Heat and Mass Transfer Impact on Differential Type Nanofluid with Carbon Nanotubes: A Study of Fractional Order System

Abstract

This paper is an analysis of flow of MHD CNTs of second grade nano-fluid under the influence of first order chemical reaction, suction, thermal generation and magnetic field. The fluid is flowing through a porous medium. 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. This shows the novelty of this work. 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 solutions are evaluated by employing Laplace transform and inversion algorithm. The flow in momentum profile due to variability in the values of parameters are graphically illustrated among C, CF and ABC models. It is concluded that fluid velocity showed decreasing behavior for $\chi$, P, ${\hslash }_2$, $M_o$, $Pr$, ℵ and $Sc$ while it showed increasing behavior for $Gr$, $Gm$, $\kappa$ and $A_o$. Moreover, ABC fractional operator presents larger memory effect than C and CF fractional operators.

1. Introduction

Nowadays, carbon nanotubes (CNTs) have fascinated researchers worldwide. Their stunning physical properties and very small dimensions makes them impressively useful in applied sciences, normal and artificial phenomena such as cooling instruments, crystal glowing, thermal exchange and various organic sciences. Palani and Abbas [1] observed MHD-free convection lambent flow over an oscillating plate. The convectional flow on a vertical conical sheet with impact of magnetic field was discussed by Kumar and Sivaraj [2]. The impact of suction/injection on nano-fluid flow from two vertical sheets has been illustrated by Das et al. [3].

Through the past thirty years, fractional derivatives have fascinated multiple researchers as compared to classical derivatives. Furthermore, fractional derivatives are more credible in mathematical modeling of real world problems. The methodology of a fractional operator involves regular derivatives and kernel of fractional operator with convolution relation. Applications of fractional calculus have not been restricted to the disciplines of engineering and physical sciences, but also in other disciplines, such as ecology, geology, viscoelasticity, economics, probability and statistics, and fluid dynamics [4,5,6,7]. Cao et al. [8] analyzed a fractional model of nano-fluid over a moving plate. Pandey et al. [9] studied the effects of viscous dissipation and suction/injection on MHD flow of a nanofluid. At room temperature, thermal conductivity of CNT’s nanofluid is six times greater than other material’s nano fluid. This result has been discussed by Murshed et al. [10]. Abro et al. [11] applied a fractional derivative with non-singular and non-local kernel on a nano-fluid under magnetism. Saqib et al. [12] highlighted the strong memory effect of the Atangana–Baleanu fractional model of CNT’s nano-fluid. The abeyance of nanoparticles in fluid airing notable enhancement of their properties at reticent nano-particle concentrations are known as nano-fluids [13]. Nanofluids parade enhanced thermal conductivity that escalates with growing volumetric fraction of nano-particles. In computers and other electronics, nano-fluids are used to cool microchips. Lin et al. [14] inspected silver nanoparticles in vibrant thermal pipes. Saqib et al. [15] applied AB derivative to MHD flow of CMC based CNT’s nano-fluid. Ikram et al. [16] diagnosed that Caputo fractional model exhibits a greater memory effect as compared to the Caputo–Fabrizio fractional model of nanofluid. Recently, Maiti et al. [17] studied the effect of the Caputo–Fabrizio derivative of fractional order model on the flow of blood in a porous tube having thermochemical properties under the magnetic and vibration mode.

The nanofluid with injection/suction, concentration and temperature are used in dissolution of garbage from nuclear reactors, filtration and absorption of chemicals by a soft medium. The motion of nanofluid under the impact of suction/injection in conical domain was discussed by Sreedevi et al. [18]. Du et al. [19] investigated the memory effect with derivative’s order. Hayat et al. [20] inspected 3D flow of nanofluid under the influence of heat generation and magnetic field. CNTs have significant application in solar system. Ghalandari et al. [21] studied how nanofluids can be used in solar system. Nanofluids have many applications in stability, phase diagram, rheology and electroosmotic flow [22,23,24]. Alawi et al. [25] analyzed the thermophysical properties and stability of carbon nanostructures and metallic oxides nanofluids.

Pandikunta et al. [26] revealed that non-Newtonian MWCNTs Tangent hyperbolic nano-fluid minimizes the friction near the stretching sheet contrasting SWCNTs. The 4th order Runge–Kutta method was applied by Aghamajidi et al. [27] to MHD nano-flow model adjacent to a spinning down pointing vertical cone. Reddy et al. [28] analyzed MHD CNTs nanofluid flow over a stretching sheet and found that by increasing the value of thermal radiation parameter, the rate of heat transfer also increases. Upreti et al. [29] canvassed the flow and heat profile of 2-D CNTs nanofluid under the influence of magnetic field and blow/suction. Upreti et al. [30] designed a MHD model of 3D Darcy–Forchheimer flow of water–CNTs nanofluid and applied the Runge–Kutta–Fehlberg method to find the effects of Ohmic heating. Hussain et al. [31] considered single and multiple walled CNTs and cross-examined that thermal generation variable is inversely proportional to wall heat flux. Alsagri et al. [32] discovered that the dimensions of nano particles, thermal efficiency and volumetric fraction are directly proportional to each other in MHD finite film flow of human blood with CNTs nanofluids. Recently, the Atangana–Baleanu fractional model that is the finest fractional MHD rate type fluid model to reveal the memory effect of velocity and thermal distribution was discovered by Kumam et al. [33]. Saqib et al. [34] investigated flow of nanofluid under the influence of heat generation, magnetic field, shape effect of nano-particles and thermal radiation. The velocity profile of Jeffery nanofluid was discussed by Roohi et al. [35] through a porous medium and under the influence of thermochemical effects. Fractional order model of thermo-solutal and magnetic nanoparticles transport for drug delivery applications has been investigated by authors [36]. Abro [37] explored a fractal–fractional derivative on ferromagnetic fluid via fractal Laplace transform.

