A comparative analysis of fractional model of second grade fluid subject to exponential heating: application of novel hybrid fractional derivative operator

Abstract

In this article, a new approach to study the fractionalized second grade fluid flow is described by the different fractional derivative operators near an exponentially accelerated vertical plate together with exponentially variable velocity, energy and mass diffusion through a porous media is critically examined. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimentional form. For the sake of better rheology of second grade fluid, developed a fractional model by applying the new definition of Constant Proportional-Caputo hybrid derivative (CPC), Atangana Baleanu in Caputo sense (ABC) and Caputo Fabrizio (CF) fractional derivative operators that describe the generalized memory effects. For seeking exact solutions in terms of Mittag-Leffler and G-functions for velocity, temperature and concentration equations, Laplace integral transformation technique is applied. For physical significance of various system parameters on fluid velocity, concentration and temperature distributions are demonstrated through various graphs by using graphical software. Furthermore, for being validated the acquired solutions, accomplished a comparative analysis with some published work. It is also analyzed that for exponential heating and non-uniform velocity conditions, the CPC fractional operator is the finest fractional model to describe the memory effect of velocity, energy and concentration profile. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, in the literature, it is observed that to derived analytical results from fractional fluid models developed by the various fractional operators, is difficult and this article contributing to answer the open problem of obtaining analytical solutions the fractionalized fluid models.

Keywords:

• Exponential heating
• fractional operator
• graphical representations
• integral Laplace transform
• second grade fluid
• special functions
• system parameters

1. Introduction

The fluid is a certain type of matter which continuously deformed when negligible amount of force is applied externally. Fluid has no specific shape, it partitioned mainly into two categories such as non-Newtonian and Newtonian fluids. The Newtonian and non-Newtonian fluids have different geometries and characteristics, but non-Newtonian fluids have more attraction for scientists and researchers as compared to Newtonian fluids. In engineering and sciences, the non-Newtonian fluids has variety of applications in the modern era and it plays a vital role in industrial sectors such as biological materials, magneto hydrodynamic flows, greases, polymer melts, clay coatings, extrusion of molten plastic, blood flow, pharmaceutical, emulsions, polymer processing, food processing industries, crude oil and gas well drilling and complex mixtures. Researchers investigated different non-Newtonian fluid models regarding physical and computational characteristics such as second grade model, viscoplastic model, power law model, Bingham plastic model, Jeffery model, Oldroyd-B fluid model, Brinkman type model, Casson model, Walters-B fluid model and Maxwell model (Kahshan et al., 2019; Khan et al., 2019; Mohebbi et al., 2019; Riaz et al., 2021a; 2021b), that different fluid models exists in the literature have various characteristics or certain limitations, for instance, the power-law model described the features of viscosity but failed to explain the impacts of elasticity, which motivate/attract the researchers and mathematicians towards the study of such complex fluids. Multiple products such as honey, soup, jelly, china clay, tomato sauce, artificial fibers, synthetic lubricants, concentrated fruit juices, pharmaceutical chemicals, paints and coal, etc. are some applied illustrations of such fluid. Systematic analysis of such fluid flow models have significantly important for theoretical studies and practical implementations in modernistic mechanization. Among such fluids, second grade fluid attracted special attention, which is the commonest non-Newtonian fluid due to its more extensive applications and substantial role in different fields serving as mechanical as well as chemical applications, bio engineering operations, metallurgy and especially in food processing industries. Lubricants that are used to lubricate the components of engine like gears, bearings, etc., are considered as differential type fluids. The study of second grade movement in the context of fluid mechanics, was explored by several mathematicians, scientists, researchers and engineers that depends upon various situations. Rajagopal and Gupta (1984) and Rajagopal (1993) examined the influence of various applications of differential type fluid, for instance, in theological problems, in biological sciences, chemical, petroleum and geophysics field.

Some Researchers and scientists are focused to investigate the flow regime of second grade fluid geometrically for configurations of many interesting features, because flow analysis of differential type fluids have wide practical applications and theoretically studies, having prominent effects in many industrial fields, for example, Erdogan (2003), Labropulu (2000), Fetecau et al. (2011), Tawari and Ravi (2009) and Islam et al. (2011) studied the unsteady second grade fluid, employed the method of separation of variables, to compute analytical solution. Rehman et al. (2021) elaborated the MHD second grade fluid flow to analyzed the effects of radiative thermal flux and computed the analytical solutions by employing Laplace integral transformation. Some significant studies regarding second grade fluid having interesting facts are described by Rashidi et al. (2014), Baranovskii (2021), Arianna and Gudrun (2005), Dinarvand et al. (2010) and Fetecau et al. (2011).

The fractional calculus, which is engaged in differential and integral operators for non integer orders, is as old branch of mathematics like conceptional calculus but currently it has been growing immensely on account of enormous significance in engineering and science. Since various daily life, real phenomena of physical problems can not be modelled by using the traditional calculus operators due to which researchers interested to searched the generalized operators that help to anticipate the preceding processes state. The fractional calculus having various fractional operators used to fractionalized the differential equations, with excellent applicable tools that are massively applied to modelled the real phenomena that appear in fluid flow problems, chemistry, dynamical processes, physics, oscillation, electricity, diffusion, mechanics, relaxation, reaction, engineering processes and many other disciplines. The main reason for exploring the numerical or exact solutions due to its significance in various daily life. To gain the numerical or exact solutions, researchers and mathematicians have been implemented numerous techniques. For instance, unified method (Osman et al., 2018), multi step approach (Al-Smadi et al., 2015; Momani et al., 2014), Riccati-Bernouli sub-ordinary diffrential equation Sub-ODE techniq (RBSODET) (Alabedalhadi et al., 2020), reproducing the kernel Hilbert space method (Al-Smadi et al., 2021; Altawallbeh et al., 2018), simple equation modification method (Islam & Akbar, 2018), residual power series method (Al-Smadi et al., 2020). Zuo (2021) suggested fractal rheological model and verified experimentally. Koo et al. (2022) to study a non-smooth boundary layer of a viscous fluid, and a fractal-fractional modification of the Blasius equation is suggested and solved analytically. He et al. (2023) investigated unsteady compressible magneto-radiative gas flow near a heated vertical wavy wall through porous medium in the presence of inclined magnetic field. Radiation effects on the flow of carbon nanotube suspended nanofluids in the presence of a magnetic field past a stretched sheet impacted by slip state studied by He and Abd Elazem (2022). Li (2023) discussed two mathematical models to describe the nanofluid flow, one is an approximate continuum model and the other is to use the conservation laws in a fractal space. Wang and He (2019) established successfully a variational principle in a fractal space by the semi-inverse method. Due to the advancement in the field of fractional calculus, scientists have suggested a couple of new techniques to interpret and established the real world problem solutions using theory of fractional calculus. To interpret and model phenomenon in different fields of sciences such as electric circuit models, fractal rheological models and fractal growth of populations models, several fractional operators have singular kernels but a lot of having non-singular kernels have been acquired, which is an important tool to analyze the rheological behavior of the physical models in fractional calculus. In literature, many researchers surprisingly work a lot in this shining field of mathematics to analyzed the fractional fluid models and derived various interesting results that are very helpful for engineers and scientists to compare their experimental results get from the govern partial differential equations with the analytical results obtained using different mathematical techniques and tools from fractional form of the non-Newtonian fluid models. Marchaud Caputo and Riemann-Liouville developed fractional integrals and described a new concept of fractional derivatives operators, that are based on singular kernels, but these fractional models have some drawbacks due to the singular kernels such as faced many difficulties during modeling process. To overcome this hurdle that occurred singularized fractional models, a new set of fractional operators have been presented that are based on non-singular kernels, such as Prabhakar fractional derivative, Caputo-Fabrizio, Yang Abdel Cattani fractional, Atangana-Baleanu fractional operators and few others for reference (Atangana & Baleanu, 2016; Rehman et al., 2021a; 2022c; Riaz et al., 2021a; 2021d; 2021e). These fractional operators having different type of non-singularized kernels, some of the kernels are mentioned here such as, Rabotnov exponential function, Exponential kernels and Mittag-Leffler functions. Yavuz et al. (2022) investigated the exact solution and a qualitative study for the fractional second-grade fluid described by a Caputo fractional operator. Rehman et al. (2022a) has been discussed heat source impact on unsteady magneto-hydro-dynamic (MHD) flows of Prabhakar-like non integer second grade fluid near an exponentially accelerated vertical plate with exponentially variable velocity, temperature and mass diffusion through a porous medium. The fractal two-scale transform method used to approximate the analytical solutions are obtained by the energy balance method by Wu et al. (2022) Comprehensive analysis of heat and mass transfer of MHD natural convection flow of water-based nano-particles in the presence of ramped conditions via Caputo–Fabrizio fractional time derivative are investigated by Rehman et al. (2022b). The fractional complex transform and the chain rule for fractional calculus are elucidated geometrically by He et al. (2012).

In the previous investigation, Haq et al. (2021) and Song et al. (2021) discussed the flow of fractional version of differential type fluid model by using different fractional operators namely CF and ABC respectively, and computed solution for each fractional model, but both respective studies presented work without analyzed the effect of mass diffusion. But in the literature fractional second grade fluid model with fractional operators CPC, CF and ABC, along with the set of non-uniform boundary conditions for velocity with exponential heating, saturated in porous media, are not investigated yet nor published. To fill this gape a new fractional second grade fluid model developed by applying the definition of recently introduced a new fractional derivative operator, namely CPC, CF and ABC operators, under effectively applied conditions for concentration, velocity field and temperature distribution. For seeking exact solution expressions in terms of G-functions by employing Laplace integral transformation method to solve the fractional models developed for velocity, concentration and temperature distribution. For physical analysis the influence of parameters like as second grade parameter ${\alpha }_2,$ dimensionless time t, fractional parameter $\alpha ,$ mass Grashof number Gm, Magnetic number M, Prandtl number Pr, Schmidt number Sc, thermal Grashof number Gr are portrayed graphically by using Mathcad software. Furthermore, for validation the current result, also accomplish the comparative analysis with different published work.

