# Generalization of Thermal and Mass Fluxes for the Flow of Differential Type Fluid with Caputo–Fabrizio Approach of Fractional Derivative

Authors: Asima Razzaque; Anam Rani; Mudassar Nazar
Journal: Complexity
 (2021)

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## Abstract

In this research work, generalized thermal and mass transports for the unsteady flow model of an incompressible differential type fluid are considered. The Caputo–Fabrizio fractional derivative is used for the respective generalization of Fourier’s and Fick’s laws. A MHD fluid flow is considered near a flat vertical surface subject to unsteady mechanical, thermal, and mass conditions at boundary. The governing equations of flow model are solved by integral transform, and closed form results for generalized momentum, thermal, and concentration fields are obtained. Generalized thermal and mass fluxes at boundary are quantified in terms of Nusselt and Sherwood numbers, respectively, and presented in tabular form. The significance of the physical parameters over the momentum, thermal, and concentration profiles is characterized by sketching the graphs.


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## Body

## 1. Introduction

Fractional calculus has been expanding rapidly in the present time for the sake of its applications in the modeling and physical explanation of natural phenomenon. The noninteger derivatives of fractional order have been applied successfully to the generalization of fundamental laws of nature specially in the transport phenomenon.Several approaches [1–4] of fractional derivatives have been proposed and utilized for the different proposes by many theorists from different fields of sciences and technology [5]. Imran et al. [6] considered two different approaches of fractional differential operators for the flow of MHD Newtonian fluid under the arbitrary boundary conditions, namely, Atangana and Caputo–Fabrizio. Kumar et al. [7] explained the Cauchy reaction diffusion equations by fractional calculus. Qureshi et al. [8] applied the fractional derivative to model a blood flow and discussed the concentration level of ethanol in blood circulation system. Hristov [9] considered a steady-state heat conduction and obtained analytical solutions by applying the Caputo–Fabrizio approach of fractional derivative. Imran et al. [10, 11] considered the Caputo time fractional derivative to discuss the slippage flow over an exponentially accelerated plate and for the flow of differential fluid past stationary heated vertical plate. Ahmad et al. [12] compared two flow models, one with the power law kernel and the other with the nonsingular kernel. Khalid et al. [13] obtained the results for flow of micropolar fluid and applied the fractional derivative for heat and mass transport. Shukla et al. [14] presented a report regarding applications of fractional calculus. Kumar et al. [15] explored the results for free convectional motion with a uniform temperature through a porous media by utilizing the power, exponential, and Mittag–Leffler kernels of fractional operator. Singh et al. [16] constructed the dynamic fractional model to explain the smoking dynamics. Sun et al. [17] presented a collection of real world applications of fractional differential operators. Nazar et al. [18] discussed the double convectional flow via two approaches of noninteger operators and compared the obtained results of thermal, mass, and momentum profiles. Gomez et al. [19] solved fractional diffusion-advection equation and obtained the analytical solution for supper diffusion. Tran et al. [20] discussed the stabilities of fractional differential equation. Recently, Tuan et al. [21] endorsed the fractional calculus to demonstrate the transition model of COVID-19. Hristov et al. [22] applied the mixed time-space derivative to obtain the result for transient flow of non-Newtonian fluid. Saqib et al. [23], Haq et al. [24], and Imran et al. [25] utilized the fractional differential operator with the nonsingular kernel to obtain the fractional result for the flow of Jeffery, and second grade fluids. Some more investigations regarding Caputo–Fabrizio fractional derivatives are found in [26–28].Hristov et al. [29] suggested the generalized transient thermal transport with damping contribution, by considering the Caputo–Fabrizio idea of nonsingular kernel. Aleem et al. [30], Sheikh et al. [31] and Ahmad et al. [32] followed the track suggested in [29] and applied it to describe the generalized heat and mass transfer flow. In light of the above motivational investigations, we are interested in discussing the generalized thermal and mass transports with heat generation and chemical reaction for the flow of second grade fluid through a porous media, with the existence of a magnetic field.


