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

Authors: Imran Siddique; Sehrish Ayaz; Dalal Alrowaili; Sohaib Abdal
Journal: Complexity
 (2021)

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## Abstract

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. The 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.


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## Body

## 1. 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 [2–8].Free convection flows are of incredible viable significance to engineers and researchers on account of its all-inclusive 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. The 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. The 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. They 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 [23–28].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. The 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 noninteger-order 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 [12–15, 46–48].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. The new fractional constitutive equations are characterized with the time-fractional Caputo derivative; hence, they give a power-law damping to the temperature and velocity. These 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. The solutions are uttered in the form of generalized G-functions of Lorenzo and Hartley [49]. The 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.


## 2. 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 distanced. The flow geometry in fixed Cartesian coordinate system x,ζ1∓andz is shown in Figure 1. The x−axis is taken along the plates in the direction of fluid flow with the velocity u∓ and the ζ1∓-axis is taken perpendicular to the plates.Figure 1
Flow geometry.Since the channel is infinitely extended in thex and z directions, we can assume that all physical entities describing the fluid motion and heat transfer are the functions of ζ1∓ and t1∓ only. At first, both the plates and the enclosed fluid are at rest at the ambient temperature Ta. After this moment, the left and right plates starts to slid in their own planes along x−direction with the velocities U0g1∓t1∓ and U0g2∓t1∓; individually, plates are maintained at temperatures Ta+Tb−Taf1∓t1∓ and Ta+Tb−Taf2∓t1∓, respectively, where U0 is a constant with dimension of velocity; the functions fi⋅ and gi⋅,i=1,2, are piecewise continuous and exponential order functions at infinity with fi0=gi0=0,i=1,2.We made the following assumptions for the fluid flows:(i)
Velocity is only a function of material coordinateζ1∓ and special coordinate t1∓, i.e., v=u1∓ζ1∓,t1∓,0,0(ii)
Joule heating and viscous dissipation in energy equations are negligibleIn the absence of a pressure gradient in the flow direction and utilizing the typical Boussinesq’s approximation, the governing equations for the unsteady free-convection flow of OBF through the vertical channel communicated by the subsequent equations [50, 51] are the linear momentum equation,(1)ρ∂u1∓ζ1∓,t1∓∂t1∓=∂τ1∓ζ1∓,t1∓∂ζ1∓+gρβTT∓ζ1∓,t1∓−Ta,the constitutive equation,(2)1+λ±∂∂t1∓τ1∓ζ1∓,t1∓=μ1+λr∓∂∂t1∓∂u1∓ζ1∓,t1∓∂ζ1∓,the thermal balance equation,(3)ρcp∂T∓ζ1∓,t1∓∂t1∓=−∂q1ζ1∓,t1∓∂ζ1∓−Q0T±ζ1∓,t1∓−Ta,and Fourier’s law,(4)q1ζ1∓,t1∓=−k∂T∓ζ1∓,t1∓∂ζ1∓,under corresponding initial and boundary conditions:(5)T∓ζ1∓,0=Ta,u1∓ζ1∓,0=0,0≤ζ1∓≤d,(6)T∓0,t1∓=Ta+Tb−Taf1∓t1∓,T∓d,t1∓=Ta+Tb−Taf2∓t1∓,t1∓≥0,(7)u1∓0,t1∓=U0g1∓t1∓,u1∓d,t1∓=U0g2∓t1∓,t1∓≥0.Introducing