A fractional study with Newtonian heating effect on heat absorbing MHD radiative flow of rate type fluid with application of novel hybrid fractional derivative operator

Abstract

The present work examines the analytical solutions of the mathematical fractional Maxwell fluid model near an infinitely vertical plate. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimensional form. For the sake of better rheology of rate type fluid, developed a fractional model by applying the new definition of Constant Proportional Caputo (CPC) fractional derivative operator that describe the generalized memory effects. For seeking exact solutions in terms of Mittag-Leffler functions for velocity and temperature, Laplace integral transformation technique is applied. For physical significance of various system parameters on fluid velocity and temperature distributions are demonstrated through various graphs by using graphical software. Furthermore, for being validated the acquired solutions, some limiting models such as ordinary Newtonian model had been recovered from fractional model. It is also analyzed that for Newtonian heating and non-uniform velocity conditions, the CPC fractional operator is the finest fractional model to describe the memory effect of velocity and energy distribution. 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:

• Analytical solution
• CPC derivative
• heat transfer
• Laplace transformation
• physical parameters system parameters

1. Introduction

The process of heat and mass transfer has a great importance from the industrial point of view. Many researchers and scientists concentrate on this area. In modern technologies and various industrial fields, the non-Newtonian fluid theory has extensive impact because Newtonian fluid model cannot express many flow characteristics. A non-Newtonian fluid that obeys the nonlinear relationships between the rate of shear strain and the shear stress. The non-Newtonian fluid theory has significant utilization in modern engineering, especially in petroleum industry used to extract crude oil from different petroleum productions. The properties of Newtonian fluid in most of the cases are not valid but scientists desire to model the complex models for non-Newtonian fluid. The importance of non-Newtonian fluid has been enlarged from the last few decades, specifically in the research field. The non-Newtonian fluids have numerous ever-increasing applications in industrial sectors, but some specific are mentioned here, such as at large-scales reducing and enhancing heating/cooling systems, biochemical and process engineering, extrusion of molten plastic in industry, reducing oil pipeline friction, polymer processing, reducing fluid friction, well drilling, flow tracers, biological materials, biomedical flow analysis, plastic foam processing, lubrication processes, food processing industries, chemical processing, all emulsions, handling of muds, slurries and complex mixtures. Many researchers and scientists focused on non-Newtonian fluid while considering different fluid geometries. Therefore, simulating and modeling the flow phenomena of non-Newtonian fluid that is facilitated and play the important role in human life. 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, Lu, & Siddiqui, 2019; Khan et al., 2019b; Rehman, Riaz, Saeed, & Yao, 2021b; Riaz et al., 2021a; Riaz, Awrejcewicz, & Rehman, 2021b), that different fluid models exists in the literature have various characteristics or certain limitations, for instance, second grade fluid model efficiently explained the elasticity but does not discuss the viscosity, 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. Systematic analysis of such fluid flow models have significantly important for theoretical studies and practical implementations in modernistic mechanization. Among several proposed mathematical models for such fluids (non-Newtonian), Maxwell fluid attracted special attention because of its simplicity, also that can be predicted stress relaxation, 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. Maxwell fluid model was initially proposed by James Clerk Maxwell in 1867, and James G. Oldroyd popularized the idea, a few years later (Adegbie, Omowaye, Disu, & Animasaun, 2015; Mackosko, 1994), with an aim to predict the visco elastic behavior of air (Maxwell, 1867). 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. The study of Maxwell fluid movement in the context of fluid mechanics, was explored by several mathematicians, scientists, researchers and engineers that depends upon various situations because of its naturalness. Khan et al., (2019a) described unsteady, MHD natural convectional transport of Maxwell model with computational aspects subject to thermal stratification. Farooq et al. (2019) has been examined the magneto hydrodynamic flow of Maxwell fluid velocity in presence of exponentially stretching surface. Megahed (2021) studied first time the Maxwell fluid movement subject to convective boundary conditions, through permeable stretching sheet and analyzed the results theoretically. Riaz, Awrejcewicz, Rehman, and Abbas (2021c) performed a systematic study to analyzed the effects of MHD Maxwell fluid, thermal flux and porosity on mass and energy transfer. Aman, Al-Mdallal, and Khan (2020) investigated the generalized peristaltic Maxwell fluid flow in a permeable channel subject to second order slip effects. Shafiq and Khalique (2020) examined the heat transfer phenomenon of MHD Maxwell fluid that passed over the stretching plate using Lie group methods.

The fractional/differential calculus is an eminent mathematical field that growing immensely due to enormous significance investigates the non-integer order behavior of integrals and derivatives as well as their applications and properties. The concept of differential calculus is old like classical calculus, first time in 1695, new idea about fractional calculus introduced when a letter from Leibniz to L ${}^{\prime }$ Hospital was written. This field attracted the attention of well-known mathematicians, researchers and scientists that proposed and built different fractional integrals and fractional derivatives. The researchers faced too much difficulties to developed a real physical phenomenon by employing the traditional calculus techniques, then the fractional differential equations have great importance for mathematicians and researchers. Since it has been investigating numerous physical models in different scientific fields such as biology, physics, chemistry, acoustic waves, finance, control theory, fractal dynamics, signal processing, hydro magnetic waves, diffusion reaction process, anomalous transport, fluid flow problems, engineering processes, oscillation, dynamical 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, Freihat, Arqub, & Shawagfeh, 2015; Momani, Freihat, & Al-Smadi, 2014), Riccati-Bernouli sub-ordinary differential equation Sub-ODE technique (RBSODET) (Alabedalhadi, Al-Smadi, Al-Omari, Baleanu, & Momani, 2020), reproducing the kernel Hilbert space method (Altawallbeh, Al-Smadi, Komashynska, & Ateiwi, 2018; Al-Smadi, Djeddi, Momani, Al-Omari, & Araci, 2021), simple equation modification method (Islam & Akbar, 2018), residual power series method (Al-Smadi, Arqub, & Hadid, 2020) and several others (Al-Smadi, Arqub, & Gaith, 2021; Hasan et al., 2021; Khan, 2021; Momani, Djeddi, Al-Smadi, & Al-Omari, 2021). 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. The effects of fractional order of derivative on the wave solutions of ZK-Burgers equation via exact and analytical approximate solution techniques are discussed by Faraz, Sadaf, Akram, Zainab, and Khan (2021). The unsteady fractional Phan Thien Tanner fluid is modeled by Faraz, Khan, and Anjum (2020). Fardi & Khan (2021) studied a finite difference-spectral method to a mobile/immobile fractal transport model formulated with the concept of a fractional derivative of Caputo–Fabrizio. 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; Riaz, Awrejcewicz, Rehman, & Akgül, 2021d, Riaz, Rehman, Awrejcewicz, & Akgül, 2021e; Rehman, Riaz, Rehman, Awrejcewicz, & Baleanu, 2022; Rehman, Shah, & Riaz, 2021a; Yang, Abdel-Aty, & Cattani, 2019). 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.