## 2. Mathematical Formulation

Suppose that the second grade fluid is lying in the vicinity of a vertical plate with ambient temperatureT∞ and concentration level C∞. The orientation of the plate in the coordinate system is placed with y-axis becoming normal to the plane of plate as shown in Figure 1. Initially, the physical system containing boundaries and fluid is in complete equilibrium. Suddenly, plate starts moving with velocity U0ft; at this moment, the temperature of plate and the concentration level near the plate rise or fall according to T∞+Tw−T∞gt and C∞+Tw−C∞ht, respectively, where f⋅, g⋅, and h⋅ are arbitrary functions and satisfy f0=0, g0=0, and h0=0, respectively. The contribution of Lorentz force is also applied for the flow of fluid. Moreover, the induced magnetic field and heat dissipation are small and can be negligible. Subject to Boussinesq’s approximation, the governing equations of respective flow take the following form [29, 30]:Figure 1
Flow geometry and coordinate system.(1)μ1+α1∂∂t∂2uy,t∂y2−ρ∂uy,t∂t−σB02uy,t−μϕK1uy,t=−ρgβCCy,t−C∞−ρgβTTy,t−T∞,(2)ρCp∂Ty,t∂t−QTy,t−T∞=k∂2Ty,t∂y2,(3)∂Cy,t∂t−KrCy,t−C∞=D∂2Cy,t∂y2.Initial and boundary conditions:(4)u=0,T=T∞,C=C∞,att=0,and,y∈0,∞,(5)u=U0ft,T=T∞+Tw−T∞gt,C=C∞+Tw−C∞ht,y=0,t>0,(6)u⟶0,T⟶T∞,C⟶C∞,asy⟶∞,t>0.


## 3. Generalized Model

The thermal balance with heat generation is expressed as follows:(7)ρCp∂Ty,t∂t=−∂q∂y+QTy,t−T∞,y,t>0,and the generalized thermal flux with damping effect [29, 30] is(8)qy,t=−k1∂Ty,t∂y−k2DCFtα1−α∂Ty,t∂y,y,t>0,where DCFtα is the fractional differential operator suggested by Caputo–Fabrizio. The constant h1 and h2 are the effective heat conduction and elastic conduction parameters, respectively.From equation (8), it is clear that, for α=1, the classical thermal flux is recovered. Similarly, the molecular balance with chemical reaction is expressed as(9)∂Cy,t∂t=−∂Jy,t∂y+KrCy,t−C∞,y,t>0,and the generalized Fick’s Law with damping effect [29, 30] is(10)Jy,t=−D1∂Cy,t∂y−D2DCFtα1−α∂Cy,t∂y,y,t>0,where D1 and D2 are the effective mass diffusion and the elastic diffusion, respectively.Introduce the relations(11)υ=uU0,η=yU0ν,τ=tU02ν,ψ=C−C∞Cw−C∞,φ=T−T∞Tw−T∞,in equations (1), (4)–(6), and (7)–(10), to get the shape free model.The momentum balance takes the following form:(12)1+α2∂∂τ∂2υη,τ∂η2−∂υη,τ∂τ−Mυη,τ−1Kυη,τ=−Grφη,τ−Gmψη,τ.Dimensionless thermal balance is(13)∂φη,τ∂t=1Pr1∂2φη,τ∂η2+1Pr2DCFtα1−α∂2φη,τ∂η2+Q0φη,τ.Dimensionless diffusion equation is(14)∂ψη,τ∂t=1Sc1∂2ψη,τ∂η2+1Sc2DCFtα1−α∂2ψη,τ∂η2+K0ψη,τ.And the corresponding initial and boundary conditions are(15)υη,0=0,φη,0=0,ψη,0=0,η>0,(16)υ0,τ=fτ,φ0,τ=gτ,ψ0,τ=hτ,τ>0,(17)υη,τ⟶0,φη,τ⟶0,ψη,τ⟶0,asη⟶0,where α2=α1U02/ν is the dimensionless second grade parameter, Gm=gβCCw−C∞/U03 is the mass Grashof number, Gr=gβTTw−T∞/U03 is the thermal Grashof number, M=νσ/ρU02 is the magnetic parameter, 1/K=ν2ϕ/K1U02 is the porosity parameter, Pr1=μCp/h1 and Pr2=μCp/h2 are Prandtl numbers, Sc1=ν/D1 and Sc2=ν/D2 are Schmidt numbers, Q0=Qν/ρCpU02 is the heat generation parameter, and K0=Krν/U02 is the chemical reaction parameter.


## 4. Solution of the Problem

The solution of generalized model is obtained by endorsing the Laplace transform.