the following nondimensionless variables, functions, and parameters,(8)y=ζ1∓d,t=vt1∓d2,u=u1∓U0,T=T∓−TaTb−Ta,q=q1dkTb−Ta,τ=τ1∓dμU0,λ=vλ∓d2,λr=vλr∓d2,Q0=Qd2k,Pr=μcpk,Gr=gβTTb−Tad2vU0,fj∓t1∓=fjd2tν,gj∓t1∓=gjd2tν,j=1,2,,into equations (1)–(7), we have(9)∂uy,t∂t=∂τy,t∂y+GrTy,t,(10)1+λ∂∂tτy,t=1+λr∂∂t∂uy,t∂y,(11)∂Ty,t∂t=−1Pr∂qy,t∂y−QTy,t,(12)qy,t=−∂Ty,t∂y.The partial differential equations (9)–(12) are taken along with the following dimensionless initial and boundary conditions:(13)uy,0=0,Ty,0=0,0≤y≤1,(14)T0,t=f1t,T1,t=f2t,t≥0,(15)u0,t=g1t,u1,t=g2t,t≥0.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:(16)1+λ∂∂tτy,t=1+λr∂∂tDCt1−α∂uy,t∂y,0<α≤1,proposed by Scott-Blair [52], respectively, and the generalized Fourier’s law,(17)qy,t=−DCt1−β∂Ty,t∂y,0<β≤1,proposed by Povstenko [53] and Hristov [54], respectively. In the above constitutive equations, DCtα⋅ denotes the time-fractional Caputo derivative defined by [55, 56]. Hence,(18)DCtαuy,t=1Γ1−α∫0tu˙y,st−s−αds=hαt∗u˙y,t,0≤α≤1,where hαt=t−α/Γ1−α is the singular power-law kernel for u˙y,s=∂uy,t/∂t|t=s.Furthermore, using the second form of the time-fractional Caputo derivative from equation (18) and one of the following properties of hα⋅ is defined as(19)Lhαt=1s1−α,h1−α∗hαt=1,h0t=L−11s=1,h1t=L−11=δt,where L⋅ denotes the Laplace transform, δ⋅ is Dirac’s distribution, and s is the transform parameter, and it is easy to show that(20)DCt0fy,t=fy,t−fy,0,DCt0fy,t=fy,t,iffy,0=0,DCt1fy,t=∂fy,t∂t.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.The generalized constitutive equation (17), written in the equivalent form qy,t=−1/Γβ∂/∂y∫0tt−sβ−1∂Ty,t/∂t|t=sds, 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. The function h1−βt=tβ−1/Γβ is the time-nonlocality kernel (the weight function).The processes with time nonlocality are called processes with memory. The 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). The 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.Eliminatingτ from equations (9) and (16) and q from equations (11) and (17), we obtain(21)1+λ∂∂t∂uy,t∂t=1+λr∂∂tDCt1−α∂2uy,t∂y2+Gr1+λ∂∂tTy,t,(22)∂Ty,t∂t=1PrDCt1−β∂2Ty,t∂y2−QTy,t.In order to obtain the equivalent forms of equations (21) and (22), we recall the time-fractional integral operator:(23)Jtαfy,t=h1−α∗ft=1Γα∫0tfy,st−sα−1ds,which is the left-inverse operator of the derivative operator DCtα⋅. Indeed, using equations (18), (19), and (23), we obtain that(24)Jtα∘DCtαfy,t=JtαDCtαfy,t=h1−α∗hα∗f˙t=h1−α∗hα∗f˙t=1∗f˙t=fy,t−fy,0,which implies(25)Jtα∘DCtαfy,t=fy,t,iffy,0=0.Moreover, using equations (18) and (23), it results that(26)Jt1−αf˙y,t=hα∗f˙t=DCtαfy,t.Now, applying the fractional integral operatorsJt1−α and Jt1−β to equations (21) and (22) and using properties (24)–(26) together with the initial condition, we obtain the following fractional differential equations:(27)1+λ∂∂tDCtαuy,t=1+λr∂∂t∂2uy,t∂y2+GrJt1−α1+λ∂∂tTy,t,(28)DCtβTy,t=1Pr∂2Ty,t∂y2−QJt1−βTy,t.To determine the solutions of equations (27) and (28), we will employ the Laplace and finite Fourier transform.The mathematical models considered in this paper describe real problems of fluid flow and heat transfer in a rectangular channel. Therefore, 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.Therefore, for the studied problems, we search solutions in the class of functions that satisfy the above properties.


