Natural convection flow of a fluid using Atangana and Baleanu fractional model

Abstract

A modified fractional model for the magnetohydrodynamic (MHD) flow of a fluid is developed utilizing Atangana–Baleanu fractional derivative (ABFD). Natural convection and wall oscillation instigate the flow over a vertical plate positioned in a porous medium. The partial differential equations (PDEs) are transmuted to ordinary differential equations (ODEs). The Laplace transform method with its inversion is employed to accomplish the exact solutions of momentum and heat equations. The final solution is expressed in terms of gamma function, modified Bessel function, and Mittag-Leffler function. The previous definitions Caputo fractional and Riemann–Liouville are rarely used by the researchers now due to their limitations. The newly introduced ABFD has got significance nowadays due to its nonlocal and nonsingular kernel. This work focuses on the oscillating boundary conditions for the viscous model in terms of ABFD. The influence of involved parameters is interpreted through plots. The velocity profile is an increasing function of fractional parameter and jumps for a higher Grashof number due to buoyancy push. Furthermore, the Atangana–Baleanu (AB) model is compared with the ordinary derivative model for limiting case and analyzed in detail. It is noted that the ordinary fluid flows faster compared to the fractional fluid.

1 Introduction

The literature of natural convection is sufficiently rich for magnetohydrodynamic oscillatory flow with classical fluid models. These natural problems related to the engineering and sciences lack the memory effect description. The fractional derivatives can describe memory effects, rheology, viscoelastic effect, diffusive transport, and fluid flow. In sciences and engineering, such as fluid mechanics, biomedical engineering, earthquake engineering, chemical engineering, and cooling processes in industries, many problems are modeled in fractional differential equations. Finding an operator that can physically describe this fact was complicated. Several scientists proposed that fractional derivatives can fully describe the memory effects, but the time and space components are the vital factors in implementing this idea. Various real-world problems follow three mathematical functions, i.e., the power law function, the exponential decay function, and the generalized Mittag-Leffler function [1]. These mentioned functions are the basis for many mathematical definitions of the fractional differential operators such as Caputo and Riemann–Liouville fractional derivatives based on the power law functions [1, 2]. These two definitions have been followed by researchers, but they have a limitation that the Caputo fractional derivative has a singular kernel and the derivative of a constant is not zero Riemann–Liouville fractional derivative [3]. These limitations can be overcome by Caputo–Fabrizio fractional derivative but due to the locality in its kernel this definition also exhibit limitations [4, 5]. Shedding some light on these two definitions, some quality literature work is worth discussing here to understand their importance. Baleanu and Fernandez [6] established a new formula consisting of Mittag-Leffler kernel in the form of a series of Riemann–Liouville fractional integrals. Several varieties of fractional definitions can be found in the literature. Fernandez et al. [7] expressed the Prabhakar fractional model and its generalized forms as a series of Riemann–Liouville integrals. In a recent study, a generalization for the existing definitions of fractional derivatives and integrals was established by Fernandez et al. [8]. They introduced an integral operator with a general kernel and expressed it as an infinite series of the Riemann–Liouville integrals. Recently, Jarad and Abdeljawad [9] established generalized noninteger derivatives and the Laplace transform to solve dynamical systems in the fractional derivatives frame. Jarad et al. [10] developed a new class of fractional operators in the Reimann–Liouville and Caputo sense. Some relevant references to the fractional integrals and their applications can be found in [11–15] and the references therein.

Azhar et al. [16] used the Caputo–Fabrizio time fractional derivative to the problem of nanofluid flow over a moving vertical plate. Fetecau et al. [17] conducted an analysis on the flow of a nanofluid over an isothermal plate Caputo-time fractional derivative. The flow of a generalized second grade fluid between parallel plates with Riemann–Liouville fractional derivative model was investigated by Wenchang and Mingyu [18]. They acquired the exact analytical solution using the Laplace transform and the Fourier transform. The flow of a second order fluid induced by a plate moving impulsively with fractional anomalous diffusion was investigated by Mingyu and Wenchang [19]. The Rayleigh–Stokes problem for a fractional second grade fluid was studied by Shen et al. [20]. The fractional Laplace transform and Fourier sine transform were employed to obtain the exact solution. Exact analytical solution unsteady flow of a generalized Maxwell fluid between two circular cylinders was determined via Laplace and Hankel transforms [21]. Recently Shen et al. [22] studied a fractional Maxwell viscoelastic nanofluid for various particle shapes. Caputo time fractional derivative was implemented by Zhang et al. [23] to acquire the numerical and analytical solutions for the problem of 2D flow of Maxwell fluid under variable pressure gradient. They used the separation of variables method to acquire analytical solution while for numerical solution the finite difference method was used. Aman et al. [24] studied a fractional Maxwell fluid for exact analytical solution with second order slip.