2. Mathematical model

Consider the MHD second grade fluid flow near an oscillating infinite vertical plate that is nested in a porous material. The plate is considered at $y=0$ and the fluid flow is restrained to $y>0,$ in the direction that is along to the plate (as exhibited in Figure 1). Initially, for time $t=0,$ the fluid and plate both are in the static mode, having ambient temperature $T_{\infty }$ and concentration $C_{\infty }.$ Later, when time $t=0^{+},$ the plate begins to oscillate and fluid starts to move with velocity $u_0\hspace{0.17em}\exp \hspace{0.17em}(\zeta t)$ where $u_0$ represents characteristic velocity, and the wall temperature is $T_w$ and concentration $C_w.$ It is presumed that temperature, velocity and concentrations are functions of y and t only. The Boussinesq equation is a mathematical model used to describe the behaviours of fluid systems. It is a partial differential equation that relates fluid flow velocity and pressure. The equation was first introduced by Joseph Valentin Boussinesq in 1872 and has been widely used in many engineering and scientific fields, including hydrology, oceanography, and geology (Wu et al., 2022). The following principal equations for second grade fluid under Boussinesq’s approximation, for velocity,concentration and energy transfer are obtained as (Ali et al., 2014; Shah & Khan, 2016): (1) $$\begin{array}{cc}\frac{\partial u(y,t)}{\partial t}=\upsilon \left(1+\frac{{\alpha }_1}{\mu }\frac{\partial }{\partial t}\right)\frac{{\partial }^{2}u(y,t)}{\partial y^2}+g{\beta }_T\left(T(y,t)-T_{\infty }\right)+g{\beta }_C\left(C(y,t)-C_{\infty }\right)\hfill & \hfill \\ -\left[\frac{{\sigma }_0{M}_0^2}{\rho }+\frac{\upsilon \varphi }{k_0}\left(1+\frac{{\alpha }_1}{\mu }\frac{\partial }{\partial t}\right)\right]u(y,t),\hfill & \hfill \end{array}$$(1) (2) $$\frac{\partial T(y,t)}{\partial t}=\frac{k}{\rho C_p}\frac{{\partial }^{2}T(y,t)}{\partial y^2},$$(2) (3) $$\frac{\partial C(y,t)}{\partial t}={\delta }_m\frac{{\partial }^{2}C(y,t)}{\partial y^2}.$$(3) where $u(y,t),T(y,t),C(y,t),\rho ,{\beta }_T,{\alpha }_1,k,$ $\upsilon ,$ $C_p,$ ${\beta }_C$ and g represents the fluid velocity, temperature, concentration, density, coefficient of volumetric thermal expansion, second grade parameter, thermal conductivity, kinematic viscosity, specific heat capacity, coefficient of volumetric expansion for concentration and gravitational acceleration respectively.with initial (ICs) and boundary (BCs) conditions are given below: (4) $$u(y,0)=0,\hspace{1em}T(y,0)=T_{\infty },\hspace{1em}C(y,0)=C_{\infty },\hspace{1em}\frac{\partial u(y,t)}{\partial t}=0,\hspace{1em}y\ge 0,$$(4) (5) $$u(0,t)=u_0\hspace{0.17em}\exp \hspace{0.17em}(\zeta t),T(0,t)=T_{\infty }+T_w(1-a_0{e}^{-b_{0}t}),C(0,t)=C_{\infty }+(C_w-C_{\infty })\exp (\zeta t)\hspace{1em}t>0,$$(5) (6) $$u(y,t)\to 0,\hspace{1em}T(y,t)\to T_{\infty }, C(y,t)\to C_{\infty } as y\to \infty .$$(6)

Figure 1. Physical geometry of the second grade fluid model.

Writing the proposed problem in terms of dimensionless form, the following dimensionless quantities are considered: (7) $$\begin{array}{ll}{u}^{*}=\frac{u}{u_0},\hspace{1em}{y}^{*}=\frac{u_0}{\upsilon }y,\hspace{1em}{t}^{*}=\frac{u_0^2}{\upsilon }t,\hspace{1em}{T}^{\ast }=\frac{T-T_{\infty }}{T_w-T_{\infty }},\hspace{1em}{G}_r=\frac{g\upsilon {\beta }_{\stackrel{\prime }{T}}(T-T_{\infty })}{u_0^3},G_m=\frac{g\upsilon {\beta }_C(C-C_{\infty })}{u_0^3},\hspace{1em}M=\frac{{\sigma }_0{M}_0^2\upsilon }{\rho u_0^2},\hspace{1em}Pr=\frac{\mu C_p}{k},\hspace{1em}Sc=\frac{\upsilon }{{\delta }_m},\hspace{1em}{\zeta }^{\ast }=\frac{\zeta \upsilon }{u_0^2},\hfill & \hfill \end{array}$$(7) $$\begin{array}{l}{C}^{\ast }=\frac{C-C_{\infty }}{C_w-C_{\infty }},\hspace{1em}\frac{1}{K}=\frac{{\upsilon }^2\varphi }{k_0{u}_0^2},\hspace{1em}{\alpha }_2=\frac{{\alpha }_1\rho u_0^2}{{\mu }^2},\hspace{1em}a=M+\frac{1}{K},b=\frac{{\alpha }_2}{K}.\hfill \end{array}$$ when substituting the Eq. (7)(7) $$\begin{array}{ll}{u}^{*}=\frac{u}{u_0},\hspace{1em}{y}^{*}=\frac{u_0}{\upsilon }y,\hspace{1em}{t}^{*}=\frac{u_0^2}{\upsilon }t,\hspace{1em}{T}^{\ast }=\frac{T-T_{\infty }}{T_w-T_{\infty }},\hspace{1em}{G}_r=\frac{g\upsilon {\beta }_{\stackrel{\prime }{T}}(T-T_{\infty })}{u_0^3},G_m=\frac{g\upsilon {\beta }_C(C-C_{\infty })}{u_0^3},\hspace{1em}M=\frac{{\sigma }_0{M}_0^2\upsilon }{\rho u_0^2},\hspace{1em}Pr=\frac{\mu C_p}{k},\hspace{1em}Sc=\frac{\upsilon }{{\delta }_m},\hspace{1em}{\zeta }^{\ast }=\frac{\zeta \upsilon }{u_0^2},\hfill & \hfill \end{array}$$(7) into Eqs. (1–3), and dropping the asterisk $\ast$ from newly obtained equations, then we have the dimensionless governing system of PDEs of the considered model as follows: (8) $$\frac{\partial u(y,t)}{\partial t}=\frac{{\partial }^{2}u(y,t)}{\partial y^2}+G_{r}T(y,t)+G_{m}C(y,t)-au(y,t)-b\frac{\partial u(y,t)}{\partial t}+{\alpha }_2\frac{{\partial }^{3}u(y,t)}{\partial t\partial y^2},$$(8) (9) $$\frac{\partial T(y,t)}{\partial t}=\frac{1}{Pr}\frac{{\partial }^{2}T(y,t)}{\partial y^2},$$(9) (10) $$\frac{\partial C(y,t)}{\partial t}=\frac{1}{Sc}\frac{{\partial }^{2}C(y,t)}{\partial y^2}.$$(10) with (11) $$u(y,0)=0,\hspace{1em}T(y,0)=0,\hspace{1em}C(y,0)=0,$$(11) (12) $$u(0,t)=e^{\zeta t},\hspace{1em}T(0,t)=1-a_0{e}^{-b_{0}t},\hspace{1em}C(0,t)=e^{\zeta t},\hspace{1em}t>0,$$(12) (13) $$u(y,t)\to 0,\hspace{1em}T(y,t)\to 0, C(y,t)\to 0 as y\to \infty .$$(13)

The fractional model for momentum, energy and concentration distribution are formulated by using Constant proportional-Caputo hybrid fractional derivative operator is described as: (14) $$\begin{array}{c}{}^{\mathit{\text{CPC}}}{D}_t^{\alpha }u(y,t)=\left(1+{\alpha }_2{}^{\mathit{\text{CPC}}}{D}_t^{\alpha }\right)\frac{{\partial }^{2}u(y,t)}{\partial y^2}+G_{r}T(y,t)+G_{m}C(y,t)-(a+b^{\mathit{\text{CPC}}}{D}_t^{\alpha })u(y,t),\hfill \end{array}$$(14) (15) $${}^{\mathit{\text{CPC}}}{D}_t^{\alpha }T(y,t)=\frac{1}{Pr}\frac{{\partial }^{2}T(y,t)}{\partial y^2},$$(15) (16) $${}^{\mathit{\text{CPC}}}{D}_t^{\alpha }C(y,t)=\frac{1}{Sc}\frac{{\partial }^{2}C(y,t)}{\partial y^2}.$$(16)

The fractional model for momentum, energy and concentration distribution are formulated by using Caputo-Fabrizio fractional derivative operator is described as: (17) $$\begin{array}{c}{}^{CF}{D}_t^{\alpha }u(y,t)=\left(1+{\alpha }_2{}^{CF}{D}_t^{\alpha }\right)\frac{{\partial }^{2}u(y,t)}{\partial y^2}+G_{r}T(y,t)+G_{m}C(y,t)-(a+b^{CF}{D}_t^{\alpha })u(y,t),\hfill \end{array}$$(17) (18) $${}^{CF}{D}_t^{\alpha }T(y,t)=\frac{1}{Pr}\frac{{\partial }^{2}T(y,t)}{\partial y^2},$$(18) (19) $${}^{CF}{D}_t^{\alpha }C(y,t)=\frac{1}{Sc}\frac{{\partial }^{2}C(y,t)}{\partial y^2}.$$(19)

Also, the fractional model for momentum, energy and concentration distribution are formulated by using Atangana-Baleanu time fractional operator is described as: (20) $$\begin{array}{c}{}^{\mathit{\text{ABC}}}{D}_t^{\alpha }u(y,t)=\left(1+{\alpha }_2{}^{\mathit{\text{ABC}}}{D}_t^{\alpha }\right)\frac{{\partial }^{2}u(y,t)}{\partial y^2}+G_{r}T(y,t)+G_{m}C(y,t)-(a+b^{\mathit{\text{ABC}}}{D}_t^{\alpha })u(y,t),\hfill \end{array}$$(20) (21) $${}^{\mathit{\text{ABC}}}{D}_t^{\alpha }T(y,t)=\frac{1}{Pr}\frac{{\partial }^{2}T(y,t)}{\partial y^2},$$(21) (22) $${}^{\mathit{\text{ABC}}}{D}_t^{\alpha }C(y,t)=\frac{1}{Sc}\frac{{\partial }^{2}C(y,t)}{\partial y^2}.$$(22) where, ${}^{\mathit{\text{CPC}}}{D}_t^{\alpha }(.,.)$ represents Constant proportional-Caputo hybrid fractional operator and its defined as: $${}^{\mathit{\text{CPC}}}{D}_t^{\alpha }f\left(y,t\right)=\frac{1}{\Gamma (1-\alpha )}{\int }_0^t(k_1(\alpha )f(y,\tau )+k_0(\alpha )\frac{\partial f(y,\tau )}{\partial \tau }){(t-\tau )}^{-\alpha }d\tau ,\hspace{1em}0<\alpha <1.$$

Laplace transformation of Constant proportional-Caputo hybrid time fractional operator is written as: $$\mathcal{L}\left({}^{\mathit{\text{CPC}}}{D}_t^{\alpha }f\left(y,t\right)\right)=\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\mathcal{L}\left(f(y,t)\right)-k_0(\alpha )s^{\alpha -1}f(y,0).$$ where $\alpha$ represents a fractional parameter.