## 3. Solution of the Problem

### 3.1. Temperature Distribution

Applying the Laplace transform to equations (28) and (14), using initial condition (13), we obtain(29)sβT¯y,s=1Pr∂2T¯y,s∂y2−Qsβ−1T¯y,s,(30)T¯0,s=F1s,T¯1,s=F2s,where T¯y,s=∫0∞T¯y,te−stdt, s is Laplace transform parameter, and Lfit=Fis;i=1,2.Applying the finite sine-Fourier transform to differential equation (29) subject to the conditions in equation (30), we obtain(31)T¯˜sn,s=nπF1s+−1n+1F2sPrsβ+Qsβ−1+nπ,where T¯˜sn,s=∫01T¯y,ssinnπydy,n=1,2,….Equation (31) can be written in the following equivalent form:(32)T¯˜sn,p=F1snπ+−1n+1F2snπ−F1s+−1n+1F2snπ×1−nπ2Pr1sβ+nπ2/Pr+QPr−1sβ−1.Taking inverse Laplace transform of equation (32), we obtain(33)T˜sn,t=f1tnπ+−1n+1f2tnπ−f1t+−1n+1f2tnπ∗δt−nπ2∑m=0∞−QmPrm+1Gβ,β−1m,m+1−nπ2Pr,t,where “∗” represents the convolution product, δt is the direct delta function, and Ga,b,c⋅ is the generalized G-function of Lorenzo and Hartley [49] defined as(34)Ga,b,cd,t=L−1pbpa−dc,Rp>0,Rac−b>0,dpa<1.Taking the inverse sine-Fourier transform of equation (33), we obtain the temperature field(35)Ty,t=f1t1−y+f2ty−2∑n=1∞1nπf1t+−1n+1f2t∗δt−nπ2∑m=0∞−QmPrm+1Gβ,β−1m,m+1−nπ2Pr,tsinnπy.For the ordinary case whenβ=1, we have(36)Ty,t=f1t1−y+f2ty−2∑n=1∞1nπf1t+−1n+1f2t∗δt−nπ2Prexp−nπ2+QPrtsinnπy.For the special case, whenQ=0, the similar solution is recovered as obtained in equation (35) in [12].


### 3.2. Velocity Field

Applying the Laplace transform to equations (27) and (15), using initial condition (13), we obtain(37)1+λssαu¯y,s=1+λrs∂2u¯y,s∂y2+Gr1+λssα−1T¯y,s,(38)u¯0,s=G1s,u¯1,s=G2s.Applying the finite sine-Fourier transform to differential equation (37) subject to the conditions in equation (38), we obtain(39)sα+λsα+1+1+λrsnπ2u¯˜sn,s=nπ1+λrsG1s+−1n+1G2s+Grsα−1+λsαT¯˜sn,s.Using equation (31) in equation (39), we write the expression in suitable form as(40)u¯˜sn,s=nπ1+λrsG1s+−1n+1G2ssα+λsα+1+1+λrsnπ2+GrnπPrsα−1+λsαF1s+−1n+1F2ssα+λsα+1+1+λrsnπ21sβ+nπ2/Pr+QPr−1sβ−1.Equation (40) can be written in the following equivalent form by using the series formula 1/z1+b=∑k=0∞−1kz1k/bk+1,z1/b<1 as(41)u¯˜sn,s=G1snπ+−1n+1G2snπ−G1s+−1n+1G2snπ×∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!sα+1k−l−1sα+λrnπ2/λl+1+λsα+1k−lsα+λrnπ2/λl+1+GrnπF1s+−1n+1F2s×∑n=0∞−1nλl+1∑k=0nl!nπ2l−kk!l−k!sα+1k−l−2sα+λrnπ2/λl+1+λsα+1k−l−1sα+λrnπ2/λl+1∑m=0∞−QmPrm+1sβ−1msβ+nπ2/Prm+1.Taking the inverse Laplace transform of equation (41), we obtain(42)u˜sn,t=g1tnπ+−1n+1g2tnπ−1nπg1t+−1n+1g2t∗∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!Gα,α+1k−l−1,l+1−λrnπ2λ,t+λGα,α+1k−l,l+1−λrnπ2λ,t+Grnπf1t+−1n+1f2t∗∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!Gα,α+1k−l−2,n+1−λrnπ2λ,t+λGα,α+1k−l−1,m+1−λrnπ2λ,t∗∑m=0∞−QmPrm+1Gβ,β−1m,m+1−nπ2Pr,t.Taking the inverse sine-Fourier transform of equation (42), we have(43)uy,t=g1t1−y+g2ty−2∑n=1∞1nπg1t+−1n+1g2t∗∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!Gα,α+1k−l−1,l+1−λrnπ2λ,t+λGα,α+1k−l,l+1−λrnπ2λ,tsinnπy+2Gr∑n=1∞f1t+−1n+1f2t∗∑l=0∞−1lλl+1∑k=0nl!nπ2l−k+1k!l−k!×Gα,α+1k−l−2,n+1−λrnπ2λ,t+λGα,α+1k−l−1,l+1−λrnπ2λ,t∗∑m=0∞−QmPrm+1Gβ,β−1m,m+1−nπ2Pr,tsinnπy.Remarks 1.
(i)
For the classical case corresponding toα=β=1, we have the ordinary velocity field of the OBF from equation (43): (44)uy,t=g1t1−y+g2ty−2∑n=1∞1nπg1t+−1n+1g2t∗∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!G1,2k−l−1,l+1−λrnπ2λ,t+λG1,2k−l,l+1−λrnπ2λ,tsinnπy+2GrnπPr∑n=1∞f1q+−1n+1f2q∗∑l=0∞−1lλl+1∑k=0nl!nπ2l−kk!l−k!G1,2k−l−2,n+1−λrnπ2λ,t+λG1,2k−l−1,l+1−λrnπ2λ,t∗∑m=0∞−QmPrm+1G1,0,m+1−nπ2Pr,tsinnπy.(ii)
For fractional and classical Maxwell fluid whenλr⟶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 whenλ⟶0,λr⟶0, and Q=0 in equation (44), w the similar solution is recovered as obtained in equation (45) and (46) in [12], respectively.