This paper reveals the study of three different fractional models to show the memory effect of CNT’s nanofluid model along with porosity, chemical reaction and suction/injection. The impact of emerging parameters for momentum, mass and energy solutions are plotted by different graphs with real justifications. Finally, a comparison has been made among C, CF and ABC fractional models. Fractional models of CNTs nanofluids have been rarely discussed due to their complexity. It has been proven in many already published articles that the heat and mass transfer do not really or always follow the classical mechanics process that is known as a memoryless process. Therefore, the model using classical differentiation based on the rate of change cannot really replicate such dynamical process very accurately; thus, a different concept of differentiation is needed to capture such processes. 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. We use these laws, which shows the novelty of our work.

2. Mathematical Model

Suppose a differential type nanofluid with carbon nanotube on a flat vertical plate under the influence of constant magnetic field $M_o$. Initially, the nanofluid and plate are at rest with ambient temperature ${\tilde{\mathrm{Y}}}_{\infty }$ and ambient concentration ${{\displaystyle \tilde{\bigwedge }}}_{\infty }$, respectively. As time increases from 0, the plate starts vibrating with velocity $U_{o}d\left(\tau \right)$. As ȷ-axis is normal to plane, therefore the flow field depends only on time $\tau$ and ȷ. The suction or injection along the direction of the fluid flow is constant, that is $\nu$ = −${\nu }_o$. The physical model of the problem can be given as follows [38] and its geometry is shown in Figure 1:

$$\begin{array}{cc}\hfill {\rho }_{nf}{\partial }_{\tau }\tilde{W}\left(\mathit{ȷ},\tau \right)-{\rho }_{nf}{\nu }_o{\partial }_{\mathit{ȷ}}\tilde{W}\left(\mathit{ȷ},\tau \right) =& {\mu }_{nf}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\tilde{W}\left(\mathit{ȷ},\tau \right)+{\mu }_{nf}\hslash {\partial }_{\tau \mathit{ȷ}\mathit{ȷ}}^3\tilde{W}\left(\mathit{ȷ},\tau \right)-\sigma M_o\tilde{W}\left(\mathit{ȷ},\tau \right)-\frac{{\mu }_{nf}\wp }{P_1}\tilde{W}\left(\mathit{ȷ},\tau \right)\hfill \\ & +g\left[{\left(\rho {\beta }_{\tilde{\mathrm{Y}}}\right)}_{nf}\left(\tilde{\mathrm{Y}}\left(\mathit{ȷ},\tau \right)-{\tilde{\mathrm{Y}}}_{\infty }\right)+{\left(\rho {\beta }_{\tilde{\mathrm{\Lambda }}}\right)}_{nf}\left(\tilde{\mathrm{\Lambda }}\left(\mathit{ȷ},\tau \right)-{\tilde{\mathrm{\Lambda }}}_{\infty }\right)\right],\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\partial }_{\tau }\tilde{\mathrm{Y}}\left(\mathit{ȷ},\tau \right)-{\nu }_o{\partial }_{\mathit{ȷ}}\tilde{\mathrm{Y}}\left(\mathit{ȷ},\tau \right) =& \frac{k_{nf}}{{\left(\rho C_p\right)}_{nf}}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\tilde{\mathrm{Y}}\left(\mathit{ȷ},\tau \right)+\frac{A_o}{\left(\rho C_p\right)}\left(\tilde{\mathrm{Y}}\left(\mathit{ȷ},\tau \right)-{\tilde{\mathrm{Y}}}_{\infty }\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\partial }_{\tau }\tilde{\mathrm{\Lambda }}\left(\mathit{ȷ},\tau \right)-{\nu }_o{\partial }_{\mathit{ȷ}}\tilde{\mathrm{\Lambda }}\left(\mathit{ȷ},\tau \right) =& D{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\tilde{\mathrm{\Lambda }}\left(\mathit{ȷ},\tau \right)+{\kappa }_r\left(\tilde{\mathrm{\Lambda }}\left(\mathit{ȷ},\tau \right)-{\tilde{\mathrm{\Lambda }}}_{\infty }\right).\hfill \end{array}$$

(3)

The suitable initial and boundary conditions are

$$\begin{array}{cc}\hfill & \tilde{W}\left(\mathit{ȷ},0\right) =0,\tilde{\mathrm{Y}}\left(\mathit{ȷ},0\right) ={\tilde{\mathrm{Y}}}_{\infty },\tilde{\mathrm{\Lambda }}\left(\mathit{ȷ},0\right) ={\tilde{\mathrm{\Lambda }}}_{\infty },\hfill \\ & \tilde{W}\left(0,\tau \right) =U_{o}d\left(\tau \right),\tilde{\mathrm{Y}}\left(0,\tau \right) ={\tilde{\mathrm{Y}}}_w,\tilde{\mathrm{\Lambda }}\left(0,\tau \right) ={\tilde{\mathrm{\Lambda }}}_w,\hfill \\ & \tilde{W}\left(\mathit{ȷ},\tau \right) =0,\tilde{\mathrm{Y}}\left(\mathit{ȷ},\tau \right) ={\tilde{\mathrm{Y}}}_{\infty },\tilde{\mathrm{\Lambda }}\left(\mathit{ȷ},\tau \right) ={\tilde{\mathrm{\Lambda }}}_{\infty },as\mathit{ȷ}\to \infty .\hfill \end{array}$$

(4)