Now to tackle the above discussed issue of local kernel, Atangana and Baleanu [25] proposed a new fractional derivative with nonlocal and nonsingular kernels, which fulfills all the mathematical principles related to fractional calculus. The ABFD definition is based on the Mittag-Leffler function and promises an improved description of dynamics of the system with the memory effects [26]. ABFD in the Caputo sense is used to investigate the model of vertical transmission and to cure a vector-borne disease [27]. They found the model very effective in curing the disease with a decrease in transmission rate. Very limited research can be found on fluid mechanics problems reporting fluid flow, heat transfer, and other world problems in applied mathematics and mathematical physics. Gomez-Aguilar et al. [28] presented a three-dimensional cancer model using Caputo–Fabrizio–Caputo and Atangana–Baleanu–Caputo fractional order derivatives. This model was proposed to study the evolution of cancer growth by the interactions of tumor cells with healthy cells and activated immune system cells. A scheme via the Laplace transform, the Sumudu–Picard integration method, and the Adams–Moulton rule was used to acquire the solutions.

Jan et al. [29] determined the solution for Brinkman type nanofluid using an Atangana–Baleanu fractional model. Owolabi and Atangana [30] analyzed the numerical simulation of an Adams–Bashforth scheme using Atangana–Baleanu–Caputo fractional derivatives. Saad et al. [31] established numerical solutions for the fractional Fisher type equation using ABFD. They employed a spectral collocation method based on Chebyshev approximations. In this research work, the spectral collocation method was implemented for the first time to solve nonlinear equation with Atangana–Baleanu derivatives. The logarithmic-KdV equation with the AB fractional operator and Mittag-Leffler type kernel was investigated by Mustafa et al. [32] using the numerical iterative method. Some plenteous literature regarding ABFD, their analytical solution, and applications can be found in [33–36] and the references therein. A few previous studies with ABFD are referenced in [34, 35, 37, 38], but mostly they are focused on non-Newtonian fluids, nanofluids, or some are formulated with constant wall velocity. The research work carried out on nanofluids connects to an entirely different formulation and outcomes as they emphasize more the thermophysical properties of the nanoparticles and fluid.

Recently research with huge practical significance was carried out on human immunodeficiency virus-tuberculosis (HIV-TB) coinfected model using ABFD and the Mittag-Leffler law by Khan et al. [39]. They carried out the simulations for the existence, uniqueness of solution, and Hyers–Ulam (HU) stability. Another interesting research was done using ABFD for human blood flow nanofluids by Khan [40]. Their target was the investigation of nanofluids with channel flow in terms of the ABFD model and via the Laplace transform technique. They considered constant wall velocity at the boundaries. There is a gap for these combinations of studies in terms of oscillating boundary conditions for the ABFD model. Thus, the proposed study will focus on the ABFD model of an oscillatory Newtonian fluid flow with magnetic and radiative heat flux.

The literature reveals the importance of ABFD in the fluid dynamics domain. There is limited literature that discusses the free convection fluid flow using an Atangana–Baleanu derivative model for oscillatory flow; therefore the authors are motivated to conduct this research. Exact solutions are to be established for the problem of natural convection flow of fluid over a vertical plate using ABFD. The analytical solution is traced out via the Laplace transform method with its inversion. The solution is particularized for limiting case, i.e., for an ordinary fluid model by making $\alpha =1$. Finally, a brief graphical interpretation has been made for various parameters and comparison with the ordinary fluid model.

The structure of the paper is arranged as follows. In Sect. 2 we derive the mathematical ABFD model for the problem under consideration. The analytical solution is given in Sect. 3, we also recover the limiting case here. The acquired results are explained briefly in Sect. 4 followed by the conclusion of the paper in Sect. 5.

2 Mathematical framework

Consider unsteady flow of a fluid over a vertical plate with heat flux taken into consideration. A uniform magnetic field is employed perpendicular to the flow of the fluid. The flow is induced by the natural buoyancy force and heat flux $q_{r}$ on the plate. The temperature on the surface of the plate is $T_{w}$, while $T_{\infty }$ is the temperature far away from the surface. The plate starts oscillations at time $t>0$. The physical interpretation of the flow is shown in Fig. 1. The governing equations of momentum and energy are given as follows:

$$\begin{aligned} & \rho _{f} \cdot \frac{\partial u}{\partial t} = \mu \frac{\partial ^{2} u}{\partial y^{2}} - \biggl( \frac{\mu }{k_{1}} + B_{0}^{2} \sigma _{f} \biggr) u + \mathrm{g} \beta _{f} \rho _{f} ( {T} - T_{\infty } ), \end{aligned}$$

(1)

$$\begin{aligned} & (\rho c_{p} )_{f} \cdot \frac{\partial T}{\partial t} = k_{f} \frac{\partial ^{2} T}{\partial y^{2}} - \frac{\partial q_{r}}{\partial y}. \end{aligned}$$

