# Fractional Study for Transient Free Convection Flow in a Channel with Mittag-Leffler Memory

Authors: Shahzad Sarwar; Maryam Aleem; Muhammad Imran Asjad; Taseer Muhammad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Fractional-order convective transient flow of viscous and incompressible fluids transiting through two infinite hot parallel upright plates is investigated analytically in the presence of chemical reaction, radiative heat flux, and mass diffusion at the boundary. A physical model for transient incompressible unsteady flow is developed with a comparatively new fractional derivative, namely, Atangana–Baleanu with nonsingular, nonlocal kernel. The developed fractional model is studied with means of an integral transform, i.e., Laplace transform method. Results obtained for the concentration, velocity, and temperature are expressed in the form of generalizedMqpy function. The impact of various physical parameters like fractional and flow parametric quantities is demonstrated diagrammatically. At last, we envisioned that for the fractional model, temperature and concentration could be enhanced for smaller fractional parameter α values while velocity for larger values of α, respectively. The proposed model gives the better results in the presence of memory effect besides the Caputo and Caputo–Fabrizio model when compared with the existing literature.


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## Body

## 1. Introduction

Processes affecting coupled mass and heat conveyance occur frequently in nature. This happens not only because of temperature gradient but also due to the concentration difference or a combination of these two differences mentioned earlier. Buoyancy forces stimulating the flow with the aggregated effect of mass and thermal diffusion have been of major concern for last three and a half decades. Nowadays, these specified problems have gained the attention of many researchers for numerous technological and engineering applications, specified as mass and heat transfer assorted with storage of nuclear fuel rubble, coal gasification below the earth’s surface, hydrology of ground water, chemical engineering, cooling of processors, and so on.Free convective flow through vertical channels had been studied extensively due to its important application in engineering and applied sciences [1]. Harris et al. [2, 3] had investigated the results of free convective transient flow transiting by an upright erect plate engrafted in poriferous media submitted to an abrupt transfer in heat flux and superficial temperature. The heat yielded by viscid dissipation gives an increment in temperature approaching the wall resulted in viscousness reduction and a substantial stratification in its profile, which bears upon H.T.R, i.e., heat transfer rate [4, 5]. Gupta underwent the study of steady, transient, free convection of an electrically transmitting fluids passing by a perpendicular plate, bearing the magnetic flux [6]. Toki applied the Laplace transform to solve transient convective rate of flow inside an oscillating poriferous. The problem is solved by [7]. Sharma et al. had enquired the oscillatory reactive hydromagnetics innate or free convective boundary levels in poriferous media with hybrid rate of flow impressions [8]. An analysis of the transient, buoyancy effectuated stream, and heat conveyance in a Darcian fluid drenched in permeable medium adjoining to an abruptly heated infinite plate was presented by Haq and Mulligan [9]. Ramanaiah and Kumaran had discussed free convection of fluid passing through a passable cylinder and cone, with radiation boundary condition [10].Results of the transient free convective flow across a propelling vertical or inclined plate and cylinder were incurred by Soundalgekar and Ganesan [11]. Takhar et al. had talked about the upshot of thermo-corporeal measures on free convective vaporous stream across an isothermic perpendicular cone in steady-state current, in which caloric conduction, absolute viscousness, and specific heat at unvarying pressure were developed as a power law magnetic fluctuation with inviolable temperature. In their research composition, they reasoned out the information of transfer of heat step-ups with suction or drop-offs with injectant [12].Transient convection contains fundamental concern in several progressive and environmental situations as in air conditioner systems, in human consolation in buildings, in atmospherical flows, in thermal regulating processes, in cooling down of electronic machines, in protection of energy systems, and so on. It is pertinent to mention here about transient flow. What it is about? Flow is said to be unsteady or transient if velocity of fluid is just not function of space variable or position inxy-plane but also depending on time. A lot of work accounted in the literature dealt with fixed velocity fields and temperature domains but merely a small figure administer with time varying boundary checks, either in natural, forced, or mixed convection [13–17].Ganesan and Muthucumaraswamy had numerically investigated incompressible, viscid, transient fluid’s flow regime passing through a semi-infinite isothermal plate satisfying natural convection [18]. Siddiqa et al. [19] had investigated transient effects on free convection of heat transfer by natural convection along a vertical wavy surface. Singh [20] and Nanda and Sharma [21] had investigated the suction effects of free convective, transient flow passing through vertical porous plate. A numerical study to analyze transient effects of natural heat and mass conveyance or free convection in power law fluids passing through upright plate embedded in poriferous medium was carried out by Nasser [22].In this modern era, the study of fractional calculus becomes hot topic because of its huge applications in all disciplines of science and engineering. Researchers have keen interest to formulate the physical problems with noninteger derivatives because fractional-order models provide the much efficient description of model[23–28]. Sarwar et al. studied the non-Newtonian fractional brinkman type fluid with AB derivative [29]. Aleem et al. had investigated channel flow of MHD Jeffrey fluid between two heated vertical parallel plates [30]. In another study, a fractional heat and mass transferal model was developed and observed that MWCT-based nanofluids are efficiently good for heat transfer as compared with SWCTs-CMC-based nanofluids [31]. For significance of time fractional operator in heat transfer analysis, refer to [32–38]. Further literature could be seen in [39–44].In this paper, our major objective is to demonstrate free convective, unsteady, flow of viscous, and incompressible fluids passing through two infinite hot parallel vertical plates. We have extended the classical derivative model to noninteger order differential Atangana–Baleanu model of orderα. The method of Laplace transform is accustomed to acquire the results. Equations of dimensionless temperature, velocity, and concentration fields have been solved analytically, and results are compared. The graphical analysis is made to envision the effect of fractional and tangible flow parameters on velocity, concentration, and temperature by MathCad and Mathematica softwares. Moreover, the impact of fractional order and other parameters is presented graphically. The obtained results are compared with the existing literature for validation.