Dimensionless variables are given below

$$\begin{array}{cc}\hfill {\mathit{ȷ}}^{\ast }& =\frac{U_o\mathit{ȷ}}{{\nu }_{nf}},{\tau }^{\ast }=\frac{U_o^2\tau }{{\nu }_{nf}},W^{\ast }=\frac{\tilde{W}}{U_o},{\mathrm{Y}}^{\ast }=\frac{\tilde{\mathrm{Y}}-{\tilde{\mathrm{Y}}}_{\infty }}{{\tilde{\mathrm{Y}}}_w-{\tilde{\mathrm{Y}}}_{\infty }},{\mathrm{\Lambda }}^{\ast }=\frac{\tilde{\mathrm{\Lambda }}-{\tilde{\mathrm{\Lambda }}}_{\infty }}{{\tilde{\mathrm{\Lambda }}}_w-{\tilde{\mathrm{\Lambda }}}_{\infty }},Pr=\frac{\mu C_p}{k},\hfill \\ & Sc=\frac{{\nu }_{nf}}{D_m},Gr=\frac{g{\left(\nu {\beta }_{\tilde{\mathrm{Y}}}\right)}_{nf}\left({\tilde{\mathrm{Y}}}_w-{\tilde{\mathrm{Y}}}_{\infty }\right)}{U_o^3},Gm=\frac{g{\left(\nu {\beta }_{\tilde{\mathrm{\Lambda }}}\right)}_{nf}\left({\tilde{\mathrm{\Lambda }}}_w-{\tilde{\mathrm{\Lambda }}}_{\infty }\right)}{U_o^3},\hfill \\ & M_o^{\ast }=\frac{\sigma M_o{\nu }_{nf}}{U_o^2},P=\frac{{\left({\nu }_{nf}\right)}^2\wp }{P_1{U}_o^2},A_o^{\ast }=\frac{A_o{\nu }_{nf}}{\rho C_p{U}_o^2},\kappa =\frac{{\kappa }_r{\nu }_{nf}}{U_o^2},\aleph =\frac{{\nu }_o}{U_o},\hfill \end{array}$$

(5)

and dimensionless set of governing equations are (ignoring *):

$$\begin{array}{cc}\hfill {\partial }_{\tau }W\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}W\left(\mathit{ȷ},\tau \right)=& {\partial }_{\mathit{ȷ}\mathit{ȷ}}^{2}W\left(\mathit{ȷ},\tau \right)+{\hslash }_2{\partial }_{\tau \mathit{ȷ}\mathit{ȷ}}^{3}W\left(\mathit{ȷ},\tau \right)+Gr\mathrm{Y}\left(\mathit{ȷ},\tau \right)\hfill \\ & +Gm\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)-M_{o}W\left(\mathit{ȷ},\tau \right)-\frac{1}{P}W\left(\mathit{ȷ},\tau \right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\partial }_{\tau }\mathrm{Y}\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}\mathrm{Y}\left(\mathit{ȷ},\tau \right)=& \frac{1}{Pr}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\mathrm{Y}\left(\mathit{ȷ},\tau \right)+A_o\mathrm{Y}\left(\mathit{ȷ},\tau \right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\partial }_{\tau }\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)=& \frac{1}{Sc}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)+\kappa \mathrm{\Lambda }\left(\mathit{ȷ},\tau \right).\hfill \end{array}$$

(8)

Furthermore, dimensionless initial and boundary conditions are:

$$\begin{array}{cc}\hfill & W\left(\mathit{ȷ},0\right) =0,\mathrm{Y}\left(\mathit{ȷ},0\right) =0,\mathrm{\Lambda }\left(\mathit{ȷ},0\right) =0,\hfill \\ & W\left(0,\tau \right) =d\left(\tau \right),\mathrm{Y}\left(0,\tau \right) =1,\mathrm{\Lambda }\left(0,\tau \right) =1,\hfill \\ & W\left(\mathit{ȷ},\tau \right) =0,\mathrm{Y}\left(\mathit{ȷ},\tau \right) =0,\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right) =0,as\mathit{ȷ}\to \infty .\hfill \end{array}$$

(9)

Preliminaries

For solutions for fractional order models, we will use the definitions given below: The Caputo time derivative is defined as [39]:

$$\begin{array}{c}\hfill {}^C{D}_{\tau }^{\chi }m\left(\mathit{ȷ},\tau \right)=\frac{1}{\mathrm{\Xi }\left(i-\chi \right)}{\int }_b^{\tau }\left(\frac{g^{\left(i\right)}\left(\varrho \right)}{{\left(\tau -\varrho \right)}^{\chi +1-i}}\right)d\varrho ,\end{array}$$

(10)

where $\mathrm{\Xi }\left(·\right)$ is gamma function.

By applying the Laplace transform to Caputo (C) derivative, we get

$$\begin{array}{c}\hfill \mathcal{L}\left({}^C{D}_{\tau }^{\chi }m\left(\mathit{ȷ},\tau \right)\right)=u^{\chi }\mathcal{L}\left(m\left(\mathit{ȷ},\tau \right)\right)-u^{\chi -1}g\left(\mathit{ȷ},0\right).\end{array}$$

(11)

The Caputo–Fabrizio (CF) time derivative is defined as [40]:

$$\begin{array}{c}\hfill {}^{CF}{D}_{\tau }^{\chi }m\left(\mathit{ȷ},\tau \right)=\frac{G\left(\chi \right)}{1-\chi }{\int }_b^{\tau }exp\left(-\frac{\chi \left(\tau -\chi \right)}{1-\chi }\right)\frac{\partial g\left(\mathit{ȷ},\varrho \right)}{\partial \varrho }d\varrho ,\end{array}$$

(12)

where $G\left(\chi \right)$ is a normalization function/constant depending on b and here we consider $G\left(\chi \right)=1$.