(2)

Here, T and u denote temperature and velocity, $\rho _{f}$, μ, $k_{1}$, $B_{0}^{2}$, $\sigma _{f}$, $\beta _{f}$, $k_{f}$, $q_{r}$ are the density, dynamic viscosity, porosity parameter, magnetic parameter, electrical conductivity, thermal expansion coefficient, thermal conductivity, and radiative flux parameter. The radiative heat flux is defined as in [16, 24]:

$$ - \frac{\partial q_{r}}{\partial y} = 4 \alpha ( T- T_{0} ). $$

The boundary conditions are

$$\begin{aligned} & T ( y,0 ) = T_{\infty }, \qquad T ( 0,t ) = T_{w},\qquad T ( \infty ,t ) = T_{\infty }, \end{aligned}$$

(3)

$$\begin{aligned} & u ( y,0 ) =0,\qquad u ( 0,t ) = U_{0} \cos \omega t,\qquad u ( \infty ,t ) =0, \end{aligned}$$

(4)

where $U_{0}$ is the amplitude and ω is the frequency of oscillations.

Figure 1

Physical geometry of the flow

The similarity variables are as follows:

$$ y^{*} = \frac{y U_{0}}{\nu },\qquad u^{*} = \frac{u}{U_{0}},\qquad t^{*} = \frac{t U_{0}^{2}}{\nu },\qquad \theta = \frac{{T} - T_{\infty }}{T_{w} - T_{\infty }} . $$

(5)

Transforming Eqs. (1)–(4) to dimensionless PDEs using Eq. (5), we get (asterisk* is omitted for convenience):

$$\begin{aligned} & \frac{\partial u}{\partial t} = \frac{\partial ^{2} u}{\partial y^{2}} - ( k+M ) u + \mathit{Gr}_{T} \theta , \end{aligned}$$

(6)

$$\begin{aligned} & \Pr \cdot \frac{\partial \theta }{\partial t} = \frac{\partial ^{2} \theta }{\partial y^{2}} + N^{2} \theta , \end{aligned}$$

(7)

with the boundary conditions

$$\begin{aligned} & \theta ( y,0 ) = 0, \qquad \theta ( 0,t ) =1, \qquad \theta ( \infty ,t ) = 0, \end{aligned}$$

(8)

$$\begin{aligned} & u ( y,0 ) =0,\qquad u ( 0,t ) = \cos \omega t,\qquad u ( \infty ,t ) =0, \end{aligned}$$

(9)

where

$$ \begin{aligned} &k= \frac{\nu ^{2}}{k_{1} U_{0}^{2}},\qquad M= \frac{\nu B_{0}^{2} \sigma _{f}}{\rho _{f} U_{0}^{2}},\qquad N^{2} = \frac{4 \alpha ^{2} \nu ^{2}}{k_{f} U_{0}^{2}}, \\ &\mathit{Gr}_{T} = \frac{\mathrm{g} \beta _{f} \nu ( T_{w} - T_{\infty } )}{U_{0}^{3}},\qquad \Pr = \frac{(\rho c_{p} )_{f} \nu }{k_{f}}, \end{aligned} $$

(10)

where k, M, $\mathit{Gr}_{T}$, Pr, N, ν, α denote the porosity, magnetic, Grashof number, Prandtl number, radiation parameter, kinematic viscosity, and mean radiation absorption coefficient respectively. Here $\nu = \frac{\mu }{\rho _{f}}$. The ABFD model of the present problem is given by

$$\begin{aligned} & {}^{\mathrm{AB}}D_{t}^{\alpha } u= \frac{\partial ^{2} u}{\partial y^{2}} - ( k+M ) u + \mathit{Gr}_{T} \theta , \end{aligned}$$

(11)

$$\begin{aligned} & \Pr\, \cdot\, {}^{\mathrm{AB}}D_{t}^{\alpha } \theta = \frac{\partial ^{2} \theta }{\partial y^{2}} + N^{2} \theta . \end{aligned}$$

(12)

Atangana and Baleanu fractional derivative [25] is defined as

$$ {}^{\mathrm{AB}}D_{t}^{\alpha } f ( y,t ) = \frac{N(\alpha )}{1-\alpha } \int _{0}^{t} E_{\alpha } \biggl( -\alpha \frac{(t-\tau )^{\alpha }}{1-\alpha } \biggr) \frac{\partial f(y,\tau )}{\partial \tau } \,d\tau , $$

(13)

where $N(\alpha )$ is a normalization function such that $N ( 0 ) =N ( 1 ) =1$ and $E_{\alpha } ( z ) = \sum_{k=0}^{\infty } \frac{z^{k}}{\varGamma ( \alpha k+1 )}$ is a Mittag-Leffler function [41]. The Laplace transform of Eq. (13) is given as follows:

$$ \mathcal{L} \bigl\{ {}^{\mathrm{AB}}D_{t}^{\alpha } f ( \mathrm{t}) \bigr\} = \frac{q^{\alpha } \mathcal{L} \{ f ( t ) \} - q^{\alpha -1} f(0)}{q^{\alpha } ( 1-\alpha ) +\alpha }, $$

(14)

where q denotes the Laplace transform operator.

