Functional Application of G-Functions of Lorenzo and Hartley on the Free Convection Flow of Oldroyd-B Fluid with Ordinary and Fractional Techniques

In this article, free convection flow of an Oldroyd-B fluid (OBF) through a vertical rectangular channel in the presence of heat generation or absorption subject to generalized boundary conditions is studied. )e fractionalized mathematical model is established by Caputo time-fractional derivative through mechanical laws (generalized shear stress constitutive equation and generalized Fourier’s law). Closed form solutions for the velocity and temperature profiles are obtained via Laplace coupled with sine-Fourier transforms and have been embedded with regards to the special functions, namely, the generalized G-functions of Lorenzo and Hartley. Solutions of the known results from recently published work (Nehad et al. Chin. J. Phy., 65, (2020) 367–376) are recovered as limiting cases. Finally, the effects of fractional and various physical parameters are graphically underlined. Furthermore, a comparison between Oldroyd-B, Maxwell and viscous fluids (fractional and ordinary) is depicted. It is found that, for short time, ordinary fluids have greater velocity as compared to the fractional fluids.


Introduction
Fluids are classified in two categories with respect to the relation of shear stress and rate of deformation called Newtonian and non-Newtonian fluids. In non-Newtonian fluids, this relation is not linear, while in Newtonian fluids the relation of shear stress is linearly related to the rate of deformation. Several fluids have non-Newtonian behavior such as molten, salt solutions, custard, ketchup, blood, toothpaste, paint, and shampoo. A significant class of non-Newtonian fluids is viscoelastic fluids which show both adaptable and viscous property and numerous models of constitutive equations have been proposed to depict the transport conduct of these fluids. Oldroyd-B fluid (OBF) [1] is an exceptional non-Newtonian fluids and its transport conduct cannot be appropriately depicted by the usual relation between the shear rate and shear stress in a straightforward shear flow. Hence, numerous models of constitutive equations have been offered for these fluids in [2][3][4][5][6][7][8].
Free convection flows are of incredible viable significance to engineers and researchers on account of its allinclusive event in various mechanical applications, for example, cooling of electronic equipment's, solar amassers, fiber protection, and geothermal frameworks [9]. Because of its event in many engineering applications, numerous authors deliberate the free convection fluid flow models between two vertical parallel plates. e free convection fluid flow between two infinite vertical parallel plates is examined by Singh et al. [10]. Narahari [11] presented critical solution for the transient free convection flow between two vertical parallel plates within the sight of steady temperature and mass diffusion by using the Laplace transform method. Na et al. [12] found the closed form solutions of Maxwell free-convection fluid flow between vertical plates with damped shear and thermal flux. Zaib et al. [13] studied the natural convection flow of second-grade fluid with thermal radiation and damped thermal flux between vertical channels. Hajizadeh et al. [14] got the exact solutions for free convection flow of nanofluids between two vertical plates with damped thermal flux. Shao et al. [15] found the semianalytical solutions for hydromagnetic free-convection flow of viscous fluid between vertical parallel plates with damped thermal and mass fluxes. Seth et al. [16] have investigated the unsteady hydromagnetic flow formation with Hall Effect due to time-dependent free stream in a rotating medium. Seth et al. [17] have studied the effect of Hall current on MHD natural convection heat and mass transfer flow of rotating fluid past a vertical plate with ramped wall temperature. e natural convective heat transfer and nanofluid flows through a rectangular vertical channel under Robin-type conditions are concentrated by Ahmed et al. [18].