In the previous investigation, Kumam et al. (2022) discussed the flow of fractional version of rate type fluid model by using different fractional operators namely Caputo, CF, ABC, and computed solution for each fractional model by jointly applied the Stehfest’s numerical algorithm and Laplace transformation, because it has efficient applications for non-uniform boundary conditions, but not computed the exact solution for the presented model. But in the literature fractional Maxwell fluid model with fractional operator Constant Proportional Caputo, along with the set of ramped conditions for velocity with Newtonian heating, saturated in porous media, are not investigated yet nor published. To fill this gape a new fractional Maxwell model developed under effectively applied ramped conditions for velocity field and Newtonian heating for energy distribution. Further, in the presented model, a new fractional operator employed to fractionalized the velocity and energy equations together with Newtonian heating and the ramped wall conditions, applying the definition of recently introduced a new fractional derivative operator, namely Constant Proportional Caputo operator having non-local and singular kernel. Owning to such interest, for better rheology of rate type fluid, developed a fractional model by employing the new definition of Constant Proportional Caputo fractional derivative operator that describe the generalized memory effects. For seeking exact solution expressions in terms of Mittag-Leffler functions, for fluid velocity and temperature, Laplace integral transformation method is used to solve the fractional model. For physical analysis the influence of parameters like as order of CPC fractional order $\alpha ,$ porosity parameter K, Maxwell fluid parameter $\lambda ,$ Prandtl parameter Pr, magnetic number M, heat injection/suction parameter Q, Grashof number Gr and the radiation parameter Nr are portrayed graphically by using Mathcad software. Furthermore, for validation the current result, limiting model such as fractional Newtonian model obtained from CPC fractional Maxwell model.

2. Mathematical model

Consider the Maxwell fluid flow over an infinite erected plate, having length infinite, that is embedded in a porous media. The plate is considered at $\varphi =0$ and the fluid flow is restrained to $\varphi >0,$ in the direction that is along to the plate (as exhibited in Figure 1). By considering the following key assumptions that are supposed to govern the Maxwell fluid flow model:

Figure 1. Schematic drawing of the flow model.

• The flow is unidirectional and one-dimensional.

• To omit the impact of an induction magnetic field Reynolds number is considered small enough.

• A uniform magnetic force of lines with magnitude $B_0$ is imposed in the direction which is perpendicular to the plate.

• It is considered that no electric force is applied to avoid the polarization influence of fluid,

• Suppose the $Q_r$ (radiative heat flux) is negligible which is in the direction that is too the plate corresponding to the radiative thermal flux that is in the normal direction of the plate.

• The energy equation without viscous dissipation term is considered.

Initially, for time $t=0,$ the fluid and plate both are in the static mode, having ambient temperature $T_{\infty }.$ Later, when time $t=0^{+},$ ramped boundary conditions for both temperature and velocity are considered at the wall such that $\varphi =0,$ the wall temperature is $T_w,$ $u(\varphi ,t)$ is taken as the velocity component along x-axis with $u_0$ is the characteristic velocity. For Maxwell fluid, ramped boundary conditions are considered for velocity, but thermal conditions are defined in terms of Newtonian heating, these conditions are assumed first time, though they have numerous applications in medical sciences and modern industrial field. Further, the velocity field satisfies the equation of continuity in the presence of these factors. By considering the all above mentioned assumptions, the following principal equations for Maxwell fluid under Boussinesq’s approximation, for velocity and energy transfer are obtained as (Riaz et al., 2021e; Rajagopal, Ruzicka, & Srinivasa, 1996):

The momentum and energy equations are given below: (1) $$\begin{array}{cc}\left(1+{\lambda }_n\frac{\partial }{\partial t}\right)\frac{\partial u(\varphi ,t)}{\partial t}=\frac{\mu }{\rho }\frac{{\partial }^{2}u(\varphi ,t)}{\partial {\varphi }^2}+g{\beta }_T\left(1+{\lambda }_n\frac{\partial }{\partial t}\right)\left(T(\varphi ,t)-T_{\infty }\right)\hfill & \hfill \\ -\frac{\sigma }{\rho }{B}_0{}^2\left(1+{\lambda }_n\frac{\partial }{\partial t}\right)u(\varphi ,t)-\frac{\mu }{\rho }\frac{\zeta }{k_p}u(\varphi ,t),\hfill & \hfill \end{array}$$(1) (2) $$\begin{array}{cc}\frac{\partial T(\varphi ,t)}{\partial t}=-\frac{1}{\rho C_p}\frac{\partial q(\varphi ,t)}{\partial \varphi }-\frac{\partial Q_r}{\partial \varphi },\hfill & \hfill \\ \left[Q_r=-\frac{4{\sigma }_1}{3k_1}\frac{\partial T^4}{\partial \varphi };T^4\approx 4T_{\infty }^{3}T-3T_{\infty }^4\right].\hfill & \hfill \end{array}$$(2)

The Fourier’s Law of thermal flux are written as: (3) $$q(\varphi ,t)=-k\frac{\partial T(\varphi ,t)}{\partial \varphi }.$$(3) with associated initial and boundary conditions (4) $$\begin{array}{l}u(\varphi ,0)=0,\hspace{1em}T(\varphi ,0)=T_{\infty },\hspace{1em}\varphi \ge 0,\\ u(0,t)=\{\begin{array}{ll}{u}_0\frac{t}{t_0},\hfill & \hspace{1em}0<t\le t_0;\hfill \\ u_0,\hfill & \hspace{1em}t>t_0\hfill \end{array},\hspace{1em}\frac{\partial T(\varphi ,t)}{\partial \varphi }{|}_{\varphi =0}=-\frac{h}{k}T(0,t),\\ u(\varphi ,t)\to 0,\hspace{1em}T(\varphi ,t)\to T_{\infty },\hspace{1em}\mathit{\text{as\hspace{1em}ϕ}}\to \infty \hspace{1em}\mathit{and}\hspace{1em}t>0.\end{array}$$(4)