Now, ${}^{CF}{D}_t^{\alpha }(.,.)$ represents Caputo-Fabrizio fractional operator and its defined as: $${}^{CF}{D}_t^{\alpha }f\left(y,t\right)=\frac{1}{1-\alpha }{\int }_0^{t}exp\left(-\frac{\alpha (t-\alpha )}{1-\alpha }\right)\frac{\partial f(y,\tau )}{\partial \tau }d\tau ,\hspace{1em}0<\alpha <1.$$

Laplace transformation of Caputo-Fabrizio time fractional operator is written as: $$\mathcal{L}\left({}^{CF}{D}_t^{\alpha }f\left(y,t\right)\right)=\frac{s\mathcal{L}\left(f(y,t)\right)-f(y,0)}{(1-\alpha )s+\alpha }.$$ where $\alpha$ represents a fractional parameter.

Also, ${}^{\mathit{\text{ABC}}}{D}_t^{\alpha }(.,.)$ represents Atangana-Baleanu time fractional operator in Caputo sense (ABC) having non-singularized and non-local kernel defined in the following way: $${}^{\mathit{\text{ABC}}}{D}_t^{\alpha }f\left(y,t\right)=\frac{1}{1-\alpha }{\int }_0^t{E}_{\alpha }\left(-\frac{\alpha (t-\alpha )}{1-\alpha }\right)\frac{\partial f(y,\tau )}{\partial \tau }d\tau ,\hspace{1em}0<\alpha <1$$ its Laplace transformation is obtained as: $$\mathcal{L}\left({}^{\mathit{\text{ABC}}}{D}_t^{\alpha }f\left(y,t\right)\right)=\frac{s^{\alpha }\mathcal{L}\left(f(y,t)\right)-s^{\alpha -1}f(y,0)}{(1-\alpha )s^{\alpha }+\alpha }.$$ where $\alpha$ is named as fractional parameter.

3. Solution of the flow problem

In this section, the analytical solution derived of the non-dimensional fractionalized second grade fluid model by employing the technique of Laplace transformation.

3.1. Temperature equation solution by using CPC derivative operator

Employing Laplace integral transformation to Eq. (15)(15) $${}^{\mathit{\text{CPC}}}{D}_t^{\alpha }T(y,t)=\frac{1}{Pr}\frac{{\partial }^{2}T(y,t)}{\partial y^2},$$(15) with conditions given in Eqs. (11–13), we have (23) $$\frac{d^2\overline{T}(y,s)}{d{y}^2}-P_r\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\overline{T}(y,s)=0.$$(23) with (24) $$\overline{T}(0,s)=\frac{1}{s}-\frac{a_0}{s+b_0}\hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{T}(y,s)\to 0 as y\to \infty .$$(24) and required solution is obtained as (25) $$\overline{T}(y,s)=c_1{e}^{y\sqrt{P_r\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}+c_2{e}^{-y\sqrt{P_r\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(25) solution for above equation with boundary conditions given in Eq. (24)(24) $$\overline{T}(0,s)=\frac{1}{s}-\frac{a_0}{s+b_0}\hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{T}(y,s)\to 0 as y\to \infty .$$(24) used to find unknown constants, we get (26) $$\overline{T}(y,s)=(\frac{1}{s}-\frac{a_0}{s+b_0})e^{-y\sqrt{P_r\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(26) it can be expressed as (27) $$\begin{array}{cc}\overline{T}(y,s)\hfill & ={\overline{T}}_a(y,s)-a_0{\overline{T}}_b(y,s).\hfill \end{array}$$(27)

Taking Laplace inverse which transform the above equation in time parameter, then exact solution of Eq. (27)(27) $$\begin{array}{cc}\overline{T}(y,s)\hfill & ={\overline{T}}_a(y,s)-a_0{\overline{T}}_b(y,s).\hfill \end{array}$$(27) is given by: (28) $$T(y,t)=T_a(y,t)-a_0{T}_b(y,t).$$(28) where $$\begin{array}{cc}{T}_a(y,t)\hfill & ={\mathcal{L}}^{-1}\left\{\frac{e^{-y\sqrt{P_r\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}}{s}\right\}.\hfill \end{array}$$

It is not easy to find $T_a(y,t)$ from exponential form, but Laplace inverse can be find if ${\overline{T}}_a(y,s)$ is written in series form, so for this purpose its series representations is equivalent to $$\begin{array}{cc}{T}_a(y,t)\hfill & ={\mathcal{L}}^{-1}\left\{\sum _{j=0}^{\infty }\frac{{(-y)}^j{(P_r{k}_0(\alpha ))}^{\frac{j}{2}}}{j!}.\frac{s^{\frac{j}{2}(\alpha -1)-1}}{{(s+d)}^{-\frac{j}{2}}}\right\}\hfill \\ \hfill & =\sum _{j=0}^{\infty }\frac{{(-y)}^j{(P_r{k}_0(\alpha ))}^{\frac{j}{2}}}{j!}.G_{1,\frac{j}{2}(\alpha -1)-1,-\frac{j}{2}}(-d,t),\hfill \\ T_b(y,t)\hfill & =(T_c\ast T_d)(t),\hfill \\ T_c(y,t)\hfill & ={\mathcal{L}}^{-1}\left\{\frac{1}{s+b_0}\right\}=e^{-b_{0}t},\hfill \\ T_d(y,t)\hfill & ={\mathcal{L}}^{-1}\left\{e^{-y\sqrt{P_r\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}\right\},\hfill \\ \hfill & ={\mathcal{L}}^{-1}\left\{\sum _{j=0}^{\infty }\frac{{(-y)}^j{(P_r{k}_0(\alpha ))}^{\frac{j}{2}}}{j!}.\frac{s^{\frac{j}{2}(\alpha -1)}}{{(s+d)}^{\frac{j}{2}}}\right\}\hfill \\ \hfill & =\sum _{j=0}^{\infty }\frac{{(-y)}^j{(P_r{k}_0(\alpha ))}^{\frac{j}{2}}}{j!}.G_{1,\frac{j}{2}(\alpha -1),-\frac{j}{2}}(-d,t).\hfill \end{array}$$

3.2. Temperature equation solution by using CF derivative operator

Employing Laplace integral transformation to Eq. (18)(18) $${}^{CF}{D}_t^{\alpha }T(y,t)=\frac{1}{Pr}\frac{{\partial }^{2}T(y,t)}{\partial y^2},$$(18) with conditions given in Eqs. (11–13), we have (29) $$\frac{d^2\overline{T}(y,s)}{d{y}^2}-P_r\frac{s}{\alpha +(1-\alpha )s}\overline{T}(y,s)=0.$$(29) with (30) $$\overline{T}(0,s)=\frac{1}{s}-\frac{a_0}{s+b_0}\mathit{ and }\overline{T}(y,s)\to 0 as y\to \infty .$$(30) and required solution is obtained as (31) $$\overline{T}(y,s)=c_1{e}^{y\sqrt{\frac{P_{r}s}{(1-\alpha )s+\alpha }}}+c_2{e}^{-y\sqrt{\frac{P_{r}s}{(1-\alpha )s+\alpha }}}.$$(31) solution for above equation with boundary conditions given in Eq. (30)(30) $$\overline{T}(0,s)=\frac{1}{s}-\frac{a_0}{s+b_0}\mathit{ and }\overline{T}(y,s)\to 0 as y\to \infty .$$(30) used to find unknown constants, we get (32) $$\overline{T}(y,s)=(\frac{1}{s}-\frac{a_0}{s+b_0})e^{-y\sqrt{\frac{P_{r}s}{(1-\alpha )s+\alpha }}}.$$(32) it can be expressed as (33) $$\begin{array}{cc}\overline{T}(y,s)\hfill & ={\overline{T}}_1(y,s)-a_0{\overline{T}}_2(y,s).\hfill \end{array}$$(33)

Taking Laplace inverse which transform the above equation in time parameter, then exact solution of Eq. (33)(33) $$\begin{array}{cc}\overline{T}(y,s)\hfill & ={\overline{T}}_1(y,s)-a_0{\overline{T}}_2(y,s).\hfill \end{array}$$(33) is given by: (34) $$T(y,t)=T_1(y,t)-a_0{T}_2(y,t).$$(34) where (35) $$\begin{array}{cc}{T}_1(y,t)={\mathcal{L}}^{-1}\left\{\frac{e^{-y\sqrt{\frac{P_{r}s}{(1-\alpha )s+\alpha }}}}{s}\right\}\hfill & \hfill \\ =1-\frac{2P_r}{\pi }{\int }_0^{\infty }\frac{\mathrm{Sin}\left(\frac{y}{\sqrt{1-\alpha }}x\right)}{x(P_r+x^2)}{e}^{-\frac{\alpha }{1-\alpha }t{x}^2}dx,\hfill & \hfill \\ T_2(y,t)=(T_3\ast T_4)(t),\hfill & \hfill \\ T_3(y,t)={\mathcal{L}}^{-1}\left\{\frac{1}{s+b_0}\right\}=e^{-b_{0}t},\hfill & \hfill \\ {\overline{T}}_4(y,s)=e^{-y\sqrt{\frac{P_{r}s}{(1-\alpha )s+\alpha }}},\hfill & \hfill \end{array}$$(35)