Applying the Laplace transform to CF derivative, we get

$$\begin{array}{c}\hfill \mathcal{L}\left({}^{CF}{D}_{\tau }^{\chi }m\left(\mathit{ȷ},\tau \right)\right)=\frac{u\mathcal{L}\left(m\left(\mathit{ȷ},\tau \right)\right)-g\left(\mathit{ȷ},0\right)}{\left(1-\chi \right)u+\chi }.\end{array}$$

(13)

The Atangana–Baleanu (ABC) time derivative in Caputo sense is defined as [41,42]:

$$\begin{array}{c}\hfill {}^{ABC}{D}_{\tau }^{\chi }m\left(\mathit{ȷ},\tau \right)=\frac{AB\left(\chi \right)}{1-\chi }{\int }_b^{\tau }{E}_{\chi }\left(-\frac{\chi \left(\tau -\chi \right)}{1-\chi }\right)\frac{\partial g\left(\mathit{ȷ},\varrho \right)}{\partial \varrho }d\varrho ,\end{array}$$

(14)

where $AB\left(\chi \right)$ is a normalization function/constant depending on a and here we consider $AB\left(\chi \right)=1$.

The Laplace transform of ABC derivative is

$$\begin{array}{c}\hfill \mathcal{L}\left({}^{ABC}{D}_{\tau }^{\chi }m\left(\mathit{ȷ},\tau \right)\right)=\frac{u^{\chi }\mathcal{L}\left(m\left(\mathit{ȷ},\tau \right)\right)-u^{\chi -1}g\left(\mathit{ȷ},0\right)}{\left(1-\chi \right)u^{\chi }+\chi }.\end{array}$$

(15)

3. Solutions via Caputo

Following are the fractional solutions of heat, mass and flow with Caputo fractional derivative.

3.1. Temperature Profile

By using definition of Caputo derivative Equation (10), we have

$$\begin{array}{c}\hfill {}^C{D}_{\tau }\mathrm{Y}\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}\mathrm{Y}\left(\mathit{ȷ},\tau \right)=\frac{1}{Pr}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\mathrm{Y}\left(\mathit{ȷ},\tau \right)+A_o\mathrm{Y}\left(\mathit{ȷ},\tau \right).\end{array}$$

(16)

Taking Laplace of Equation (16) by using Equation (11)

$$\begin{array}{c}\hfill u^{\chi }{\overline{\mathrm{Y}}}_C\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{\mathrm{Y}}}_C\left(\mathit{ȷ},u\right)=\frac{1}{Pr}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{\mathrm{Y}}}_C\left(\mathit{ȷ},u\right)+A_o{\overline{\mathrm{Y}}}_C\left(\mathit{ȷ},u\right).\end{array}$$

(17)

Solution of Equation (17) is

$$\begin{array}{c}\hfill {\overline{\mathrm{Y}}}_C\left(\mathit{ȷ},u\right)=\frac{1}{u}{e}^{-\mathit{ȷ}\left[\frac{\aleph Pr}{2}+\sqrt{\frac{{\left(\aleph Pr\right)}^2+Pr\left(u^{\chi }-A_o\right)}{4}}\right]}.\end{array}$$

(18)

3.2. Concentration Profile

Concentration field with C time-fractional derivative is given by applying Equation (10), we have

$$\begin{array}{c}\hfill {}^C{D}_{\tau }\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)=\frac{1}{Sc}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)+\kappa \mathrm{\Lambda }\left(\mathit{ȷ},\tau \right).\end{array}$$

(19)

Taking Laplace of Equation (19) by using Equation (11), we get

$$\begin{array}{c}\hfill u^{\chi }{\overline{\mathrm{\Lambda }}}_C\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{\mathrm{\Lambda }}}_C\left(\mathit{ȷ},u\right)=\frac{1}{Sc}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{\mathrm{\Lambda }}}_C\left(\mathit{ȷ},u\right)+\kappa {\overline{\mathrm{\Lambda }}}_C\left(\mathit{ȷ},u\right).\end{array}$$

(20)

Solution of Equation (20) is

$$\begin{array}{c}\hfill {\overline{\mathrm{\Lambda }}}_C\left(\mathit{ȷ},u\right)=\frac{1}{u}{e}^{-\mathit{ȷ}\left[\frac{\aleph Sc}{2}+\sqrt{\frac{{\left(\aleph Sc\right)}^2+Sc\left(u^{\chi }-\kappa \right)}{4}}\right]}.\end{array}$$

(21)

3.3. Velocity Profile

Velocity field with C time-fractional derivative is given by applying Equation (10), we have

$$\begin{array}{c}\hfill {}^C{D}_{\tau }W\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}W\left(\mathit{ȷ},\tau \right)={\partial }_{\mathit{ȷ}\mathit{ȷ}}^{2}W\left(\mathit{ȷ},\tau \right)+{\hslash }_2{}^C{D}_{\tau }{\partial }_{\mathit{ȷ}\mathit{ȷ}}^{2}W\left(\mathit{ȷ},\tau \right)+Gr\mathrm{Y}\left(\mathit{ȷ},\tau \right)\\ \hfill +Gm\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)-M_{o}W\left(\mathit{ȷ},\tau \right)-\frac{1}{P}W\left(\mathit{ȷ},\tau \right).\end{array}$$

(22)

Taking Laplace of Equation (22) by using Equation (11), we get

$$\begin{array}{c}\hfill u^{\chi }{\overline{W}}_C\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{W}}_C\left(\mathit{ȷ},u\right)={\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{W}}_C\left(\mathit{ȷ},u\right)+{\hslash }_2{u}^{\chi }{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{W}}_C\left(\mathit{ȷ},u\right)+Gr{\overline{\mathrm{Y}}}_C\left(\mathit{ȷ},u\right)+Gm{\overline{\mathrm{\Lambda }}}_C\left(\mathit{ȷ},u\right)\\ \hfill -M_o{\overline{W}}_C\left(\mathit{ȷ},u\right)-\frac{1}{P}{\overline{W}}_C\left(\mathit{ȷ},u\right),\end{array}$$

