begin{array}{c}\hfill \tilde{\tau }\ge 0,\tilde{u}\left(\tilde{\eta },\tilde{\tau }\right)\to 0,\tilde{T}\left(\tilde{\eta },\tilde{\tau }\right)\to {\tilde{T}}_{\infty },\tilde{C}\left(\tilde{\eta },\tilde{\tau }\right)\to {\tilde{C}}_{\infty },\tilde{\eta }\to \infty .\end{array}$$

(7)

Figure 2. Flow chart of the method.

Figure 2. Flow chart of the method.

For the simplification of Equations (1)–(7), we introduce the dimensionless variables given below:

$$\begin{array}{c}\hfill \zeta =\frac{u_0{\tilde{\tau }}_0^2}{\upsilon }\tilde{\eta },w=\frac{\tilde{u}}{{\tilde{\tau }}_0^2{u}_0},\theta =\frac{\tilde{T}-{\tilde{T}}_{\infty }}{{\tilde{T}}_w-{\tilde{T}}_{\infty }},C=\frac{\tilde{C}-{\tilde{C}}_{\infty }}{{\tilde{C}}_w-{\tilde{C}}_{\infty }},M^2=\frac{\sigma B_0^2{\tau }_0}{\rho },K_r={\tau }_0\widetilde{k_2},\end{array}$$

(8)

$$\begin{array}{c}\hfill S_c=\frac{\upsilon }{D_M},G_r=\frac{g\widetilde{{\beta }_T}\left({\tilde{T}}_w-{\tilde{T}}_{\infty }\right)}{{\tilde{u}}_0{\tau }_0},G_m=\frac{g\widetilde{{\beta }_C}\left({\tilde{C}}_w-{\tilde{C}}_{\infty }\right)}{{\tilde{u}}_0{\tau }_0},{\tau }_0={\left(\frac{\upsilon }{{\tilde{u}}_0^2}\right)}^{\frac{1}{5}},{\gamma }_1=\frac{\gamma }{\rho \upsilon {\tau }_0},\end{array}$$

(9)

$$\begin{array}{c}\hfill S_r=\frac{D_T\left({\tilde{T}}_w-{\tilde{T}}_{\infty }\right)}{\upsilon \left({\tilde{C}}_w-{\tilde{C}}_{\infty }\right)},P_r=\frac{\rho \upsilon C_p}{k},R=\frac{16{\sigma }^{*}{\tilde{T}}_{\infty }^3}{3k{k}^{*}},\frac{1}{k_1}=\frac{u_0\upsilon {\tau }_0^2\mathrm{\Phi }}{\widetilde{k_1}},H=\frac{Q_0\upsilon {\tau }_0}{k},\end{array}$$

(10)

$$\begin{array}{c}\hfill t=\frac{\tilde{\tau }}{{\tau }_0^2}h=\frac{1}{u_0{\tau }_0\widetilde{k_1}},c=1+{\gamma }_{1}h,b=M^2+h.\end{array}$$

(11)

Therefore, the dimensionless momentum, energy and mass equations are [75]

$$\begin{array}{c}\hfill c\left(\frac{\partial w\left(\zeta ,t\right)}{\partial t}\right)=\frac{{\partial }^{2}w}{\partial {\zeta }^2}+{\gamma }_1\left(\frac{{\partial }^{3}w\left(\zeta ,t\right)}{\partial t\partial {\zeta }^2}\right)+G_r\theta \left(\zeta ,t\right)+G_{m}C\left(\zeta ,t\right)-bw\left(\zeta ,t\right),\end{array}$$

(12)

$$\begin{array}{c}\hfill P_r\left(\frac{\partial \theta \left(\zeta ,t\right)}{\partial t}\right)=\left(1+R\right)\frac{{\partial }^2\theta \left(\zeta ,t\right)}{\partial {\zeta }^2}+H\theta \left(\zeta ,t\right),\end{array}$$

(13)

$$\begin{array}{c}\hfill \frac{\partial C\left(\zeta ,t\right)}{\partial t}=\frac{1}{S_c}\left(\frac{{\partial }^{2}C\left(\zeta ,t\right)}{\partial {\zeta }^2}\right)+S_r\left(\frac{{\partial }^2\theta \left(\zeta ,t\right)}{\partial {\zeta }^2}\right)-K_{r}C\left(\zeta ,t\right),\end{array}$$

(14)

with initial and boundary conditions

$$\begin{array}{c}\hfill t\le 0,w\left(\zeta ,0\right)=0,\theta \left(\zeta ,0\right)=0,C\left(\zeta ,0\right)=0,\zeta \ge 0,\end{array}$$

(15)

$$t>0,w\left(\zeta ,t\right)=t^2,\theta (0,t)=\left\{\begin{array}{cc}t\hfill & 0<t\le 1;\hfill \\ 1\hfill & t>1,\hfill \end{array}\right.,C(0,t)=\left\{\begin{array}{cc}t\hfill & 0<t\le 1;\hfill \\ 1\hfill & t>1,\hfill \end{array}\right.,\zeta =0,$$

(16)

$$\begin{array}{c}\hfill t>0,w\left(\zeta ,t\right)\to 0,\theta \left(\zeta ,t\right)\to 0,C\left(\zeta ,t\right)\to 0,\zeta \to \infty .\end{array}$$

(17)

3. Solution for Caputo Fractional Operator

The Caputo (C) fractional time derivative and its Laplace transform [76] are given below:

$$\begin{array}{c}\hfill {}^C{D}_{\tau }^{\kappa }N\left(\zeta ,\tau \right)=\frac{1}{\Gamma (n-\kappa )}{\int }_0^{\tau }\left(\frac{N^{\left(n\right)}\left(\xi \right)}{{(\tau -\xi )}^{\kappa +1-n}}\right)d\xi ,\end{array}$$

(18)

$$\begin{array}{c}\hfill \mathcal{L}\left({}^C{D}_{\tau }^{\kappa }N\left(\zeta ,\tau \right)\right)=s^{\kappa }\mathcal{L}\left(N(\zeta ,\tau )\right)-s^{\kappa -1}N(\zeta ,0).\end{array}$$

(19)

3.1. Temperature Field

The following Caputo derivative form of temperature Equation (13) is developed by using Equation (18):

$$\begin{array}{c}\hfill P_r{}^C{D}_t^{\kappa }\theta (\zeta ,t)=\left(1+R\right)\frac{{\partial }^2\theta }{\partial {\zeta }^2}+H\theta .\end{array}$$