3 Analytical solution

Applying the Laplace transform to Eqs. (8)–(12) and utilizing the definition given in Eq. (14), we get

$$\begin{aligned} & \frac{q^{\alpha } \overline{u}}{q^{\alpha } ( 1-\alpha ) +\alpha } = \frac{\partial ^{2} \overline{u}}{\partial y^{2}} - ( k+M ) \overline{u} + \mathit{Gr}_{T} \overline{\theta }, \end{aligned}$$

(15)

$$\begin{aligned} & \frac{\Pr q^{\alpha } \overline{\theta }}{q^{\alpha } ( 1-\alpha ) +\alpha } = \frac{\partial ^{2} \overline{\theta }}{\partial y^{2}} + N^{2} \overline{ \theta }, \end{aligned}$$

(16)

$$\begin{aligned} & \overline{\theta } ( 0,q ) = \frac{1}{q},\qquad \overline{\theta } ( \infty ,0 ) = 0 \end{aligned}$$

(17)

$$\begin{aligned} & \overline{u} ( 0,q ) = \frac{q}{q^{2} + \omega ^{2}},\qquad \overline{u} ( \infty ,0 ) =0. \end{aligned}$$

(18)

Thus

$$\begin{aligned} & \overline{\theta } (y,q)= \biggl( \frac{1}{q^{1-\alpha }} \biggr) \biggl( \frac{1}{q^{\alpha }} \biggr) \mathrm{e}^{-\mathrm{y} \sqrt{b_{3}} \sqrt{\frac{q^{\alpha } + b_{2}}{q^{\alpha } + b_{1}}}}, \end{aligned}$$

(19)

$$\begin{aligned} & \overline{u} ( y,q ) = \biggl( \frac{q}{q^{2} + \omega ^{2}} \biggr) \mathrm{e}^{-\mathrm{y} \sqrt{b_{4}} \sqrt{\frac{q^{\alpha } + b_{6}}{q^{\alpha } + b_{1}}}} + \frac{\mathit{Gr}_{T}}{ b_{7}} \biggl( \frac{q^{\alpha } + b_{1}}{ b_{3} ( q^{\alpha } + b_{2} ) - ( b_{4} q^{\alpha } + b_{5} )} \biggr) \mathrm{e}^{-\mathrm{y} \sqrt{b_{4}} \sqrt{\frac{q^{\alpha } + b_{6}}{q^{\alpha } + b_{1}}}} \\ &\hphantom{\overline{u} ( y,q ) ={}}{}- \frac{\mathit{Gr}_{T}}{ b_{7}} \biggl( \frac{q^{\alpha } + b_{1}}{ b_{3} ( q^{\alpha } + b_{2} ) - ( b_{4} q^{\alpha } + b_{5} )} \biggr) \mathrm{e}^{-\mathrm{y} \sqrt{b_{3}} \sqrt{\frac{q^{\alpha } + b_{2}}{q^{\alpha } + b_{1}}}}, \end{aligned}$$

(20)

where

$$ \begin{aligned} &b_{0} = \frac{1}{ ( 1-\alpha )},\qquad b_{1} = b_{0} \alpha ,\qquad b_{2} = \frac{b_{1} N^{2}}{b_{3}},\qquad b_{3} = b_{0} \Pr + N^{2}, \\ &b_{4} = (b_{0} +k+M ),\qquad b_{5} = b_{1} ( k+M ), \qquad b_{6} = \frac{b_{5}}{b_{4}}, \\ &b_{7} = b_{3} - b_{4},\qquad b_{8} = \frac{b_{2} b_{3} - b_{4} b_{5}}{b_{7}}. \end{aligned} $$

(21)

Equations (19) and (20) can be written as follows:

$$\begin{aligned} & \overline{\theta } ( y,q ) = \overline{h} ( q,\alpha ) \overline{\varPhi } ( y \sqrt{b_{3}}, q;0, b_{2}, b_{1} ), \end{aligned}$$

(22)

$$\begin{aligned} & \overline{u} ( y,q ) = \biggl( \frac{q}{q^{2} + \omega ^{2}} \biggr) \overline{\varPhi 1} ( y \sqrt{b_{4}}, q;0, b_{6}, b_{1} ) + \frac{\mathit{Gr}_{T}}{ b_{7}} \biggl( h ( q,\alpha ) + ( b_{1} - b_{8} ) \biggl( \frac{q^{\alpha -1}}{q^{\alpha } + b_{8}} \biggr) \biggr) \\ &\hphantom{\overline{u} ( y,q ) ={}}{}\times \bigl[ \overline{\varPhi } ( y \sqrt{b_{4}}, q;0, b_{6}, b_{1} ) - \overline{\varPhi } ( y \sqrt{b_{3}}, q;0, b_{2}, b_{1} ) \bigr], \end{aligned}$$