(23)

Solution of Equation (23) is

$$\begin{array}{c}\hfill {\overline{W}}_C\left(\mathit{ȷ},u\right)=d\left(u^{\chi }\right)e^{-\mathit{ȷ}{F}_1}+Gr\left[\frac{e^{-\mathit{ȷ}{F}_1}-e^{-\mathit{ȷ}{A}_1}}{u\left[\left(1+{\hslash }_2{u}^{\chi }\right)A_1^2+\aleph A_1-\left(M_o+\frac{1}{P}+u^{\chi }\right)\right]}\right]\\ \hfill +Gm\left[\frac{e^{-\mathit{ȷ}{F}_1}-e^{-\mathit{ȷ}{B}_1}}{u\left[\left(1+{\hslash }_2{u}^{\chi }\right)B_1^2+\aleph B_1-\left(M_o+\frac{1}{P}+u^{\chi }\right)\right]}\right],\end{array}$$

(24)

where

$$\begin{array}{c}\hfill F_1\left(u^{\chi }\right)=\frac{1}{2}\left[\frac{\aleph }{1+{\hslash }_2{u}^{\chi }}+\sqrt{{\left(\frac{\aleph }{1+{\hslash }_2{u}^{\chi }}\right)}^2+\frac{4}{1+{\hslash }_2{u}^{\chi }}\left(M_o+\frac{1}{P}+u^{\chi }\right)}\right],\end{array}$$

(25)

$$\begin{array}{c}\hfill A_1\left(u^{\chi }\right)=\frac{1}{2}\left[\aleph Pr+\sqrt{{\left(\aleph Pr\right)}^2+4Pr\left(u^{\chi }-A_o\right)}\right],\end{array}$$

(26)

$$\begin{array}{c}\hfill B_1\left(u^{\chi }\right)=\frac{1}{2}\left[\aleph Sc+\sqrt{{\left(\aleph Sc\right)}^2+4Sc\left(u^{\chi }-\kappa \right)}\right].\end{array}$$

(27)

4. Solutions via Caputo–Fabrizio

The non-integer order solutions via singular kernel Caputo–Fabrizio fractional derivative for temperature, concentration and velocity are as follow:

4.1. Temperature Profile

Applying definition of Caputo–Fabrizio (12) to non-dimensional heat equation

$$\begin{array}{c}\hfill {}^{CF}{D}_{\tau }\mathrm{Y}\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}\mathrm{Y}\left(\mathit{ȷ},\tau \right)=\frac{1}{Pr}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\mathrm{Y}\left(\mathit{ȷ},\tau \right)+A_o\mathrm{Y}\left(\mathit{ȷ},\tau \right),\end{array}$$

(28)

and applying Laplace Equation (13) to above Equation (28)

$$\begin{array}{c}\hfill \frac{u}{\left(1-\chi \right)u+\chi }{\overline{\mathrm{Y}}}_{CF}\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{\mathrm{Y}}}_{CF}\left(\mathit{ȷ},u\right)=\frac{1}{Pr}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{\mathrm{Y}}}_{CF}\left(\mathit{ȷ},u\right)+A_o{\overline{\mathrm{Y}}}_{CF}\left(\mathit{ȷ},u\right),\end{array}$$

(29)

Solution of Equation (29) is

$$\begin{array}{c}\hfill {\overline{\mathrm{Y}}}_{CF}\left(\mathit{ȷ},u\right)=\frac{1}{u}{e}^{-\mathit{ȷ}\left[\frac{\aleph Pr}{2}+\sqrt{\frac{{\left(\aleph Pr\right)}^2+Pr\left(\frac{u}{\left(1-\chi \right)u+\chi }-A_o\right)}{4}}\right]}.\end{array}$$

(30)

4.2. Concentration Profile

Applying definition of Caputo–Fabrizio (12) to non-dimensional mass equation

$$\begin{array}{c}\hfill {}^{CF}{D}_{\tau }\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)=\frac{1}{Sc}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)+\kappa \mathrm{\Lambda }\left(\mathit{ȷ},\tau \right),\end{array}$$

(31)

and applying Laplace Equation (13) to above Equation (31)

$$\begin{array}{c}\hfill \frac{u}{\left(1-\chi \right)u+\chi }{\overline{\mathrm{\Lambda }}}_{CF}\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{\mathrm{\Lambda }}}_{CF}\left(\mathit{ȷ},u\right)=\frac{1}{Sc}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{\mathrm{\Lambda }}}_{CF}\left(\mathit{ȷ},u\right)+\kappa {\overline{\mathrm{\Lambda }}}_{CF}\left(\mathit{ȷ},u\right),\end{array}$$

(32)

The solution of Equation (32) is

$$\begin{array}{c}\hfill {\overline{\mathrm{\Lambda }}}_{CF}\left(\mathit{ȷ},u\right)=\frac{1}{u}{e}^{-\mathit{ȷ}\left[\frac{\aleph Sc}{2}+\sqrt{\frac{{\left(\aleph Sc\right)}^2+Sc\left(\frac{u}{\left(1-\chi \right)u+\chi }-\kappa \right)}{4}}\right]}.\end{array}$$

(33)

4.3. Velocity Profile

Applying definition of Caputo–Fabrizio (12) to non-dimensional flow equation

$$\begin{array}{c}\hfill {}^{CF}{D}_{\tau }W\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}W\left(\mathit{ȷ},\tau \right)={\partial }_{\mathit{ȷ}\mathit{ȷ}}^{2}W\left(\mathit{ȷ},\tau \right)+{\hslash }_2{}^{CF}{D}_{\tau }{\partial }_{\mathit{ȷ}\mathit{ȷ}}^{2}W\left(\mathit{ȷ},\tau \right)+Gr\mathrm{Y}\left(\mathit{ȷ},\tau \right)\\ \hfill +Gm\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)-M_{o}W\left(\mathit{ȷ},\tau \right)-\frac{1}{P}W\left(\mathit{ȷ},\tau \right).\end{array}$$