### 4.1. Generalized Temperature Field

Endorsing the Laplace transform to equation (16), an ordinary differential equation is obtained as(18)qφ¯η,q=1Pr1∂2φ¯η,s∂η2+1Pr21−αqq1−α+α∂2φ¯η,q∂η2+Q0φ¯η,q,with the following transformed boundary conditions:(19)φ¯η,q=Gq,andφ¯η,q⟶0,asη⟶∞.Equation (18) is solved with conditions (19), and its solution is expressed as(20)φ¯η,q=Gqexp−ηq−a0q−Q0a1q+a2,where a0=α/1−α, a1=Pr1+Pr2/Pr1Pr2, and a2=a0Pr2/Pr1+Pr2.Equation (20) is complicated, and it is not possible to invert the temperature field in t-domain by ordinary formula of Laplace inverse. Therefore, the inversion algorithms, namely, Stehfest and Tzou, are utilized to invert the transformed temperature profile, and the obtained results are presented in Figure 2(a) for gt=1.Figure 2
Profiles of inverted temperature and concentration with Stehfest’s and Tzou’s algorithms for (a)gt=1 and (b) ht=1.
(a)(b)


### 4.2. Generalized Concentration Field

Again endorsing the Laplace transform to equation (14), an ordinary differential equation is obtained as(21)qψ¯η,q=1Sc1∂2ψ¯η,s∂η2+1Sc21−αqq1−α+α∂2ψ¯η,q∂η2+K0ψ¯η,q,(22)ψ¯η,q=Hq,andψ¯η,q⟶0,asη⟶∞.Equation (21) is solved with conditions (22), and its solution is expressed as(23)ψ¯η,q=Hqexp−ηq−a0q−K0b1q+b2,where b1=Sc1+Sc2/Sc1Sc2 and b2=b0Sc2/Sc1+Sc2.Equation (23) is complicated, and it is not possible to invert the temperature field in t-domain by ordinary formula of Laplace inverse. Therefore, the inversion algorithms are utilized to obtain the temperature profile for ht=1, and the obtained results are presented in Figure 2(b).


### 4.3. Velocity Field with Generalized Thermal and Mass Transport

Equation (13) is converted to an ordinary differential equation via Laplace transform as(24)1+α2q∂2υ¯η,q∂η2−qυ¯η,q−Mυ¯η,q−1Kυ¯η,q=−Grφ¯η,q−Gmψ¯η,q,with corresponding transformed boundary conditions(25)υ¯0,q=Fq,andυ¯η,q⟶0asη⟶∞.Equation (24), subject to condition (25), is solved for velocity field in q-domain as follows:(26)υ¯η,q=Fqexp−yq+M0q+α3+Grα3Gqexp−yq+M0/q+α3−exp−yq+a0q−Q0/a1q+a2q+α3q−a0q−Q0−a1q+a2q+M0+Gmα3Hqexp−yq+M0/q+α3−exp−yq+b0q−K0/b1q+b2q+α3q−b0q−K0−b1q+b2q+M0,where α3=1/α2 and M0=M+1/K. Equation (26) is also complex, and it is inverted in t-domain with the help of inversion algorithms. The inverse Laplace of velocity for ft=1, and ft=t is demonstrated in Figures 3(a) and 3(b).Figure 3
Profile of inverted velocity with Stehfest’s and Tzou’s algorithms for (a)ft=1 and (b) ft=t.
(a)(b)


## 4.1. Generalized Temperature Field

Endorsing the Laplace transform to equation (16), an ordinary differential equation is obtained as(18)qφ¯η,q=1Pr1∂2φ¯η,s∂η2+1Pr21−αqq1−α+α∂2φ¯η,q∂η2+Q0φ¯η,q,with the following transformed boundary conditions:(19)φ¯η,q=Gq,andφ¯η,q⟶0,asη⟶∞.Equation (18) is solved with conditions (19), and its solution is expressed as(20)φ¯η,q=Gqexp−ηq−a0q−Q0a1q+a2,where a0=α/1−α, a1=Pr1+Pr2/Pr1Pr2, and a2=a0Pr2/Pr1+Pr2.Equation (20) is complicated, and it is not possible to invert the temperature field in t-domain by ordinary formula of Laplace inverse. Therefore, the inversion algorithms, namely, Stehfest and Tzou, are utilized to invert the transformed temperature profile, and the obtained results are presented in Figure 2(a) for gt=1.Figure 2
Profiles of inverted temperature and concentration with Stehfest’s and Tzou’s algorithms for (a)gt=1 and (b) ht=1.
(a)(b)