To reduce the number of involving parameters, introducing the following set of unit-free quantities (5) $$\begin{array}{cc}{t}^{\ast }=\upsilon {\left(\frac{h}{k}\right)}^{2}t,\hspace{1em}{\varphi }^{\ast }=\left(\frac{h}{k}\right)\varphi ,\hspace{1em}{u}^{\ast }=\frac{u}{u_0},\hspace{1em}{\lambda }_n{}^{\ast }=\upsilon {\left(\frac{h}{k}\right)}^2{\lambda }_n,\hfill & \hfill \\ q^{\ast }=\frac{q}{q_0},\hspace{1em}{T}^{\ast }=\frac{T-T_{\infty }}{T_{\infty }},\hspace{1em}{q}_0=\frac{k{u}_0{T}_{\infty }}{\upsilon }.\hfill & \hfill \end{array}$$(5) when substituting the Equation (5)(5) $$\begin{array}{cc}{t}^{\ast }=\upsilon {\left(\frac{h}{k}\right)}^{2}t,\hspace{1em}{\varphi }^{\ast }=\left(\frac{h}{k}\right)\varphi ,\hspace{1em}{u}^{\ast }=\frac{u}{u_0},\hspace{1em}{\lambda }_n{}^{\ast }=\upsilon {\left(\frac{h}{k}\right)}^2{\lambda }_n,\hfill & \hfill \\ q^{\ast }=\frac{q}{q_0},\hspace{1em}{T}^{\ast }=\frac{T-T_{\infty }}{T_{\infty }},\hspace{1em}{q}_0=\frac{k{u}_0{T}_{\infty }}{\upsilon }.\hfill & \hfill \end{array}$$(5) into Equation (1)(1) $$\begin{array}{cc}\left(1+{\lambda }_n\frac{\partial }{\partial t}\right)\frac{\partial u(\varphi ,t)}{\partial t}=\frac{\mu }{\rho }\frac{{\partial }^{2}u(\varphi ,t)}{\partial {\varphi }^2}+g{\beta }_T\left(1+{\lambda }_n\frac{\partial }{\partial t}\right)\left(T(\varphi ,t)-T_{\infty }\right)\hfill & \hfill \\ -\frac{\sigma }{\rho }{B}_0{}^2\left(1+{\lambda }_n\frac{\partial }{\partial t}\right)u(\varphi ,t)-\frac{\mu }{\rho }\frac{\zeta }{k_p}u(\varphi ,t),\hfill & \hfill \end{array}$$(1) and Equation (2)(2) $$\begin{array}{cc}\frac{\partial T(\varphi ,t)}{\partial t}=-\frac{1}{\rho C_p}\frac{\partial q(\varphi ,t)}{\partial \varphi }-\frac{\partial Q_r}{\partial \varphi },\hfill & \hfill \\ \left[Q_r=-\frac{4{\sigma }_1}{3k_1}\frac{\partial T^4}{\partial \varphi };T^4\approx 4T_{\infty }^{3}T-3T_{\infty }^4\right].\hfill & \hfill \end{array}$$(2) , and dropping the asterisk $\ast$ from newly obtained equations, then we have the dimensionless governing system of PDEs of the considered model as follows: (6) $$\begin{array}{ll}(1+{\lambda }_n\frac{\partial }{\partial t})\frac{\partial u(\varphi ,t)}{\partial t}\hfill & =\frac{{\partial }^{2}u(\varphi ,t)}{\partial {\varphi }^2}-[M(1+{\lambda }_n\frac{\partial }{\partial t})+\frac{1}{K}]u(\varphi ,t)+Gr(1+{\lambda }_n\frac{\partial }{\partial t})T(\varphi ,t),\hfill \end{array}$$(6) (7) $$\frac{\partial T(\varphi ,t)}{\partial t}=-\left(\frac{1+Nr}{Pr}\right)\frac{\partial q(\varphi ,t)}{\partial \varphi },$$(7) (8) $$q(\varphi ,t)=-\frac{\partial T(\varphi ,t)}{\partial \varphi },$$(8)

Along with the analogous initial and boundary conditions when applied the new quantities mentioned in Equation (5)(5) $$\begin{array}{cc}{t}^{\ast }=\upsilon {\left(\frac{h}{k}\right)}^{2}t,\hspace{1em}{\varphi }^{\ast }=\left(\frac{h}{k}\right)\varphi ,\hspace{1em}{u}^{\ast }=\frac{u}{u_0},\hspace{1em}{\lambda }_n{}^{\ast }=\upsilon {\left(\frac{h}{k}\right)}^2{\lambda }_n,\hfill & \hfill \\ q^{\ast }=\frac{q}{q_0},\hspace{1em}{T}^{\ast }=\frac{T-T_{\infty }}{T_{\infty }},\hspace{1em}{q}_0=\frac{k{u}_0{T}_{\infty }}{\upsilon }.\hfill & \hfill \end{array}$$(5) for non-dimensionalization process, are stated as: (9) $$u(\varphi ,0)=0,\hspace{1em}T(\varphi ,0)=0,\hspace{1em}\mathit{for}\hspace{1em}\varphi \ge 0,$$(9) (10) $$u(0,t)=\{\begin{array}{cc}t\hfill & \hspace{1em}0<t\le 1\hfill \\ 1\hfill & \hspace{1em}t>1\hfill \end{array},\hspace{1em}\frac{\partial T(\varphi ,t)}{\partial \varphi }{|}_{\varphi =0}=-\left[T(0,t)+1\right],$$(10) (11) $$u(\varphi ,t)\to 0,\hspace{1em}T(\varphi ,t)\to 0\mathit{\text{\hspace{1em}as\hspace{1em}ϕ}}\to \infty \hspace{1em}\mathit{and}\hspace{1em}t>0.$$(11) where $$\begin{array}{c}Gr=\frac{g{\beta }_T{T}_{\infty }}{\mu u_0}{\left(\frac{h}{k}\right)}^2,\hspace{1em}Pr=\frac{\mu C_p}{k},\hspace{1em}M=\frac{\sigma B_0^2}{\upsilon \rho }{\left(\frac{h}{k}\right)}^2,\\ N_r=\frac{16{\sigma }_1{T}_{\infty }^3}{3k{k}_1},\hspace{1em}\theta =\frac{P_r}{1+N_r},\hspace{1em}\frac{1}{K}=\frac{\zeta }{k_p}{\left(\frac{h}{k}\right)}^2.\end{array}$$ where, Gr represents Grashof number, Pr is denoted by Prandtl number, Nr is radiation parameter, M represents magnetic number, kp is permeability, k is thermal conductivity, $k_1$ is the coefficient of Rosseland absorption, ${\sigma }_1$ Stefan-Boltzmann constant, $\zeta$ is porosity, Qr is radiative heat flux and K is defined as porosity.

3. Preliminaries

The Constant Proportional Caputo (CPC) hybrid fractional operator used in this work, that is developed recently by Dumitru et al. (Baleanu, Fernandez, & Akgül, 2020). This newly developed fractional operator is a linear combination of two fractional operators, namely Constant proportional and Caputo fractional derivative operator, due to this reason it is also called hybrid fractional operator. CPC-fractional derivative operator of order $\alpha$ is described as: (12) $${}^{\mathit{CPC}}{D}_{\eta }^{\alpha }f\left(\psi ,\eta \right)=\frac{1}{\Gamma (1-\alpha )}{\int }_0^{\eta }(k_1(\alpha )f(\psi ,\tau )+k_0(\alpha )\frac{\partial f(\psi ,\tau )}{\partial \tau }){(\eta -\tau )}^{-\alpha }d\tau ,\hspace{1em}0<\alpha <1.$$(12)

Laplace transformation of Constant proportional Caputo hybrid time fractional operator is written as: (13) $$\mathcal{L}\left({}^{\mathit{CPC}}{D}_{\eta }^{\alpha }f\left(\psi ,\eta \right)\right)=\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\mathcal{L}\left(f(\psi ,\eta )\right)-k_0(\alpha )s^{\alpha -1}f(\psi ,0).$$(13) where $\alpha$ is used to represent the fractional parameter and Laplace transform parameter is denoted by s.