(34)

Taking Laplace of Equation (34), we get

$$\begin{array}{cc}\hfill \frac{u}{\left(1-\chi \right)u+\chi }{\overline{W}}_{CF}\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{W}}_{CF}\left(\mathit{ȷ},u\right) =& {\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{W}}_{CF}\left(\mathit{ȷ},u\right)+{\hslash }_2\frac{u}{\left(1-\chi \right)u+\chi }{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{W}}_{CF}\left(\mathit{ȷ},u\right)\hfill \\ & +Gr{\overline{\mathrm{Y}}}_{CF}\left(\mathit{ȷ},u\right)+Gm\overline{{\mathrm{\Lambda }}_{CF}}\left(\mathit{ȷ},u\right)-M_o\overline{W_{CF}}\left(\mathit{ȷ},u\right)\hfill \\ & -\frac{1}{P}{\overline{W}}_{CF}\left(\mathit{ȷ},u\right),\hfill \\ \hfill \end{array}$$

(35)

The solution of Equation (35) is

$$\begin{array}{cc}\hfill {\overline{W}}_{CF}\left(\mathit{ȷ},u\right)=& d\left(u^{\chi }\right)e^{-\mathit{ȷ}{F}_2}+Gr\left[\frac{e^{-\mathit{ȷ}{F}_2}-e^{-\mathit{ȷ}{A}_2}}{u\left[\frac{bu+\chi }{\left(1-\chi \right)u+\chi }{A}_2^2+\aleph A_2-\left(M_o+\frac{1}{P}+\frac{u}{\left(1-\chi \right)u+\chi }\right)\right]}\right]\hfill \\ & +Gm\left[\frac{e^{-\mathit{ȷ}{F}_2}-e^{-\mathit{ȷ}{B}_2}}{u\left[\frac{bu+\chi }{\left(1-\chi \right)u+\chi }{B}_2^2+\aleph B_2-\left(M_o+\frac{1}{P}+u^{\chi }\frac{u}{\left(1-\chi \right)u+\chi }\right)\right]}\right],\hfill \\ \hfill \end{array}$$

(36)

where

$$\begin{array}{cc}\hfill F_2\left(u^{\chi }\right)=& \frac{1}{2}\left[\aleph \frac{\left(1-\chi \right)u+\chi }{bu+\chi }+\sqrt{{\left(\aleph \frac{\left(1-\chi \right)u+\chi }{bu+\chi }\right)}^2+4\aleph \frac{\left(1-\chi \right)u+\chi }{bu+\chi }\left(M_o+\frac{1}{P}+\frac{u}{\left(1-\chi \right)u+\chi }\right)}\right],\hfill \end{array}$$

(37)

$$\begin{array}{c}\hfill A_2\left(u^{\chi }\right)=\frac{1}{2}\left[\aleph Pr+\sqrt{{\left(\aleph Pr\right)}^2+4Pr\left(\frac{u}{\left(1-\chi \right)u+\chi }-A_o\right)}\right],\end{array}$$

(38)

$$\begin{array}{c}\hfill B_2\left(u^{\chi }\right)=\frac{1}{2}\left[\aleph Sc+\sqrt{{\left(\aleph Sc\right)}^2+4Sc\left(\frac{u}{\left(1-\chi \right)u+\chi }-\kappa \right)}\right].\end{array}$$

(39)

5. Solutions via Atangana Baleanu

The non-integer order solutions via Atangana–Baleanu derivative of temperature, concentration and velocity are as follow:

5.1. Temperature Profile

Applying definition of Atangana–Baleanu (14) to non-dimensional heat equation

$$\begin{array}{c}\hfill {}^{ABC}{D}_{\tau }\mathrm{Y}\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}\mathrm{Y}\left(\mathit{ȷ},\tau \right)=\frac{1}{Pr}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\mathrm{Y}\left(\mathit{ȷ},\tau \right)+A_o\mathrm{Y}\left(\mathit{ȷ},\tau \right),\end{array}$$

(40)

and applying Laplace (15) to above Equation (40)

$$\begin{array}{c}\hfill \frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }{\overline{\mathrm{Y}}}_{ABC}\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{\mathrm{Y}}}_{ABC}\left(\mathit{ȷ},u\right)=\frac{1}{Pr}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{\mathrm{Y}}}_{ABC}\left(\mathit{ȷ},u\right)+A_o{\overline{\mathrm{Y}}}_{ABC}\left(\mathit{ȷ},u\right),\end{array}$$

(41)

Solution of Equation (41) is

$$\begin{array}{c}\hfill {\overline{\mathrm{Y}}}_{ABC}\left(\mathit{ȷ},u\right)=\frac{1}{u}{e}^{-\mathit{ȷ}\left[\frac{\aleph Pr}{2}+\sqrt{\frac{{\left(\aleph Pr\right)}^2+Pr\left(\frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }-A_o\right)}{4}}\right]}.\end{array}$$

(42)

5.2. Concentration Profile

Applying definition of Atangana–Baleanu (14) to non-dimensional mass equation

$$\begin{array}{c}\hfill {}^{ABC}{D}_{\tau }\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)=\frac{1}{Sc}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)+\kappa \mathrm{\Lambda }\left(\mathit{ȷ},\tau \right),\end{array}$$