## 2. Problem Formulation

Let us undertake an incompressible, unsteady, free convective liquid passing through two parallel upright plates possessing the traits of temperature gradient and mass diffusion. We have fixed the plates inxy− plane of the Cartesian coordinate system. One of the plate is fixed along x−axis as shown in figure while y-axis is normal to the plate. The following assumptions have been made:(1)
At timet≤0, the plates and the fluent possess ambient concentration Cd and temperature Td(2)
At timet>0, concentration and temperature of the fluid at y=0 are changed to Cw and Tw, respectively(3)
The change in temperature and concentration has created free convection flows, presented in Figure1.Figure 1
Problem orientation.(1)ut=νuyy+gβT−Td+gβ∗C−Cd,(2)ρCpTt=kTyy−qry,y,t>0,Equations governing unsteady flow are obtained by Boussinesq’s approximations [1].where qr is radiative heat flux and by Rosseland approximation [45], the above equation will get the form [46] as follows:(3)Tt=kρCp1+16σ∗Td33kk∗Tyy,y,t>0,where σ∗ is the Stefan–Boltzmann constant, ρ is the density, k is the thermic conduction, k∗ represents the mean value of heat absorption parameter, and Cp stands for the specific heat with invariant pressure.(4)Ct=DCyy−KC−Cd.K is chemical reaction parameter. Appropriate IBCs are(5)t=0:ut,y=0,Tt,y=Td,Ct,y=Cd,0≤y≤d,(6)0<t:ut,y=0,Tt,y=Tw,Ct,y=Cw,aty=0,(7)0<t:ut,y=0,Tt,y=Td,Ct,y=Cd,aty=d.In order to make problem dimensionless, the following variables or parameters are used:(8)t∗=νtd2,y∗=yd,u∗=νud2gβTw−Td,Gr=gβTw−Tdd3ν2,N=GmGr,θ=T−TdTw−Td,Φ=C−CdCw−Cd,Gm=gβ∗Cw−Cdd3ν2,Sc=νD,Pr=μCpκ,Nr=16σ∗Td33kk∗,Preff=Pr1+Nr,Kr=Kd2ν.Gm, Gr, Preff, Nr, and Kr are dimensionless mass and thermal Grashof numbers, effective Prandtl number, and conduction-radiation and chemical reaction parameters, respectively.Equations (1), (3), and (4) become(9)ut−θ+NC+uyy=0,(10)Preffθt−θyy=0,(11)ScCt−Φyy+KrC=0,with dimensionless IBCs (initial and boundary conditions) given in equations (5)–(7).(12)t=0:ut,y=0,θt,y=0,Ct,y=0,for0≤y≤1,(13)0<t:ut,y=0,θt,y=1,Ct,y=1,aty=0,(14)0<t:ut,y=0,θt,y=0,Ct,y=0,aty=1.Equations (9)–(11) are nonhomogenous PDE’s of second order. The fractional model is obtained by changing time derivative with fractional derivative given as follows:(15)ABCDtαut,y−θy,t+NCt,y+uyyt,y=0,(16)ABCDtαθt,y−1Preffθyyt,y=0,(17)Cyy−ScABCDtαCt,y+KrCt,y=0.In 2016, Atangana and Baleanu introduced a new fractional-order operator of differentiation in Riemann–Liouville and Caputo sense, which is called Atangana–Baleanu (AB) derivatives. This new AB derivative is nonsingular and has nonlocal kernel and possesses the long memory due to the existence of Mittag-Lefler kernel which is the generalization of the exponential kernel. This new AB fractional derivative in Caputo sense of orderα>0 is defined as follows [47, 48]:(18)ABCDa,tαft=Mα1−α∫atEα−αt−τα1−αf′τdτ,t>a,where Eαx=∑k=1∞xk/Γαk+1 is Mittag-Leffler function and Mα is normalization function satisfying the conditions M0=M1=0.The Laplace transformation ofABC fractional derivative is(19)LABCDtαft,y=sαLft,y−sα−1fy,0sα1−α+α.


## 3. Fractional-Order IBV Problem of Fluid

In this section, we develop the fractional model of equations (15)–(17) by using relation given in equation (19), and we get partial differential equations with associated initial and boundary conditions:(20)sαξsα+ξαu¯y,s=∂2u¯y,s∂y2+θ¯y,s+NC¯y,s,(21)Preffsαξsα+ξαθ¯y,s=∂2θ¯y,s∂y2,(22)Scsαξsα+ξαC¯y,s=∂2C¯y,s∂y2−KrC¯y,s,associated with initial and boundary conditions:(23)u¯y,0=0,T¯y,0=0,C¯y,0=0,(24)u¯0,s=0,T¯0,s=1s,C¯0,s=1s,(25)u¯1,s=0,T¯1,s=0,C¯1,s=0,where ξ=1/1−α.


## 4. Analytical Solution of Fractional-Order IBV Problem of Fluid

Equations (21) and (22) are solved separately, and then their results are substituted in equation (20), i.e., momentum equation.