On the contrart, heat generation/absorption effects become relevant in many processes involving chemical reaction and dissociating fluids, since the flow and thermal fields can be greatly influenced by the heat source/sink term. Jha et al. [19] examined the effect of heat generation/absorption on flow formation in a parallel plate channel with ramped temperatures. In a related work, Jha and Aina [20] obtained the exact solution for heat generating/absorbing fluid in a cylinder under time periodic boundary conditions in the presence of magnetic field. ey found out that the rate of heat transfer is significantly declined with increase in heat sink, while the contrast is true in presence of heat source. Jha et al. [21] presented the exact solution responsible for the mixed convection in an annular geometry when the fluid is either heat generating or heat absorbing. In another work, Saba et al. [22] presented a study on thermal analysis of nanofluid in the presence of internal heat generation/absorption. Other related discussion can be found in [23][24][25][26][27][28].
Fractional calculus [29,30] has been utilized effectively in the depiction of viscoelasticity. Mostly, these constitutive equations are obtained from known models via substituting time ordinary derivatives of stress and strain by derivatives of fractional order. e generality permits one to express indeed noninteger-order integral or derivatives. Recently, numerous specialists have considered various issues associated to such fluids. Hyder Ali Muttaqi Shah [31] examined the flows of generalized OBF between two side walls perpendicular to the plate. Akgül et al. [32,33] analyzed the MHD Couette flows by fractal-fractional differential operators. Some accelerated flows of a generalized OBF are discussed by Fetecau et al. [34]. Siddique and Akgül [35] examined the blood liquor model via nonlocal and singular constant proportional Caputo hybrid differential operator. Khan et al. [36] examined the MHD flow of a generalized OBF in a round funnel. Siddique and Sajid [37] founnd the exact solutions for the unsteady axial flow of non-Newtonian fluids through a circular cylinder. MHD OBF through fractional calculus approach under several conditions has been examined by Liu et al. [38]. Solutions for OBF between two oscillating plates with transverse magnetic field have been investigated by Bose and Basu [39]. Zhao et al. [40] carried out numerical solutions for fractional OBF in porous medium under the influence of heat transfer. Zafar et al. [41] contemplated an OBF for circular cylinders with nonintegerorder derivatives. Zhang et al. [42] examined the analytical solutions for time-fractional OBF utilizing the definition of Caputo derivative. As of late, Riaz and Saeed [43] evaluated the conduct of MHD OBF under slip condition with the assistance of integer order, Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives. Wang et al. [44] acquired the semianalytical solutions for velocity field and tangential stress correspond to fractional Oldroyd-B fluid in an annulus by Laplace transforms and modified Bessel equation. Kamran et al. [45] find the exact solutions for the unsteady rotational flow of an Oldroyd-B fluid with fractional derivative through an infinite circular cylinder by means of the finite Hankel and Laplace transforms. In all these studies the ordinary derivatives are falsely supplanted by fractional derivatives. Some recent works were done by specialists, and they introduced fractional derivatives through mechanical laws to governing equations [12][13][14][15][46][47][48].
In the current paper, as uniqueness, we build up a nonlocal mathematical model in which the thermal transport is depicted by the fractional Fourier's law.
e new fractional constitutive equations are characterized with the time-fractional Caputo derivative; hence, they give a powerlaw damping to the temperature and velocity. ese realities were the significant reason of the lack of literature on the study of OBFs between two-side walls under general boundary conditions. In this work, we acquire the closed form solutions of the generalized convection flows of OBF between two parallel plates under general boundary conditions by utilizing the Laplace transform coupled with the finite sine-Fourier transform. e solutions are uttered in the form of generalized G-functions of Lorenzo and Hartley [49]. e impacts of fractional and physical parameters are graphically outlined. In addition, the limiting cases of the current outcomes compare to the solutions for fractional Maxwell, ordinary Maxwell, Oldroyd-B, and Newtonian fluids.