(43)

and applying Laplace (15) to above Equation (43)

$$\begin{array}{c}\hfill \frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }{\overline{\mathrm{\Lambda }}}_{ABC}\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{\mathrm{\Lambda }}}_{ABC}\left(\mathit{ȷ},u\right)=\frac{1}{Sc}{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{\mathrm{\Lambda }}}_{ABC}\left(\mathit{ȷ},u\right)+\kappa {\overline{\mathrm{\Lambda }}}_{ABC}\left(\mathit{ȷ},u\right),\end{array}$$

(44)

The solution of Equation (44) is

$$\begin{array}{c}\hfill {\overline{\mathrm{\Lambda }}}_{ABC}\left(\mathit{ȷ},u\right)=\frac{1}{u}{e}^{-\mathit{ȷ}\left[\frac{\aleph Sc}{2}+\sqrt{\frac{{\left(\aleph Sc\right)}^2+Sc\left(\frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }-\kappa \right)}{4}}\right]}.\end{array}$$

(45)

5.3. Velocity Profile

Applying definition of Atangana–Baleanu (14) to non-dimensional flow equation

$$\begin{array}{cc}\hfill {}^{ABC}{D}_{\tau }W\left(\mathit{ȷ},\tau \right)-\aleph {\partial }_{\mathit{ȷ}}W\left(\mathit{ȷ},\tau \right)=& {\partial }_{\mathit{ȷ}\mathit{ȷ}}^{2}W\left(\mathit{ȷ},\tau \right)+{\hslash }_2{}^{ABC}{D}_{\tau }{\partial }_{\mathit{ȷ}\mathit{ȷ}}^{2}W\left(\mathit{ȷ},\tau \right)+Gr\mathrm{Y}\left(\mathit{ȷ},\tau \right)\hfill \\ & +Gm\mathrm{\Lambda }\left(\mathit{ȷ},\tau \right)-M_{o}W\left(\mathit{ȷ},\tau \right)-\frac{1}{P}W\left(\mathit{ȷ},\tau \right).\hfill \\ \hfill \end{array}$$

(46)

Taking Laplace of Equation (46), we get

$$\begin{array}{cc}\hfill \frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }{\overline{W}}_{ABC}\left(\mathit{ȷ},u\right)-\aleph {\partial }_{\mathit{ȷ}}{\overline{W}}_{ABC}\left(\mathit{ȷ},u\right)=& {\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{W}}_{ABC}\left(\mathit{ȷ},u\right)+{\hslash }_2\frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }{\partial }_{\mathit{ȷ}\mathit{ȷ}}^2{\overline{W}}_{ABC}\left(\mathit{ȷ},u\right)\hfill \\ & +Gr{\overline{\mathrm{Y}}}_{ABC}\left(\mathit{ȷ},u\right)+Gm{\overline{\mathrm{\Lambda }}}_{ABC}\left(\mathit{ȷ},u\right)-M_o{\overline{W}}_{ABC}\left(\mathit{ȷ},u\right)\hfill \\ & -\frac{1}{P}{\overline{W}}_{ABC}\left(\mathit{ȷ},u\right),\hfill \\ \hfill \end{array}$$

(47)

Solution of Equation (47) is

$$\begin{array}{cc}\hfill {\overline{W}}_{ABC}\left(\mathit{ȷ},u\right)=& d\left(u^{\chi }\right)e^{-\mathit{ȷ}{F}_3}+Gr\left[\frac{e^{-\mathit{ȷ}{F}_3}-e^{-\mathit{ȷ}{A}_3}}{u\left[\frac{b{u}^{\chi }+\chi }{\left(1-\chi \right)u^{\chi }+\chi }{A}_3^2+\aleph A_3-\left(M_o+\frac{1}{P}+\frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }\right)\right]}\right]\hfill \\ & +Gm\left[\frac{e^{-\mathit{ȷ}{F}_3}-e^{-\mathit{ȷ}{B}_3}}{u\left[\frac{b{u}^{\chi }+\chi }{\left(1-\chi \right)u^{\chi }+\chi }{B}_3^2+\aleph B_3-\left(M_o+\frac{1}{P}+\frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }\right)\right]}\right],\hfill \\ \hfill \end{array}$$

(48)

where

$$\begin{array}{cc}\hfill F_3\left(u^{\chi }\right)=& \frac{1}{2}\left[\aleph \frac{\left(1-\chi \right)u^{\chi }+\chi }{b{u}^{\chi }+\chi }+\sqrt{{\left(\aleph \frac{\left(1-\chi \right)u^{\chi }+\chi }{b{u}^{\chi }+\chi }\right)}^2+4\aleph \frac{\left(1-\chi \right)u^{\chi }+\chi }{b{u}^{\chi }+\chi }\left(M_o+\frac{1}{P}+\frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }\right)}\right]\hfill \\ \hfill \end{array}$$

(49)

$$\begin{array}{c}\hfill A_3\left(u^{\chi }\right)=\frac{1}{2}\left[\aleph Pr+\sqrt{{\left(\aleph Pr\right)}^2+4Pr\left(\frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }-A_o\right)}\right],\end{array}$$

(50)

$$\begin{array}{c}\hfill B_3\left(u^{\chi }\right)=\frac{1}{2}\left[\aleph Sc+\sqrt{{\left(\aleph Sc\right)}^2+4Sc\left(\frac{u^{\chi }}{\left(1-\chi \right)u^{\chi }+\chi }-\kappa \right)}\right].\end{array}$$

(51)

The results obtained for velocity, temperature and concentration after solving the fractionalized models are complex and generalized in nature. As the models were fractionalized with power Law, exponential and non-local kernel to see the significance of memory effects. These are generalized results, and as the fractional parameter goes to 1, we recovered the result calculated for integer order case. The detail is as by taking $\chi \to 1$ in Equations (18), (30) and (42), we will get similar results as for integer order obtained by Ahmad et al. [38] [Equation (12)]. As $\chi$ approaches 1 in Equations (21), (33) and (45), identical results exist in Ahmad et al. [38] [Equation (17)]. Furthermore, as $\chi \to 1$ in Equations (24), (36) and (48), results for fractional order reduce to integer order presented by Ahmad et al. [38] [Equation (22)]. This shows worth of our results. As we mentioned early the roots are complex in nature in order to find their inverses we use inversion algorithm namely Stehfest’s formula [43], one of the simplest algorithm we use to sort out the inverse Laplace transform.