(20)

By implementing the Laplace transform on Equation (20), we obtain

$$\begin{array}{c}\hfill \frac{{\partial }^2{\overline{\theta }}_c(\zeta ,s)}{\partial {\zeta }^2}-\left(\frac{P_r}{1+R}\right)\left(s^{\kappa }-\frac{H}{P_r}\right){\overline{\theta }}_c(\zeta ,s)=0,\end{array}$$

(21)

where the homogenous solution of the above equation is

$$\begin{array}{c}\hfill {\overline{\theta }}_c(\zeta ,s)=c_1{e}^{-\zeta \sqrt{\left(\frac{P_r}{1+R}\right)\left(s^{\kappa }-\frac{H}{P_r}\right)}}+c_2{e}^{\zeta \sqrt{\left(\frac{P_r}{1+R}\right)\left(s^{\kappa }-\frac{H}{P_r}\right)}}.\end{array}$$

(22)

$c_1$ and $c_2$ can be determined by employing Equations (15)–(17), and the required solution is given below:

$$\begin{array}{c}\hfill {\overline{\theta }}_c(\zeta ,s)=\left(\frac{1-e^{-s}}{s^2}\right)e^{-\zeta \sqrt{\left(\frac{P_r}{1+R}\right)\left(s^{\kappa }-\frac{H}{P_r}\right)}}.\end{array}$$

(23)

3.2. Concentration Field

The following Caputo derivative form of concentration Equation (14) is developed by using Equation (18):

$$\begin{array}{c}\hfill {}^C{D}_t^{\kappa }C(\zeta ,t)=\frac{1}{S_c}\frac{{\partial }^{2}C(\zeta ,t)}{\partial {\zeta }^2}+S_r\frac{{\partial }^2\theta (\zeta ,t)}{\partial {\zeta }^2}-K_{r}C(\zeta ,t).\end{array}$$

(24)

By implementing the Laplace transform on Equation (24),

$$\begin{array}{c}\hfill \left(s^{\kappa }+K_r\right){\overline{C}}_c(\zeta ,s)=\frac{1}{S_c}\frac{{\partial }^2{\overline{C}}_c(\zeta ,s)}{\partial {\zeta }^2}+S_r\frac{{\partial }^2{\overline{\theta }}_c(\zeta ,s)}{\partial {\zeta }^2},\end{array}$$

(25)

the homogenous solution of the above equation is

$$\begin{array}{c}\hfill {\overline{C}}_h(\zeta ,s)=c_1{e}^{-\zeta \sqrt{S_c\left(\frac{s^{\kappa }}{(1-\kappa )s+\kappa }+K_r\right)}}+c_2{e}^{\zeta \sqrt{S_c\left(\frac{s^{\kappa }}{(1-\kappa )s+\kappa }+K_r\right)}},\end{array}$$

(26)

the particular solution of Equation (25) is

$$\begin{array}{cc}\hfill {\overline{C}}_p(\zeta ,s)=& -\frac{S_c{S}_r(1-e^{-s})\left\{\left(P_r-H+H\kappa \right)s^{\kappa }-H\kappa \right\}}{s^2\left\{\left(P_r-H(1-\kappa )-S_c(1+R)-S_c{K}_r(1+R)(1-\kappa )\right)s^{\kappa }-(H\kappa +S_c{K}_r\kappa (1+R))\right\}}\hfill \\ & e^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s^{\kappa }}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}},\hfill \end{array}$$

(27)

and the required solution is given below:

$$\begin{array}{cc}\hfill {\overline{C}}_c(\zeta ,s)& =\left(\frac{1-e^{-s}}{s^2}-A_1\right)e^{-\zeta \sqrt{S_c\left(s^{\kappa }+K_r\right)}}+A_1{e}^{-\zeta \sqrt{\left(\frac{P_r}{1+R}\right)\left(s^{\kappa }-\frac{H}{P_r}\right)}}.\hfill \end{array}$$

(28)

where

$$\begin{array}{cc}\hfill A_1& =\frac{S_c{S}_r\left(\frac{P_r}{1+R}\right)\left(s^{\kappa }-\frac{H}{P_r}\right)\left(1-e^{-s}\right)}{\left(\frac{H}{1+R}\right)+S_c{K}_r}\times \hfill \\ & \left(\frac{(\frac{P_r}{1+R}-S_c)}{s^{\kappa }\left(\frac{H}{1+R}\right)+S_c{K}_r}+\frac{1}{s^2}-\frac{1}{(\frac{H}{1+R}+S_c{K}_r)\left(s^{\kappa }\left(\frac{P_r}{1+R}-S_c\right)-\left(\frac{H}{1+R}+S_c{K}_r\right)\right)}\right).\hfill \end{array}$$

(29)

3.3. Velocity Field

The following Caputo derivative form of velocity Equation (12) is developed by using Equation (18):

$$\begin{array}{c}\hfill \left(1+h{\gamma }_1\right){(}^C{D}_t^{\kappa })w(\zeta ,t)=\left(1+{\gamma }_1^C{D}_t^{\kappa }\right)\frac{{\partial }^{2}w}{\partial {\zeta }^2}+G_r\theta +G_{m}C-\left(M^2+h\right)w.\end{array}$$

(30)

By implementing the Laplace transform on Equation (30),

$$\begin{array}{c}\hfill \left(1+{\gamma }_1{s}^{\kappa }\right)\frac{{\partial }^2{\overline{w}}_c(\zeta ,s)}{\partial {\zeta }^2}-\left(c{s}^{\kappa }+b\right){\overline{w}}_c(\zeta ,s)=-G_r{\overline{\theta }}_c(\zeta ,s)+G_m{\overline{C}}_c(\zeta ,s),\end{array}$$

(31)

the homogenous solution of above equation is

$$\begin{array}{c}\hfill {\overline{w}}_h(\zeta ,s)=c_1{e}^{-\zeta \sqrt{\left(1+\frac{{\gamma }_1}{k_1}\right)\left(\frac{s^{\kappa }}{(1-\kappa )s+\kappa }\right)+\left(M^2+k_1\right)}}+c_2{e}^{\zeta \sqrt{\left(1+\frac{{\gamma }_1}{k_1}\right)\left(\frac{s^{\kappa }}{(1-\kappa )s+\kappa }\right)+\left(M^2+k_1\right)}},\end{array}$$