## 3.1. Temperature Distribution

Applying the Laplace transform to equations (28) and (14), using initial condition (13), we obtain(29)sβT¯y,s=1Pr∂2T¯y,s∂y2−Qsβ−1T¯y,s,(30)T¯0,s=F1s,T¯1,s=F2s,where T¯y,s=∫0∞T¯y,te−stdt, s is Laplace transform parameter, and Lfit=Fis;i=1,2.Applying the finite sine-Fourier transform to differential equation (29) subject to the conditions in equation (30), we obtain(31)T¯˜sn,s=nπF1s+−1n+1F2sPrsβ+Qsβ−1+nπ,where T¯˜sn,s=∫01T¯y,ssinnπydy,n=1,2,….Equation (31) can be written in the following equivalent form:(32)T¯˜sn,p=F1snπ+−1n+1F2snπ−F1s+−1n+1F2snπ×1−nπ2Pr1sβ+nπ2/Pr+QPr−1sβ−1.Taking inverse Laplace transform of equation (32), we obtain(33)T˜sn,t=f1tnπ+−1n+1f2tnπ−f1t+−1n+1f2tnπ∗δt−nπ2∑m=0∞−QmPrm+1Gβ,β−1m,m+1−nπ2Pr,t,where “∗” represents the convolution product, δt is the direct delta function, and Ga,b,c⋅ is the generalized G-function of Lorenzo and Hartley [49] defined as(34)Ga,b,cd,t=L−1pbpa−dc,Rp>0,Rac−b>0,dpa<1.Taking the inverse sine-Fourier transform of equation (33), we obtain the temperature field(35)Ty,t=f1t1−y+f2ty−2∑n=1∞1nπf1t+−1n+1f2t∗δt−nπ2∑m=0∞−QmPrm+1Gβ,β−1m,m+1−nπ2Pr,tsinnπy.For the ordinary case whenβ=1, we have(36)Ty,t=f1t1−y+f2ty−2∑n=1∞1nπf1t+−1n+1f2t∗δt−nπ2Prexp−nπ2+QPrtsinnπy.For the special case, whenQ=0, the similar solution is recovered as obtained in equation (35) in [12].