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

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

6. Results and Discussion

MHD CNT’s flow of nanofluid has been studied, in this paper for three particular time derivative (C, CF and ABC). Fractional models of flow, heat and mass are introduced. Graphs has been drawn to ellaborate the effects of $\chi$, $Gr$, $Gm$, ℵ, $Pr$, $Sc$, P, $M_o$, ${\hslash }_2$, $A_o$ and $\kappa$ for fractional models.

6.1. Effect of $\chi$

From Figure 2, we can observe that velocity of nanofluid decreases as fractional parameter increases. The fractional models C, CF, and ABC reduce to integer order model as $\chi$ tends to 1. It is worth mentioning that fractional order models are best to explain the history (memory) of the fluids. As there is non-local and non-singular kernel present in Atangana–Baleanu time derivative, velocity is highest for ABC.

6.2. Effect of $Gm$

From Figure 3, velocity of nano-fluid in CNT escalates as $Gm$ increases. Apparently, augmenting buoyancy forces leads to decrease viscous forces and this causes reduction in frictional forces. Due to this physical phenomena, velocity increase. It is lowest for Caputo.

6.3. Effect of $Gr$

From Figure 4, one can see that as $Gr$ increases, the flow of nano-fluid increases. Density of fluid decreases when buoyancy forces increases, so there is no or very less viscous forces which yields the increase in velocity profile. Among all three fractional derivatives, ABC shows maximum velocity.

6.4. Effect of P

Figure 5 shows that as permittivity of medium increases, flow profile decreases. Apparently, P is directly proportional to porosity. Hence, when medium becomes more porous, flow will reduce its speed. Among all three fractional derivatives, ABC fractional model shows highest velocity as permittivity increases simultaneously as time varies.

6.5. Effect of ${\hslash }_2$

From Figure 6, it is obvious that CNT flow of a nano-fluid decreases with increase in ${\hslash }_2$. Moreover, unlike other fractional parameters, it shows a strange behavior, i.e., at certain point on ȷ-axis, ${\hslash }_2$ starts increasing and converges as ȷ approaches infinity. It is minimum, moderate and maximum for C, CF and ABC, respectively.

6.6. Effect of $\kappa$

The fluid flow increases the way $\kappa$ escalates as shown in Figure 7. It is greatest for ABC fractional MHD model. The resultant velocity for ABC model is good as compared to C and CF.

6.7. Effect of $M_o$

Figure 8 illustrates that as magnetic field amplifies, fluid flow de-escalates. Lorentz force is a frictional force caused by magnetic field. When magnetic field maximizes, there is an increase in Lorentz force at outermost layer of fluid. this leads into decrease in velocity of nano-fluid. ABC fractional model bears highest flow rate as compared to Caputo and Caputo–Fabrizio fractional MHD models.

6.8. Effect of $Pr$

The flow profile declines, providing the values of $Pr$ rise as displayed in Figure 9. $Pr$ is directly and inversely proportional to momentum and thermal diffusivity, respectively. Fluid’s viscosity increases, granting a decrease in thermal diffusivity. In response, the flow field decreases and it is minimal for Caputo.

6.9. Effect of $A_o$

Figure 10 elaborates the effect of $A_o$ on velocity. Flow profile increases if there is an increment in heat generation. Velocity is maximal, average and minimal for ABC, CF and C, respectively.

6.10. Effect of ℵ

In Figure 11, velocity field lessens when suction scales up. It is utmost for ABC and lowest for C. While making comparison, velocity for the Atangana–Baleanu model is greatest because it has a non-local kernel. Velocity for CF is greater than C. This is because CF has a non-singular kernel that imitates C with the singular kernel.

6.11. Effect of $Sc$

As the values of $Sc$ increases, it increases kinematic viscosity, but decreases mass diffusivity. This successively descends flow profile of nano-fluid as presented in Figure 12. Velocity field for ABC, CF and C is supreme, adequate and least, respectively.

7. Conclusions

This article inquires time-dependent, MHD CNTs flow of nanofluid. The C, CF and ABC operators are used to construct equation for heat, mass and velocity. Solutions of model equations are presented by Laplace transform. Several graphs are given to illustrate impact of incipient parameters for solutions. Significant findings of this study are remarked as follows:

• Fluid flow descends for $\chi$, P, ${\hslash }_2$, $Pr$, ℵ and $Sc$.
• Presence of resistive forces due to $M_o$, fluid flow decelerates.
• Increment in flow field has been generated when $Gr$, $Gm$, $\kappa$ and $A_o$.
• For $\aleph =0$, velocity of nano-fluid escalates. Whereas for $\aleph >0$, flow’s velocity de-escalates.
• Velocity of flow increases with increment in volumetric fraction for Caputo, Caputo–Fabrizio and Atangana–Baleanu models.
• Velocity profile is maximum when $d\left(\tau \right)=e^{\tau }$ and minimum when $d\left(\tau \right)=1$.
• Among C, CF and ABC, flow profile is maximum for ABC.

Recently, there are some new techniques have been introduced in order to study fractional order models like modified generalized Taylor fractional series method (MGTFSM), fractional natural decomposition method (FNDM) and fractional Shehu transform, etc. In future, researchers can compare their results by the results we obtained in our research work using Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional approach.