Analysis of mathematical model of fractional viscous fluid through a vertical rectangular channel

Analysis of mathematical model of fractional viscous fluid through a vertical rectangular channel

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Analysis of mathematical model of fractional viscous fluid through a vertical rectangular channel Maryam Aleem, Muhammad Imran Asjad, Muhammad S.R. Chowdhury, Abid Hussanan PII: DOI: Reference:

S0577-9073(19)30903-7 https://doi.org/10.1016/j.cjph.2019.08.014 CJPH 926

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

31 July 2018 20 May 2019 30 August 2019

Please cite this article as: Maryam Aleem, Muhammad Imran Asjad, Muhammad S.R. Chowdhury, Abid Hussanan, Analysis of mathematical model of fractional viscous fluid through a vertical rectangular channel, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.08.014

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Highlights • Mathematical model is introduced to study the viscous fluid in a rectangular channel. • Fractional model is developed with Caputo and Caputo-Fabrizio time derivatives. • Laplace transform method is used to obtain the analytical solutions. • The liquid with Caputo-Fabrizio derivatives flow faster than Caputo.

1

Analysis of mathematical model of fractional viscous fluid through a vertical rectangular channel

Maryam Aleem1 , Muhammad Imran Asjad2 , Muhammad S. R. Chowdhury3 , Abid Hussanan4

∗

1,2

Department of Mathematics, University of Management and Technology, Lahore, 54770 Pakistan 3 Department of Mathematics and Statistics, University of Lahore, 54590 Pakistan 4 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, 700000 Viet Nam 4 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, 700000 Viet Nam

Abstract Unsteady motion of a viscous fluid passing through a vertical channel with MHD (magnetohydrodynamics) effect has been analyzed in this manuscript. Fractional model is developed by two approaches, namely; Caputo fractional time derivatives with singular kernel and Caputo-Fabrizio fractional time derivatives with non-singular kernel. Analytical solutions have been obtained via Laplace transform method after converting the governing equations into dimensionless form and presented graphical analysis of the obtained results in terms of comparison. The effect of fractional and other flow parameters on temperature, concentration and velocity fields is seen, respectively. As a result, we have found that for the physical model with fractional derivative of Caputo-Fabrizio, temperature, concentration and velocity have greater values in comparison with Caputo one. It is also noted that velocity has shown dual nature for large and small time. Further, rates of heat and mass transfer and skin friction can also be enhanced for the small values of non-integer order parameter and are presented in table 1. Key words: Heat and mass transfer flow; MHD (magnetohydrodynamics); Fractional modeling; Channel flow. Nomenclature: u Velocity field [ms−1 ] g Acceleration due to gravity [ms−2 ] β1 Volumetric coefficient of thermal expansion [K −1 ] β2 Volumetric coefficient of diffusion expansion [K −1 ] T Temperature of the fluid [K] ∗

Corresponding author E-mail: [email protected]

2

Td C Cd Cp Gr Gm D Nu Pr Sc β, γ µ ν ρ Cw Tw C CF βo

1

Fluid’s temperature at the plate [K] Concentration of the fluid [Kgm−3 ] Fluid’s concentration at the plate [Kgm−3 ] Specific heat at constant pressure [jKg −1 K −1 ] Thermal Grashof number Mass Grashof number Mass diffusion coefficient Nusselt number Prandtl number Schimdt number Fractional parameters Dynamic viscosity [Kgm−1 s−1 ] Kinematic viscosity [m2 s−1 ] Fluid’s density [Kgm−3 ] Fluid’s concentration at the plate Fluid’s temperature at the plate Caputo time fractional derivative Caputo-Fabrizio time fractional derivative Magnetic field parameter