(23)

where

$$\begin{aligned}& \overline{h} ( q,\alpha ) = \biggl( \frac{1}{q^{1-\alpha }} \biggr), \qquad \overline{ \varPhi 1} ( y \sqrt{b_{4}}, q;0, b_{6}, b_{1} ) = \mathrm{e}^{-\mathrm{y} \sqrt{b_{4}} \sqrt{\frac{q^{\alpha } + b_{6}}{q^{\alpha } + b_{1}}}}, \\& \overline{\varPhi } ( y \sqrt{b_{3}}, q;0, b_{2}, b_{1} ) = \biggl( \biggl( \frac{1}{q^{\alpha }} \biggr) \mathrm{e}^{-\mathrm{y} \sqrt{b_{3}} \sqrt{\frac{q^{\alpha } + b_{2}}{q^{\alpha } + b_{1}}}} \biggr), \\& \overline{\varPhi } ( y \sqrt{b_{4}}, q;0, b_{6}, b_{1} ) = \biggl( \biggl( \frac{1}{q^{\alpha }} \biggr) \mathrm{e}^{-\mathrm{y} \sqrt{b_{4}} \sqrt{\frac{q^{\alpha } + b_{6}}{q^{\alpha } + b_{1}}}} \biggr). \end{aligned}$$

Employing the inverse Laplace transform to Eqs. (22) and (23), utilizing the convolution theorem (A1) and Appendix (A2)–(A5), the final expression is obtained as follows:

$$\begin{aligned} & \theta ( y,t ) = \int _{0}^{t} h ( t-s,\alpha ) \varPhi ( y \sqrt{b_{3}}, s;0, b_{2}, b_{1} ) \,ds, \end{aligned}$$

(24)

$$\begin{aligned} & u ( y,t ) = \int _{0}^{t} \cos \bigl( \omega ( t-s ) \bigr) \varPhi 1 ( y \sqrt{b_{4}}, s;0, b_{6}, b_{1} ) \,ds \\ &\hphantom{u ( y,t ) ={}}{}+ \frac{\mathit{Gr}}{ b_{7}} \biggl( \int _{0}^{t} h ( t-s,\alpha ) \varPhi ( y \sqrt{b_{4}}, s;0, b_{6}, b_{1} ) \,ds \\ &\hphantom{u ( y,t ) ={}}{}+ ( b_{1} - b_{8} ) \int _{0}^{t} E_{\alpha ,1} \bigl( - b_{8} (t-s)^{\alpha } \bigr) \varPhi ( y \sqrt{b_{4}}, s;0, b_{6}, b_{1} ) \,ds \biggr) \\ &\hphantom{u ( y,t ) ={}}{}- \frac{\mathit{Gr}}{ b_{7}} \biggl( \int _{0}^{t} h ( t-s,\alpha ) \varPhi ( y \sqrt{b_{3}}, s;0, b_{2}, b_{1} ) \,ds \\ &\hphantom{u ( y,t ) ={}}{} + ( b_{1} - b_{8} ) \int _{0}^{t} E_{\alpha ,1} \bigl( - b_{8} ( t-s )^{\alpha } \bigr) \varPhi ( y \sqrt{b_{3}}, s;0, b_{2}, b_{1} ) \,ds \biggr), \end{aligned}$$

(25)