### 4.1. Results of Fractional Temperature Field

Utilizing the Laplace transform to equation (21), we obtain(26)∂2∂y2−Preffsαξsα+ξαθ¯y,s=0,θ¯y,s=c3exp−yPreffsαξsα+ξα+c4expyPreffsαξsα+ξα.Using conditions from equations (24) and (25) in equation (26), we have(27)θ¯y,s=1ssinh1−yPreffsαξ/sα+ξαsinhPreffsαξ/sα+ξα,θ¯y,s=1s∑m=0∞e−2m+yPreffsαξ/sα+ξα−e−2m+2−yPreffsαξ/sα+ξα.Equation (27) can be written in suitable form as(28)θ¯y,s=1s+∑m=0∞∑m1=1∞∑m2=0∞−2m+ym1Preffξm1/2−ξαm2m1!m2!s1+αm2Γm1+m2Γm1−∑m=0∞∑m3=1∞∑m4=0∞y−2m−2m3Preffξm3/2−ξαm4m3!m4!s1+αm4Γm3+m4Γm3.Now, by applying inverse Laplace transform to equation (28),(29)θt,y=1+∑m=0∞∑m1=1∞∑m2=0∞−2m+ym1Preffξm1/2−ξαm2m1!m2!tαm2Γm1+m2Γm1Γ1+αm2−∑m=0∞∑m3=1∞∑m4=0∞y−2m−2m3Preffξm3/2−ξαm4m3!m4!tαm4Γm3+m4Γm3Γ1+αm4.Further, the solution can be expressed in more general M-function [49] form as(30)θt,y=1+∑m=0∞∑m1=1∞−2m+ym1Preffξm1/2m1!M21−ξαtα|m1,0,1,αm1,1−∑m=0∞∑m3=1∞y−2m−2m3Preffξm3/2m3!M21−ξαtα|m3,0,1,αm3,1.In case of ordinary temperature, whenα⟶1 and Nr=0, our solution is reduced to the known results obtained in [1], and for the validation, Figure 2 is presented.Figure 2
Comparison forθt,y.


### 4.2. Results of Fractional Concentration Field

Solving equation (22) by using conditions from (24) and (25) in a similar pattern as presented for temperature field, solution for concentration field is(31)Ct,y=1+∑l=0∞∑l1=1∞∑l2=0∞−2l+yl1Krl1/2−l2Scξl2l1!l2!M21−ξαtα|l2,0,1,αl2,1−−∑l=0∞∑l4=1∞∑l5=0∞y−2l−2l4Krl4/2−l5Scξl5l4!l5!M21−ξαtα|l5,0,1,αl5,1.In case of ordinary concentration whenα⟶1 and Kr=0, our solution is reduced to the known results obtained in [1], and for validation, Figure 3 is presented.Figure 3
Comparison forCt,y.