Introduction

The study of channel flow has attracted many researchers due to its practicality in many fields such as in aerospace, civil, chemical, mechanical and bio mechanical engineering and in environmental sciences. Flows in the river and open or closed channels are an example of such flows. Theoretical analysis of MHD (magnetohydrodynamics) channel streams is of keen concern because it has widespread practical applications in projecting engine cooling systems with liquid metallic element, in MHD generators or accelerators, tickers and in flow meters, etc. The MHD (magnetohydrodynamics) phenomenon represents an interaction between the electromagnetic field and hydrodynamic boundary layer [1]. Newly there has been a progressive involvement in liquid flowing through MHD channel. The results of MHD flow rate inside a duct has practical applications in numerous engineering problems. An extended theoretic study has been accomplished about the hydromagnetic fluid flow inside a duct under the homogenous magnetic field [2]. Morely et al. [3], Joneidi [4], Mansoor et al.[5], examined the fully formulated liquified metallic alloy in MHD channel flow. Misra et al. [6] had analyzed the viscoelastic MHD liquid flowing through a channel with stretching walls and drawn numerical solutions of the problem. Convective heat transfer across the MHD channel with radiation effects and heat absorption embedded in porous medium was examined by Uwanta et al. [7]. Moreau [8] had drawn off the analysis of MHD flows for different types of fluids. For the flow of electrically conducting fluid in various geometries, magnetic field acts as a flow controller parametric quantity and a substantial modification in flow rate is carried referable to the another dimensionless 3

parametric quantity, i.e., the Hartman number. Later on Hartman and Lazarus [9] initiated working in the magnetic field at the laminar flow of viscous fluids between two parallel vertical plates. Many investigators examined the effects of magnetic field in a lot of additional flow geometries admitting converging or departing channels [10-13]. Hayat et al. [14] had investigated heat transfer effects on second grade fluid flow in divergent or convergent chambers. In another paper, he got the solution for channel flow of a (MHD) Jeffrey fluid with variable viscosity in series form [15]. Misra et al. [16-19] had developed the numerical model to carry out an experiment on mixed convection of hydro magnetic fluid over non linear, inclined porous sheet and also worked out the solutions for unsteady flow and temperature fields in a diseased capillary. Recently, the fractional calculus had been a purely mathematical tool without apparent applications. Currently, fractional dynamical equations play a major role in the modeling of anomalous behavior and memory effects, which are common characteristics of natural phenomena [20-23]. The fact that fractional derivatives introduce a convolution integral with a power-law memory kernel makes the fractional differential equations an important model to describe memory effects in complex systems. Thus, it is seen that fractional derivatives or integrals appear naturally when modeling long-term behaviors, especially in the areas of viscoelastic materials and viscous fluid dynamics. Fractional calculus is believed to be as a practical method in several branches of science disciplines [24]. An arising bit of works in scientific discipline and engineering science consider the dynamic system modeled by the set of equations of non-integer order that require derivatives and integrals of non-integer order [25-27]. These newly developed models are usually much more adequate than antecedently applied integer order frameworks. As fractional order differentials and integrals account for the memory and complex substances examined by Zaslavsky [28] and Poddulony [29]. These are the virtually significant advantage of non-integer order in comparing with integer order, in which such effects are overlooked. In the linguistic context of flow through porous medium fractional space differentials framework macroscopic motions by highly semiconducting fractures or layers, whereas fractional time differentials depict particles which stays inactive for prolonged time period studied by Meerscheart et al. [30]. Applications of fractional derivative models can be seen in the frameworks of polymers in the glass state, wire and fibre coating, chemically processed equipments, in the designing of several heat engines or heat exchangers. Many researchers are generalizing the classical fluid models to fractional models by applying different approaches like Caputo (2005)(C), Caputo-Fabrizio (CF)(2015) or the latest definition of Atangana-Baleanu (AB) (2016). Khan and Shah [31] latterly applied the CF derivatives to the problem formulated due to transfer of heat in second grade viscoelastic fluid. Ali et al. [32], applied the CF differentials and examined free convective MHD flow of a generalized Walter’s-B fluid framework. Imran et al. [33-36] developed heat transfer problems by applying fractional derivatives and drawn comparison between the results obtained by applying Caputo derivatives with singular local kernel and Caputo-Fabriozio time derivative with non-singular exponential kernel. In all above mentioned works no one has formulated the fractional model of viscous fluid for channel flow with MHD effect, heat transfer and chemical reaction. In this research article, we have formulated a fractional model of viscous incompressible Newtonian fluids with heat source, MHD and chemical reaction parameters. Fractional fluid model is developed by applying two fractional approaches namely; Caputo time fractional derivatives and 4