4. Solution of the problem

In the present article, introducing a novel mathematical model named as CPC fractional operator which generalized the thermal memory effects. The time-fractional Maxwell fluid equation for velocity and energy based on Constant Proportional Caputo derivative operator are given as: (14) $$\begin{array}{c}\left(1+{\lambda }_n^{\mathit{CPC}}{D}_t^{\alpha }\right)\frac{\partial u(\varphi ,t)}{\partial t}\hfill \\ =\frac{{\partial }^{2}u(\varphi ,t)}{\partial {\varphi }^2}-\left[M\left(1+{\lambda }_n^{\mathit{CPC}}{D}_t^{\alpha }\right)+\frac{1}{K}\right]u(\varphi ,t)\hfill \\ +Gr\left(1+{\lambda }_n^{\mathit{CPC}}{D}_t^{\alpha }\right)T(\varphi ,t),\hfill & \hfill \end{array}$$(14) (15) $${}^{\mathit{CPC}}{D}_t^{\alpha }T(\varphi ,t)=\frac{1}{\theta }\frac{{\partial }^{2}T(\varphi ,t)}{\partial {\varphi }^2}.$$(15) where ${}^{\mathit{CPC}}{D}_t^{\alpha }$ represents Constant Proportional Caputo derivative operator and further results with properties regarding CPC derivative operator are given in (Baleanu et al., 2020).

4.1. Solution of temperature equation by using CPC derivative operator

To derive the solution for energy Equation (15)(15) $${}^{\mathit{CPC}}{D}_t^{\alpha }T(\varphi ,t)=\frac{1}{\theta }\frac{{\partial }^{2}T(\varphi ,t)}{\partial {\varphi }^2}.$$(15) with appropriate non-dimensional conditions Equations (9)–(11)(11) $$u(\varphi ,t)\to 0,\hspace{1em}T(\varphi ,t)\to 0\mathit{\text{\hspace{1em}as\hspace{1em}ϕ}}\to \infty \hspace{1em}\mathit{and}\hspace{1em}t>0.$$(11) , employing the technique of Laplace transformation in view of Equation (13)(13) $$\mathcal{L}\left({}^{\mathit{CPC}}{D}_{\eta }^{\alpha }f\left(\psi ,\eta \right)\right)=\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\mathcal{L}\left(f(\psi ,\eta )\right)-k_0(\alpha )s^{\alpha -1}f(\psi ,0).$$(13) as follows: (16) $$\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\overline{T}(\varphi ,s)-k_0(\alpha )s^{\alpha -1}\overline{T}(\varphi ,0)=\frac{1}{\theta }\frac{{\partial }^2\overline{T}(\varphi ,s)}{\partial {\varphi }^2}.$$(16) with transformed initial and boundary conditions are (17) $$\overline{T}(\varphi ,0)=0,\hspace{1em}\frac{\partial \overline{T}(\varphi ,s)}{\partial \varphi }{|}_{\varphi =0}=-\left[\overline{T}(0,s)+\frac{1}{s}\right]\hspace{1em}\mathit{and}\hspace{1em}\overline{T}(\varphi ,s)\to 0\mathit{\text{\hspace{1em}as\hspace{1em}ϕ}}\to \infty .$$(17)

The definition of Laplace integral transformation of any function $X(\varphi ,t)$ is denoted by $\overline{X}(\varphi ,s),$ are expressed mathematically as: $$\overline{X}(\varphi ,s)={\int }_0^{\infty }X(\varphi ,t)e^{-st}dt.$$

The solution for energy Equation (16)(16) $$\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\overline{T}(\varphi ,s)-k_0(\alpha )s^{\alpha -1}\overline{T}(\varphi ,0)=\frac{1}{\theta }\frac{{\partial }^2\overline{T}(\varphi ,s)}{\partial {\varphi }^2}.$$(16) is written as: (18) $$\overline{T}(\varphi ,s)=e_1{e}^{-\varphi \sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}+e_2{e}^{\varphi \sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(18)

The temperature solution of Equation (18)(18) $$\overline{T}(\varphi ,s)=e_1{e}^{-\varphi \sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}+e_2{e}^{\varphi \sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}.$$(18) which satisfies the boundary conditions Equation (17)(17) $$\overline{T}(\varphi ,0)=0,\hspace{1em}\frac{\partial \overline{T}(\varphi ,s)}{\partial \varphi }{|}_{\varphi =0}=-\left[\overline{T}(0,s)+\frac{1}{s}\right]\hspace{1em}\mathit{and}\hspace{1em}\overline{T}(\varphi ,s)\to 0\mathit{\text{\hspace{1em}as\hspace{1em}ϕ}}\to \infty .$$(17) is given by (19) $$\begin{array}{cc}\overline{T}(\varphi ,s)=\frac{1}{s}\frac{1}{\left(\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1\right)}{e}^{-\varphi \sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}},\hfill & \hfill \\ =\overline{T_1}(\varphi ,s)\ast \overline{T_2}(\varphi ,s).\hfill & \hfill \end{array}$$(19)

The inverse Laplace of the above Equation (19)(19) $$\begin{array}{cc}\overline{T}(\varphi ,s)=\frac{1}{s}\frac{1}{\left(\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1\right)}{e}^{-\varphi \sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}},\hfill & \hfill \\ =\overline{T_1}(\varphi ,s)\ast \overline{T_2}(\varphi ,s).\hfill & \hfill \end{array}$$(19) , the required temperature field solution is given by (20) $$T(\varphi ,t)={\int }_0^t{T}_1(\varphi ,\tau )T_2(\varphi ,t-\tau )d\tau .$$(20) where $T_1(\varphi ,t)$ and $T_2(\varphi ,t)$ are obtained as $$\begin{array}{cc}{T}_1(\varphi ,t)={\mathcal{L}}^{-1}\left\{\overline{T_1}(\varphi ,s)\right\}={\mathcal{L}}^{-1}\left\{\frac{1}{\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1}\right\},\hfill & \hfill \\ T_1(\varphi ,t)=-{\mathcal{L}}^{-1}\left\{\sum _{n=0}^{\infty }{(\theta )}^{\frac{n}{2}}{\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]}^{\frac{n}{2}}{s}^{\frac{n\alpha }{2}}\right\},\hfill & \hfill \\ =-\sum _{n=0}^{\infty }{(\theta )}^{\frac{n}{2}}{(k_0(\alpha ))}^{\frac{n}{2}}{t}^{\frac{-n\alpha }{2}-1}{E}_{1,\frac{-n\alpha }{2}}^{-\frac{n}{2}}\left(-\frac{k_1(\alpha )}{k_0(\alpha )}t\right),\hfill & \hfill \\ T_2(\varphi ,t)={\mathcal{L}}^{-1}\left\{\overline{T_2}(\varphi ,s)\right\}={\mathcal{L}}^{-1}\left\{\frac{1}{s}{e}^{-\varphi \sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}\right\},\hfill & \hfill \\ ={\mathcal{L}}^{-1}\left\{\sum _{n=0}^{\infty }\frac{{(-\varphi )}^n{(\theta )}^{\frac{n}{2}}{(k_0(\alpha ))}^{\frac{n}{2}}}{n!}.\frac{1}{s^{\frac{n\alpha }{2}+1}{\left(1+\frac{k_1(\alpha )}{k_0(\alpha )}{s}^{-1}\right)}^{\frac{-n}{2}}}\right\},\hfill & \hfill \\ =\sum _{n=0}^{\infty }\frac{{(-\varphi )}^n{(\theta )}^{\frac{n}{2}}{(k_0(\alpha ))}^{\frac{n}{2}}}{n!}.t^{\frac{n\alpha }{2}}{E}_{1,\frac{n\alpha }{2}+1}^{-\frac{n}{2}}\left(-\frac{k_1(\alpha )}{k_0(\alpha )}t\right).\hfill & \hfill \end{array}$$ by using ${\mathcal{L}}^{-1}\left\{\frac{s^{-\beta }}{{(1-\wp s^{-\alpha })}^{\gamma }}\right\}=t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }(\wp t^{\alpha })$