### 4.3. Results of Fractional Velocity Field

Solving equation (20) after substituting equations (27) and (31), we get(32)∂2∂y2−sαξsα+ξαu¯y,s=−1s∑m=0∞e−2m+yPreffsαξ/sα+ξα−e−2m+2−yPreffsαξ/sα+ξα−Ns∑l=0∞e−2l+yScsαξ/sα+ξα+Kr−e−2l+2−yScsαξ/sα+ξα+Kr.Solving equation (32) subject to conditions (24) and (25),(33)u¯y,s=Ns∑l=0∞∑n=0∞e−2l+yScsαξ/sα+ξα+Kr−e−2l+2−yScsαξ/sα+ξα+KrKr+Sc−1sαξ/sα+ξαe−2n+ysαξ/sα+ξα−ey−2n−2sαξ/sα+ξα+1s∑m=0∞∑n=0∞e−2m+yPreffsαξ/sα+ξα−e−2m+2−yPreffsαξ/sα+ξαPreff−1sαξ/sα+ξαe−2n+ysαξ/sα+ξα−ey−2n−2sαξ/sα+ξα−Ns∑l=0∞e−2l+yScsαξ/sα+ξα+Kr−e−2l+2−yScsαξ/sα+ξα+KrKr+Sc−1sαξ/sα+ξα−1s∑m=0∞e−2m+yPreffsαξ/sα+ξα−e−2m+2−yPreffsαξ/sα+ξαPreff−1sαξ/sα+ξα.Equation (33) can be written as(34)u¯y,s=∑l=0∞∑n=0∞∑l1=0∞∑l2=0∞∑l3=0∞∑l7=0∞∑l8=0∞N∑n1=0∞∑n2=0∞Φ¯1−N∑n3=0∞∑n4=0Φ¯2−∑l=0∞∑n=0∞∑l4=0∞∑l5=0∞∑l6=0∞∑l7=0∞∑l8=0∞N∑n1=0∞∑n2=0∞Φ¯3−N∑n3=0∞∑n4=0∞Φ¯4+∑m=0∞∑n=0∞∑m1=0∞∑m2=0∞∑m5=0∞∑n1=0∞∑n2=0∞Φ¯5−∑n3=0∞∑n4=0∞Φ¯6−∑m=0∞∑n=0∞∑m3=0∞∑m4=0∞∑m5=0∞∑n1=0∞∑n2=0∞Φ¯7+∑n3=0∞∑n4=0∞Φ¯8−N∑l=0∞∑l1=1∞∑l2=0∞∑l3=0∞∑l7=0∞∑l8=0∞−2l+yl1Scξl2ξ1−Scl7−ξαl3+l8l1!l2!l3!l8!Kr1−l1/2+l2+l7s1+αl3+αl8Γl2+l3Γl7+l8Γl2Γl7+N∑l=0∞∑l4=1∞∑l5=0∞∑l6=0∞∑l7=0∞∑l8=0∞y−2l−2l4Scξl5ξ1−Scl7−ξαl6+l8l4!l5!l6!l8!Kr1−l4/2+l5+l7s1+αl6+αl8Γl5+l6Γl7+l8Γl5Γl7−∑m=0∞∑m1=1∞∑m2=0∞∑m5=0∞−2m+ym1Preffξm1/2−ξαm2+m5m1!m2!Preff−1s1+αm2+αm5Γm1+m2Γm1+∑m=0∞∑m3=1∞∑m4=0∞∑m5=0∞y−2m−2m3Preffξm3/2−ξαm4+m5m3!m4!s1+αm4+αm5Γm3+m4Γm3,where(35)Φ¯1=−2l+yl1Scξl2−ξαl3+l8+n2l1!l2!l3!l8!n1!n2!Kr1−l1/2+l2+l7ξ1−Scl7−2n+yn1ξn1/2Γl2+l3Γl7+l8Γn1/2+n2s1+αl3+αl8+αn2Γl2Γl7Γn1/2,Φ¯2=−2l+yl1Scξl2−ξαl3+l8+n4l1!l2!l3!l8!n3!n4!Kr1−l1/2+l2+l7ξ1−Scl7y−2n−2n3ξn3/2Γl2+l3Γl7+l8Γn3/2+n4s1+αl3+αl8+αn4Γl2Γl7Γn3/2,Φ¯3=y−2l−2l4Scξl5−ξαl6+l8+n2l4!l5!l6!l8!n1!n2!Kr1−l4/2+l5+l7ξ1−Scl7−2n+yn1ξn1/2Γl5+l6Γl7+l8Γn1/2+n2s1+αl6+αl8+αn2Γl5Γl7Γn1/2,Φ¯4=y−2l−2l4Scξl5−ξαl6+l8+n4l4!l5!l6!l8!n3!n4!Kr1−l4/2+l5+l7ξ1−Scl7y−2n−2n3ξn3/2Γl5+l6Γl7+l8Γn1/2+n2s1+αl6+αl8+αn4Γl5Γl7Γn1/2,Φ¯5=−2m+ym1Preffξm1/2−ξαm2+m5+n2m1!m2!n1!n2!Preff−1−2n+yn1ξn1/2Γm1+m2Γn1/2+n2s1+αm2+αm5+αn2Γm1Γn1/2,Φ¯6=−2m+ym1Preffξm1/2−ξαm2+m5+n4m1!m2!n3!n4!Preff−1y−2n−2n3ξn3/2Γm1+m2Γn3/2+n4s1+αm2+αm5+αn4Γm1Γn3/2,Φ¯7=y−2m−2m3Preffξm3/2−ξαm4+m5+n2m3!m4!n1!n2!Preff−1−2n+yn1ξn1/2Γm3+m4Γn1/2+n2s1+αm4+αm5+αn2Γm3Γn1/2,Φ¯8=y−2m−2m3Preffξm3/2−ξαm4+m5+n4m3!m4!n3!n4!Preff−1y−2n−2n3ξn3/2Γm3+m4Γn3/2+n4s1+αm4+αm5+αn4Γm3Γn3/2.Taking Laplace inverse, we have(36)uy,t=∑l=0∞∑n=0∞∑l1=0∞∑l2=0∞∑l3=0∞∑l7=0∞∑l8=0∞N∑n1=0∞∑n2=0∞Φ1−N∑n3=0∞∑n4=0Φ2−∑l=0∞∑n=0∞∑l4=0∞∑l5=0∞∑l6=0∞∑l7=0∞∑l8=0∞N∑n1=0∞∑n2=0∞Φ3−N∑n3=0∞∑n4=0∞Φ4+∑m=0∞∑n=0∞∑m1=0∞∑m2=0∞∑m5=0∞∑n1=0∞∑n2=0∞Φ5−∑n3=0∞∑n4=0∞Φ6−∑m=0∞∑n=0∞∑m3=0∞∑m4=0∞∑m5=0∞∑n1=0∞∑n2=0∞Φ7+∑n3=0∞∑n4=0∞Φ8−N∑l=0∞∑l1=1∞∑l2=0∞∑l3=0∞∑l7=0∞∑l8=0∞−2l+yl1Scξl2ξ1−Scl7−ξαl3+l8tαl3+αl8l1!l2!l3!l8!Kr1−l1/2+l2+l7×Γl2+l3Γl7+l8Γ1+αl3+αl8Γl2Γl7+N∑l=0∞∑l4=1∞∑l5=0∞∑l6=0∞∑l7=0∞∑l8=0∞y−2l−2l4Scξl5ξ1−Scl7−ξαl6+l8tαl6+αl8l4!l5!l6!l8!Kr1−l4/2+l5+l7×Γl5+l6Γl7+l8Γ1+αl6+αl8Γl5Γl7−∑m=0∞∑m1=1∞∑m2=0∞∑m5=0∞−2m+ym1Preffξm1/2−ξαm2+m5m1!m2!Preff−1tαm2+αm5Γm1+m2Γ1+αm2+αm5Γm1+∑m=0∞∑m3=1∞∑m4=0∞∑m5=0∞y−2m−2m3Preffξm3/2−ξαm4+m5m3!m4!tαm4+αm5Γm3+m4Γ1+αm4+αm5Γm3,where(37)Φ1=−2l+yl1Scξl2−ξαl3+l8+n2l1!l2!l3!l8!n1!n2!Kr1−l1/2+l2+l7×ξ1−Scl7−2n+yn1ξn1/2tαl3+αl8+αn2Γl2+l3Γl7+l8Γn1/2+n2Γ1+αl3+αl8+αn2Γl2Γl7Γn1/2,Φ2=−2l+yl1Scξl2−ξαl3+l8+n4l1!l2!l3!l8!n3!n4!Kr1−l1/2+l2+l7×ξ1−Scl7y−2n−2n3ξn3/2tαl3+αl8+αn4Γl2+l3Γl7+l8Γn3/2+n4Γ1+αl3+αl8+αn4Γl2Γl7Γn3/2,Φ3=y−2l−2l4Scξl5−ξαl6+l8+n2l4!l5!l6!l8!n1!n2!Kr1−l4/2+l5+l7×ξ1−Scl7−2n+yn1ξn1/2tαl6+αl8+αn2Γl5+l6Γl7+l8Γn1/2+n2Γ1+αl6+αl8+αn2Γl5Γl7Γn1/2,Φ4=y−2l−2l4Scξl5−ξαl6+l8+n4l4!l5!l6!l8!n3!n4!Kr1−l4/2+l5+l7×ξ1−Scl7y−2n−2n3ξn3/2tαl6+αl8+αn4Γl5+l6Γl7+l8Γn1/2+n2Γ1+αl6+αl8+αn4Γl5Γl7Γn1/2,Φ5=−2m+ym1Preffξm1/2−ξαm2+m5+n2m1!m2!n1!n2!Preff−1×−2n+yn1ξn1/2tαm2+αm5+αn2Γm1+m2Γn1/2+n2Γ1+αm2+αm5+αn2Γm1Γn1/2,Φ6=−2m+ym1Preffξm1/2−ξαm2+m5+n4m1!m2!n3!n4!Preff−1×y−2n−2n3ξn3/2tαm2+αm5+αn4Γm1+m2Γn3/2+n4Γ1+αm2+αm5+αn4Γm1Γn3/2,Φ7=y−2m−2m3Preffξm3/2−ξαm4+m5+n2m3!m4!n1!n2!Preff−1×−2n+yn1ξn1/2tαm4+αm5+αn2Γm3+m4Γn1/2+n2Γ1+αm4+αm5+αn2Γm3Γn1/2,Φ8=y−2m−2m3Preffξm3/2−ξαm4+m5+n4m3!m4!n3!n4!Pr−1×y−2n−2n3ξn3/2tαm4+αm5+αn4Γm3+m4Γn3/2+n4Γ1+αm4+αm5+αn4Γm3Γn3/2.In case of ordinary velocity whenα⟶1, Nr=0, and Kr=0, our solution is reduced to the known results obtained in [1], and for validation, Figure 4 is presented.Figure 4
Comparison forut,y.