Caputo-Fabrizio time fractional derivatives [37-41]. The obtained results are compared for heat and mass transfer effects by solving energy, diffusion and velocity fields respectively, via Laplace transform method. Inverse Laplace transform will be found numerically [42, 43] for the practical accuracy of the current problem. At the end, the impact of fractional as well as material parameters was seen graphically and determined that fluid velocity can be raised with Caputo-Fabrizio fractional model.

2

Mathematical Formulation

Let us consider the unsteady free convection of an incompressible viscous fluid flowing between two vertical parallel plates with constant concentration and temperature gradient. The x-axis is taken along one of the plate which is fixed in the vertically upward direction and y-axis is normal to the plate. Initially, at time t ≤ 0, both the plates and the fluid are considered to be at the temperature Td and concentration Cd . At time t > 0, the temperature and the concentration of the fluid at y = 0 are raised to Tw and Cw respectively, causing the flow of free convection currents as shown in the Fig. 1. According to usual Boussinesq’s approximation the governing equations for unsteady flow are [1]

Figure 1: Geometry of the problem ∂u(y, t) ∂ 2 u(y, t) σβo2 =ν + gβ1 (T (y, t) − Td ) + gβ2 (C(y, t) − Cd ) − u(y, t), ∂t ∂t2 ρ ρCp

∂T (y, t) ∂ 2 T (y, t) =k , ∂t ∂y 2 5

(1) (2)

∂C(y, t) ∂ 2 C(y, t) =D , ∂t ∂y 2

(3)

appropriate initial and boundary conditions are t≤0:

u(y, t) = 0,

T (y, t) = Td ,

C(y, t) = Cd ,

0 ≤ y ≤ d,

(4)

t>0:

u(y, t) = 0,

T (y, t) = Tw ,

C(y, t) = Cw ,

at y = 0,

(5)

T (y, t) = Td ,

C(y, t) = Cd ,

at y = d.

(6)

t>0:

2.1

u(y, t) = 0,

Basic Definitions

The Caputo fractional time derivative of order α ∈ [0, 1) is defined [36] Z t 1 ∂f (y, τ ) C α dτ. Dt f (t) = (t − τ )−α Γ(1 − α) 0 ∂τ The Laplace transform of Caputo time derivative is  L C Dtα f (y, t) = sα L {f (y, s)} − sα−1 f (y, 0).

The Caputo-Fabrizio time fractional derivative of order α ∈ [0, 1) is defined [37]   Z t 1 α(t − τ ) ∂f (y, τ ) CF α Dt f (y, t) = exp − dτ. 1−α 0 1−α ∂τ

(7)

(8)

(9)

The Laplace transform of Caputo-Fabrizio time derivative is L

3

CF

sL {f (y, t)} − f (y, 0) Dtα f (y, t) = . (1 − α)s + α

(10)

Modeling with Caputo time fractional derivative

To make the problem dimension less we introduce the following set of variables y y∗ = , d

t∗ =

νt , d2

u∗ =

νu d2 gβ1 (Tw

− Td )

,

θ=

T − Td C − Cd , C= . Tw − Td Cw − Cd

(11)

Momentum equation (1) in dimensionless form by making use of non-dimensional variables provided in Eq. (11) ∂u(y, t) ∂ 2 u(y, t) = + θ(y, t) + N C(y, t) − M u(y, t), (12) ∂t ∂y 2 2 2

3

3

d )d d )d od where M = σβνρ , Gr = gβ1 (Twν−T , Gm = gβ2 (Cwν−C , N = Gm are the magnetic field parameter, 2 2 Gr thermal and mass Grashof numbers and the ratio of mass to thermal Grashof numbers respectively.