(32)

the particular solution for Equation (31) is

$$\begin{array}{cc}\hfill {\overline{w}}_p(\zeta ,s)=& -\frac{G_r(1-e^{-s}){\left(s^{\kappa }+b_2\right)}^2}{s^2\left\{(a_{4}s-a_5)(a_{12}s+b_2)-(a_{13}s+a_{14})(s+b_2)\right\}}{e}^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s^{\kappa }}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}}\hfill \\ & -\frac{G_m(1-e^{-s}){\left(s^{\kappa }+b_2\right)}^2}{s^2\left\{(a_7{s}^{\kappa }-a_8)(a_{12}s+b_2)-(a_{13}{s}^{\kappa }+a_{14})(s^{\kappa }+b_2)\right\}}\times \hfill \\ & \frac{S_c{S}_r\left\{\left(P_r-H+H\kappa \right)s^{\kappa }-H\kappa \right\}}{s^2\left\{\left(P_r-H(1-\kappa )-S_c(1+R)-S_c{K}_r(1+R)(1-\kappa )\right)s^{\kappa }-(H\kappa +S_c{K}_r\kappa (1+R))\right\}}\hfill \\ & \left(\frac{1-e^{-s}}{s^2}\right)e^{-\zeta \sqrt{S_c\left(\frac{s^{\kappa }}{(1-\kappa )s^{\kappa }+\kappa }+K_r\right)}}-\frac{G_m(1-e^{-s}){\left(s^{\kappa }+b_2\right)}^2}{s^2\left\{(a_7{s}^{\kappa }-a_8)(a_{12}{s}^{\kappa }+b_2)-(a_{13}{s}^{\kappa }+a_{14})(s^{\kappa }+b_2)\right\}}\hfill \\ & \times \left\{e^{-\zeta \sqrt{S_c\left(\frac{s^{\kappa }}{(1-\kappa )s+\kappa }+K_r\right)}}-e^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s^{\kappa }}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}}\right\}\hfill \end{array}$$

(33)

and the required solution is given below:

$$\begin{array}{cc}\hfill {\overline{w}}_c(\zeta ,s)=& \frac{2}{s^3}{e}^{-\zeta \sqrt{\frac{c{s}^{\kappa }+b}{1+{\gamma }_1{s}^{\kappa }}}}+\frac{G_r(1-e^{-s})}{s^2\left\{\left(\frac{P_r}{1+R}\right)(s^{\kappa }-\frac{H}{P_r})(1+{\gamma }_1{s}^{\kappa })-(c{s}^{\kappa }+b)\right\}}\times \hfill \\ & \left\{e^{-\zeta \sqrt{\frac{c{s}^{\kappa }+b}{1+{\gamma }_1{s}^{\kappa }}}}-e^{-\zeta \sqrt{\left(\frac{P_r}{1+R}\right)(s^{\kappa }-\frac{H}{P_r})}}\right\}\hfill \\ & +\frac{G_m}{S_c\left(s^{\kappa }+K_r\right)\left(1+{\gamma }_1{s}^{\kappa }\right)-\left(c{s}^{\kappa }+b\right)}\times \hfill \\ & \left\{\left(\frac{1-e^{-s}}{s^2}\right)e^{-\zeta \sqrt{\frac{c{s}^{\kappa }+b}{1+{\gamma }_1{s}^{\kappa }}}}-\left(\frac{1-e^{-s}}{s^2}\right)e^{-\zeta \sqrt{S_c\left(s^{\kappa }+K_r\right)}}+A_1{A}_2\right\},\hfill \end{array}$$

(34)

where

$$\begin{array}{cc}\hfill & A_2=e^{-\zeta \sqrt{S_c(s^{\kappa }+K_r)}}-e^{-\zeta \sqrt{\left(\frac{P_r}{1+R}\right)(s^{\kappa }-\frac{H}{P_r})}},b_1=\frac{1}{1-\kappa },b_2=\kappa b_1,\hfill \\ & a_1=\frac{1+R}{P_r},a_2=\frac{H}{P_r},a_3=\frac{1}{a_1},a_4=a_3{b}_1-a_3{a}_2,a_5=a_3{a}_2{b}_2,\hfill \\ & b_4=a_3-S_c,b_5=a_3{a}_2+S_c{K}_r,a_6=\frac{b_5}{b_4},a_7=S_c(b_1+K_r),\hfill \\ & a_8=S_c{K}_r{b}_2,a_9=a_4-a_7,a_{10}=a_5+a_8,a_{11}=\frac{a_{10}}{b_9},\hfill \\ & a_{12}=\frac{\gamma }{1-\kappa }+1,a_{13}=\frac{c}{1-\kappa }+b,a_{14}=\frac{b\kappa }{1-\kappa }+b.\hfill \end{array}$$

(35)

4. Solution for Caputo–Fabrizio Fractional Operator

The CF derivative and its Laplace transform [77] are given below:

$$\begin{array}{c}\hfill {}^{CF}{D}_{\tau }^{\kappa }N\left(\zeta ,\tau \right)=\frac{1}{1-\kappa }{\int }_0^{\tau }exp\left(-\frac{\kappa (\tau -\xi )}{1-\kappa }\right)N^{/}\left(\xi \right)d\xi ,0<\kappa <1,\end{array}$$

(36)

$$\begin{array}{c}\hfill \mathcal{L}\left({}^{CF}{D}_{\tau }^{\kappa }N\left(\zeta ,\tau \right)\right)=\frac{s\mathcal{L}\left(N(\zeta ,\tau )\right)-N(\zeta ,0)}{(1-\kappa )s+\kappa }.\end{array}$$

(37)

4.1. Temperature Field

The following CF derivative form of temperature Equation (13) is developed by using Equation (36):

$$\begin{array}{c}\hfill P_r{}^{CF}{D}_t^{\kappa }\theta (\zeta ,t)=\left(1+R\right)\frac{{\partial }^2\theta }{\partial {\zeta }^2}+H\theta .\end{array}$$

(38)

By implementing the Laplace transform on Equation (38),

$$\begin{array}{c}\hfill \frac{{\partial }^2{\overline{\theta }}_{cf}(\zeta ,s)}{\partial {\zeta }^2}-\frac{P_r}{1+R}\left(\frac{s}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right){\overline{\theta }}_{cf}(\zeta ,s)=0,\end{array}$$

(39)

the homogenous solution of above equation is

$$\begin{array}{c}\hfill {\overline{\theta }}_{cf}(\zeta ,s)=c_1{e}^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}}+c_2{e}^{\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}},\end{array}$$