It is not easy to find $T_4(y,t)$ from exponential form, but Laplace inverse can be find if ${\overline{T}}_4(y,s)$ is written in series form, so for this purpose its series representations is equivalent to (36) $${\overline{T}}_4(y,s)=\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(P_r)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{1}{s^j}.$$(36)

Applying inverse transformation, we get (37) $$T_4(y,t)=\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(P_r)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{t^{j-1}}{\Gamma (j)}.$$(37)

3.3. Temperature equation solution by using ABC derivative operator

Employing Laplace integral transformation to Eq. (21)(21) $${}^{\mathit{\text{ABC}}}{D}_t^{\alpha }T(y,t)=\frac{1}{Pr}\frac{{\partial }^{2}T(y,t)}{\partial y^2},$$(21) with conditions given in Eqs. (11–13), we have (38) $$\frac{d^2\overline{T}(y,s)}{d{y}^2}-P_r\frac{s^{\alpha }}{\alpha +(1-\alpha )s^{\alpha }}\overline{T}(y,s)=0.$$(38) with (39) $$\overline{T}(0,s)=\frac{1}{s}-\frac{a_0}{s+b_0}\hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{T}(y,s)\to 0 as y\to \infty .$$(39) and required solution is obtained as (40) $$\overline{T}(y,s)=c_1{e}^{y\sqrt{\frac{P_r{s}^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}+c_2{e}^{-y\sqrt{\frac{P_r{s}^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}.$$(40) solution for above equation with boundary conditions given in Eq. (30)(30) $$\overline{T}(0,s)=\frac{1}{s}-\frac{a_0}{s+b_0}\mathit{ and }\overline{T}(y,s)\to 0 as y\to \infty .$$(30) used to find unknown constants, we get (41) $$\overline{T}(y,s)=(\frac{1}{s}-\frac{a_0}{s+b_0})e^{-y\sqrt{\frac{P_r{s}^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}.$$(41) it can be expressed as (42) $$\begin{array}{cc}\overline{T}(y,s)\hfill & ={\overline{T}}_5(y,s)-a_0{\overline{T}}_6(y,s).\hfill \end{array}$$(42)

Taking Laplace inverse which transform the above equation in time parameter, then exact solution of Eq. (42)(42) $$\begin{array}{cc}\overline{T}(y,s)\hfill & ={\overline{T}}_5(y,s)-a_0{\overline{T}}_6(y,s).\hfill \end{array}$$(42) is given by: (43) $$T(y,t)=T_5(y,t)-a_0{T}_6(y,t).$$(43) where (44) $$\begin{array}{cc}{T}_5(y,t)\hfill & ={\mathcal{L}}^{-1}\left\{\frac{e^{-y\sqrt{\frac{P_r{s}^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}}{s}\right\}.\hfill \end{array}$$(44)

It is not easy to find $T_5(y,t)$ from exponential form, but Laplace inverse can be find if ${\overline{T}}_5(y,s)$ is written in series form, so for this purpose its series representations is equivalent to (45) $${\overline{T}}_4(y,s)=\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(P_r)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{1}{s^{j\alpha +1}}.$$(45)

Applying inverse transformation, we get (46) $$T_4(y,t)=\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(P_r)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{t^{j\alpha }}{\Gamma (j\alpha +1)}.$$(46) (47) $$\begin{array}{cc}{T}_6(y,t)=(T_7\ast T_8)(t),\hfill & \hfill \\ T_7(y,t)={\mathcal{L}}^{-1}\left\{\frac{1}{s+b_0}\right\}=e^{-b_{0}t},\hfill & \hfill \\ {\overline{T}}_8(y,s)=e^{-y\sqrt{\frac{P_r{s}^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}},\hfill & \hfill \end{array}$$(47)

It is not easy to find $T_8(y,t)$ from exponential form, but Laplace inverse can be find if ${\overline{T}}_8(y,s)$ is written in series form, so for this purpose its series representations is equivalent to (48) $${\overline{T}}_8(y,s)=\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(P_r)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{1}{s^{j\alpha }}.$$(48)

Applying inverse transformation, we get (49) $$T_8(y,t)=\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(P_r)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{t^{j\alpha -1}}{\Gamma (j\alpha )}.$$(49)

3.4. Concentration equation solution by using CPC derivative operator

Solving Eq. (16)(16) $${}^{\mathit{\text{CPC}}}{D}_t^{\alpha }C(y,t)=\frac{1}{Sc}\frac{{\partial }^{2}C(y,t)}{\partial y^2}.$$(16) using Eqs. (11–13), then it becomes (50) $$\frac{d^2\overline{C}(y,s)}{d{y}^2}-\left(S_c\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)\overline{C}(y,s)=0.$$(50) with conditions transformed as (51) $$\overline{C}(y,s)\to 0 as y\to \infty \hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{C}(0,s)=\frac{1}{s-\zeta }.$$(51) the solution for concentration is described as: (52) $$\overline{C}(y,s)=c_3{e}^{y\sqrt{S_c\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}+c_4{e}^{-y\sqrt{S_c\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(52)

Applying conditions given in Eq. (58)(58) $$\overline{C}(y,s)\to 0 as y\to \infty \hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{C}(0,s)=\frac{1}{s-\zeta }.$$(58) for concentration then the above solution has the form (53) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(53)

For concentration solution, taking Laplace inverse of Eq. (53)(53) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(53) , so finally it takes the form (54) $$\begin{array}{cc}C(y,t)\hfill & ={\mathcal{L}}^{-1}\left\{\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}\right\}\hfill \end{array}$$(54) (55) $$={\mathcal{L}}^{-1}\left\{\frac{1}{s-\zeta }\sum _{j=0}^{\infty }\frac{{(-y)}^j{(S_c{k}_0(\alpha ))}^{\frac{j}{2}}}{j!}.\frac{s^{\frac{j}{2}(\alpha -1)}}{{(s+d)}^{\frac{j}{2}}}\right\}$$(55) (56) $$=e^{\zeta t}\ast \sum _{j=0}^{\infty }\frac{{(-y)}^j{(S_c{k}_0(\alpha ))}^{\frac{j}{2}}}{j!}.G_{1,\frac{j}{2}(\alpha -1),-\frac{j}{2}}(-d,t).$$(56)

3.5. Concentration equation solution by using CF derivative operator

Solving Eq. (19)(19) $${}^{CF}{D}_t^{\alpha }C(y,t)=\frac{1}{Sc}\frac{{\partial }^{2}C(y,t)}{\partial y^2}.$$(19) using Eqs. (11–13), then it becomes (57) $$\frac{d^2\overline{C}(y,s)}{d{y}^2}-\left(S_c\frac{s}{(1-\alpha )s+\alpha }\right)\overline{C}(y,s)=0.$$(57) with conditions transformed as (58) $$\overline{C}(y,s)\to 0 as y\to \infty \hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{C}(0,s)=\frac{1}{s-\zeta }.$$(58) the solution for concentration is described as: (59) $$\overline{C}(y,s)=c_3{e}^{y\sqrt{S_c\frac{s}{(1-\alpha )s+\alpha }}}+c_4{e}^{-y\sqrt{S_c\frac{s}{(1-\alpha )s+\alpha }}}.$$(59)

Applying conditions given in Eq. (58)(58) $$\overline{C}(y,s)\to 0 as y\to \infty \hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{C}(0,s)=\frac{1}{s-\zeta }.$$(58) for concentration then the above solution has the form (60) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\frac{s}{(1-\alpha )s+\alpha }}}.$$(60)

For concentration solution, taking Laplace inverse of Eq. (60)(60) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\frac{s}{(1-\alpha )s+\alpha }}}.$$(60) , so finally it takes the form (61) $$\begin{array}{cc}C(y,t)\hfill & ={\mathcal{L}}^{-1}\left\{\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\frac{s}{(1-\alpha )s+\alpha }}}\right\}\hfill \end{array}$$(61) (62) $$={\mathcal{L}}^{-1}\left\{\frac{1}{s-\zeta }\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(S_c)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{1}{s^j}\right\}$$(62) (63) $$=e^{\zeta t}\ast \sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(S_c)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{t^{j-1}}{\Gamma (j)}.$$(63)

3.6. Concentration equation solution by using ABC derivative operator

Solving Eq. (22)(22) $${}^{\mathit{\text{ABC}}}{D}_t^{\alpha }C(y,t)=\frac{1}{Sc}\frac{{\partial }^{2}C(y,t)}{\partial y^2}.$$(22) using Eqs. (11–13), then it becomes (64) $$\frac{d^2\overline{C}(y,s)}{d{y}^2}-\left(S_c\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }\right)\overline{C}(y,s)=0.$$(64) with conditions transformed as (65) $$\overline{C}(y,s)\to 0 as y\to \infty \hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{C}(0,s)=\frac{1}{s-\zeta }.$$(65) the solution for concentration is described as: (66) $$\overline{C}(y,s)=c_3{e}^{y\sqrt{S_c\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}+c_4{e}^{-y\sqrt{S_c\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}.$$(66)

Applying conditions given in Eq. (65)(65) $$\overline{C}(y,s)\to 0 as y\to \infty \hspace{1em}\mathit{\text{and}}\hspace{1em}\overline{C}(0,s)=\frac{1}{s-\zeta }.$$(65) for concentration then the above solution has the form (67) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}.$$(67)