Mathematical Modeling
Let us consider the unsteady free-convection Oldroyd-B, incompressible, and electrically conducting fluid flow between two infinite nonconducting parallel vertical plates separated by a distance d.
e flow geometry in fixed Cartesian coordinate system x, ζ ∓ 1 and z is shown in Figure 1. e x− axis is taken along the plates in the direction of fluid flow with the velocity u ∓ and the ζ ∓ 1 -axis is taken perpendicular to the plates.
Since the channel is infinitely extended in the x and z directions, we can assume that all physical entities describing the fluid motion and heat transfer are the functions of ζ ∓ 1 and t ∓ 1 only. At first, both the plates and the enclosed fluid are at rest at the ambient temperature T a . After this moment, the left and right plates starts to slid in their own planes along x− direction with the velocities U 0 g ∓ 1 (t ∓ 1 ) and U 0 g ∓ 2 (t ∓ 1 ); individually, plates are maintained at temperatures under corresponding initial and boundary conditions: Introducing the following nondimensionless variables, functions, and parameters, into equations (1)-(7), we have e partial differential equations (9)-(12) are taken along with the following dimensionless initial and boundary conditions: In the following, we develop a fractional model in which the classical constitutive equations (10) and (12) are generalized by using the constitutive shear stress equation: proposed by Scott-Blair [52], respectively, and the generalized Fourier's law, proposed by Povstenko [53] and Hristov [54], respectively. In the above constitutive equations, C D α t (·) denotes the time-fractional Caputo derivative defined by [55,56]. Hence, where h α (t) � (t − α /Γ(1 − α)) is the singular power-law kernel for _ u(y, s) � (zu(y, t)/zt)| t�s . Furthermore, using the second form of the time-fractional Caputo derivative from equation (18) and one of the following properties of h α (·) is defined as where L · { } denotes the Laplace transform, δ(·) is Dirac's distribution, and s is the transform parameter, and it is easy to show that 4 Complexity Using properties (20), for α � β � 1, equations (16) and (17) reduce to the classical forms (10) and (12).
It is important to specify that the models based on fractional constitutive equations are more general than those described by constitutive equations with integer derivatives.
For this, let us analyze the constitutive equations for the thermal flux (12) and (17). Equation (12) corresponding to Fourier's law is a local form in time and space. Indeed, the value of the heat flux in a position y at time t is determined by the value of the temperature gradient in the same spatial position and at the same time.
e generalized constitutive equation (17), written in the equivalent form q(y, t) � − (1/Γ(β))(z/zy) t 0 (t − s) β− 1 zT(y, t)/zt| t�s ds, shows that the value of the thermal flux in position y at time t is influenced by the temperature history; therefore, the values of the time derivative of temperature for all past and present times determine the thermal flux at the moment t. Such types of constitutive equations describe the thermal processes with time nonlocality. e function h 1− β (t) � (t β− 1 /Γ(β)) is the time-nonlocality kernel (the weight function). e processes with time nonlocality are called processes with memory. e classical Fourier's law describes processes with "instantaneous memory" whose time nonlocality kernel being Dirac's distribution. If the weight function is constant, the process is called the "full memory" process, i.e., there is no fading of memory.
A similar discussion can be made regarding the generalized constitutive equation (16). e difference, in this case, is about the memory of the material deformation. It can be seen from equation (16) that the history of the strain rate influences the stress at the moment t.
Eliminating τ from equations (9) and (16) and q from equations (11) and (17), we obtain In order to obtain the equivalent forms of equations (21) and (22), we recall the time-fractional integral operator: which is the left-inverse operator of the derivative operator C D α t (·). Indeed, using equations (18), (19), and (23), we obtain that which implies Moreover, using equations (18) and (23), it results that Now, applying the fractional integral operators J 1− α t and J 1− β t to equations (21) and (22) and using properties (24)-(26) together with the initial condition, we obtain the following fractional differential equations: To determine the solutions of equations (27) and (28), we will employ the Laplace and finite Fourier transform. e mathematical models considered in this paper describe real problems of fluid flow and heat transfer in a rectangular channel. erefore, we consider for the study problems in which the temperature and velocity fields are functions that are at least twice differentiable in relation to the variables on which they depend. It is also considered that the functions that describe the movement and heat transfer have additional properties so that the existence of the Laplace and Fourier transforms to be provided. erefore, for the studied problems, we search solutions in the class of functions that satisfy the above properties.
Applying the finite sine-Fourier transform to differential equation (29) subject to the conditions in equation (30), we obtain where T s (n, s) � 1 0 T(y, s)sin(nπy)dy, n � 1, 2, . . .. Equation (31) can be written in the following equivalent form: Taking inverse Laplace transform of equation (32), we obtain where " * " represents the convolution product, δ(t) is the direct delta function, and G a,b,c (·) is the generalized G-function of Lorenzo and Hartley [49] defined as Taking the inverse sine-Fourier transform of equation (33), we obtain the temperature field For the ordinary case when β � 1, we have 6 Complexity 2 Pr exp − (nπ) 2 + Q Pr t sin(nπy).