(40)

and $c_1$ and $c_2$ can be determined by employing Equations (15)–(17). The required solution is given below:

$$\begin{array}{c}\hfill {\overline{\theta }}_{cf}(\zeta ,s)=\left(\frac{1-e^{-s}}{s^2}\right)e^{-\zeta \sqrt{B_2}}.\end{array}$$

(41)

4.2. Concentration Field

The following CF derivative form of concentration Equation (13) is developed by using Equation (36):

$$\begin{array}{c}\hfill {}^{CF}{D}_t^{\kappa }C(\zeta ,t)=\frac{1}{S_c}\frac{{\partial }^{2}C(\zeta ,t)}{\partial {\zeta }^2}+S_r\frac{{\partial }^2\theta (\zeta ,t)}{\partial {\zeta }^2}-K_{r}C(\zeta ,t).\end{array}$$

(42)

By implementing the Laplace transform on Equation (42),

$$\begin{array}{c}\hfill \left(\frac{s}{(1-\kappa )s+\kappa }+K_r\right)C_{cf}(\zeta ,s)=\frac{1}{S_c}\frac{{\partial }^2{\overline{C}}_{cf}(\zeta ,s)}{\partial {\zeta }^2}+S_r\frac{{\partial }^2{\overline{\theta }}_{cf}(\zeta ,s)}{\partial {\zeta }^2},\end{array}$$

(43)

the homogenous solution of the above equation is

$$\begin{array}{c}\hfill {\overline{C}}_h(\zeta ,s)=c_1{e}^{-\zeta \sqrt{S_c\left(\frac{s}{(1-\kappa )s+\kappa }+K_r\right)}}+c_2{e}^{\zeta \sqrt{S_c\left(\frac{s}{(1-\kappa )s+\kappa }+K_r\right)}},\end{array}$$

(44)

the particular solution of Equation (44) is

$$\begin{array}{cc}\hfill {\overline{C}}_p(\zeta ,s)=& -\frac{S_c{S}_r(1-e^{-s})\left\{\left(P_r-H+H\kappa \right)s-H\kappa \right\}}{s^2\left\{\left(P_r-H(1-\kappa )-S_c(1+R)-S_c{K}_r(1+R)(1-\kappa )\right)s-(H\kappa +S_c{K}_r\kappa (1+R))\right\}}\hfill \\ & e^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}},\hfill \end{array}$$

(45)

and the required solution is given below:

$$\begin{array}{cc}\hfill {\overline{C}}_{cf}(\zeta ,s)& =\left(\frac{1-e^{-s}}{s^2}-A_3\right)e^{-\zeta \sqrt{B_3}}+A_3{e}^{-\zeta \sqrt{B_2}},\hfill \end{array}$$

(46)

where

$$\begin{array}{cc}\hfill A_3& =\frac{S_c{S}_r\left(\left(\frac{P_r}{1+R}\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)\right)s-\left(\frac{H}{1+R}\right)\left(\frac{\kappa }{1-\kappa }\right)\right)\left(1-e^{-s}\right)}{\left(\frac{H}{1+R}+S_c{K}_r\right)\left(\frac{\kappa }{1-\kappa }\right)}\times \hfill \\ & \left(\frac{\left(\frac{P_r}{1+R}\right)\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)-S_c\left(\frac{1}{1-\kappa }+K_r\right)}{s(\frac{H}{1+R}+S_c{K}_r)\left(\frac{\kappa }{1-\kappa }\right)}+\frac{1}{s^2}-A_4\right),\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill A_4& =\frac{1}{(\frac{H}{1+R}+S_c{K}_r)\left(\frac{\kappa }{1-\kappa }\right)\left(s\left(\frac{P_r}{1+R}\right)\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)-\left(\frac{H}{1+R}+S_c{K}_r\right)\left(\frac{\kappa }{1-\kappa }\right)\right)},\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill B_3& =\frac{S_c\left(\frac{1}{1-\kappa }+K_r\right)s+S_c{K}_r\left(\frac{\kappa }{1-\kappa }\right)}{s+\frac{\kappa }{1-\kappa }},\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill B_2& =\frac{\left(\frac{P_r}{1+R}\right)\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)s-\left(\frac{H}{1+R}\right)\left(\frac{\kappa }{1-\kappa }\right)}{s+\frac{\kappa }{1-\kappa }}.\hfill \end{array}$$

(50)

4.3. Velocity Field

The following CF derivative form of velocity Equation (12) is developed by using Equation (36):

$$\begin{array}{c}\hfill \left(1+h{\gamma }_1\right){(}^{CF}{D}_t^{\kappa })w(\zeta ,t)=\left(1+{\gamma }_1^{CF}{D}_t^{\kappa }\right)\frac{{\partial }^{2}w}{\partial {\zeta }^2}+G_r\theta +G_{m}C-\left(M^2+h\right)w.\end{array}$$

(51)

By implementing the Laplace transform on Equation (51),

$$\begin{array}{cc}\hfill \left(\frac{\left(\frac{c}{1-\kappa }+b\right)s+\frac{b\kappa }{1-\kappa }}{s+\frac{\kappa }{1-\kappa }}\right){\overline{w}}_{cf}(\zeta ,s)& =\left(\frac{\left(\frac{{\gamma }_1}{1-\kappa }+1\right)s+\frac{\kappa }{1-\kappa }}{s+\frac{\kappa }{1-\kappa }}\right)\frac{{\partial }^2{\overline{w}}_{cf}(\zeta ,s)}{\partial {\zeta }^2}\hfill \\ & +G_r{\overline{\theta }}_{cf}(\zeta ,s)+G_m{\overline{C}}_{cf}(\zeta ,s),\hfill \end{array}$$

(52)

the homogenous solution of the above equation is

$$\begin{array}{c}\hfill {\overline{w}}_h(\zeta ,s)=c_1{e}^{-\zeta \sqrt{\left(1+\frac{{\gamma }_1}{k_1}\right)\left(\frac{s}{(1-\kappa )s+\kappa }\right)+\left(M^2+k_1\right)}}+c_2{e}^{\zeta \sqrt{\left(1+\frac{{\gamma }_1}{k_1}\right)\left(\frac{s}{(1-\kappa )s+\kappa }\right)+\left(M^2+k_1\right)}}.\end{array}$$

(53)