where

$$ \begin{aligned} & \varPhi ( y \sqrt{b_{3}}, t;0, b_{2}, b_{1} ) \\ &\quad = \frac{1}{\pi } \int _{0}^{\infty } \int _{0}^{\infty } \psi ( y \sqrt{b_{3}}, u;0, b_{2}, b_{1} ) \\ &\qquad {}\times \exp \bigl( -\tau -r-u r^{\alpha } \cos \alpha \pi \bigr) \bigl( u r^{\alpha } \sin \alpha \pi \bigr) \,dr\,du, \\ &\varPhi ( y \sqrt{b_{4}}, t;0, b_{6}, b_{1} ) \\ &\quad = \frac{1}{\pi } \int _{0}^{\infty } \int _{0}^{\infty } \psi ( y \sqrt{b_{4}}, u;0, b_{6}, b_{1} ) \\ &\qquad {}\times \exp \bigl( -\tau -r-u r^{\alpha } \cos \alpha \pi \bigr) \bigl( u r^{\alpha } \sin \alpha \pi \bigr) \,dr\,du, \\ &\overline{\varPhi 1} ( y \sqrt{b_{4}}, t;0, b_{6}, b_{1} ) \\ &\quad = \frac{1}{\pi } \int _{0}^{\infty } \int _{0}^{\infty } \psi ( y \sqrt{b_{4}}, u;0, b_{6}, b_{1} ) \\ &\qquad {}\times \exp \bigl( -tr-u r^{\alpha } \cos \alpha \pi \bigr) \sin \bigl( u r^{\alpha } \sin \alpha \pi \bigr) \,dr\,du, \\ &\psi ( y \sqrt{b_{3}}, t;0, b_{2}, b_{1} ) \\ &\quad = e^{-y \sqrt{b_{3}}} - \frac{y \sqrt{b_{2} - b_{1}}}{2\pi } \\ &\qquad {}\times \int _{0}^{\infty } \int _{0}^{t} \frac{1}{\sqrt{t}} e^{ ( - b_{1} t- \frac{y^{2}}{4u} -u )} I_{1} \bigl( 2 \sqrt{ ( b_{2} - b_{1} ) ut} \bigr) \,dt\,du, \\ &E_{\alpha ,1} \bigl( - b_{8} t^{\alpha } \bigr) = \sum _{k=0}^{\infty } \frac{( - b_{8} ( t )^{\alpha } )^{k}}{\varGamma ( \alpha k+1 )},\qquad I_{n} ( x ) = \sum_{k=0}^{\infty } \frac{(-1)^{k}}{k!(n+k)!} \biggl( \frac{x}{2} \biggr)^{n+2k};\quad n \geq 0. \end{aligned} $$

(26)

Here $I_{1} ( x )$ is a Bessel function of the first kind.

3.1 Limiting case

Taking $\alpha =1$ in Eq. (14), we acquire the limiting case by recovering the solution for the classical model of the present problem:

$$\begin{aligned} \lim_{\alpha \rightarrow 1} D_{t}^{\alpha } \overline{f} ( y,t ) &= \lim_{\alpha \rightarrow 1} \mathcal{L}^{-1} \biggl\{ \frac{q^{\alpha } \mathcal{L} \{ f ( y,t ) \} - q^{\alpha -1} f ( y,0 )}{q^{\alpha } ( 1-\alpha ) +\alpha } \biggr\} \\ &= \mathcal{L}^{-1} \biggl\{ \lim _{\alpha \rightarrow 1} \frac{q^{\alpha } \overline{f} ( y,q ) - q^{\alpha -1} f ( y,0 )}{q^{\alpha } ( 1-\alpha ) +\alpha } \biggr\} \\ & = \mathcal{L}^{-1} \bigl\{ q \overline{f} ( y,q ) -f(y,0) \bigr\} . \end{aligned}$$

(27)

Thus, the solution we recovered is given as follows:

$$\begin{aligned} &\theta ( y,t ) = \frac{1}{2} \biggl[ e^{-y(\Pr \sqrt{- N^{2}} )}\cdot erfc \biggl( \frac{y \sqrt{\Pr }}{2 \sqrt{t}} - \sqrt{ \biggl( - \frac{N^{2}}{\Pr } \biggr) t} \biggr) \\ &\hphantom{\theta ( y,t ) ={}}{} + e^{y(\Pr \sqrt{- N^{2}} )} \cdot erfc \biggl( \frac{y \sqrt{\Pr }}{2 \sqrt{t}} + \sqrt{ \biggl( - \frac{N^{2}}{\Pr } \biggr) t} \biggr) \biggr], \end{aligned}$$

(28)

$$\begin{aligned} & u ( y,t ) = \int _{0}^{t} \cos \bigl( \omega ( t-s ) \bigr) \varPhi 1 ( y \sqrt{b_{4}}, s;0, b_{6}, b_{1} ) \,ds \\ &\hphantom{u ( y,t ) ={}}{}+ \frac{\mathit{Gr}}{ b_{7}} \biggl( \int _{0}^{t} h ( t-s,\alpha ) \varPhi ( y \sqrt{b_{4}}, s;0, b_{6}, b_{1} ) \,ds \\ &\hphantom{u ( y,t ) ={}}{}+ ( b_{1} - b_{8} ) \int _{0}^{t} E_{\alpha ,1} \bigl( - b_{8} (t-s)^{\alpha } \bigr) \varPhi ( y \sqrt{b_{4}}, s;0, b_{6}, b_{1} ) \,ds \biggr). \end{aligned}$$

(29)

4 Results and discussion

This section highlights the principal features and acquired results for the analysis of oscillatory flow of a fluid under free convection and radiation. The analytical solution for the present problem in terms of the Atangana–Baleanu fractional model has been established via the Laplace transform method. Gamma function, modified Bessel function, and Mittag-Leffler functions are used to express the final general solution for momentum and heat equation. The flow and heat are analyzed for the influence of the parameters such as fractional parameter, radiation parameter, magnetic parameter, porosity parameter, and Grashof number. At the end, a comparison analysis has been accomplished between the fractional and ordinary models. The brief findings in the present analysis are stated below.