(36)
For the special case, when Q � 0, the similar solution is recovered as obtained in equation (35) in [12].

Complexity
Taking the inverse Laplace transform of equation (41), we obtain Taking the inverse sine-Fourier transform of equation (42), we have Pr , t sin(nπy).

Complexity
Remarks 1 (i) For the classical case corresponding to α � β � 1, we have the ordinary velocity field of the OBF from equation (43): (ii) For fractional and classical Maxwell fluid when λ r ⟶ 0 and Q � 0 in equation (44), the similar solution is recovered as obtained in equations (43) and (44) in [12], respectively. (iii) For fractional and classical viscous fluid when λ ⟶ 0, λ r ⟶ 0, and Q � 0 in equation (44), w the similar solution is recovered as obtained in equation (45) and (46) in [12], respectively.

Numerical Results and Discussion
In order to examine the communicative changes on the thermal transport and flow profiles arising due to change of physical parameters, numerical estimations of the fluid temperature and velocity are figured and shown graphically in Figures 2-14.
In Figure 2, we exposed the effect of fractional parameter β at different values of time t and considering the functions f 1 (t) � − sin(πt/4) and f 2 (t) � sin(πt/4) at left and right plates, respectively. It is pointed out that, for small values of time t, the temperature is decreasing by increasing the values of fractional parameter β; after critical point, the influence becomes reversed for the large value of time. In Figure 3, we presented the effect of fractional parameter β at different values of time t and considering constant temperature at both plates. It has the similar influence like Figure 2. Also, we observed from this figure that the value of temperature at the Complexity middle of the channel is low.
e impact of absorption coefficient Q on temperature profile is presented in Figure 4. We observed that temperature is an increasing function of Q for small and large values of time t. Figures 5 and 6 are presented to show the critical point for Figures 2 and 3, respectively.
In Figure 7, we presented the effect of fractional parameter α at different values of time t, and f 1 (t) � H(t), f 2 (t) � e − 5t , and g 1 (t) � H(t) � g 2 (t). From Figures 7(a) and 7(b), we observed that velocity profile increasing for small time t corresponds to the large values of α, while velocity profile increasing for large values of time t corresponds to small values of α. e effect of fractional parameter β at different values of time t, f 1 (t) � f 1 (t) � H(t) � g 1 (t) � g 2 (t) and f 1 (t) � − H(t), and f 2 (t) � H(t) � g 1 (t) � g 2 (t) is presented in Figures 8 and 9, respectively.     In order to find the critical value of time at which the influence becomes reversed, it is presented in Figure 10. e effects of relaxation and retardation parameters λ and λ r and absorption coefficient Q on velocity profile are presented in Figures 11-13. We observed that the velocity is an increasing function of λ for small and large values of time t because the relaxation time is the time needed by the fluid particles to adjust the flow motion of the fluid, while velocity is a decreasing function of λ r and Q for small and large values of time t. e comparison between fractional Oldroyd-B, ordinary Oldroyd-B, fractional Maxwell, ordinary Maxwell, fractional viscous, and ordinary viscous fluids is presented in Figure 14. It is investigated that, for small time, ordinary fluids have greater velocity as compared to fractional fluids while for large time vice versa. Figures 15 and 16 are plotted to see the validity of our obtained results for temperature and velocity profiles by comparing to the Na et al. [12] outcomes. It can be seen from these figures that, by ignoring   Figure 10: Profiles of dimensionless velocity versus y for variation of α at altered values of time t, Gr � 5, λ � 2, λ r � 3, Pr � 20, Q � 0.5, and  Figure 11: Profiles of dimensionless velocity versus y for variation of λ at altered values of time t, Gr � 5, λ r � 3, Pr � 20, Q � 0.5, and   Figure 13: Profiles of dimensionless velocity versus y for variation of Q at altered values of time t, Gr � 5, λ � 2, λ r � 3, Pr � 20, and  Our results when Q=0 Figure 15: Temperature profiles of our models compared with the temperature profile of [12] f 1 (t) � f 1 (t) � H(t) � g 1 (t) � g 2 (t). 18 Complexity the effects of Q and λ r , our results are identical to those obtained by [12].

Conclusions
e aim of this work is to study the free-convection flow of an OBF between two vertical parallel plates in the presence of heat generation or absorption subject to generalized boundary conditions. e Caputo time-fractional derivative is introduced by means of the generalized constitutive shear stress and damped thermal flux by using generalized Fourier's law. Closed form solutions of fractional differential equations are found by employing the Laplace and sine-

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. Result [12,Eq. 43] Our result when Q=0, λr=0 Figure 16: Velocity profiles of our models compared with the velocity profile of [12] f 1 (t) � f 1 (t) � H(t) � g 1 (t) � g 2 (t).