6

Thermal balance equation is ∂T (y, t) ∂q =− , (13) ∂t ∂y where q is the heat flux and Cp is specific heat at constant pressure. The generalized fractional Fourier’s law suggested by Hristov [43] and Henry et al. [44]   ∂T (y, t) C 1−β , 0 < β ≤ 1, (14) q(y, t) = −κβ Dt ∂y  Kgm  where κβ is coefficient of generalised thermal conductivity with dimensions Ks β+2 . By substituting Eq. (14) into (13) and making result dimensionless through Eq. (11)   1 C 1−β ∂ 2 θ(y, t) ∂θ(y, t) = , (15) Dt ∂t Pr ∂y 2 ρCp

p where Pr = µC is the generalized Prandtl number. Applying the fractional integral operator It1−β κβ to Eq. (15) we get 1 ∂ 2 θ(y, t) C β Dt θ(y, t) = , (16) Pr ∂y 2 The diffusion balance equation is ∂J ∂C(y, t) =− , (17) ∂t ∂y

with generalized constitutive equation of mass diffusion obtained by Fick’s law [43]   ∂C(y, t) C 1−γ J(y, t) = −Dγ Dt , 0 < γ ≤ 1, ∂y

(18)

where Dγ is the generalised coefficient of molecular diffusion. Substituting Eq. (18) into Eq. (17) and making the result dimensionless by applying Eq. (7)   ∂C(y, t) 1 C 1−γ ∂ 2 C(y, t) = Dt , (19) ∂t Sc ∂y 2 where Sc = Dνγ is the generalised Schimdt number. Applying the fractional integral operator It1−γ to Eq. (19) we get 1 ∂ 2 C(y, t) C γ . (20) Dt C(y, t) = Sc ∂y 2 Associated dimensionless initial and boundary conditions are: t≤0:

u(y, t) = 0,

θ(y, t) = 0,

C(y, t) = 0,

0 ≤ y ≤ 1,

(21)

t>0:

u(y, t) = 0,

θ(y, t) = 1,

C(y, t) = 1,

at

y = 0,

(22)

t>0:

u(y, t) = 0,

θ(y, t) = 0,

C(y, t) = 0,

at

y = 1.

(23)

7

3.1

Calculation of temperature field with Caputo time fractional derivative

Energy equation (16) and diffusion equation (20) are not coupled with second order momentum equation (12), so we need to solve Eqs. (16) and (20) separately by applying Laplace transform technique and then the momentum equation. Applying Laplace transform to Eq. (16) we get ¯ s) ∂ 2 θ(y, ¯ s), = Prsβ θ(y, ∂y 2

(24)

satisfies the conditions

¯ s) = 1 θ(0, s The solution of Eq. (24) subject to Eq. (25) θ¯C (y, s) =

¯ s) = 0. and θ(1,

∞ X e−(y+2k) k=0

√

√

Prsβ

−

s

(25)

∞ X e(y−2−2k) k=0

Prsβ

.

s

(26)

with inverse Laplace of Eq. (26) θC (y, t) =

∞ X k=0

[∆1 (y, t, Pr, k, u) − ∆2 (y, t, Pr, k, u)] ,

(27)

where   √ β − β2 ∆1 (y, t, Pr, k, u) = φ 1, − 2 ; −(y + 2k) P rt ,   √ β ∆2 (y, t, Pr, k, u) = φ 1, − β2 ; −(y − 2 − 2k) P rt− 2 , for 0 < β ≤ 1 and φ is Wright’s function and is defined in the Appendix.