Figure 2(a)–(b) shows the temperature field at $t=2$, with the variation of fractional parameter α. The maximizing values of fractional parameter α cause a rise in the temperature profile for $\alpha =0.1, 0.7, 0.9$. In Fig. 2(a), the temperature profile for $N=0.5$ is low, while in Fig. 2(b) the temperature shows a unique rising behavior at $\alpha =0.7$ and $\alpha =0.9$ for greater radiation parameter, i.e., $N=2$. Figure 3(a)–(b) depicts the temperature variation under the effect of radiation parameter N. The results are plotted at two different values of α. The temperature profile of the fluid increases significantly for $N=0.1, 1.5, 2$ in Figs. 3(a) and 3(b) at $\alpha =0.1$ and $\alpha =0.9$. The temperature rise is more significant at the higher value of fractional parameter for $\Pr =5$, $t=1$.

Figure 2

Temperature profile variation with fractional parameter α and radiation parameter N

Figure 3

Temperature profile variation with radiation parameter N and fractional parameter α

Figure 4(a)–(b) shows that the fluid velocity increases when the fractional parameter value varies as $\alpha =0.1, 0.2, 0.3$, while other parameters are kept constant, i.e., $N=0.5$, $\Pr =5$, $M=0.01$, $k=0.2$. The results are acquired at two different values of the Grashof number Gr to interpret the simultaneous influence of the Grashof number and the fractional parameter. For low buoyancy forces, $\mathit{Gr}=0.02$, the flow gradually rises in Fig. 4(a) with variation of α. A noteworthy increasing behavior can be seen for $\mathit{Gr}=5.5$ in Fig. 4(b) due to a huge push of buoyancy forces. All these observations are accomplished at $t=2$.

Figure 4

Velocity profile variation with fractional parameter α and Grashof number Gr

In Fig. 5(a)–(b), the fluid velocity increases with the varying Grashof number $\mathit{Gr}=0, 1, 2$ for other constant parameters, i.e., $N=0.5$, $t=3$, $\Pr =5$, $M=0.05$, $k=0.2$. Here, the results are plotted at $\alpha =0.1$ and $\alpha =0.8$ to examine the parallel influence of α with Gr. At some small fractional parameter $\alpha =0.1$, the flow variation is not much significant in Fig. 5(a), while at a value $\alpha =0.8$ there is a considerable rise in Fig. 5(b) with variation of α. The variation in velocity is remarkable at a greater value of the fractional parameter for the same variation in the Grashof number.

Figure 5

Velocity profile variation with Grashof number Gr and fractional parameter α

Figure 6(a)–(b) shows that the fluid velocity increases with the increasing porosity parameter $k=0, 0.2, 0.4$, while other parameters are constant, i.e., $N=0.5$, $\Pr =5$, $M=0.01$, $\mathit{Gr}=0.5$. For small fractional parameter $\alpha =0.1$, the flow gradually rises in Fig. 6(a) with variation of k. The increasing behavior for $\alpha =0.9$ in Fig. 6(b) is considerable compared to that in case of smaller α due to a huge push of buoyancy forces. All these observations are carried out at $t=2$.

Figure 6

Velocity profile variation with porosity parameter k and fractional parameter α

In Fig. 7(a)–(b), the fluid velocity is observed at two different values of α, i.e., $\alpha = 0.2$, $\alpha =0.9$, for the variation of magnetic parameter $M=0.01, 0.2, 0.4$. The remaining parameters are considered constant for this observation, i.e., $N=0.5$, $\Pr =5$, $t=2$, $k=0.2$, $\mathit{Gr}=0.5$. Fluid velocity tends to decrease gradually in Figs. 7(a) and 7(b), but the effect is more significant with higher velocity range in case of a greater value of α.

Figure 7

Velocity profile variation with magnetic parameter M and fractional parameter α

In Fig. 8(a)–(b), the velocity profile is analyzed for two different values of the Grashof number, i.e., $\mathit{Gr} =1.5$, $\mathit{Gr} =3.5$, with the variation of radiation parameter $N=0.1, 3, 7$. The greater buoyancy forces lead to a significant variation in the velocity profile for various values of N, while for smaller value of Gr, the velocity variation with radiation parameter is not much obvious [Figs. 8(a)–8(b)]. These inspections are carried out for constant values $M=0.06$, $\Pr =5$, $t=2$, $k=0.2$, $\alpha =0.8$. It is obvious from all the results that for greater values of Gr and α the variation with any other parameter becomes significant and higher compared to that of smaller values of Gr and α. The higher buoyancy forces push the fluid to flow faster.