For concentration solution, taking Laplace inverse of Eq. (67)(67) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}.$$(67) , so finally it takes the form (68) $$\begin{array}{cc}C(y,t)\hfill & ={\mathcal{L}}^{-1}\left\{\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}\right\}\hfill \end{array}$$(68) (69) $$={\mathcal{L}}^{-1}\left\{\frac{1}{s-\zeta }\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(P_r)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{1}{s^{j\alpha }}\right\}$$(69) (70) $$=e^{\zeta t}\ast \sum _{k=0}^{\infty }\sum _{k=0}^{\infty }\sum _{j=0}^{\infty }\frac{{(-1)}^j{(-y)}^k{(P_r)}^{\frac{k}{2}}{(\alpha )}^j\Gamma (\frac{k}{2}+j)}{k!j!{(1-\alpha )}^{\frac{k}{2}+j}\Gamma \left(\frac{k}{2}\right)}.\frac{t^{j\alpha -1}}{\Gamma (j\alpha )}.$$(70)

3.7. Exact solution of fluid velocity by using CF derivative operator

Solution of Eq. (17)(17) $$\begin{array}{c}{}^{CF}{D}_t^{\alpha }u(y,t)=\left(1+{\alpha }_2{}^{CF}{D}_t^{\alpha }\right)\frac{{\partial }^{2}u(y,t)}{\partial y^2}+G_{r}T(y,t)+G_{m}C(y,t)-(a+b^{CF}{D}_t^{\alpha })u(y,t),\hfill \end{array}$$(17) with the application of Laplace integral transformation, then its solution in the following form (71) $$\left(a+(1+b)\frac{s}{(1-\alpha )s+\alpha }\right)\overline{u}(y,s)=\left(1+{\alpha }_2\frac{s}{(1-\alpha )s+\alpha }\right)\frac{d^2\overline{u}(y,s)}{d{y}^2}+G_r\overline{T}(y,s)+G_m\overline{C}(y,s),$$(71) rearrange the above equation then it can be written as: (72) $$\frac{d^2\overline{u}(y,s)}{d{y}^2}-\left(\frac{b_{1}s+a_1}{b_{2}s+\alpha }\right)\overline{u}(y,s)=-\left(\frac{b_{3}s+\alpha }{b_{2}s+\alpha }\right)\left(G_r\overline{T}(y,s)+G_m\overline{C}(y,s)\right).$$(72) by using the Eq. (32)(32) $$\overline{T}(y,s)=(\frac{1}{s}-\frac{a_0}{s+b_0})e^{-y\sqrt{\frac{P_{r}s}{(1-\alpha )s+\alpha }}}.$$(32) and Eq. (60)(60) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\frac{s}{(1-\alpha )s+\alpha }}}.$$(60) in Eq. (72)(72) $$\frac{d^2\overline{u}(y,s)}{d{y}^2}-\left(\frac{b_{1}s+a_1}{b_{2}s+\alpha }\right)\overline{u}(y,s)=-\left(\frac{b_{3}s+\alpha }{b_{2}s+\alpha }\right)\left(G_r\overline{T}(y,s)+G_m\overline{C}(y,s)\right).$$(72) then the solution in general form is represented as: (73) $$\begin{array}{l}\overline{u}(y,s)=c_5{e}^{y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}+c_6{e}^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}-\left(\frac{G_r{\left(b_{3}s+\alpha \right)}^2\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)e^{-y\sqrt{\frac{P_{r}s}{b_{3}s+\alpha }}}}{s\left(b_{2}s+\alpha \right)P_r-(b_{1}s+a_1)(b_{3}s+\alpha )}\right)\hfill \\ -\left(\frac{G_m{\left(b_{3}s+\alpha \right)}^2\left(\frac{1}{s}\right)e^{-y\sqrt{\frac{S_{c}s}{b_{3}s+\alpha }}}}{s\left(b_{2}s+\alpha \right)S_c-(b_{1}s+a_1)(b_{3}s+\alpha )}\right).\hfill \end{array}$$(73) to determine unknowns $c_5$ and $c_6,$ using $\overline{u}(0,s)=\frac{1}{s-\zeta }$ and $\overline{u}(y,s)\to 0$ as $y\to \infty ,$ we get (74) $$\begin{array}{l}\overline{u}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}+\left(\frac{G_r{\left(b_{3}s+\alpha \right)}^2\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)}{s\left(b_{2}s+\alpha \right)P_r-(b_{1}s+a_1)(b_{3}s+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}-e^{-y\sqrt{\frac{P_{r}s}{b_{3}s+\alpha }}}\right]\hfill \\ +\left(\frac{G_m{\left(b_{3}s+\alpha \right)}^2\left(\frac{1}{s}\right)}{s\left(b_{2}s+\alpha \right)S_c-(b_{1}s+a_1)(b_{3}s+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}-e^{-y\sqrt{\frac{S_{c}s}{b_{3}s+\alpha }}}\right].\hfill \end{array}$$(74) the above expression can be expressed in the following way (75) $$\begin{array}{cc}\overline{u}\left(y,s\right)=\overline{E}\left(y,s\right)\overline{F}\left(y,s\right)+G_r\overline{P}\left(y,s\right)\left[\overline{D}\left(y,s\right)-a_0\overline{L}\left(y,s\right)-\overline{T}\left(y,s\right)\right]\hfill & \hfill \\ +G_m\overline{R}\left(y,s\right)\left[\overline{D}\left(y,s\right)-\overline{C}\left(y,s\right)\right].\hfill & \hfill \end{array}$$(75)

Applying Laplace inverse to write the solution for momentum equation as: (76) $$\begin{array}{cc}u\left(y,t\right)=\left(E\ast F\right)(t)+G_r\left[\left(P\ast D\right)(t)-a_0\left(P\ast L\right)(t)-\left(P\ast T\right)(t)\right]\hfill & \hfill \\ +G_m\left[\left(R\ast D\right)(t)-\left(R\ast C\right)(t)\right].\hfill & \hfill \end{array}$$(76) where (77) $$\begin{array}{cc}\overline{F}\left(y,s\right)\hfill & =e^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}.\hfill \end{array}$$(77) first express the $\overline{F}(y,s)$ in series form, (78) $$\overline{F}(y,s)=\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{1}{s^{p+h}}.$$(78)

Employing Laplace inverse then it takes the form as: (79) $$F(y,t)=\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{t^{p+h-1}}{\Gamma (p+h)}.$$(79)

Similarly, $D(y,t)$ is calculated as: (80) $$\begin{array}{l}\overline{D}\left(y,s\right)=\frac{1}{s}\overline{F}\left(y,s\right),\hfill \\ =\frac{1}{s}\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{1}{s^{p+h}},\hfill \\ =\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{1}{s^{p+h+1}},\hfill \\ D(y,t)=\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{t^{p+h}}{\Gamma (p+h+1)}.\hfill \end{array}$$(80) and (81) $$\begin{array}{cc}E\left(y,t\right)={\mathcal{L}}^{-1}\left\{\frac{1}{s-\zeta }\right\}=e^{\zeta t},\hfill & \hfill \\ J\left(y,t\right)={\mathcal{L}}^{-1}\left\{\frac{1}{s+b_0}\right\}=e^{-b_{0}t},\hfill & \hfill \\ \overline{L}\left(y,s\right)=\overline{J}\left(y,s\right)\overline{F}\left(y,s\right),\hfill & \hfill \\ L\left(y,t\right)=(J\ast F)(t)={\int }_0^{t}F(\tau )J(t-\tau )d\tau ,\hfill & \hfill \end{array}$$(81)

Next consider $\overline{P}(y,s)$ to calculate $P(y,t),$ we have (82) $$\begin{array}{l}\overline{P}\left(y,s\right)={(b_{3}s+\alpha )}^2\frac{1}{(Pr{b}_2-b_1{b}_3)s^2-(a_1{b}_3+b_1\alpha -P_r\alpha )s-a_1\alpha },\hfill \\ ={(b_{3}s+\alpha )}^2\frac{1}{\beta s^2-\gamma s-\delta },\hfill \\ =b_3^2\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}\frac{{(s)}^{-n+1}}{{(s-\frac{\gamma }{\beta })}^{n+1}}+2b_3\alpha \sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}\frac{{(s)}^{-n}}{{(s-\frac{\gamma }{\beta })}^{n+1}}+{\alpha }^2\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}\frac{{(s)}^{-n-1}}{{(s-\frac{\gamma }{\beta })}^{n+1}},\hfill \\ P\left(y,t\right)=\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}\left[b_3^2{G}_{1,-n+1,n+1}(\frac{\gamma }{\beta },t)+2b_3\alpha G_{1,-n,n+1}(\frac{\gamma }{\beta },t)+{\alpha }^2{G}_{1,-n-1,n+1}(\frac{\gamma }{\beta },t)\right],\hfill \end{array}$$(82)

Similarly, we can calculate $R(y,t),$ so consider (83) $$\begin{array}{c}\overline{R}\left(y,s\right)={(b_{3}s+\alpha )}^2\frac{1}{(S_c{b}_2-b_1{b}_3)s^2-(a_1{b}_3+b_1\alpha -S_c\alpha )s-a_1\alpha },\hfill \\ ={(b_{3}s+\alpha )}^2\frac{1}{\chi s^2-\xi s-\delta },\hfill \\ =b_3^2\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\chi )}^{n+1}}\frac{{(s)}^{-n+1}}{{(s-\frac{\xi }{\chi })}^{n+1}}+2b_3\alpha \sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\chi )}^{n+1}}\frac{{(s)}^{-n}}{{(s-\frac{\xi }{\chi })}^{n+1}}+{\alpha }^2\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\chi )}^{n+1}}\frac{{(s)}^{-n-1}}{{(s-\frac{\xi }{\chi })}^{n+1}},\hfill \\ R\left(y,t\right)=\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}\left[b_3^2{G}_{1,-n+1,n+1}(\frac{\xi }{\chi },t)+2b_3\alpha G_{1,-n,n+1}(\frac{\xi }{\chi },t)+{\alpha }^2{G}_{1,-n-1,n+1}(\frac{\xi }{\chi },t)\right],\hfill \end{array}$$(83)