3.2

Calculation of concentration field with Caputo time fractional derivative

Applying Laplace transform to Eq. (20) with Caputo fractional time derivative we get ¯ s) 1 ∂ 2 C(y, ¯ s), = sγ C(y, 2 Sc ∂y

(28)

satisfies the conditions

¯ s) = 1 C(0, s The solution of Eq. (28) subject to Eq. (29) C¯C (y, s) =

√

∞ X e−(y+2k) k=0

¯ s) = 0. C(1,

and

Scsγ

−

s

8

∞ X e(y−2−2k) k=0

s

(29)

√

Scsγ

,

(30)

with inverse Laplace transform of Eq. (30) is given as CC (y, t) =

∞ X k=0

[∆3 (y, t, Sc, k, u) − ∆4 (y, t, Sc, k, u)] ,

(31)

where   √ γ ∆3 (y, t, Sc, k, u) = φ 1, − γ2 ; −(y + 2k) Sct− 2 ,  √ −γ  γ ∆4 (y, t, Sc, k, u) = φ 1, − 2 ; −(y − 2 − 2k) Sct 2 , for 0 < γ ≤ 1.

3.3

Calculation of velocity field with Caputo derivative

Applying Laplace transform to Eq. (12) and using the expressions from Eqs. (26) and (30) we get h i √ √ ∂ 2 u(y, s) 1 y√Prsβ 1 (1+y) Prsβ (1−y) Prsβ √ u(y, s) = − e e − − (s + M ) + − e ∂y 2 s 2s sinh Prsβ h i √ √ N √ γ N γ γ √ − ey Scs + e(1+y) Scs − e(1−y) Scs , s 2s sinh Scsγ

(32)

satisfies the conditions u¯(0, s) = 0

and u¯(1, s) = 0.

(33)

The solution of Eq. (32) subject to Eq. (33) is given as, n √ o o n √ √ √ β γ  √ o N ey s+M − ey Scs ey s+M − ey Prs + − 2Λ(s) sinh y s + M + u¯C (y, s) = s (Prsβ − s − M ) s (Scsγ − s − M ) n n o o √ √ √ √ (1+y) Prsβ (1+y) Scsγ (1−y) Prsβ (1−y) Scsγ e N e −e −e √  √  + + 2s sinh Prsβ (Prsβ − s − M ) 2s sinh Scsγ (Scsγ − s − M ) where

Λ(y) =

1 √ 2 sinh( s+M )

"

e

√ √ s+M −e Prsβ

s(Prsβ −s−M )



+

n √ o √ γ N e s+M −e Scs s(Scsγ −s−M )

+

 √  β e2 Prs −1 √ 2s sinh( Prsβ )(Prsβ −s−M )

+

(34)

# n √ o γ N e2 Scs −1 √ . 2s sinh( Scsγ )(Scsγ −s−M )

The inverse Laplace transform of Eq. (34) will be found numerically by applying Stehfest’s and Tzou’s algorithms [42,43].

4

Modeling with Caputo-Fabrizio time fractional derivative

Transient heat conduction with damping term expressed through Caputo-Fabrizio fractional derivative [Eq.(15), 44] is ∂θ(y, t) ∂ 2 θ(y, t) ∂ 2 θ(y, t) CF β = ω1 + ω2 Dt (1 − β) , (35) ∂t ∂y 2 ∂y 2

9

where β is fractional parameter and ω1 =

κ1 , ρCp

ω2 =

κ2 . ρCp

The dimensionless form of Eq. (35) is

∂θ(y, t) 1 ∂ 2 θ(y, t) 1 CF β ∂ 2 θ(y, t) = + D (1 − β) , t ∂t Pr1 ∂y 2 Pr2 ∂y 2

(36)

In a similar way, the constitutive equation for mass diffusion can be written in dimensionless form as ∂C(y, t) 1 ∂ 2 C(y, t) 1 CF γ ∂ 2 C(y, t) = + D (1 − γ) , (37) t ∂t Sc1 ∂y 2 Sc2 ∂y 2 where Pr1 =