## 3.2. Velocity Field

Applying the Laplace transform to equations (27) and (15), using initial condition (13), we obtain(37)1+λssαu¯y,s=1+λrs∂2u¯y,s∂y2+Gr1+λssα−1T¯y,s,(38)u¯0,s=G1s,u¯1,s=G2s.Applying the finite sine-Fourier transform to differential equation (37) subject to the conditions in equation (38), we obtain(39)sα+λsα+1+1+λrsnπ2u¯˜sn,s=nπ1+λrsG1s+−1n+1G2s+Grsα−1+λsαT¯˜sn,s.Using equation (31) in equation (39), we write the expression in suitable form as(40)u¯˜sn,s=nπ1+λrsG1s+−1n+1G2ssα+λsα+1+1+λrsnπ2+GrnπPrsα−1+λsαF1s+−1n+1F2ssα+λsα+1+1+λrsnπ21sβ+nπ2/Pr+QPr−1sβ−1.Equation (40) can be written in the following equivalent form by using the series formula 1/z1+b=∑k=0∞−1kz1k/bk+1,z1/b<1 as(41)u¯˜sn,s=G1snπ+−1n+1G2snπ−G1s+−1n+1G2snπ×∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!sα+1k−l−1sα+λrnπ2/λl+1+λsα+1k−lsα+λrnπ2/λl+1+GrnπF1s+−1n+1F2s×∑n=0∞−1nλl+1∑k=0nl!nπ2l−kk!l−k!sα+1k−l−2sα+λrnπ2/λl+1+λsα+1k−l−1sα+λrnπ2/λl+1∑m=0∞−QmPrm+1sβ−1msβ+nπ2/Prm+1.Taking the inverse Laplace transform of equation (41), we obtain(42)u˜sn,t=g1tnπ+−1n+1g2tnπ−1nπg1t+−1n+1g2t∗∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!Gα,α+1k−l−1,l+1−λrnπ2λ,t+λGα,α+1k−l,l+1−λrnπ2λ,t+Grnπf1t+−1n+1f2t∗∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!Gα,α+1k−l−2,n+1−λrnπ2λ,t+λGα,α+1k−l−1,m+1−λrnπ2λ,t∗∑m=0∞−QmPrm+1Gβ,β−1m,m+1−nπ2Pr,t.Taking the inverse sine-Fourier transform of equation (42), we have(43)uy,t=g1t1−y+g2ty−2∑n=1∞1nπg1t+−1n+1g2t∗∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!Gα,α+1k−l−1,l+1−λrnπ2λ,t+λGα,α+1k−l,l+1−λrnπ2λ,tsinnπy+2Gr∑n=1∞f1t+−1n+1f2t∗∑l=0∞−1lλl+1∑k=0nl!nπ2l−k+1k!l−k!×Gα,α+1k−l−2,n+1−λrnπ2λ,t+λGα,α+1k−l−1,l+1−λrnπ2λ,t∗∑m=0∞−QmPrm+1Gβ,β−1m,m+1−nπ2Pr,tsinnπy.Remarks 1.
(i)
For the classical case corresponding toα=β=1, we have the ordinary velocity field of the OBF from equation (43): (44)uy,t=g1t1−y+g2ty−2∑n=1∞1nπg1t+−1n+1g2t∗∑l=0∞−1lλl+1∑k=0ll!nπ2l−kk!l−k!G1,2k−l−1,l+1−λrnπ2λ,t+λG1,2k−l,l+1−λrnπ2λ,tsinnπy+2GrnπPr∑n=1∞f1q+−1n+1f2q∗∑l=0∞−1lλl+1∑k=0nl!nπ2l−kk!l−k!G1,2k−l−2,n+1−λrnπ2λ,t+λG1,2k−l−1,l+1−λrnπ2λ,t∗∑m=0∞−QmPrm+1G1,0,m+1−nπ2Pr,tsinnπy.(ii)
For fractional and classical Maxwell fluid whenλr⟶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 whenλ⟶0,λr⟶0, and Q=0 in equation (44), w the similar solution is recovered as obtained in equation (45) and (46) in [12], respectively.