## 4.1. Results of Fractional Temperature Field

Utilizing the Laplace transform to equation (21), we obtain(26)∂2∂y2−Preffsαξsα+ξαθ¯y,s=0,θ¯y,s=c3exp−yPreffsαξsα+ξα+c4expyPreffsαξsα+ξα.Using conditions from equations (24) and (25) in equation (26), we have(27)θ¯y,s=1ssinh1−yPreffsαξ/sα+ξαsinhPreffsαξ/sα+ξα,θ¯y,s=1s∑m=0∞e−2m+yPreffsαξ/sα+ξα−e−2m+2−yPreffsαξ/sα+ξα.Equation (27) can be written in suitable form as(28)θ¯y,s=1s+∑m=0∞∑m1=1∞∑m2=0∞−2m+ym1Preffξm1/2−ξαm2m1!m2!s1+αm2Γm1+m2Γm1−∑m=0∞∑m3=1∞∑m4=0∞y−2m−2m3Preffξm3/2−ξαm4m3!m4!s1+αm4Γm3+m4Γm3.Now, by applying inverse Laplace transform to equation (28),(29)θt,y=1+∑m=0∞∑m1=1∞∑m2=0∞−2m+ym1Preffξm1/2−ξαm2m1!m2!tαm2Γm1+m2Γm1Γ1+αm2−∑m=0∞∑m3=1∞∑m4=0∞y−2m−2m3Preffξm3/2−ξαm4m3!m4!tαm4Γm3+m4Γm3Γ1+αm4.Further, the solution can be expressed in more general M-function [49] form as(30)θt,y=1+∑m=0∞∑m1=1∞−2m+ym1Preffξm1/2m1!M21−ξαtα|m1,0,1,αm1,1−∑m=0∞∑m3=1∞y−2m−2m3Preffξm3/2m3!M21−ξαtα|m3,0,1,αm3,1.In case of ordinary temperature, whenα⟶1 and Nr=0, our solution is reduced to the known results obtained in [1], and for the validation, Figure 2 is presented.Figure 2
Comparison forθt,y.


## 4.2. Results of Fractional Concentration Field

Solving equation (22) by using conditions from (24) and (25) in a similar pattern as presented for temperature field, solution for concentration field is(31)Ct,y=1+∑l=0∞∑l1=1∞∑l2=0∞−2l+yl1Krl1/2−l2Scξl2l1!l2!M21−ξαtα|l2,0,1,αl2,1−−∑l=0∞∑l4=1∞∑l5=0∞y−2l−2l4Krl4/2−l5Scξl5l4!l5!M21−ξαtα|l5,0,1,αl5,1.In case of ordinary concentration whenα⟶1 and Kr=0, our solution is reduced to the known results obtained in [1], and for validation, Figure 3 is presented.Figure 3
Comparison forCt,y.