The particular solution of Equation (52) is

$$\begin{array}{cc}\hfill {\overline{w}}_p(\zeta ,s)=& -\frac{G_r(1-e^{-s}){\left(s+b_2\right)}^2}{s^2\left\{(a_{4}s-a_5)(a_{12}s+b_2)-(a_{13}s+a_{14})(s+b_2)\right\}}{e}^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}}\hfill \\ & -\frac{G_m(1-e^{-s}){\left(s+b_2\right)}^2}{s^2\left\{(a_{7}s-a_8)(a_{12}s+b_2)-(a_{13}s+a_{14})(s+b_2)\right\}}\times \hfill \\ & \frac{S_c{S}_r\left\{\left(P_r-H+H\kappa \right)s-H\kappa \right\}}{s^2\left\{\left(P_r-H(1-\kappa )-S_c(1+R)-S_c{K}_r(1+R)(1-\kappa )\right)s-(H\kappa +S_c{K}_r\kappa (1+R))\right\}}\hfill \\ & \left(\frac{1-e^{-s}}{s^2}\right)e^{-\zeta \sqrt{S_c\left(\frac{s}{(1-\kappa )s+\kappa }+K_r\right)}}-\frac{G_m(1-e^{-s}){\left(s+b_2\right)}^2}{s^2\left\{(a_{7}s-a_8)(a_{12}s+b_2)-(a_{13}s+a_{14})(s+b_2)\right\}}\hfill \\ & \times \left\{e^{-\zeta \sqrt{S_c\left(\frac{s}{(1-\kappa )s+\kappa }+K_r\right)}}-e^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}}\right\}\hfill \end{array}$$

(54)

and the required solution is given below:

$$\begin{array}{cc}\hfill {\overline{w}}_{cf}(\zeta ,s)=& \frac{2}{s^3}{e}^{-\zeta \sqrt{B_1}}+\frac{G_r(1-e^{-s}){\left(s+\frac{\kappa }{1-\kappa }\right)}^2}{s^2{A}_7}\left(e^{-\zeta \sqrt{B_1}}-e^{-\zeta \sqrt{B_2}}\right)\hfill \\ & +\frac{G_m{\left(s+\frac{\kappa }{1-\kappa }\right)}^2}{s^2{A}_8}\left(e^{-\zeta \sqrt{B_1}}-e^{-\zeta \sqrt{B_3}}\right)+A_5\left(e^{-\zeta \sqrt{B_1}}-e^{-\zeta \sqrt{B_2}}\right)\hfill \end{array}$$

(55)

where

$$\begin{array}{cc}\hfill A_7& =\left(\left(\frac{P_r}{1+R}\right)\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)s-\left(\frac{H}{1+R}\right)\left(\frac{\kappa }{1-\kappa }\right)\right)\left(\left(\frac{{\gamma }_1}{1+\kappa }+1\right)s+\frac{\kappa }{1-\kappa }\right)\hfill \\ & -\left(\left(\frac{c}{1-\kappa }+b\right)s+\frac{b\kappa }{1-\kappa }\right)\left(s+\frac{\kappa }{1-\kappa }\right)\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill A_8& =\left(S_c\left(\frac{1}{1-\kappa }-K_r\right)s+S_c{K}_r\left(\frac{\kappa }{1-\kappa }\right)\right)\left(\left(\frac{{\gamma }_1}{1+\kappa }+1\right)s+\frac{\kappa }{1-\kappa }\right)\hfill \\ & -\left(\left(\frac{c}{1-\kappa }+b\right)s+\frac{b\kappa }{1-\kappa }\right)\left(s+\frac{\kappa }{1-\kappa }\right)\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill B_1& =\frac{\left(\frac{c}{1-\kappa }+b\right)s+\frac{b\kappa }{1-\kappa }}{\left(\frac{\gamma 1}{1-\kappa }+1\right)s+\frac{\kappa }{1-\kappa }}\hfill \end{array}$$

(58)

5. Solution for Atangana–Baleanu Fractional Operator

The ABC derivative and its Laplace transform [78] are given below:

$$\begin{array}{c}\hfill {}^{ABC}{D}_{\tau }^{\alpha }f(\xi ,\tau )=\frac{1}{1-\alpha }{\int }_0^{\tau }{\mathbf{E}}_{\alpha }\left(-\frac{\alpha {(t-\tau )}^{\alpha }}{1-\alpha }\right)\frac{\partial f(\xi ,\tau )}{\partial \tau }d\tau ,\end{array}$$

(59)

$$\begin{array}{c}\hfill \mathcal{L}\left({}^{ABC}{D}_t^{\kappa }f\left(\xi ,\tau \right)\right)=\frac{s^{\kappa }\mathcal{L}\left(f(\xi ,\tau )\right)-s^{\kappa -1}f(\xi ,0)}{(1-\kappa )s^{\kappa }+\kappa }.\end{array}$$

(60)

5.1. Temperature Field

The following ABC derivative form of temperature Equation (13) is developed by using Equation (59):

$$\begin{array}{c}\hfill P_r{}^{ABC}{D}_t^{\kappa }\theta (\zeta ,t)=\left(1+R\right)\frac{{\partial }^2\theta }{\partial {\zeta }^2}+H\theta ,\end{array}$$

(61)

The Laplace transform of Equation (61) is

$$\begin{array}{c}\hfill \frac{{\partial }^2{\overline{\theta }}_{abc}(\zeta ,s)}{\partial {\zeta }^2}-\frac{P_r}{1+R}\left(\frac{s^{\kappa }}{(1-\kappa )s^{\kappa }+\kappa }-\frac{H}{P_r}\right){\overline{\theta }}_{abc}(\zeta ,s)=0,\end{array}$$

(62)

the homogenous solution of above equation is

$$\begin{array}{c}\hfill {\overline{\theta }}_{abc}(\zeta ,s)=c_1{e}^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s^{\kappa }}{(1-\kappa )s^{\kappa }+\kappa }-\frac{H}{P_r}\right)}}+c_2{e}^{\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s^{\kappa }}{(1-\kappa )s^{\kappa }+\kappa }-\frac{H}{P_r}\right)}}.\end{array}$$

(63)

and $c_1$ and $c_2$ can be determined by employing Equations (15)–(17). The required solution is given below:

$$\begin{array}{c}\hfill {\overline{\theta }}_{abc}(\zeta ,s)=\left(\frac{1-e^{-s}}{s^2}\right)e^{-\zeta \sqrt{B_2}}.\end{array}$$