4.2. Solution of velocity field by using CPC derivative operator

To derive the solution for velocity field Equation (14)(14) $$\begin{array}{c}\left(1+{\lambda }_n^{\mathit{CPC}}{D}_t^{\alpha }\right)\frac{\partial u(\varphi ,t)}{\partial t}\hfill \\ =\frac{{\partial }^{2}u(\varphi ,t)}{\partial {\varphi }^2}-\left[M\left(1+{\lambda }_n^{\mathit{CPC}}{D}_t^{\alpha }\right)+\frac{1}{K}\right]u(\varphi ,t)\hfill \\ +Gr\left(1+{\lambda }_n^{\mathit{CPC}}{D}_t^{\alpha }\right)T(\varphi ,t),\hfill & \hfill \end{array}$$(14) with appropriate non-dimensional conditions Equations (9)–(11)(11) $$u(\varphi ,t)\to 0,\hspace{1em}T(\varphi ,t)\to 0\mathit{\text{\hspace{1em}as\hspace{1em}ϕ}}\to \infty \hspace{1em}\mathit{and}\hspace{1em}t>0.$$(11) , employing the Laplace transformation method in view of Equation (13)(13) $$\mathcal{L}\left({}^{\mathit{CPC}}{D}_{\eta }^{\alpha }f\left(\psi ,\eta \right)\right)=\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\mathcal{L}\left(f(\psi ,\eta )\right)-k_0(\alpha )s^{\alpha -1}f(\psi ,0).$$(13) as follows: (21) $$\begin{array}{cc}\left(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)s\frac{\partial \overline{u}(\varphi ,s)}{\partial s}=\frac{d^2\overline{u}(\varphi ,s)}{d{\varphi }^2}\hfill & \hfill \\ -\left[M\left(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)+\frac{1}{K}\right]\overline{u}(\varphi ,s)\hfill & \hfill \\ +Gr\left(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)\overline{T}(\varphi ,s),\hfill & \hfill \end{array}$$(21) with (22) $$\overline{u}(\varphi ,0)=0,\hspace{1em}\overline{u}(0,s)=\frac{1-e^{-s}}{s^2}\hspace{1em}\mathit{and}\hspace{1em}\overline{u}(\varphi ,s)\to 0\mathit{\text{\hspace{1em}as\hspace{1em}ϕ}}\to \infty .$$(22) substituting the value of $\overline{T}(\varphi ,s)$ from Equation (19)(19) $$\begin{array}{cc}\overline{T}(\varphi ,s)=\frac{1}{s}\frac{1}{\left(\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1\right)}{e}^{-\varphi \sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}},\hfill & \hfill \\ =\overline{T_1}(\varphi ,s)\ast \overline{T_2}(\varphi ,s).\hfill & \hfill \end{array}$$(19) in Equation (21)(21) $$\begin{array}{cc}\left(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)s\frac{\partial \overline{u}(\varphi ,s)}{\partial s}=\frac{d^2\overline{u}(\varphi ,s)}{d{\varphi }^2}\hfill & \hfill \\ -\left[M\left(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)+\frac{1}{K}\right]\overline{u}(\varphi ,s)\hfill & \hfill \\ +Gr\left(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }\right)\overline{T}(\varphi ,s),\hfill & \hfill \end{array}$$(21) , then after manipulation the solution written in the form (23) $$\begin{array}{cc}\overline{u}(\varphi ,s)=e_5{e}^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}+e_6{e}^{\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}\hfill & \hfill \\ +\frac{Gr}{s}\left(\frac{1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}{\sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}-1}\right)\times \hfill & \hfill \\ \left[\frac{e^{-\varphi \sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}}}{((M+s)(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha })+\frac{1}{K})-\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}\right].\hfill & \hfill \end{array}$$(23)

The velocity solution of Equation (23)(23) $$\begin{array}{cc}\overline{u}(\varphi ,s)=e_5{e}^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}+e_6{e}^{\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}\hfill & \hfill \\ +\frac{Gr}{s}\left(\frac{1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}{\sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}-1}\right)\times \hfill & \hfill \\ \left[\frac{e^{-\varphi \sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}}}{((M+s)(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha })+\frac{1}{K})-\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}\right].\hfill & \hfill \end{array}$$(23) which satisfies the boundary conditions Equation (22)(22) $$\overline{u}(\varphi ,0)=0,\hspace{1em}\overline{u}(0,s)=\frac{1-e^{-s}}{s^2}\hspace{1em}\mathit{and}\hspace{1em}\overline{u}(\varphi ,s)\to 0\mathit{\text{\hspace{1em}as\hspace{1em}ϕ}}\to \infty .$$(22) is given by (24) $$\begin{array}{cc}\overline{u}(\varphi ,s)=(\frac{1-e^{-s}}{s^2})e^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}\hfill & \hfill \\ +\frac{Gr}{s}\left(\frac{1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}{\sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}-1}\right)\times \hfill & \hfill \\ \left[\frac{e^{-\varphi \sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}}-e^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}}{((M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K})-\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}\right].\hfill & \hfill \end{array}$$(24)

To find Laplace inverse of the above Equation (24)(24) $$\begin{array}{cc}\overline{u}(\varphi ,s)=(\frac{1-e^{-s}}{s^2})e^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}\hfill & \hfill \\ +\frac{Gr}{s}\left(\frac{1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}{\sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}-1}\right)\times \hfill & \hfill \\ \left[\frac{e^{-\varphi \sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}}-e^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}}{((M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K})-\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}\right].\hfill & \hfill \end{array}$$(24) , first we write it in the following form: (25) $$\overline{u}(\varphi ,s)=\overline{\Omega }(\varphi ,s)+Gr[\overline{\Phi }(\varphi ,s)+\overline{\Psi }(\varphi ,s)][\overline{T}(\varphi ,s)-\overline{T_1}(\varphi ,s)\overline{\varpi }(\varphi ,s)].$$(25) and (26) $$\overline{\Omega }(\varphi ,s)={\overline{\Omega }}_1(\varphi ,s)-e^{-s}{\overline{\Omega }}_1(\varphi ,s).$$(26)