## 4.2. Generalized Concentration Field

Again endorsing the Laplace transform to equation (14), an ordinary differential equation is obtained as(21)qψ¯η,q=1Sc1∂2ψ¯η,s∂η2+1Sc21−αqq1−α+α∂2ψ¯η,q∂η2+K0ψ¯η,q,(22)ψ¯η,q=Hq,andψ¯η,q⟶0,asη⟶∞.Equation (21) is solved with conditions (22), and its solution is expressed as(23)ψ¯η,q=Hqexp−ηq−a0q−K0b1q+b2,where b1=Sc1+Sc2/Sc1Sc2 and b2=b0Sc2/Sc1+Sc2.Equation (23) is complicated, and it is not possible to invert the temperature field in t-domain by ordinary formula of Laplace inverse. Therefore, the inversion algorithms are utilized to obtain the temperature profile for ht=1, and the obtained results are presented in Figure 2(b).


## 4.3. Velocity Field with Generalized Thermal and Mass Transport

Equation (13) is converted to an ordinary differential equation via Laplace transform as(24)1+α2q∂2υ¯η,q∂η2−qυ¯η,q−Mυ¯η,q−1Kυ¯η,q=−Grφ¯η,q−Gmψ¯η,q,with corresponding transformed boundary conditions(25)υ¯0,q=Fq,andυ¯η,q⟶0asη⟶∞.Equation (24), subject to condition (25), is solved for velocity field in q-domain as follows:(26)υ¯η,q=Fqexp−yq+M0q+α3+Grα3Gqexp−yq+M0/q+α3−exp−yq+a0q−Q0/a1q+a2q+α3q−a0q−Q0−a1q+a2q+M0+Gmα3Hqexp−yq+M0/q+α3−exp−yq+b0q−K0/b1q+b2q+α3q−b0q−K0−b1q+b2q+M0,where α3=1/α2 and M0=M+1/K. Equation (26) is also complex, and it is inverted in t-domain with the help of inversion algorithms. The inverse Laplace of velocity for ft=1, and ft=t is demonstrated in Figures 3(a) and 3(b).Figure 3
Profile of inverted velocity with Stehfest’s and Tzou’s algorithms for (a)ft=1 and (b) ft=t.
(a)(b)