## 4.3. Results of Fractional Velocity Field

Solving equation (20) after substituting equations (27) and (31), we get(32)∂2∂y2−sαξsα+ξαu¯y,s=−1s∑m=0∞e−2m+yPreffsαξ/sα+ξα−e−2m+2−yPreffsαξ/sα+ξα−Ns∑l=0∞e−2l+yScsαξ/sα+ξα+Kr−e−2l+2−yScsαξ/sα+ξα+Kr.Solving equation (32) subject to conditions (24) and (25),(33)u¯y,s=Ns∑l=0∞∑n=0∞e−2l+yScsαξ/sα+ξα+Kr−e−2l+2−yScsαξ/sα+ξα+KrKr+Sc−1sαξ/sα+ξαe−2n+ysαξ/sα+ξα−ey−2n−2sαξ/sα+ξα+1s∑m=0∞∑n=0∞e−2m+yPreffsαξ/sα+ξα−e−2m+2−yPreffsαξ/sα+ξαPreff−1sαξ/sα+ξαe−2n+ysαξ/sα+ξα−ey−2n−2sαξ/sα+ξα−Ns∑l=0∞e−2l+yScsαξ/sα+ξα+Kr−e−2l+2−yScsαξ/sα+ξα+KrKr+Sc−1sαξ/sα+ξα−1s∑m=0∞e−2m+yPreffsαξ/sα+ξα−e−2m+2−yPreffsαξ/sα+ξαPreff−1sαξ/sα+ξα.Equation (33) can be written as(34)u¯y,s=∑l=0∞∑n=0∞∑l1=0∞∑l2=0∞∑l3=0∞∑l7=0∞∑l8=0∞N∑n1=0∞∑n2=0∞Φ¯1−N∑n3=0∞∑n4=0Φ¯2−∑l=0∞∑n=0∞∑l4=0∞∑l5=0∞∑l6=0∞∑l7=0∞∑l8=0∞N∑n1=0∞∑n2=0∞Φ¯3−N∑n3=0∞∑n4=0∞Φ¯4+∑m=0∞∑n=0∞∑m1=0∞∑m2=0∞∑m5=0∞∑n1=0∞∑n2=0∞Φ¯5−∑n3=0∞∑n4=0∞Φ¯6−∑m=0∞∑n=0∞∑m3=0∞∑m4=0∞∑m5=0∞∑n1=0∞∑n2=0∞Φ¯7+∑n3=0∞∑n4=0∞Φ¯8−N∑l=0∞∑l1=1∞∑l2=0∞∑l3=0∞∑l7=0∞∑l8=0∞−2l+yl1Scξl2ξ1−Scl7−ξαl3+l8l1!l2!l3!l8!Kr1−l1/2+l2+l7s1+αl3+αl8Γl2+l3Γl7+l8Γl2Γl7+N∑l=0∞∑l4=1∞∑l5=0∞∑l6=0∞∑l7=0∞∑l8=0∞y−2l−2l4Scξl5ξ1−Scl7−ξαl6+l8l4!l5!l6!l8!Kr1−l4/2+l5+l7s1+αl6+αl8Γl5+l6Γl7+l8Γl5Γl7−∑m=0∞∑m1=1∞∑m2=0∞∑m5=0∞−2m+ym1Preffξm1/2−ξαm2+m5m1!m2!Preff−1s1+αm2+αm5Γm1+m2Γm1+∑m=0∞∑m3=1∞∑m4=0∞∑m5=0∞y−2m−2m3Preffξm3/2−ξαm4+m5m3!m4!s1+αm4+αm5Γm3+m4Γm3,where(35)Φ¯1=−2l+yl1Scξl2−ξαl3+l8+n2l1!l2!l3!l8!n1!n2!Kr1−l1/2+l2+l7ξ1−Scl7−2n+yn1ξn1/2Γl2+l3Γl7+l8Γn1/2+n2s1+αl3+αl8+αn2Γl2Γl7Γn1/2,Φ¯2=−2l+yl1Scξl2−ξαl3+l8+n4l1!l2!l3!l8!n3!n4!Kr1−l1/2+l2+l7ξ1−Scl7y−2n−2n3ξn3/2Γl2+l3Γl7+l8Γn3/2+n4s1+αl3+αl8+αn4Γl2Γl7Γn3/2,Φ¯3=y−2l−2l4Scξl5−ξαl6+l8+n2l4!l5!l6!l8!n1!n2!Kr1−l4/2+l5+l7ξ1−Scl7−2n+yn1ξn1/2Γl5+l6Γl7+l8Γn1/2+n2s1+αl6+αl8+αn2Γl5Γl7Γn1/2,Φ¯4=y−2l−2l4Scξl5−ξαl6+l8+n4l4!l5!l6!l8!n3!n4!Kr1−l4/2+l5+l7ξ1−Scl7y−2n−2n3ξn3/2Γl5+l6Γl7+l8Γn1/2+n2s1+αl6+αl8+αn4Γl5Γl7Γn1/2,Φ¯5=−2m+ym1Preffξm1/2−ξαm2+m5+n2m1!m2!n1!n2!Preff−1−2n+yn1ξn1/2Γm1+m2Γn1/2+n2s1+αm2+αm5+αn2Γm1Γn1/2,Φ¯6=−2m+ym1Preffξm1/2−ξαm2+m5+n4m1!m2!n3!n4!Preff−1y−2n−2n3ξn3/2Γm1+m2Γn3/2+n4s1+αm2+αm5+αn4Γm1Γn3/2,Φ¯7=y−2m−2m3Preffξm3/2−ξαm4+m5+n2m3!m4!n1!n2!Preff−1−2n+yn1ξn1/2Γm3+m4Γn1/2+n2s1+αm4+αm5+αn2Γm3Γn1/2,Φ¯8=y−2m−2m3Preffξm3/2−ξαm4+m5+n4m3!m4!n3!n4!Preff−1y−2n−2n3ξn3/2Γm3+m4Γn3/2+n4s1+αm4+αm5+αn4Γm3Γn3/2.Taking