The inverse Laplace of the above Equation (26)(26) $$\overline{\Omega }(\varphi ,s)={\overline{\Omega }}_1(\varphi ,s)-e^{-s}{\overline{\Omega }}_1(\varphi ,s).$$(26) , is obtained as: (27) $$\Omega (\varphi ,t)={\Omega }_1(\varphi ,t)-{\Omega }_1(\varphi ,t)P(t-1).$$(27)

In the above expression $P(t-1)$ represents a Heaviside function.

where $$\begin{array}{l}{\Omega }_1(\varphi ,t)={\mathcal{L}}^{-1}\left\{{\overline{\Omega }}_1(\varphi ,s)\right\}={\mathcal{L}}^{-1}\left\{\frac{1}{s^2}{e}^{-\varphi \sqrt{(a+s)+(M+s)({\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })}}\right\},\\ ={\mathcal{L}}^{-1}\left\{\sum _{n=0}^{\infty }\sum _{j=0}^{\infty }\sum _{m=0}^{\infty }\sum _{r=0}^{\infty }\frac{{(-\varphi )}^n{({\lambda }_n{k}_0(\alpha ))}^j{(a)}^{\frac{n}{2}-j-r}{(M)}^{j-m}}{n!r!m!\Gamma (\frac{n}{2}-j-r+1)\Gamma (j-m+1)}.\frac{\Gamma (\frac{n}{2}+1)}{s^{2-(m+r+\alpha j)}{(1+\frac{k_1(\alpha )}{k_0(\alpha )}{s}^{-1})}^{-j}}\right\}\\ =\sum _{n=0}^{\infty }\sum _{j=0}^{\infty }\sum _{m=0}^{\infty }\sum _{r=0}^{\infty }\frac{{(-\varphi )}^n{({\lambda }_n{k}_0(\alpha ))}^j{(a)}^{\frac{n}{2}-j-r}{(M)}^{j-m}\Gamma (\frac{n}{2}+1)}{n!r!m!\Gamma (\frac{n}{2}-j-r+1)\Gamma (j-m+1)}{t}^{1-(m+r+\alpha j)}{E}_{1,2-(m+r+\alpha j)}^{-j}(-\frac{k_1(\alpha )}{k_0(\alpha )}t)\\ \Phi (\varphi ,t)={\mathcal{L}}^{-1}\left\{\overline{\Phi }(\varphi ,s)\right\}={\mathcal{L}}^{-1}\left\{\frac{1}{(a+s)+({\lambda }_{n}s-b)[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}\right\},\\ ={\mathcal{L}}^{-1}\left\{\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\sum _{r=0}^{\infty }\frac{{(-1)}^{n+m}{(-b)}^{n-r}{({\lambda }_n)}^r{(a)}^{1-n-m}\Gamma (n+m+1)}{m!r!\Gamma (n-r+1)\Gamma (n-m+1)}.\frac{{(k_0(\alpha ))}^n}{s^{-(m+r+n\alpha )}{(1+\frac{k_1(\alpha )}{k_0(\alpha )}{s}^{-1})}^{-n}}\right\},\\ =\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\sum _{r=0}^{\infty }\frac{{(-1)}^{n+m}{(-b)}^{n-r}{({\lambda }_n)}^r{(a)}^{1-n-m}{(k_0(\alpha ))}^n\Gamma (n+m+1)}{m!r!\Gamma (n-r+1)\Gamma (n-m+1)}\\ \times t^{-(m+r+n\alpha +1)}{E}_{1,-(m+r+n\alpha )}^{-n}(-\frac{k_1(\alpha )}{k_0(\alpha )}t),\\ \Psi (\varphi ,t)={\mathcal{L}}^{-1}\left\{\overline{\Psi }(\varphi ,s)\right\}={\mathcal{L}}^{-1}\left\{\frac{{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}{(a+s)+({\lambda }_{n}s-b)[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}\right\},\\ ={\mathcal{L}}^{-1}\left\{\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\sum _{r=0}^{\infty }\frac{{(-1)}^{n+m}{(-b)}^{n-r}{({\lambda }_n)}^{r+1}{(a)}^{1-n-m}}{m!r!\Gamma (n-r+1)\Gamma (n-m+1)}.\frac{\Gamma (n+m+1){(k_0(\alpha ))}^{n+1}}{s^{-(m+r+(n+1)\alpha )}{(1+\frac{k_1(\alpha )}{k_0(\alpha )}{s}^{-1})}^{-(n+1)}}\right\},\\ =\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\sum _{r=0}^{\infty }\frac{{(-1)}^{n+m}{(-b)}^{n-r}{({\lambda }_n)}^r{(a)}^{1-n-m}{(k_0(\alpha ))}^n\Gamma (n+m+1)}{m!r!\Gamma (n-r+1)\Gamma (n-m+1)}.\end{array}$$ $$\begin{array}{l}\times t^{-(m+r+(n+1)\alpha +1)}{E}_{1,-(m+r+(n+1)\alpha )}^{-(n+1)}(-\frac{k_1(\alpha )}{k_0(\alpha )}t),\\ \varpi (\varphi ,t)={\mathcal{L}}^{-1}\left\{\overline{\varpi }(\varphi ,s)\right\}={\mathcal{L}}^{-1}\left\{\frac{1}{s}{e}^{-\varphi \sqrt{(a+s)+(M+s)({\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })}}\right\},\\ ={\mathcal{L}}^{-1}\left\{\sum _{n=0}^{\infty }\sum _{j=0}^{\infty }\sum _{m=0}^{\infty }\sum _{r=0}^{\infty }\frac{{(-\varphi )}^n{({\lambda }_n{k}_0(\alpha ))}^j{(a)}^{\frac{n}{2}-j-r}{(M)}^{j-m}}{n!r!m!\Gamma (\frac{n}{2}-j-r+1)\Gamma (j-m+1)}.\frac{\Gamma (\frac{n}{2}+1)}{s^{1-(m+r+\alpha j)}{(1+\frac{k_1(\alpha )}{k_0(\alpha )}{s}^{-1})}^{-j}}\right\},\\ =\sum _{n=0}^{\infty }\sum _{j=0}^{\infty }\sum _{m=0}^{\infty }\sum _{r=0}^{\infty }\frac{{(-\varphi )}^n{({\lambda }_n{k}_0(\alpha ))}^j{(a)}^{\frac{n}{2}-j-r}{(M)}^{j-m}\Gamma (\frac{n}{2}+1)}{n!r!m!\Gamma (\frac{n}{2}-j-r+1)\Gamma (j-m+1)}{t}^{-(m+r+\alpha j)}{E}_{1,1-(m+r+\alpha j)}^{-j}(-\frac{k_1(\alpha )}{k_0(\alpha )}t).\end{array}$$