(64)

5.2. Concentration Field

The following ABC derivative form of concentration Equation (14) is developed by using Equation (59):

$$\begin{array}{c}\hfill {}^{ABC}{D}_t^{\kappa }{C}_{(}\zeta ,t)=\frac{1}{S_c}\frac{{\partial }^{2}C(\zeta ,t)}{\partial {\zeta }^2}+S_r\frac{{\partial }^2\theta (\zeta ,t)}{\partial {\zeta }^2}-K_{r}C(\zeta ,t).\end{array}$$

(65)

The Laplace transform of Equation (65) is

$$\begin{array}{c}\hfill \left(\frac{s^{\kappa }}{(1-\kappa )s^{\kappa }+\kappa }+K_r\right){\overline{C}}_{abc}(\zeta ,s)=\frac{1}{S_c}\frac{{\partial }^2{\overline{C}}_{abc}(\zeta ,s)}{\partial {\zeta }^2}+S_r\frac{{\partial }^2{\overline{\theta }}_{abc}(\zeta ,s)}{\partial {\zeta }^2},\end{array}$$

(66)

and the homogenous solution of the above equation is

$$\begin{array}{c}\hfill {\overline{C}}_h(\zeta ,s)=c_1{e}^{-\zeta \sqrt{S_c\left(\frac{s^{\kappa }}{(1-\kappa )s^{\kappa }+\kappa }+K_r\right)}}+c_2{e}^{\zeta \sqrt{S_c\left(\frac{s^{\kappa }}{(1-\kappa )s^{\kappa }+\kappa }+K_r\right)}}.\end{array}$$

(67)

The particular solution of Equation (66) is

$$\begin{array}{cc}\hfill {\overline{C}}_p(\zeta ,s)=& -\frac{S_c{S}_r(1-e^{-s})\left\{\left(P_r-H+H\kappa \right)s^{\kappa }-H\kappa \right\}}{s^2\left\{\left(P_r-H(1-\kappa )-S_c(1+R)-S_c{K}_r(1+R)(1-\kappa )\right)s^{\kappa }-(H\kappa +S_c{K}_r\kappa (1+R))\right\}}\hfill \\ & e^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s^{\kappa }}{(1-\kappa )s^{\kappa }+\kappa }-\frac{H}{P_r}\right)}},\hfill \end{array}$$

(68)

and the required solution is given below:

$$\begin{array}{cc}\hfill {\overline{C}}_{abc}(\zeta ,s)& =\left(\frac{1-e^{-s}}{s^2}-A_5\right)e^{-\zeta \sqrt{B_6}}+A_5{e}^{-\zeta \sqrt{B_5}},\hfill \end{array}$$

(69)

where

$$\begin{array}{cc}\hfill A_5& =\frac{S_c{S}_r\left(\left(\frac{P_r}{1+R}\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)\right)s^{\kappa }-\left(\frac{H}{1+R}\right)\left(\frac{\kappa }{1-\kappa }\right)\right)\left(1-e^{-s}\right)}{\left(\frac{H}{1+R}+S_c{K}_r\right)\left(\frac{\kappa }{1-\kappa }\right)}\times \hfill \\ & \left(\frac{\left(\frac{P_r}{1+R}\right)\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)-S_c\left(\frac{1}{1-\kappa }+K_r\right)}{s^{\kappa }(\frac{H}{1+R}+S_c{K}_r)\left(\frac{\kappa }{1-\kappa }\right)}+\frac{1}{s^2}-A_6\right),\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill A_6& =\frac{1}{(\frac{H}{1+R}+S_c{K}_r)\left(\frac{\kappa }{1-\kappa }\right)\left(s^{\kappa }\left(\frac{P_r}{1+R}\right)\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)-\left(\frac{H}{1+R}+S_c{K}_r\right)\left(\frac{\kappa }{1-\kappa }\right)\right)},\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill B_6& =\frac{S_c\left(\frac{1}{1-\kappa }+K_r\right)s^{\kappa }+S_c{K}_r\left(\frac{\kappa }{1-\kappa }\right)}{s^{\kappa }+\frac{\kappa }{1-\kappa }},\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill B_5& =\frac{\left(\frac{P_r}{1+R}\right)\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)s^{\kappa }-\left(\frac{H}{1+R}\right)\left(\frac{\kappa }{1-\kappa }\right)}{s^{\kappa }+\frac{\kappa }{1-\kappa }}.\hfill \end{array}$$

(73)

5.3. Velocity Field

The following ABC derivative form of velocity Equation (12) is developed by using Equation (59):

$$\begin{array}{c}\hfill \left(1+h{\gamma }_1\right){(}^{ABC}{D}_t^{\kappa })w(\zeta ,t)=\left(1+{\gamma }_1^{ABC}{D}_t^{\kappa }\right)\frac{{\partial }^{2}w}{\partial {\zeta }^2}+G_r\theta +G_{m}C-\left(M^2+h\right)w.\end{array}$$

(74)

The Laplace transform of Equation (74) is

$$\begin{array}{cc}\hfill \left(\frac{\left(\frac{c}{1-\kappa }+b\right)s^{\kappa }+\frac{b\kappa }{1-\kappa }}{s^{\kappa }+\frac{\kappa }{1-\kappa }}\right){\overline{w}}_{abc}(\zeta ,s)& =\left(\frac{\left(\frac{{\gamma }_1}{1-\kappa }+1\right)s^{\kappa }+\frac{\kappa }{1-\kappa }}{s^{\kappa }+\frac{\kappa }{1-\kappa }}\right)\frac{{\partial }^2{\overline{w}}_{abc}(\zeta ,s)}{\partial {\zeta }^2}\hfill \\ & +G_r{\overline{\theta }}_{abc}(\zeta ,s)+G_m{\overline{C}}_{abc}(\zeta ,s),\hfill \end{array}$$

(75)

the homogenous solution of the above equation is

$$\begin{array}{c}\hfill {\overline{w}}_h(\zeta ,s)=c_1{e}^{-\zeta \sqrt{\left(1+\frac{{\gamma }_1}{k_1}\right)\left(\frac{s}{(1-\kappa )s+\kappa }\right)+\left(M^2+k_1\right)}}+c_2{e}^{\zeta \sqrt{\left(1+\frac{{\gamma }_1}{k_1}\right)\left(\frac{s}{(1-\kappa )s+\kappa }\right)+\left(M^2+k_1\right)}},\end{array}$$