Laplace inverse, we have(36)uy,t=∑l=0∞∑n=0∞∑l1=0∞∑l2=0∞∑l3=0∞∑l7=0∞∑l8=0∞N∑n1=0∞∑n2=0∞Φ1−N∑n3=0∞∑n4=0Φ2−∑l=0∞∑n=0∞∑l4=0∞∑l5=0∞∑l6=0∞∑l7=0∞∑l8=0∞N∑n1=0∞∑n2=0∞Φ3−N∑n3=0∞∑n4=0∞Φ4+∑m=0∞∑n=0∞∑m1=0∞∑m2=0∞∑m5=0∞∑n1=0∞∑n2=0∞Φ5−∑n3=0∞∑n4=0∞Φ6−∑m=0∞∑n=0∞∑m3=0∞∑m4=0∞∑m5=0∞∑n1=0∞∑n2=0∞Φ7+∑n3=0∞∑n4=0∞Φ8−N∑l=0∞∑l1=1∞∑l2=0∞∑l3=0∞∑l7=0∞∑l8=0∞−2l+yl1Scξl2ξ1−Scl7−ξαl3+l8tαl3+αl8l1!l2!l3!l8!Kr1−l1/2+l2+l7×Γl2+l3Γl7+l8Γ1+αl3+αl8Γl2Γl7+N∑l=0∞∑l4=1∞∑l5=0∞∑l6=0∞∑l7=0∞∑l8=0∞y−2l−2l4Scξl5ξ1−Scl7−ξαl6+l8tαl6+αl8l4!l5!l6!l8!Kr1−l4/2+l5+l7×Γl5+l6Γl7+l8Γ1+αl6+αl8Γl5Γl7−∑m=0∞∑m1=1∞∑m2=0∞∑m5=0∞−2m+ym1Preffξm1/2−ξαm2+m5m1!m2!Preff−1tαm2+αm5Γm1+m2Γ1+αm2+αm5Γm1+∑m=0∞∑m3=1∞∑m4=0∞∑m5=0∞y−2m−2m3Preffξm3/2−ξαm4+m5m3!m4!tαm4+αm5Γm3+m4Γ1+αm4+αm5Γm3,where(37)Φ1=−2l+yl1Scξl2−ξαl3+l8+n2l1!l2!l3!l8!n1!n2!Kr1−l1/2+l2+l7×ξ1−Scl7−2n+yn1ξn1/2tαl3+αl8+αn2Γl2+l3Γl7+l8Γn1/2+n2Γ1+αl3+αl8+αn2Γl2Γl7Γn1/2,Φ2=−2l+yl1Scξl2−ξαl3+l8+n4l1!l2!l3!l8!n3!n4!Kr1−l1/2+l2+l7×ξ1−Scl7y−2n−2n3ξn3/2tαl3+αl8+αn4Γl2+l3Γl7+l8Γn3/2+n4Γ1+αl3+αl8+αn4Γl2Γl7Γn3/2,Φ3=y−2l−2l4Scξl5−ξαl6+l8+n2l4!l5!l6!l8!n1!n2!Kr1−l4/2+l5+l7×ξ1−Scl7−2n+yn1ξn1/2tαl6+αl8+αn2Γl5+l6Γl7+l8Γn1/2+n2Γ1+αl6+αl8+αn2Γl5Γl7Γn1/2,Φ4=y−2l−2l4Scξl5−ξαl6+l8+n4l4!l5!l6!l8!n3!n4!Kr1−l4/2+l5+l7×ξ1−Scl7y−2n−2n3ξn3/2tαl6+αl8+αn4Γl5+l6Γl7+l8Γn1/2+n2Γ1+αl6+αl8+αn4Γl5Γl7Γn1/2,Φ5=−2m+ym1Preffξm1/2−ξαm2+m5+n2m1!m2!n1!n2!Preff−1×−2n+yn1ξn1/2tαm2+αm5+αn2Γm1+m2Γn1/2+n2Γ1+αm2+αm5+αn2Γm1Γn1/2,Φ6=−2m+ym1Preffξm1/2−ξαm2+m5+n4m1!m2!n3!n4!Preff−1×y−2n−2n3ξn3/2tαm2+αm5+αn4Γm1+m2Γn3/2+n4Γ1+αm2+αm5+αn4Γm1Γn3/2,Φ7=y−2m−2m3Preffξm3/2−ξαm4+m5+n2m3!m4!n1!n2!Preff−1×−2n+yn1ξn1/2tαm4+αm5+αn2Γm3+m4Γn1/2+n2Γ1+αm4+αm5+αn2Γm3Γn1/2,Φ8=y−2m−2m3Preffξm3/2−ξαm4+m5+n4m3!m4!n3!n4!Pr−1×y−2n−2n3ξn3/2tαm4+αm5+αn4Γm3+m4Γn3/2+n4Γ1+αm4+αm5+αn4Γm3Γn3/2.In case of ordinary velocity whenα⟶1, Nr=0, and Kr=0, our solution is reduced to the known results obtained in [1], and for validation, Figure 4 is presented.Figure 4
Comparison forut,y.