## 5. Results and Parametric Discussion

This article is designed for the analysis of generalized thermal and mass transport flow of differential type fluid under generalized boundary conditions. The effects of magnetic field, heat source, and chemical reaction are also considered for flow model. The respective governing equations of flow model are solved analytically via integral transform method, and closed form expressions for field variables are attained.The effects of potent parameters are also discussed graphically by plotting some graphs of for variation of appeared parameters. Figures4(a) and 4(b) are sketched to explain the effect of fractional parameter α and Pr over the temperature profile, and it is noted that the temperature of fluid is raised with the developing values of α. Also, it is detected that the temperature falls down with the increasing values of Pr, because the fluid is thick, and momentum diffusivity is dominant to the thermal diffusivity for the greater Pr; therefore, fluid velocity slows down for increasing Pr. Figure 4(c) is drawn for three specification of gt, and it is noted that temperature profiles satisfy the boundary conditions.Figure 4
Sketch thermal profile for changingα, Pr, and gt.
(a)(b)(c)In Figure5, the subjectivity of concentration is explained for variations of α, Sc, and three specification for the gt. The same behavior of concentration is seen for respective parameters as seen in temperature profiles.Figure 5
Sketch concentration profile for changingα, Sc, and ht.
(a)(b)(c)Velocity profile is outlined in Figure6(a) due variation of fractional parameters α. The figure pattern shows that the fluid gains more and more momentum as α tends to increase. The influence of Pr and Sc is signified in Figures 6(b) and 6(c). As Pr and Sc are quantified by the ratios of momentum diffusivity to thermal diffusivity and molecular diffusion coefficients, respectively, and for enhancing values of Pr and Sc referred to as the dominant momentum diffusivity, the fluid slows down for developing values of Pr and Sc.Figure 6
Sketch velocity profile for changingα, Pr, and Sc.
(a)(b)(c)Figure7(a) is drawn to see the significance role of Gr for supporting the flow, and it is noted that the speed of fluid is elevated for developing values of Gr. As Gr is quantified by relative buoyancy force induced by the variations in temperature differences to the retarding force generates by the virtue of viscosity of the fluid, hence, for larger values of Gr, there is more convectional current, so fluid speeds up. The effects of thermal generation parameter Q0, and chemical reaction parameters K0 are discussed in Figures 7(b) and 7(c), and it is concluded that velocity profiles rise for enhancing values of Q0 and K0.Figure 7
Sketch velocity profile for changing Gr,Q0, and K0.
(a)(b)(c)The subjectivity of magnetic parameterM is highlighted in Figure 8(a) and from this figure, it is seen that profile lowers down for increasing values of M, because strong magnetic field creates more hindrance to the flow of fluid. Figure 8(b) shows the effect of K and it is seen that fluid speeds up with the increasing values of K. The increasing values of K refer to the decreasing effect of porosity, and hence, fluid velocity increases with the enhancing values of K. Figure 8(c) is plotted for three specifications of ft; from the profiles, it is clear that velocity profiles satisfy the boundary conditions for different ft. The present results for velocity and temperature are also compared with the existing results obtained by Sheh et al. [27] in Figures 9(a) and 9(b). The overlapping profiles confirm the validity of our results.Figure 8
Sketch of velocity profile for changing values ofM, K, and for ft.
(a)(b)(c)Figure 9
Sketch of velocity profileft=Htexpiwt,gt=1,Gm=0 a comparison with Shah et al. [27].
(a)(b)Further, heat and mass transfer at plate is discussed numerically in terms of Nusselt and Sherwood numbers, and results are presented in Tables1 and 2. From these tables, it is clear that both Nusselt and Sherwood numbers are grown with elevating fractional parameter α for small time, while there is an opposite behavior for large time.Table 1
Subjectivity of Nusselt number due toα variation.
αPr = 2.0,t = 0.05Pr = 2.0,t = 0.5Pr = 2.0,t = 0.3Q0 = 0.3, t = 0.05Q0 = 0.3, t = 0.5Q0 = 0.3, t = 30.11.285346841.633624041.587603511.3669437410.3673302810.100951360.22.016633811.767766951.085763151.8951174880.4587359130.106637810.32.939723681.913246750.834121602.7482086350.5430717300.093713720.44.473588652.344310090.480299443.2492590120.6806038720.084091790.55.499719412.700308620.275971293.5938112910.8984292530.070356240.66.285393613.234494790.109502503.8490017091.0222617760.047434160.76.922693213.673872670.074431124.0469196591.1433055990.036091990.87.457159814.050462940.048323454.2054610891.2460114750.103660660.97.915448254.381372910.033670674.3355843981.3352012300.11544229Table 2
Subjectivity of Sherwood number due toα variation.
αSc = 1.5,t = 0.05Sc = 1.5,t = 0 .5Sc = 1.5,t = 3K0 = 0.5, t = 0.05K0 = 0.5, t = 0.5K0 = 0.5, t = 30.11.366943741.074129760.219089021.166943741.354339280.210951360.21.895117481.176458460.309838671.222631651.428935930.206637810.32.748208631.811473830.438178052.458072531.543571050.203713720.43.593811292.180287610.489897952.906225611.906030350.150910790.53.593811292.434997660.579655073.214402542.504252410.119356240.63.849001792.624800010.619677343.442651862.645612320.107434160.74.046919692.772882040.726636083.619675012.743305590.086091990.84.205461092.892143740.819756063.761478752.946011470.013660660.94.335584382.990498480.903327183.877864563.002201230.00544229


## 6. Conclusion

This investigation is designed to discuss the generalized thermal and mass transports and flow modeling for MHD second grade fluid subject to arbitrary conditions with the effect of heat generation and chemical reaction through a porous medium. The mathematical model is solved by integral transform method, and closed form relations for temperature, concentrations, and velocity fields are obtained. The effects of parameters for thermal and mass flow are discussed graphically. Also, thermal and mass fluxes at boundary of flow domain are explained numerically for the due variation ofα, and obtained results are given in the tabular form.Some concluded bullets of this study are as follows:(i)
Temperature of the fluid is raised with incremental variation ofα and Q0, while it falls for the incremental variation of both Pr1 and Pr2(ii)
The thermal boundary conditions are also satisfied by temperature for different specifications ofgt(iii)
The concentration level of the fluid is raised with incremental variation ofα and K0, whereas the level falls for the incremental variation of both Sc1 and Sc2(iv)
The concentration boundary conditions are also satisfied by the concentration for different specification ofht(v)
Velocity profile shows a growing trend for incremental increase in the values ofα, Gr, Gm, Q0, K0, and K where as it retards for developing Pr1Pr2, Sc1, Sc2, and M(vi)
The Nusselt number is boosted with the increasing values ofα by taking large values of time, while it falls down with the increasing α for small time(vii)
The Sherwood number grows with the increasing values ofα, for the large values of time, while it falls down with the increasing α for small time



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