3.8. Exact solution of fluid velocity by using ABC derivative operator

Solution of Eq. (20)(20) $$\begin{array}{c}{}^{\mathit{\text{ABC}}}{D}_t^{\alpha }u(y,t)=\left(1+{\alpha }_2{}^{\mathit{\text{ABC}}}{D}_t^{\alpha }\right)\frac{{\partial }^{2}u(y,t)}{\partial y^2}+G_{r}T(y,t)+G_{m}C(y,t)-(a+b^{\mathit{\text{ABC}}}{D}_t^{\alpha })u(y,t),\hfill \end{array}$$(20) with the application of Laplace integral transformation, then its solution in the following form (84) $$\left(a+(1+b)\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }\right)\overline{u}(y,s)=\left(1+{\alpha }_2\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }\right)\frac{d^2\overline{u}(y,s)}{d{y}^2}+G_r\overline{T}(y,s)+G_m\overline{C}(y,s),$$(84) rearrange the above equation then it can be written as: (85) $$\frac{d^2\overline{u}(y,s)}{d{y}^2}-\left(\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }\right)\overline{u}(y,s)=-\left(\frac{b_3{s}^{\alpha }+\alpha }{b_2{s}^{\alpha }+\alpha }\right)\left(G_r\overline{T}(y,s)+G_m\overline{C}(y,s)\right).$$(85) by using the Eq. (41)(41) $$\overline{T}(y,s)=(\frac{1}{s}-\frac{a_0}{s+b_0})e^{-y\sqrt{\frac{P_r{s}^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}.$$(41) and Eq. (67)(67) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\frac{s^{\alpha }}{(1-\alpha )s^{\alpha }+\alpha }}}.$$(67) in Eq. (85)(85) $$\frac{d^2\overline{u}(y,s)}{d{y}^2}-\left(\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }\right)\overline{u}(y,s)=-\left(\frac{b_3{s}^{\alpha }+\alpha }{b_2{s}^{\alpha }+\alpha }\right)\left(G_r\overline{T}(y,s)+G_m\overline{C}(y,s)\right).$$(85) then the solution in general form is represented as: (86) $$\begin{array}{c}\overline{u}(y,s)=c_{11}{e}^{y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}+c_{12}{e}^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}-\left(\frac{G_r{\left(b_3{s}^{\alpha }+\alpha \right)}^2\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)e^{-y\sqrt{\frac{P_r{s}^{\alpha }}{b_3{s}^{\alpha }+\alpha }}}}{s^{\alpha }\left(b_2{s}^{\alpha }+\alpha \right)P_r-(b_1{s}^{\alpha }+a_1)(b_3{s}^{\alpha }+\alpha )}\right)\hfill \\ -\left(\frac{G_m{\left(b_3{s}^{\alpha }+\alpha \right)}^2\left(\frac{1}{s}\right)e^{-y\sqrt{\frac{S_c{s}^{\alpha }}{b_3{s}^{\alpha }+\alpha }}}}{s^{\alpha }\left(b_2{s}^{\alpha }+\alpha \right)S_c-(b_1{s}^{\alpha }+a_1)(b_3{s}^{\alpha }+\alpha )}\right).\hfill \end{array}$$(86) to determine unknowns $c_{11}$ and $c_{12},$ using $\overline{u}(0,s)=\frac{1}{s-\zeta }$ and $\overline{u}(y,s)\to 0$ as $y\to \infty ,$ we get (87) $$\begin{array}{c}\overline{u}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}+\left(\frac{G_r{\left(b_3{s}^{\alpha }+\alpha \right)}^2\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)}{s^{\alpha }\left(b_2{s}^{\alpha }+\alpha \right)P_r-(b_1{s}^{\alpha }+a_1)(b_3{s}^{\alpha }+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}-e^{-y\sqrt{\frac{P_r{s}^{\alpha }}{b_3{s}^{\alpha }+\alpha }}}\right]\hfill \\ +\left(\frac{G_m{\left(b_3{s}^{\alpha }+\alpha \right)}^2\left(\frac{1}{s}\right)}{s^{\alpha }\left(b_2{s}^{\alpha }+\alpha \right)S_c-(b_1{s}^{\alpha }+a_1)(b_3{s}^{\alpha }+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}-e^{-y\sqrt{\frac{S_c{s}^{\alpha }}{b_3{s}^{\alpha }+\alpha }}}\right].\hfill \end{array}$$(87) the above expression can be expressed in the following way (88) $$\begin{array}{cc}\overline{u}\left(y,s\right)=\overline{E_1}\left(y,s\right)\overline{F_1}\left(y,s\right)+G_r\overline{P_1}\left(y,s\right)\left[\overline{D_1}\left(y,s\right)-a_0\overline{L_1}\left(y,s\right)-\overline{T}\left(y,s\right)\right]\hfill & \hfill \\ +G_m\overline{R_1}\left(y,s\right)\left[\overline{D_1}\left(y,s\right)-\overline{C}\left(y,s\right)\right].\hfill & \hfill \end{array}$$(88)

Applying Laplace inverse to write the solution for momentum equation as: (89) $$\begin{array}{cc}u\left(y,t\right)=\left(E_1\ast F_1\right)(t)+G_r\left[\left(P_1\ast D_1\right)(t)-a_0\left(P_1\ast L_1\right)(t)-\left(P_1\ast T\right)(t)\right]\hfill & \hfill \\ +G_m\left[\left(R_1\ast D_1\right)(t)-\left(R_1\ast C\right)(t)\right].\hfill & \hfill \end{array}$$(89) where (90) $$\begin{array}{cc}\overline{F_1}\left(y,s\right)\hfill & =e^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}.\hfill \end{array}$$(90) first express the $\overline{F_1}(y,s)$ in series form, (91) $$\overline{F_1}(y,s)=\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{1}{s^{\alpha p+\alpha h}}.$$(91)

Employing Laplace inverse then it takes the form as: (92) $$F(y,t)=\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{t^{\alpha p+\alpha h-1}}{\Gamma (\alpha p+\alpha h)}.$$(92)

Similarly, $D_1(y,t)$ is calculated as: (93) $$\begin{array}{c}\overline{D_1}\left(y,s\right)=\frac{1}{s}\overline{F_1}\left(y,s\right),\hfill \\ =\frac{1}{s}\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{1}{s^{\alpha p+\alpha h}},\hfill \\ =\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{1}{s^{\alpha p+\alpha h+1}},\hfill \\ D_1(y,t)=\sum _{v=1}^{\infty }\sum _{p=0}^{\infty }\sum _{h=0}^{\infty }\frac{{(-1)}^h{(-y)}^v{(a_2)}^p{(b_4)}^{\frac{v}{2}-p}{(\alpha )}^h\Gamma (\frac{v}{2}+1)\Gamma (p+h)}{v!\hspace{1em}p!\hspace{1em}h!\hspace{1em}{(b_2)}^{p+h}\hspace{1em}\Gamma (\frac{v}{2}-p+1)\Gamma (h)}.\frac{t^{\alpha p+\alpha h}}{\Gamma (\alpha p+\alpha h+1)}.\hfill \end{array}$$(93) and (94) $$\begin{array}{cc}{E}_1\left(y,t\right)={\mathcal{L}}^{-1}\left\{\frac{1}{s-\zeta }\right\}=e^{\zeta t},\hfill & \hfill \\ J_1\left(y,t\right)={\mathcal{L}}^{-1}\left\{\frac{1}{s+b_0}\right\}=e^{-b_{0}t},\hfill & \hfill \\ \overline{L_1}\left(y,s\right)=\overline{J_1}\left(y,s\right)\overline{F}\left(y,s\right),\hfill & \hfill \\ L_1\left(y,t\right)=(J_1\ast F_1)(t)={\int }_0^t{F}_1(t)J_1(t-t)dt,\hfill & \hfill \end{array}$$(94)

Next consider $\overline{P_1}(y,s)$ to calculate $P_1(y,t),$ we have (95) $$\begin{array}{c}\overline{P_1}\left(y,s\right)={(b_3{s}^{\alpha }+\alpha )}^2\frac{1}{(Pr{b}_2-b_1{b}_3)s^{2\alpha }-(a_1{b}_3+b_1\alpha -P_r\alpha )s^{\alpha }-a_1\alpha },\hfill \\ ={(b_3{s}^{\alpha }+\alpha )}^2\frac{1}{\beta s^{2\alpha }-\gamma s^{\alpha }-\delta },\hfill \\ =b_3^2\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}\frac{{(s)}^{-n\alpha +\alpha }}{{(s^{\alpha }-\frac{\gamma }{\beta })}^{n+1}}+2b_3\alpha \sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}\frac{{(s)}^{-n\alpha }}{{(s^{\alpha }-\frac{\gamma }{\beta })}^{n+1}}+{\alpha }^2\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}\frac{{(s)}^{-n\alpha -\alpha }}{{(s^{\alpha }-\frac{\gamma }{\beta })}^{n+1}},\hfill \\ P_1\left(y,t\right)=\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\beta )}^{n+1}}[b_3^2{G}_{1,-n\alpha +\alpha ,n+1}(\frac{\gamma }{\beta },t)+2b_3\alpha G_{1,-n\alpha ,n+1}(\frac{\gamma }{\beta },t)+{\alpha }^2{G}_{1,-n\alpha -\alpha ,n+1}(\frac{\gamma }{\beta },t)],\hfill \end{array}$$(95)