## 5. Graphical Results and Discussion

Transient, viscid, incompressible, free convective flow between upright parallel plates of infinite length bearing the traits of mass dissemination and invariant temperature has been studied in this manuscript. The obtained model is solved analytically by Laplace transform technique executing all boundary and initial conditions. The results for temperature, concentration, and velocity expressions are envisioned graphically.Figure5 visualises the effect of fractional parameter α on fluid’s concentration regime. By fixing other parameters as constant and increasing the value of α, concentration field the associated boundary layer thickness is increased. Figure 6 shows the effect of Sc on concentration domain. It is observed, with the increase in Sc, the denseness of fluent decreases which affects the rate of molecular diffusion and it tends to minimise thickness of boundary layer. Figure 7 is presented to envision the impact of chemical reaction parameter Kr. It can be seen clearly that as Kr increases, fluent concentration decreases. Figure 8 shows the consequences of time fractional-order concentration and observes that fluid concentration increases with time.Figure 5
Ct,y profiles for α.Figure 6
Ct,y profiles for Sc.Figure 7
Ct,y profiles for Kr.Figure 8
Ct,y profiles for t.In Figure9, we vary α and fix other parameters on temperature domain. As α increases, liquid temperature decreases and the boundary level heaviness increases. Figure 10 shows varied effective Prandtl number Preff against y by limiting t=0.01 and α=0.65. Graphically, it is to the point that as Prandtl number Preff heightens, liquid temperature minimises. As anticipated, raising Preff, abbreviates the thermic conduction because the high viscosity reduces the thickness of the caloric boundary stratum. Figure 11 is drawn by fixing Preff=1.5 and α=0.65, and effect of fractional time is observed on θt,y.Figure 9
θt,y profiles for α.Figure 10
θt,y profiles for Pr.Figure 11
θt,y profiles for t.Figure12 is drawn for different values of alpha by keeping other parameters fixed. It is a well-known fact that the integer order differential operator is a local operator whereas the fractional-order differential operator (ABC derivative) is nonlocal in the sense that the next stage of the system depends not only upon its current stage but also upon all of its proceeding stages. It describes the behaviour of the function in better way because it holds Mittag-Leffler kernel which stores memory factor and therefore better in describing fluid’s flow fields. It is rather visible from the graphical record that velocity increases as α value upshots, and when fractional parameter approaches to one, its velocity will be closer to the classical values. The effects on velocity by Preff are shown in Figure 13. One can see that by increasing Preff, the fluid velocity decreases, and likely the stratum boundary thickness shrinks because Preffincreases the viscousness and decreases thermic stratum boundary in slower movement. Figure 14 is given for respective measures of the buoyancy ratio parametric quantity N which interprets the ratio between thermic and aggregative buoyancy drives. For N=0, there is no mass transference, and buoyancy effect is primarily attributable to caloric diffusion rate. Therefore, for dissimilar N values, the thickness of boundary stratum and velocity get enhanced.Figure 12
θt,y profiles for α.Figure 13
ut,y profiles for Pr.Figure 14
ut,y profiles for N.Figure15 shows the effect of Sc number on fluid speed by fixing the parameters. It is observed that increase in Sc decreases the velocity. Figure 16 is given to experience the temperature of time on velocity when Sc=0.22, Preff=7, N=0.015, and α=0.65. As time t increments the fluent velocity, likewise the boundary stratum thickness diminishes as seen from graph. Figures 14–16 are drawn to ascertain the validness of obtained for concentration, velocity, and temperature disciplines compared with [1] in the absence of radiative heat flux and chemical reaction parameter. The overlapping arcs distinctly signal the integration of found results. Another comparison is drawn to validate our obtained results. Results are compared with [26], in the absence of MHD, radiative heat flux, and chemical reaction. Heat and mass transfer model with three fractional approaches such as (i) Caputo, (ii) Caputo–Fabrizio, and (iii) Atangana–Baleanu is compared graphically and presented in Figures 17–22. How fractional parameter controls fluid flow can be seen in this comparison? From these displays, we concluded that the Caputo–Fabrizio model exhibits similar behaviour as the classical model and Atangana–Baleanu model is well suited in stimulating the history function due to Mittag-Leffler kernel which can describe full memory effect for a given system. By increasing the value of fractional parameter, all three profiles display decreasing behaviour. The overlapping curves depicts the integrity of three models when fractional parameter ⟶1.Figure 15
ut,y profiles for Sc.Figure 16
ut,y profiles for t.Figure 17
Comparison forθt,y.Figure 18
Comparison forθt,y.Figure 19
Comparison forCt,y.Figure 20
Comparison forCt,y.Figure 21
Comparison forut,y.Figure 22
Comparison forut,y.


## 6. Conclusion

Free convective transient flow of viscous and incompressible fluids transiting through two infinite hot parallel upright plates is investigated analytically in the presence of chemical reaction, radiative heat flux, and mass diffusion at the boundary. A fractional model is developed with comparatively newABC derivative. The following observations are made:(i)
In this model, the fractional-order parameterα observes the concentration thickness of thermal and momentum boundary layers.(ii)
The fluid temperature is enhanced for smaller values ofα and larger values of time t while decreases for greater Preff.(iii)
Concentration grows by decreasing the values of fractional parameterα and for larger values of time t. Additionally, concentration abbreviates by raising Sc and Kr likewise the boundary layer thickness.(iv)
Velocity increases by raising the parametric values ofN, α, Nr, and time t whereas it decreases by raising Pr, Kr, and Sc values, respectively. The proposed model with ABC derivative gives results as compared with Caputo and Caputo–Fabrizio model because of memory effect.



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