## 4. 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 Figures2–14.Figure 2
Profiles of dimensionless temperature versusy for variation of β at altered values of time t, Pr=20andQ=0.5, and f1t=−sinπt/4andf2t=sinπt/4.
(a)(b)(c)(d)Figure 3
Profiles of dimensionless temperature versusy for variation of β at altered values of time t, Pr=20andQ=0.5, and f1t=Ht=f2t.
(a)(b)(c)(d)Figure 4
Profiles of dimensionless temperature versusy for variation of Q at altered values of time t, Pr=20, and f1t=Ht=f2t.
(a)(b)Figure 5
Profiles of dimensionless temperature versest for β at two values of y for Pr=20andQ=0.5 and f1t=−sinπt/4andf2t=sinπt/4.
(a)(b)Figure 6
Profiles of dimensionless temperature versest for β at two values of y for Pr=20,Q=0.5, and f1t=Ht=f2t.
(a)(b)Figure 7
Profiles of dimensionless velocity versusy for variation of α at altered values of time t, Gr=5,λ=2,λr=3,Pr=20,andQ=0.5, and f1t=Ht,f2t=e−5t,andg1t=Ht=g2t.
(a)(b)(c)(d)Figure 8
Profiles of dimensionless velocity versusy for variation of α at altered values of time t, Gr=5,λ=2,λr=3,Pr=20,andQ=0.5, and f1t=−Htandf2t=Ht=g1t=g2t.
(a)(b)(c)(d)Figure 9
Profiles of dimensionless velocity versusy for variation of β at altered values of time t, Gr=5,λ=2,λr=3,Pr=20,Q=0.5, and f1t=f1t=Ht=g1t=g2t.
(a)(b)(c)(d)Figure 10
Profiles of dimensionless velocity versusy for variation of α at altered values of time t, Gr=5,λ=2,λr=3,Pr=20,Q=0.5, and f1t=f1t=Ht=g1t=g2t.
(a)(b)Figure 11
Profiles of dimensionless velocity versusy for variation of λ at altered values of time t, Gr=5,λr=3,Pr=20,Q=0.5, and f1t=f1t=Ht=g1t=g2t.
(a)(b)Figure 12
Profiles of dimensionless velocity versusy for variation of λ at altered values of time t, Gr=5,λ=2,Pr=20,Q=0.5, and f1t=f1t=Ht=g1t=g2t.
(a)(b)Figure 13
Profiles of dimensionless velocity versusy for variation of Q at altered values of time t, Gr=5,λ=2,λr=3,Pr=20, and f1t=f1t=Ht=g1t=g2t.
(a)(b)Figure 14
Comparisons between fractional Oldroy-B, ordinary Oldroyd-B, fractional Maxwell, ordinary Maxwell, fractional viscous, and ordinary viscous fluids atGr=5,Pr=20,Q=0.5, and f1t=f1t=Ht=g1t=g2t.
(a)(b)In Figure2, we exposed the effect of fractional parameter β at different values of time t and considering the functions f1t=−sinπt/4andf2t=sinπt/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 middle of the channel is low. The 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.Figures5 and 6 are presented to show the critical point for Figures 2 and 3, respectively.In Figure7, we presented the effect of fractional parameter α at different values of time t, and f1t=Ht, f2t=e−5t,andg1t=Ht=g2t. 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. The effect of fractional parameter β at different values of time t, f1t=f1t=Ht=g1t=g2t and f1t=−Ht, and f2t=Ht=g1t=g2t is presented in Figures 8 and 9, respectively. From Figures 7–9, it is observed that, for small values of time velocity decreases by increasing the values of fractional parameters α, while the influence is opposite for large values of time.In order to find the critical value of time at which the influence becomes reversed, it is presented in Figure10. The effects of relaxation and retardation parameters λ and λr 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 λr and Q for small and large values of time t. The 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 the effects of Q and λr, our results are identical to those obtained by [12].Figure 15
Temperature profiles of our models compared with the temperature profile of [12] f1t=f1t=Ht=g1t=g2t.Figure 16
Velocity profiles of our models compared with the velocity profile of [12] f1t=f1t=Ht=g1t=g2t.


## 5. Conclusions

The 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. The 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-Fourier transforms which are suitable for the boundary conditions. The obtained solutions are expressed in the form of generalized G-functions of Lorenzo and Hartley. The effects of fractional and physical parameters are graphically presented in Figures2–14. Some key findings of our work are(i)
For small values of time, temperature is decreasing by increasing the values of fractional parameter and vice versa(ii)
Velocity decreases by increasing the values of fractional parameters for small values of time and vice versa(iii)
Velocity is an increasing and decreasing functions ofλ and λr for small and large values of time t, respectively(iv)
Velocity is a decreasing functions of absorption coefficientQ for small and large values of time t(v)
Velocity of Maxwell ordinary fluid is higher as compared to ordinary viscous and ordinary OBF for small times, while for large times velocity of fractional Maxwell fluid is higher as compared to fractional viscous and fractional OBF(vi)
Our obtained results for temperature and velocity profiles are identical to those obtained by [12]



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