The inverse Laplace of the above Equation (25)(25) $$\overline{u}(\varphi ,s)=\overline{\Omega }(\varphi ,s)+Gr[\overline{\Phi }(\varphi ,s)+\overline{\Psi }(\varphi ,s)][\overline{T}(\varphi ,s)-\overline{T_1}(\varphi ,s)\overline{\varpi }(\varphi ,s)].$$(25) , the required velocity solution is given by (28) $$u(\varphi ,t)=\Omega (\varphi ,t)+Gr\left[\Phi (\varphi ,t)+\Psi (\varphi ,t)\right]\ast \left[T(\varphi ,t)-T_1(\varphi ,t)\ast \varpi (\varphi ,t)\right].$$(28) where $a=M+\frac{1}{K}$ and $b=\theta -M{\lambda }_n.$

4.3. Nusselt number and skin friction

To estimate the heat transfer rate at solid–fluid interface under Newtonian heating condition for CPC fractional operator, the Nusselt number is calculated as: (29) $$\begin{array}{cc}{N}_u=-\frac{\partial T(\varphi ,t)}{\partial \varphi }{|}_{\varphi =0},\hfill & \hfill \\ =-\frac{\partial }{\partial \varphi }{\mathcal{L}}^{-1}\left\{\overline{T}(\varphi ,s)\right\}{|}_{\varphi =0},\hfill & \hfill \\ =-{\mathcal{L}}^{-1}\left\{\frac{\partial \overline{T}(\varphi ,s)}{\partial \varphi }{|}_{\varphi =0}\right\},\hfill & \hfill \\ ={\mathcal{L}}^{-1}\left\{\frac{\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}{s(\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1)}\right\},\hfill & \hfill \end{array}$$(29)

To estimate the shear stress at enclosing boundary, skin friction coefficient is derived as: (30) $$\begin{array}{l}{C}_f=\left[\frac{1}{1+{\lambda }^{\mathit{CPC}}{D}_t^{\alpha }}\right]\frac{\partial \overline{u}(\varphi ,s)}{\partial \varphi }{|}_{\varphi =0},\hfill \\ =\left[\frac{-1}{1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}\right]\left[(\frac{1-e^{-s}}{s^2})\sqrt{(M+s)(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha })+\frac{1}{K}}\right]\hfill \\ +\frac{Gr}{s(\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1)}\times \hfill \\ \left[\frac{\sqrt{(M+s)(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha })+\frac{1}{K}}-\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}{((M+s)(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha })+\frac{1}{K})-\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}\right].\hfill \end{array}$$(30)

Since the solutions given in Equation (29)(29) $$\begin{array}{cc}{N}_u=-\frac{\partial T(\varphi ,t)}{\partial \varphi }{|}_{\varphi =0},\hfill & \hfill \\ =-\frac{\partial }{\partial \varphi }{\mathcal{L}}^{-1}\left\{\overline{T}(\varphi ,s)\right\}{|}_{\varphi =0},\hfill & \hfill \\ =-{\mathcal{L}}^{-1}\left\{\frac{\partial \overline{T}(\varphi ,s)}{\partial \varphi }{|}_{\varphi =0}\right\},\hfill & \hfill \\ ={\mathcal{L}}^{-1}\left\{\frac{\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}{s(\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1)}\right\},\hfill & \hfill \end{array}$$(29) and Equation (30)(30) $$\begin{array}{l}{C}_f=\left[\frac{1}{1+{\lambda }^{\mathit{CPC}}{D}_t^{\alpha }}\right]\frac{\partial \overline{u}(\varphi ,s)}{\partial \varphi }{|}_{\varphi =0},\hfill \\ =\left[\frac{-1}{1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}\right]\left[(\frac{1-e^{-s}}{s^2})\sqrt{(M+s)(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha })+\frac{1}{K}}\right]\hfill \\ +\frac{Gr}{s(\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1)}\times \hfill \\ \left[\frac{\sqrt{(M+s)(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha })+\frac{1}{K}}-\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}}{((M+s)(1+{\lambda }_n\left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha })+\frac{1}{K})-\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}\right].\hfill \end{array}$$(30) contains complex terms of Laplace parameter s. To derive the solution in a real time t, we applied numerical inversion method known as Durbin Method (Durbin, 1974).

4.3.1. Limiting models

Some special cases are discussed here that derived from current problem to analyze the influence on solutions for different cases arises in the absence of some parameters.

4.3.1.1. Solution in the absence of Maxwell parameter

In this case supposed that the Maxwell fluid parameter ${\lambda }_n$ is taking as zero , i.e. ${\lambda }_n=0,$ then the behavior of non-Newtonian fluid reduced into Newtonian fluid and the velocity Equation (24)(24) $$\begin{array}{cc}\overline{u}(\varphi ,s)=(\frac{1-e^{-s}}{s^2})e^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}\hfill & \hfill \\ +\frac{Gr}{s}\left(\frac{1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}{\sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}-1}\right)\times \hfill & \hfill \\ \left[\frac{e^{-\varphi \sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}}-e^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}}{((M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K})-\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}\right].\hfill & \hfill \end{array}$$(24) is turn out as: (31) $$\overline{u}(\varphi ,s)=(\frac{1-e^{-s}}{s^2})e^{-\varphi \sqrt{a+s}}+\frac{Gr}{s}(\frac{1}{\sqrt{\theta \left[\frac{k_1(\alpha )}{s}+k_0(\alpha )\right]s^{\alpha }}-1})[\frac{e^{-\varphi \sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}}-e^{-\varphi \sqrt{a+s}}}{(a+s)-\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}].$$(31)

4.3.1.2. Solution in the absence of MHD and porosity parameter

In this case supposed that $M=0$ and $\frac{1}{K}=0$ in the velocity Equation (24)(24) $$\begin{array}{cc}\overline{u}(\varphi ,s)=(\frac{1-e^{-s}}{s^2})e^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}\hfill & \hfill \\ +\frac{Gr}{s}\left(\frac{1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}{\sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}-1}\right)\times \hfill & \hfill \\ \left[\frac{e^{-\varphi \sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}}-e^{-\varphi \sqrt{(M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K}}}}{((M+s)(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })+\frac{1}{K})-\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}\right].\hfill & \hfill \end{array}$$(24) that reduced in the following form: (32) $$\begin{array}{cc}\overline{u}(\varphi ,s)=(\frac{1-e^{-s}}{s^2})e^{-\varphi \sqrt{s(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })}}+\frac{Gr}{s}(\frac{1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}{\sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}-1})\times \hfill & \hfill \\ [\frac{e^{-\varphi \sqrt{\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}}-e^{-\varphi \sqrt{s(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })}}}{s(1+{\lambda }_n[\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha })-\theta [\frac{k_1(\alpha )}{s}+k_0(\alpha )]s^{\alpha }}].\hfill & \hfill \end{array}$$(32)

5. Results and discussion

The present work examines the analytical solutions of the mathematical fractional Maxwell fluid model that flow through porous plate near an infinitely vertical plate, saturated in porous media. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimensional form with suitable new non-dimensional variables. For the sake of better rheology of rate type fluid, developed a fractional model by employing the new definition of Constant Proportional Caputo fractional derivative operator that describe the generalized memory effects. For seeking exact solution expressions in terms of Mittag-Leffler functions, for Maxwell fluid velocity and Maxwell fluid temperature, Laplace integral transformation method is used to solve the fractional model. For several physical significance of various fluidic parameters involved in the problem such as fractional parameter $\alpha ,$ Prandtl number Pr, Maxwell parameter $\lambda ,$ thermal Grashof number Gr, magnetic parameter M, chemical reaction rate Nr and porosity parameter K on the Maxwell fluid velocity and Maxwell fluid temperature are evaluated and executed graphically in Figures 2–12 by using graphical Math-cad Software. New aspect of this presented work is the use of the Constant Proportional Caputo fractional operator for Maxwell fluid with ramped conditions under Newtonian heating effect investigated.