Figure 8

Velocity profile variation with radiation parameter N and Grashof number Gr

In Fig. 9, the temperature profile of the current fractional problem is compared with the ordinary model of the problem by making $\alpha =1$ in the heat equation (Eq. (2)). The deviation at some points is clear at $t=1$ with constant parameters $\Pr =5$, $N=1.5$. It is worth noting that both ordinary and fractional models have qualitatively identical behavior of fluid temperature. Figure 10 depicts the comparison for velocity profile of fractional derivative $0> \alpha >1$ and the classical derivative model $\alpha =1$. The fluid with classical model has a unique and greater velocity than the fractional model. These results are accomplished at constant parameters, i.e., $M=0.06$, $\Pr =5$, $t=2$, $k=0.2$, $\mathit{Gr}=1.5$, $N=2$. It is pointed out that the fractional fluid moves slower in comparison with the ordinary model. The numerical values for fractional and ordinary fluid models are depicted in Table 1, corresponding to the results in Figs. 9–10. These numerical values are evaluated for different distances of fluid from the plate, i.e., $0\leq y\leq 2$. In consistency with these plots, the numerical values signify the abrupt changes when $\alpha =1$, i.e., the fractional fluid model is replaced by the ordinary one. For the fractional fluid, the highest value of velocity $u ( y,t )$ is 0.422, while for the ordinary fluid it is 0.173.

Figure 9

Temperature profile for comparison of fractional and ordinary models

Figure 10

Velocity profile for comparison of fractional and ordinary models

Table 1 Numerical values of $\theta ( y,t )$ and $u ( y,t )$ for fractional and ordinary fluid models

5 Conclusion

In this section, the concluding remarks for the analysis of a fluid under free convection and radiation are emphasized. The ABFD model has been extended for oscillatory flow and heat transfer model for which the analytical solution has been established in terms of Mittag-Leffler function via the attractive technique “Laplace transform method”. Limiting case is recovered for an ordinary fluid when fractional parameter is taken to be one, i.e., $\alpha =1$. The main outcomes and concluding remarks are given as follows. Fluid velocity increases with increasing effect of the radiation parameter. But for the same constant parameters and varying radiation parameter, the velocity acquires a significant increase for greater value of the Grashof number. Fluid flows faster when the value of porosity parameter increases. The more permeable the medium is, the faster the fluid flows. The maximizing range of fractional parameter gives a unique jump to the velocity profile when the Grashof number is as high as 5.5 or above. A similar behavior is found in case of temperature for greater value of the radiation parameter. For the ordinary fluid, the temperature profile has identical behavior to the fractional model. The velocity profile is higher for the ordinary fluid compared to the fractional fluid.

Appendix

$$\begin{aligned}& f ( t ) * \mathrm{g} ( y,t ) = \int _{0}^{t} f ( t-s ) \mathrm{g} ( y,s ) \,ds, \end{aligned}$$

(A1)

$$\begin{aligned}& \mathcal{L}^{-1} \biggl( \frac{q^{-1}}{q^{\alpha } + c_{0}} \biggr) = t^{\alpha } E_{\alpha ,\alpha +1} \bigl( - c_{0} t^{\alpha } \bigr);\quad q >0 , \end{aligned}$$

(A2)

$$\begin{aligned}& \mathcal{ L}^{-1} \bigl( \psi ( y \sqrt{a}, q;a, b, c ) \bigr) \\& \quad = \psi ( y \sqrt{a}, t;a, b, c ) \\& \quad = e^{-at-y} - \frac{y \sqrt{b-c}}{2\pi } \int _{0}^{\infty } \int _{0}^{t} \frac{e^{-at}}{\sqrt{t}} e^{ ( at-ct- \frac{y^{2}}{4u} -u )} I_{1} \bigl( 2 \sqrt{ ( b-c ) ut} \bigr) \,dt\,du , \end{aligned}$$

(A3)

$$\begin{aligned}& \varPhi ( y \sqrt{a}, t;a, b, c ) \\& \quad = \frac{1}{\pi } \int _{0}^{\infty } \int _{0}^{\infty } \psi ( y \sqrt{a}, u;a, b, c ) \\& \qquad {}\times \exp \bigl( -tr-u r^{\alpha } \cos \alpha \pi \bigr) \sin \bigl( u r^{\alpha } \sin \alpha \pi \bigr) \,dr\,du, \end{aligned}$$

(A4)

$$\begin{aligned}& \mathcal{L}^{-1} \bigl( h ( q,\alpha ) \bigr) =h ( t,\alpha ) = \biggl( \frac{1}{t^{\alpha } \varGamma (1-\alpha )} \biggr), \qquad \mathcal{L}^{-1} \biggl( \frac{q}{q^{2} + a^{2}} \biggr) = \cos (at) . \end{aligned}$$

(A5)