4.1

µCp , κ1

Pr2 =

µCp , κ2

Sc1 =

ν D1

and Sc2 =

ν . D2

Calculation of temperature field with Caputo-Fabrizio derivative

Applying Laplace transform method to Eq. (36) we get   2¯ ¯ s) 1 ∂ 2 θ(y, 1 s ∂ θ(y, s) ¯ sθ(y, s) = (1 − β) + , 2 Pr1 ∂y Pr2 s(1 − β) + β ∂y 2

(38)

satisfies the conditions

¯ s) = 1 θ(0, s The solution of Eq. (38) subject to Eq. (39) 1 y θ¯CF (y, s) = e s

r

s(s+a1 ) a2 (s+a3 )

−

2s sinh

¯ s) = 0. and θ(1,

1 q

s(s+a1 ) a2 (s+a3 )

"

e

(39)

r s(s+a ) (1+y) a (s+a1 ) 2

3

# r s(s+a ) (1−y) a (s+a1 )

−e

2

3

,

(40)

β Pr2 2 , a2 = PrPr11+Pr and a3 = Pra11+Pr . The inverse Laplace transform of Eq. (40) will be where a1 = 1−β Pr2 2 found numerically by applying Stehfest’s and Tzou’s algorithms [42-43].

4.2

Calculation of concentration field with Caputo-Fabrizio derivative

Applying Laplace transform method to Eq. (37) with Caputo-Fabrizio fractional time derivative we get   2¯ 2¯ C(y, s) 1 ∂ 1 s ∂ C(y, s) ¯ s) = sC(y, + (1 − γ) , (41) 2 Sc1 ∂y Sc2 s(1 − γ) + γ ∂y 2 satisfies the conditions ¯ s) = 1 ¯ s) = 0. C(0, and C(1, (42) s The solution of Eq. (41) subject to Eq. (42) # " r r r s(s+b ) s(s+b ) s(s+b ) y b (s+b1 ) (1−y) b (s+b1 ) (1+y) b (s+b1 ) 1 1 2 3 2 3 q C¯CF (y, s) = e 2 3 − e −e , (43) s(s+b1 ) s 2s sinh b2 (s+b3 )

γ Sc2 2 where b1 = 1−γ , b2 = ScSc11+Sc and b3 = Scb11+Sc . The inverse Laplace transform of Eq. (43) will be Sc2 2 found numerically by applying Stehfest’s and Tzou’s algorithms [42-43].

10

4.3

Calculation of velocity field with Caputo-Fabrizio derivative

Applying Laplace transform to Eq. (12) and using the expressions from Eqs. (40) and (43) we get ∂ 2 u¯(y, s) 1 y (s + M )¯ u(y, s) = + e ∂y 2 s " r (1+y)

× e −

2s sinh

satisfies the conditions

s(s+a1 ) a2 (s+a3 )

1 q

r

s(s+a1 ) a2 (s+a3 )

1 q

× 1) 2s sinh as(s+a 2 (s+a3 ) # r r s(s+a ) (1−y) a (s+a1 ) N y s(s+b1 ) 2 3 + e b2 (s+b3 ) − −e s # " r r s(s+b1 ) b2 (s+b3 )

(1+y)

s(s+b1 ) b2 (s+b3 )

e

u¯(0, s) = 0

−

−e

(1−y)

s(s+b1 ) b2 (s+b3 )

,

(44)

and u¯(1, s) = 0.

(45)

The solution of Eq. (44) subject to Eq. (45) is given as, h i √ √ −(y+1) s+M (y−1) s+M N e −e e −e √ √ u¯CF (y, s) = + − 2s [A − (s + M )] sinh s + M 2s [B − (s + M )] sinh s + M h √ √ i √ √ √ −y s+M y B N e − e −y s+M y A sinh y s + M e −e √ + + −λ(s) + s[A − (s + M )] s[B − (s + M )] sinh s + M h √ √ i √ √ (y+1) B (1−y) B N e −e e(y+1) A − e(1−y) A √ + √ , + 2s [A − (s + M )] sinh A 2s [B − (s + M )] sinh B √ −(y+1) s+M

√

√ (y−1) s+M

(46)