(76)

the particular solution of Equation (75) is

$$\begin{array}{cc}\hfill {\overline{w}}_p(\zeta ,s)=& -\frac{G_r(1-e^{-s}){\left(s+b_2\right)}^2}{s^2\left\{(a_{4}s-a_5)(a_{12}s+b_2)-(a_{13}s+a_{14})(s+b_2)\right\}}{e}^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}}\hfill \\ & -\frac{G_m(1-e^{-s}){\left(s+b_2\right)}^2}{s^2\left\{(a_{7}s-a_8)(a_{12}s+b_2)-(a_{13}s+a_{14})(s+b_2)\right\}}\times \hfill \\ & \frac{S_c{S}_r\left\{\left(P_r-H+H\kappa \right)s-H\kappa \right\}}{s^2\left\{\left(P_r-H(1-\kappa )-S_c(1+R)-S_c{K}_r(1+R)(1-\kappa )\right)s-(H\kappa +S_c{K}_r\kappa (1+R))\right\}}\hfill \\ & \left(\frac{1-e^{-s}}{s^2}\right)e^{-\zeta \sqrt{S_c\left(\frac{s}{(1-\kappa )s+\kappa }+K_r\right)}}-\frac{G_m(1-e^{-s}){\left(s+b_2\right)}^2}{s^2\left\{(a_{7}s-a_8)(a_{12}s+b_2)-(a_{13}s+a_{14})(s+b_2)\right\}}\hfill \\ & \times \left\{e^{-\zeta \sqrt{S_c\left(\frac{s}{(1-\kappa )s+\kappa }+K_r\right)}}-e^{-\zeta \sqrt{\frac{P_r}{1+R}\left(\frac{s}{(1-\kappa )s+\kappa }-\frac{H}{P_r}\right)}}\right\}\hfill \end{array}$$

(77)

and the required solution is given below

$$\begin{array}{cc}\hfill {\overline{w}}_{abc}(\zeta ,s)=& \frac{2}{s^3}{e}^{-\zeta \sqrt{B_4}}+\frac{G_r(1-e^{-s}){\left(s^{\kappa }+\frac{\kappa }{1-\kappa }\right)}^2}{s^2{A}_9}\left(e^{-\zeta \sqrt{B_4}}-e^{-\zeta \sqrt{B_5}}\right)\hfill \\ & +\frac{G_m{\left(s^{\kappa }+\frac{\kappa }{1-\kappa }\right)}^2}{s^2{A}_{10}}\left(e^{-\zeta \sqrt{B_4}}-e^{-\zeta \sqrt{B_6}}\right)+A_6\left(e^{-\zeta \sqrt{B_4}}-e^{-\zeta \sqrt{B_5}}\right).\hfill \end{array}$$

(78)

where

$$\begin{array}{cc}\hfill A_9& =\left(\left(\frac{P_r}{1+R}\right)\left(\frac{1}{1-\kappa }-\frac{H}{P_r}\right)s^{\kappa }-\left(\frac{H}{1+R}\right)\left(\frac{\kappa }{1-\kappa }\right)\right)\left(\left(\frac{{\gamma }_1}{1+\kappa }+1\right)s^{\kappa }+\frac{\kappa }{1-\kappa }\right)\hfill \\ & -\left(\left(\frac{c}{1-\kappa }+b\right)s^{\kappa }+\frac{b\kappa }{1-\kappa }\right)\left(s^{\kappa }+\frac{\kappa }{1-\kappa }\right),\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill A_{10}& =\left(S_c\left(\frac{1}{1-\kappa }-K_r\right)s^{\kappa }+S_c{K}_r\left(\frac{\kappa }{1-\kappa }\right)\right)\left(\left(\frac{{\gamma }_1}{1+\kappa }+1\right)s^{\kappa }+\frac{\kappa }{1-\kappa }\right)\hfill \\ & -\left(\left(\frac{c}{1-\kappa }+b\right)s^{\kappa }+\frac{b\kappa }{1-\kappa }\right)\left(s^{\kappa }+\frac{\kappa }{1-\kappa }\right),\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill B_4& =\frac{\left(\frac{c}{1-\kappa }+b\right)s^{\kappa }+\frac{b\kappa }{1-\kappa }}{\left(\frac{\gamma 1}{1-\kappa }+1\right)s^{\kappa }+\frac{\kappa }{1-\kappa }}.\hfill \end{array}$$

(81)

As $\kappa \to 1$ in Equations (23), (41) and (64) for temperature, Equations (28), (46) and (69) for concentration and Equations (34), (55) and (78) for velocity, we recuperate results for temperature, concentration and velocity profile for integer order shown in Kataria and Hari (Equations (13)–(15)) [75], respectively.

Stehfest’s formula [79] is one of the simplest algorithms we use to sort out the inverse Laplace transform.

$$w(r,t)=\frac{e^{4.7}}{t}\left[\frac{1}{2}\overline{w}\left(r,\frac{4.7}{t}\right)+Re\left\{\sum _{k=1}^{N_1}{(-1)}^k\overline{w}\left(r,\frac{4.7+k\pi i}{t}\right)\right\}\right],$$

where Re(.) is the real part, i is the imaginary unit and $N_1$ is a natural number.

6. Results and Discussion

This article shows the effect of heat and mass transfer in MHD SGF past a vertical plate. Three fractional models C, CF and ABC for flow, energy and mass equations are presented. Fractional derivatives and Laplace transform are applied to examine solutions for non-dimension fractional models. The limiting cases of fractional models are discussed. The impact of several parameters on momentum, heat and mass profiles are compared and studied by graphs.

Figure 3 highlights the behavior of momentum curves for $\kappa$. We see the velocity accelerates by raising $\kappa$. The reason is by an increment in $\kappa$, the thickness of the boundary layer will enhance, so the velocity increases. Moreover, velocity is highest for the ABC.

Figure 4 reveals the deviation in velocity distribution under the MHD condition. As M increases, the frictional force rises and hence fluid velocity decreases. For different values of M, an increase in Lorentz force effectively decreases flow accelerating forces; as a result, velocity is decelerated. Fluid velocity is maximum, moderate and minimum for ABC, CF and C models, respectively.

Figure 5 analyzes the influence of $G_r$ on momentum profile. Physically, large values respond to significant buoyancy force as it is related to strong convection currents. As $G_r$ increases, all buoyancy forces are dominant frictional forces and the hence momentum profile becomes amplified.