Similarly, we can calculate $R(y,t),$ so consider (96) $$\begin{array}{c}\overline{R_1}\left(y,s\right)={(b_3{s}^{\alpha }+\alpha )}^2\frac{1}{(S_c{b}_2-b_1{b}_3)s^{2\alpha }-(a_1{b}_3+b_1\alpha -S_c\alpha )s^{\alpha }-a_1\alpha },\hfill \\ ={(b_3{s}^{\alpha }+\alpha )}^2\frac{1}{\chi s^{2\alpha }-\xi s^{\alpha }-\delta },\hfill \\ =b_3^2\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\chi )}^{n+1}}\frac{{(s)}^{-n\alpha +\alpha }}{{(s^{\alpha }-\frac{\xi }{\chi })}^{n+1}}+2b_3\alpha \sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\chi )}^{n+1}}\frac{{(s)}^{-n\alpha }}{{(s^{\alpha }-\frac{\xi }{\chi })}^{n+1}}+{\alpha }^2\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\chi )}^{n+1}}\frac{{(s)}^{-n\alpha -\alpha }}{{(s^{\alpha }-\frac{\xi }{\chi })}^{n+1}},\hfill \\ R_1\left(y,t\right)=\sum _{n=0}^{\infty }\frac{{(\delta )}^n}{{(\chi )}^{n+1}}[b_3^2{G}_{1,-n\alpha +\alpha ,n+1}(\frac{\xi }{\chi },t)+2b_3\alpha G_{1,-n\alpha ,n+1}(\frac{\xi }{\chi },t)+{\alpha }^2{G}_{1,-n\alpha -\alpha ,n+1}(\frac{\xi }{\chi },t)],\hfill \end{array}$$(96) where (97) $$\begin{array}{cc}\beta =P_r{b}_2-b_1{b}_3,\hspace{1em}\gamma =a_1{b}_3+b_1\alpha -P_r\alpha ,\hspace{1em}\delta =a_1\alpha ,\hfill & \hfill \\ \chi =S_c{b}_2-b_1{b}_3,\hspace{1em}\xi =a_1{b}_3+b_1\alpha -S_c\alpha ,\hspace{1em}{a}_1=a\alpha ,\hfill & \hfill \end{array}$$(97) (98) $$\begin{array}{c}{b}_1=a(1-\alpha )+(1+b),\hspace{1em}{b}_2={\alpha }_2+(1-\alpha ),\hspace{1em}{b}_3=1-\alpha ,\hfill \end{array}$$(98) (99) $$b_4=\frac{b_1}{b_2},\hspace{1em}{a}_2=a_1-\alpha \frac{b_1}{b_2}\hspace{1em}\mathit{\text{and}}\hspace{1em}(\wp \ast \chi )(t)={\int }_0^t\wp (t)\chi (t-\tau )d\tau .$$(99)

3.9. Exact solution of fluid velocity by using CPC derivative operator

Solution of Eq. (14)(14) $$\begin{array}{c}{}^{\mathit{\text{CPC}}}{D}_t^{\alpha }u(y,t)=\left(1+{\alpha }_2{}^{\mathit{\text{CPC}}}{D}_t^{\alpha }\right)\frac{{\partial }^{2}u(y,t)}{\partial y^2}+G_{r}T(y,t)+G_{m}C(y,t)-(a+b^{\mathit{\text{CPC}}}{D}_t^{\alpha })u(y,t),\hfill \end{array}$$(14) with the application of Laplace integral transformation, then its solution in the following form (100) $$\begin{array}{cc}\left(a+(1+b)\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)\overline{u}(y,s)\hfill & =\left(1+{\alpha }_2\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)\frac{d^2\overline{u}(y,s)}{d{y}^2}\hfill \\ +G_r\overline{T}(y,s)+G_m\overline{C}(y,s),\hfill & \hfill \end{array}$$(100) rearrange the above equation then it can be written as: (101) $$\frac{d^2\overline{u}(y,s)}{d{y}^2}-\left(\frac{a+d_{1}A(s)}{1+{\alpha }_{2}A(s)}\right)\overline{u}(y,s)=-\left(\frac{1}{1+{\alpha }_{2}A(s)}\right)\left(G_r\overline{T}(y,s)+G_m\overline{C}(y,s)\right).$$(101) where $A(s)=\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }$ and $d_1=1+b$by using the Eq. (26)(26) $$\overline{T}(y,s)=(\frac{1}{s}-\frac{a_0}{s+b_0})e^{-y\sqrt{P_r\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(26) and Eq. (53)(53) $$\overline{C}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{S_c\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(53) in Eq. (101)(101) $$\frac{d^2\overline{u}(y,s)}{d{y}^2}-\left(\frac{a+d_{1}A(s)}{1+{\alpha }_{2}A(s)}\right)\overline{u}(y,s)=-\left(\frac{1}{1+{\alpha }_{2}A(s)}\right)\left(G_r\overline{T}(y,s)+G_m\overline{C}(y,s)\right).$$(101) then the solution in general form is represented as: (102) $$\begin{array}{cc}\overline{u}(y,s)=c_5{e}^{y\sqrt{\frac{a+d_{1}A(s)}{1+{\alpha }_{2}A(s)}}}+c_6{e}^{-y\sqrt{\frac{a+d_{1}A(s)}{1+{\alpha }_{2}A(s)}}}-\left(\frac{G_r\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)e^{-y\sqrt{P_{r}A(s)}}}{P_{r}A(s)(1+{\alpha }_{2}A(s))-(a+d_{1}A(s))}\right)\hfill & \hfill \\ -\left(\frac{G_m\left(\frac{1}{s}\right)e^{-y\sqrt{S_{c}A(s)+\lambda }}}{(S_{c}A(s)+\lambda )(1+{\alpha }_{2}A(s))-(a+d_{1}A(s))}\right).\hfill & \hfill \end{array}$$(102) to determine unknowns $c_5$ and $c_6,$ using $\overline{u}(0,s)=\frac{1}{s-\zeta }$ and $\overline{u}(y,s)\to 0$ as $y\to \infty ,$ we get (103) $$\begin{array}{c}\overline{u}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{\frac{a+d_{1}A(s)}{1+{\alpha }_{2}A(s)}}}+\left(\frac{G_r\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)}{P_{r}A(s)(1+{\alpha }_{2}A(s))-(a+d_{1}A(s))}\right)\left[e^{-y\sqrt{\frac{a+d_{1}A(s)}{1+{\alpha }_{2}A(s)}}}-e^{-y\sqrt{P_{r}A(s)}}\right]\hfill \\ +\left(\frac{G_m\left(\frac{1}{s}\right)}{(S_{c}A(s)+\lambda )(1+{\alpha }_{2}A(s))-(a+d_{1}A(s))}\right)\left[e^{-y\sqrt{\frac{a+d_{1}A(s)}{1+{\alpha }_{2}A(s)}}}-e^{-y\sqrt{S_{c}A(s)+\lambda }}\right].\hfill \end{array}$$(103)

Similarly, we find the Laplace inverse of $\overline{u}(y,s),$ to obtain $u(y,t),$ as we have computed for CF and ABC fractional derivative operators. The function $G_{h,b,l}(.,\tau )$ is known as G-function and is defined as $\frac{s^b}{{(s^h-j)}^l}=\mathcal{L}\left\{G_{h,b,l}(j,\tau )\right\}$ with $Re(hl-b)>0,$ $Re(s)>0,$ $\left|\frac{j}{s^h}\right|<1.$

For validation purpose of the current study, we recovered the same velocity field equations Eq. (74)(74) $$\begin{array}{l}\overline{u}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}+\left(\frac{G_r{\left(b_{3}s+\alpha \right)}^2\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)}{s\left(b_{2}s+\alpha \right)P_r-(b_{1}s+a_1)(b_{3}s+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}-e^{-y\sqrt{\frac{P_{r}s}{b_{3}s+\alpha }}}\right]\hfill \\ +\left(\frac{G_m{\left(b_{3}s+\alpha \right)}^2\left(\frac{1}{s}\right)}{s\left(b_{2}s+\alpha \right)S_c-(b_{1}s+a_1)(b_{3}s+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}-e^{-y\sqrt{\frac{S_{c}s}{b_{3}s+\alpha }}}\right].\hfill \end{array}$$(74) as Haq et al. (2021) investigated with exponential heating by removing the term Gm, i.e. $Gm=0.$ Also, we obtained the same velocity field equations Eq. (87)(87) $$\begin{array}{c}\overline{u}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}+\left(\frac{G_r{\left(b_3{s}^{\alpha }+\alpha \right)}^2\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)}{s^{\alpha }\left(b_2{s}^{\alpha }+\alpha \right)P_r-(b_1{s}^{\alpha }+a_1)(b_3{s}^{\alpha }+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}-e^{-y\sqrt{\frac{P_r{s}^{\alpha }}{b_3{s}^{\alpha }+\alpha }}}\right]\hfill \\ +\left(\frac{G_m{\left(b_3{s}^{\alpha }+\alpha \right)}^2\left(\frac{1}{s}\right)}{s^{\alpha }\left(b_2{s}^{\alpha }+\alpha \right)S_c-(b_1{s}^{\alpha }+a_1)(b_3{s}^{\alpha }+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}-e^{-y\sqrt{\frac{S_c{s}^{\alpha }}{b_3{s}^{\alpha }+\alpha }}}\right].\hfill \end{array}$$(87) as Song et al. (2021) investigated with exponential heating by removing the term Gm, i.e. $Gm=0.$ Further, in all of three cases CPC, CF and ABC, we get the classical case when $\alpha \to 1.$This prove the authenticity of our results.

4. Results and discussion

The unsteady second grade fluid with natural convective flow over flat plate of infinite length has been examined under exponential heating. Fractional model developed for non-dimensional velocity, concentration and energy equations by using CPC, CF and ABC fractional operators. Exact solution expressions obtained in terms of generalized G-function by simplifying the fractionalized model analytically with initial boundary conditions are provided for the proposed problem. For physical significance of various system parameters involved in the problem such as fractional parameter $\alpha ,$ thermal Grashof number Gr, Prandtl parameter Pr, Schmidt number Sc and mass Grashof number Gm on fluid velocity, concentration and temperature are evaluated and executed graphically in Figures 2–9 by using graphical Mathcad-Software, taking four values of fractional parameter $\alpha$ lies between 0 and 1, by considering $a_0=0.70,$ $b_0=0.10,$ $k_0=0.3,$ $k_1=0.5,$ $t=3,$ $M=0.6,$ $\lambda =0.7$ and ${\alpha }_2=0.5$

Figure 2. Representation of temperature profile for varying the values of Pr via CF, ABC and CPC.

Figure 3. Representation of concentration profile via CF, ABC and CPC for distinct values of Sc.

Figure 4. Representation of second grade fluid velocity via CF, ABC and CPC. for distinct values of Gm.

Figure 5. Representation of second grade fluid velocity via CF, ABC and CPC for distinct values of Gr.

Figure 6. Representation of second grade fluid velocity via CF, ABC and CPC for distinct values of Pr.