Figure 2. Representation of Maxwell fluid temperature against $\varphi$ for distinct values of $\alpha .$

Figure 3. Representation of Maxwell fluid temperature against $\varphi$ for distinct values of Pr.

Figure 4. Representation of Maxwell fluid temperature against $\varphi$ for distinct values of Nr.

Figure 5. Representation of Maxwell fluid velocity against $\varphi$ for distinct values of $\alpha .$

Figure 6. Representation of Maxwell fluid velocity against $\varphi$ for distinct values of Pr.

Figure 7. Representation of Maxwell fluid velocity against $\varphi$ for distinct values of Gr.

Figure 8. Representation of Maxwell fluid velocity against $\varphi$ for distinct values of K.

Figure 9. Representation of Maxwell fluid velocity against $\varphi$ for distinct values of M.

Figure 10. Representation of velocity and temperature against $\varphi$ when $\alpha$ Tends to 1.

Figure 11. Comparison of CPC fractional operator and Caputo fractional operator against $\varphi .$

Figure 12. Comparison of CPC fractional operator and Caputo fractional operator against $\varphi .$

Figures 2 portrays the impact of fractional parameter $\alpha$ on fluid temperature against $\varphi$ and a significant effect on fluid temperature within the boundary layer is investigated. For distinct values of fractional parameter, increasing values of $\alpha ,$ and consequently, fluid temperature decreased for $t=0.7,$ but the inverse trends of temperature distribution noticed for $t=2.2.$ Also, it is observed that fractional parameter $\alpha$ has significant influence on heat flux when the values of time have been taken small and large. Figure 3 displays the Prandtl number $P_r$ effect on Maxwell fluid temperature distribution against $\varphi ,$ for different values of $P_r,$ at four different values of fractional parameter $\alpha .$ 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. Figures 4 illustrates the behavior of Nr on temperature profile of Maxwell fluid by taking its dissimilar values. From the curves it is analyzed that energy profile elevated for large values of Nr, where the values of fractional parameters have been supposed between 0 and 1. Physically, change of heat flux increases but $k_1$ reduces along the plate which is in normal direction, this implies that the more amount of heat radiation is absorbed to the fluid which cause to increase the temperature profile. Figure 5 depicts the graphical behavior of fractional parameter $\alpha ,$ and significant effect noticed on Maxwell fluid velocity curve together with fluidic parameters. For $t=2.2,$ it is noteworthy to point out that the graphical view of the fluid velocity for small values of fractional parameter is higher and the curves of fluid velocity reduced continuously corresponding to increasing the values of fractional parameter. Further, it is observed that velocity curves declines with enhancing values of fractional parameter. For $t=0.7,$ the inverse trends of velocity distribution noticed. This happened due to time difference that effectively influences the momentum boundary layer and produces a significant variation in it. Hence, the consequent velocity profiles follow inverse trends. Figure 6 exhibits the effect of Prandtl number $P_r$ on Maxwell fluid velocity corresponding to $\varphi ,$ for different values of $P_r,$ at four different values of fractional parameter $\alpha .$ 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. Figure 7 portrays the influence of thermal Grashof number Gr on Maxwell fluid flow for ramped plate. Since Gr describes the fraction of buoyancy force to viscous force, as a result, with an increasing in Gr cause a remarkable increasing impact on the Maxwell fluid velocity. have appeared due to boost in the value of Gr. Physically, an increasing the values of thermal Grashof number that leads to decrease in viscous hydrodynamic forces, and as a result, the momentum of the Maxwell fluid is higher. Figure 8 elucidates the influence of permeability parameter K on the velocity graphs for Maxwell fluid against $\varphi ,$ by choosing the distinct values of K for taking the values of $\alpha$ small and large. An increase in the porosity of medium cause to weak the resistive force and consequently, the flow regime enhances due to momentum development. It is depicted that the elevation in the velocity profile with an increasing values of K under ramped conditions. Figure 9 interprets the impact of M on the momentum profile against $\varphi ,$ when assigned different values of M in the velocity expression to exemplify the physical behavior of Maxwell fluid velocity corresponding to distinct values of fractional parameter. It is established that the decline in both magnitude of boundary layer thickness and velocity when the strong magnetic field applied. Subsequently, this explanation justifies the fluid gets slowed down corresponding to an increase in magnetic number because dragging forces cause to dominates the flow supporting forces. Eventually, decay in the velocity profile with an enhancement in values of magnetic number. Figure 10 the flow performance of Maxwell fluid velocity and temperature are analyzed under the variation of fractional parameter $\alpha$ together with fluidic parameters. It is observed from the graphical view, the fluid velocity and temperature, both the curves of fluid velocity and temperature approaches continuously to classical model corresponding to increasing the values of fractional parameter. Figures 11 and 12 are the comparative illustration of temperature distribution and velocity field for constant proportional Caputo and Caputo fractional operators. It also demonstrates the temperature and velocity profile at two different levels of time. In case of Newtonian heating and ramped velocity, it is witnessed that CPC time fractional operator produces the higher energy and velocity profiles as compared to Caputo fractional model.

6. Conclusion

In this article, the analytical solutions of the mathematical fractional Maxwell fluid that flow through vertical porous plate under ramped conditions is investigated. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimensional form. For the sake of better rheology of rate type fluid, developed a fractional model by employing the new definition of Constant Proportional Caputo fractional derivative operator that describe the generalized memory effects. For seeking exact solution expressions in terms of Mittag-Leffler functions, for Maxwell fluid velocity and Maxwell fluid temperature, Laplace integral transformation method is used to solve the fractional model. For several physical significance of various fluidic 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:

• The velocity and temperature decreases with rising values of fractional parameter $\alpha .$

• An increase in the values of Pr causes to decline velocity and temperature.

• The accumulative values of Nr and Q escalates the temperature graphs.

• Higher values of Gr and K enhancing the Maxwell fluid velocity.

• The increasing variation of magnetic number M, decay in the velocity is observed.

In future, the researcher can be investigated the same problem by employing various fractional derivative operators with singular and non-singular kernels.

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.