To highlight the velocity behavior for $G_m$, we present Figure 6. Physically, the increment in buoyancy forces reduces the viscous force that leads to augmenting the flow raise with higher values of $G_m$. The velocity curves show maximum behavior for the ABC model as compared to the other two models.

Figure 3. Velocity curves corresponding to C, CF and ABC with variable $\kappa$ where $P_r=7$, $S_c=0.66$, $G_r=10$, $G_m=5$, $H=3$, $R=5$, $S_r=3$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 3. Velocity curves corresponding to C, CF and ABC with variable $\kappa$ where $P_r=7$, $S_c=0.66$, $G_r=10$, $G_m=5$, $H=3$, $R=5$, $S_r=3$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 4. Velocity curves corresponding to C, CF and ABC with variable M where $P_r=7$, $S_c=0.66$, $G_r=10$, $G_m=5$, $H=3$, $R=5$, $S_r=3$, $K_r=2$, $\kappa =0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 4. Velocity curves corresponding to C, CF and ABC with variable M where $P_r=7$, $S_c=0.66$, $G_r=10$, $G_m=5$, $H=3$, $R=5$, $S_r=3$, $K_r=2$, $\kappa =0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 5. Velocity curves corresponding to C, CF and ABC with variable $G_r$ where $P_r=7$, $S_c=0.66$, $\kappa =0.5$, $G_m=5$, $H=3$, $R=5$, $S_r=3$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 5. Velocity curves corresponding to C, CF and ABC with variable $G_r$ where $P_r=7$, $S_c=0.66$, $\kappa =0.5$, $G_m=5$, $H=3$, $R=5$, $S_r=3$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 6. Velocity curves corresponding to C, CF and ABC with variable $G_m$ where $P_r=7$, $S_c=0.66$, $\kappa =0.5$, $G_r=10$, $H=3$, $R=5$, $S_r=3$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 6. Velocity curves corresponding to C, CF and ABC with variable $G_m$ where $P_r=7$, $S_c=0.66$, $\kappa =0.5$, $G_r=10$, $H=3$, $R=5$, $S_r=3$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

The significant impact of $K_r$ on momentum and mass profiles is shown in Figure 7. Both the fluid flow and the concentration decay with the rise in the $K_r$. The presence of chemical reaction reduces the buoyancy effects, which decreases and hence weakens the flow field. Clearly, the ABC model shows the highest velocity and concentration.

Figure 8 describes the behavior of momentum and mass profile with increasing values of $S_r$. As the velocity profile increases, the momentum boundary layer becomes thicker. Physically, large values of $S_r$ respond with a significant increase in mass buoyancy force; as a result, momentum and mass profile are raised. The effect is greatest for the ABC fractional MHD SGF model.

By raising thermal radiation parameter, momentum and heat profiles accelerate as shown in Figure 9. As the thermal radiation parameter increases, heat generation through flow increases, and as a result, bonds between fluid particles split which causes fluid to flow fast.

Figure 10 depicts velocity curves corresponding to C, CF and ABC for $\kappa =0.6$. Clearly, the ABC model shows significant behavior as compared to the other two curves. The reason is that Atangana and Baleanu propounded an advanced fractional operator by utilizing the generalized Mittag–Leffler function as a non-local and non-singular kernel. Comparison of Nusselt number with ref. [72] at Pr = 0.71 is given in

Table 1. Comparison of Nusselt number with ref. [72] at $P_r$ = 0.71.

R	$\mathit{\varphi }$	t	$\mathbf{Nu}$ (Ref. [75]) for Ramped Temp	$\mathbf{Nu}$ (C) for Ramped Temp	$\mathbf{Nu}$ (CF) for Ramped Temp	$\mathbf{Nu}$ (ABC) for Ramped Temp
2	3	0.3	0.3837	0.384	0.385	0.386
2	3	0.5	0.5583	0.557	0.558	0.559
2	3	0.7	0.7289	0.727	0.728	0.729
2	3	0.5	0.4498	0.447	0.448	0.449
2	3	0.5	0.5583	0.557	0.558	0.559
2	5	0.5	0.6521	0.653	0.654	0.655
2	3	0.5	0.5583	0.557	0.558	0.559
4	3	0.5	0.4324	0.433	0.434	0.435
6	3	0.5	0.3655	0.366	0.367	0.368

.

Figure 9. Velocity and temperature curves corresponding to C, CF and ABC with variable R where $P_r=7$, $S_c=0.66$, $\kappa =0.5$, $G_m=5$, $H=3$, $S_r=3$, $G_r=10$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 9. Velocity and temperature curves corresponding to C, CF and ABC with variable R where $P_r=7$, $S_c=0.66$, $\kappa =0.5$, $G_m=5$, $H=3$, $S_r=3$, $G_r=10$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 10. Velocity curves corresponding to fractional models with $\kappa =0.6$ and $P_r=7$, $S_c=0.66$, $\kappa =0.5$, $G_m=5$, $H=3$, $S_r=3$, $G_r=10$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

Figure 10. Velocity curves corresponding to fractional models with $\kappa =0.6$ and $P_r=7$, $S_c=0.66$, $\kappa =0.5$, $G_m=5$, $H=3$, $S_r=3$, $G_r=10$, $K_r=2$, $M=0.5$, ${\gamma }_1=0.1$, $h=0.5$, $c=1.05$ and $b=0.75$.

7. Conclusions

This article is about the study of SGF with radiation and chemical reaction. Three fractional operators are applied to establish momentum, heat and mass profiles. Laplace transform and Stehfest’s formula are utilized for the solutions of the mentioned equations. Several graphs are presented to illustrate the impact of incipient parameters for the solutions. Some main results are given below:

• Velocity curves are increasing for greater values of $\kappa$, $G_r$ and $G_m$.
• Fluid flow descends for $P_r$ and M.
• Velocity and concentration curves show a decreasing behavior under the influence of $K_r$.
• Fluid velocity accelerates under the impact of $S_r$ and R.
• Heat and mass profiles for $S_r$ and R are show an increasing behavior.
• Curves show prominent behavior for ABC among C, CF and ABC.

Extending the work in this article as suggested below will be an interesting endeavor.

We have studied MHD SGF in the presence of ramped concentration and temperature with thermodiffusion. This study can be further carried out by considering more complex OBF models. Additionally, this study can be carried out by a rotational fluid model. Moreover, the present work can be extended by selecting different non-dimensional parameters and quantities.