Figure 7. Representation of second grade fluid velocity via CF, ABC and CPC for distinct values of Sc.

Figure 8. Trace of dimensionless velocity for comparison of CPC, ABC and CF models.

Figure 9. Comparison of the present velocity graphs, taking Gm = 0, with Haq et al. (2021) and Song et al. (2021) velocity graphs.

Figure 2 displays the Prandtl number $P_r$ effect on second grade fluid temperature distribution against y via CF, CPC and ABC operator, for different values of $P_r.$ It is noticed that a decreasing effect on temperature in the boundary layer when the values of the Prandtl number enlarged. Physically, an increasing the values of Prandtl number that leads to an increases the fluid viscosity, because of this fluid becomes thicker due to viscosity increased, and as a result, fluid temperature decreased.

Figure 3 illustrates the behaviour of Sc on concentration profile of second grade fluid by taking the values of other parameters fixed, via CF, CPC and ABC operators. From the curves it is analysed that concentration profile reduced for large values of Sc, where the values of fractional parameters have been supposed between 0 and 1. Physically, boundary layer of concentration is declined due to change of Schmidt number from small to large.

Figures 4, 5 portrays the influence of mass Grashof number Gm and thermal Grashof numbers Gr on second grade fluid flow against y via CF, CPC and ABC operators. Since Gr describes the fraction of thermal buoyancy force to viscous force, but Gm describes the fraction of species buoyancy force to viscous force, that are acting on the fluid transportation, as a result, with an increasing in Gr or Gm cause a remarkable increasing impact on the second grade fluid velocity. Physically, an increasing the values of thermal or mass Grashof numbers that leads to decrease in viscous hydrodynamic forces, and as a result, the momentum of the second grade fluid is higher.

Figure 6 exhibits the effect of Prandtl number $P_r$ on velocity corresponding to y, for different values of $P_r$ via CF, CPC and ABC operators. It is noticeable that a decreasing effect on velocity in the boundary layer when the values of the Prandtl number enlarged. Physically, an increasing the values of Prandtl number that causes to an increases the fluid viscosity, because of this fluid becomes thicker due to viscosity increased, and as a result, fluid velocity decreased.

The behavior of Sc on velocity curve is depicted in Figure 7, against y, for different values of Sc, via CF, CPC and ABC operators. It is noticed that a decreasing effect on concentration in the boundary layer when the values of the Schmidt number enlarged. Physically, the relative influence of of momentum diffusivity to species diffusivity is the definition of Schmidt numberSc. It is noticed that, momentum diffusivity is quicker than species diffusivity when Sc is greater than one $(Sc>1),$ but it is reverse when Sc is less than one $(Sc<1),$ and in case of $(Sc=1),$ both species and momentum boundary layers have magnitude of the same order.

Figure 8, displays the modification in velocity field, for comparative study among the three best fractional operators, namely a novel hybrid fractional operator CPC, CF and ABC. The graphical behaviour and significant effect noticed on fluid velocity curve for $t=0.3$ and $t=3.0$ together with various parameters represents. The velocity graph is higher for CPC fractional operator as compared to CF and ABC, also the lowest velocity curves is observed for CF fractional operator for small time. It is noteworthy to point out that the graphical view of the fluid velocity, the ABC and CF fractional derivatives follow the velocity profile of CPC operator in respective order, but reverse order of operators observed for large time.

Figure 9, depicts the graphical behaviour for comparative study, taking different values of t, and significant effect noticed on fluid velocity curve together with different parameters. It is noteworthy to point out that the graphical view of the fluid velocity is higher. Further, it is observed that the same velocity field equations Eq. (74)(74) $$\begin{array}{l}\overline{u}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}+\left(\frac{G_r{\left(b_{3}s+\alpha \right)}^2\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)}{s\left(b_{2}s+\alpha \right)P_r-(b_{1}s+a_1)(b_{3}s+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}-e^{-y\sqrt{\frac{P_{r}s}{b_{3}s+\alpha }}}\right]\hfill \\ +\left(\frac{G_m{\left(b_{3}s+\alpha \right)}^2\left(\frac{1}{s}\right)}{s\left(b_{2}s+\alpha \right)S_c-(b_{1}s+a_1)(b_{3}s+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_{1}s+a_1}{b_{2}s+\alpha }}}-e^{-y\sqrt{\frac{S_{c}s}{b_{3}s+\alpha }}}\right].\hfill \end{array}$$(74) as Haq et al. (2021) investigated with exponential heating by removing the term Gm, i.e. $Gm=0.$ Also, we obtained the same velocity field equations Eq. (87)(87) $$\begin{array}{c}\overline{u}(y,s)=\frac{1}{s-\zeta }{e}^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}+\left(\frac{G_r{\left(b_3{s}^{\alpha }+\alpha \right)}^2\left(\frac{1}{s}-\frac{a_0}{s+b_0}\right)}{s^{\alpha }\left(b_2{s}^{\alpha }+\alpha \right)P_r-(b_1{s}^{\alpha }+a_1)(b_3{s}^{\alpha }+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}-e^{-y\sqrt{\frac{P_r{s}^{\alpha }}{b_3{s}^{\alpha }+\alpha }}}\right]\hfill \\ +\left(\frac{G_m{\left(b_3{s}^{\alpha }+\alpha \right)}^2\left(\frac{1}{s}\right)}{s^{\alpha }\left(b_2{s}^{\alpha }+\alpha \right)S_c-(b_1{s}^{\alpha }+a_1)(b_3{s}^{\alpha }+\alpha )}\right)\left[e^{-y\sqrt{\frac{b_1{s}^{\alpha }+a_1}{b_2{s}^{\alpha }+\alpha }}}-e^{-y\sqrt{\frac{S_c{s}^{\alpha }}{b_3{s}^{\alpha }+\alpha }}}\right].\hfill \end{array}$$(87) as Song et al. (2021) investigated with exponential heating by removing the term Gm, i.e. $Gm=0.$

5. Conclusion

In this paper, unsteady second grade fluid flow on an exponentially accelerated vertical plate along with exponentially variable velocity, energy and mass diffusion, embedded in a permeable medium is analyzed. For the sake of better rheology of differential type fluid, developed a fractional model by employing the new definition of CPC, CF and ABC fractional derivative operators and exact solution obtained by using Laplace integral transformation. For several physical significance of various system parameters, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, in the literature, it is observed that to derived analytical results from fractional fluid models developed by the various fractional operators, is difficult and this article contributing to answer the open problem of obtaining analytical solutions the fractionalized fluid models. Some essential major concluding observations obtained from the graphical analysis are summarized as follows:

• Mass concentration profile represented the decreasing behaviour as the values of Sc increases, this is because of the relative thickness of the hydrodynamic layer.

• The temperature distribution corresponding to smaller and larger values of Pr has disclosed quick and thicker heat diffusivity;

• The increasing values of Sc that causes to decline the velocity field.

• The growing values of Gr and Gm stimulates the velocity field.

• An increase in the values of Pr declines the velocity contour.

• It can be seen that fluid velocity graphs represents dual behavior for small and large time.

• It can be noticed that the second grade fluid velocity represents same behavior for small and large values of α via CPC, CF and ABC

• In comparison with Atangana Baleanu and Caputo Fabrizio fractional derivatives, the CPC fractional model is exceptionally suitable for simulating the history of velocity function.

• It is depicted that the fluid’s velocity for constant caputo-proportional hybrid derivative operator is greater than Atangana Baleanu and Caputo Fabrizio models respectively for small time but reverse order observed for large time.

Nomenclature
Symbol	=	Quantity (Units)
u	=	Non-dimensional velocity $(-)$
T	=	Dimensionless temperature $(-)$
C	=	Dimensionless concentration $(-)$
$T_w$	=	Temperature of the plate $(K)$
$T_{\infty }$	=	Temperature of fluid far away from the plat $(K)$
$C_w$	=	Concentration of the fluid near the plate $(kg{m}^{-3})$
$C_{\infty }$	=	Concentration of the fluid far away from the plate $(kg{m}^{-3})$
${\beta }_T$	=	Thermal expansion coefficient $(K^{-1})$
${\beta }_C$	=	Volumetric coefficient of concentration expansion $(K^{-1})$
$Pr$	=	Prandtl number $(-)$
$Sc$	=	Schmidt number $(-)$
$\mu$	=	Dynamic viscosity $(Kg{m}^{-1}{s}^{-1})$
$\upsilon$	=	Kinematic coefficient of viscosity $(m^2{s}^{-1})$
$Gr$	=	Thermal Grashof number $(-)$
$Gm$	=	Mass Grashof number $(-)$
g	=	Acceleration due to gravity $(m.s^{-2})$
$\rho$	=	Fluid density $(Kg{m}^{-3})$
$\sigma$	=	Electrical conductivity $(s{m}^{-1})$
${\alpha }_2$	=	Second grade fluid parameter $(-)$
M	=	Total Magnetic field $(-)$
$C_p$	=	Specific heat at constant pressure $(jK{g}^{-1}{K}^{-1})$
${\delta }_m$	=	Mass diffusivity $(m^2{s}^{-1})$
$u_0$	=	Characteristic velocity $(m{s}^{-1})$
$M_0$	=	Imposed Magnetic field $(W{m}^{-2})$
t	=	Time $(s)$
s	=	Laplace transform parameter $(s^{-1})$
k	=	Thermal conductivity of the fluid $(W{m}^{-2}{K}^{-1})$
P	=	Pressure $(N{m}^{-2})$
$k_1$	=	Coefficient of Rosseland absorption $(-)$
${\sigma }_1$	=	Stefan-Boltzmann constant $(W{m}^{-2}{K}^{-4})$
K	=	Permeability of porous medium $(m^2)$
$\alpha$	=	Fractional parameter $(-)$

Acknowledgement

The Author Muhammad Bilal Riaz is highly thankful to the Ministry of Education, Youth and Sports of the Czech Republic for their support through the e-INFRA CZ (ID:90254).

Disclosure statement

All the authors affirmed that they have no conflicts of interest.

Data availability statement

The data used in current study, available within the article that support the findings of the present research work.