√

√

N [e2 B−1 ] s(s+a1 ) e2 A −1 Ne B 1) √ − √ , A = where λ(s) = 2s[A−(s+M + , B = bs(s+b , s[B−(s+M )] a2 (s+a3 ) )] sinh A 2s[B−(s+M )] sinh B 2 (s+b3 ) β γ Pr2 Sc2 1 2 2 a1 = 1−β , a2 = PrPr11+Pr , a3 = Pra11+Pr . b1 = 1−γ , b2 = ScSc11+Sc and b3 = Scb11+Sc , c1 = 1−α , c2 Pr2 Sc2 2 2

=

α . 1−α

The inverse Laplace transform of Eq. (46) will be found numerically by means of [42, 43].

4.4

Calculation of Nessult number, Sherwood number and Skin friction

The rates of heat and mass transfer as well as skin friction will be computed numerically by using numerical techniques; Stehfest’s and Tzou’s [42,43] by using relations given in Eqs. (47) and (48) respectively N uC = − N uCF = −

∂θC (y, t) |y=0 , ∂y

∂θCF (y, t) |y=0 , ∂y

ShC = − ShCF = −

∂C C (y, t) |y=0 , ∂y

∂C CF (y, t) |y=0 , ∂y 11

(Cf )C = − (Cf )CF = −

∂uC (y, t) |y=0 , ∂y

(47)

∂uCF (y, t) |y=0 . ∂y

(48)

4.5

Graphical results and discussions

Transient free convective flow of viscous, incompressible fluid flowing between two vertical parallel plates of infinite length in the presence of MHD is studied in this paper. Fractional fluid model is solved analytically by means of Laplace transform method satisfying initial and boundary conditions. Temperature, concentration and velocity expressions are obtained and displayed graphically. The influence of material and flow parameters are shown graphically. The inverse Laplace transform is obtained by using Stehfest’s algorithm as well as Tzou’s algorithm to check the validity of our obtained results. Figure 2 is plotted to see the impact of fractional time variable γ on concentration field. By adding up the value of γ the fluid’s concentration and boundary layer thickness increases. Furthermore CF fractional model possesses higher concentration level then C fractional model. Figure 3 is plotted to interpret the impact of Schmidt number Sc on concentration field. It is noticed that with an increment in the value of Sc the concentration of fluid decreases because as the value of Sc increases, molecular diffusivity decreases and it tends to reduce the boundary layer thickness of concentration. Figure 4 is plotted to see the influence of time on concentration field and concluded that the concentration of the fluid increases with time. Figure 5 is plotted to see the variation of fractional parameter on temperature field by taking other parametric values as constants. By increasing the value of of fractional parameter, it is found that temperature of of the fluid decreases but the boundary layer thickness increases. Further the temperature field of CF fractional model is higher than the C fractional model. Figure 6 is plotted for altered values of Prandtl number Pr versus y by fixing other parameters. Graphically it is pertinent that as the values of Prandtl number Pr enhances, fluid’s temperature decreases. As expected, increasing Pr reduces the thermal conductivity and enhances the viscousness of the fluid which results in reduction of thermal boundary layer thickness. Figure 7 is plotted to see the impact of time t on the temperature. It is evident from the graph that for the lager values of time t, temperature increases. Thermal boundary layer thickness for CF fractional model is more than the C fractional model. Figures 8 is plotted for different values of β for smaller time and for larger values of time. It is quite obvious from the graph that velocity is an increasing function of β for larger value of time, but a decreasing function of β for smaller time. Figure 9 is plotted for altered values of M, the magnetic field parameter. As expected, for larger values of M, fluid velocity decreases . Moreover, it is more obvious that the boundary layer thickness and decreases in general free stream domain. Physically, it is because of drag force acting in opposite direction to the fluid flow and hence reduces the fluid flow. Figure 10 is plotted for various values of N, the buoyancy ratio parameter representing the ratio between thermal Grashof Gr and mass Grashof Gm. When the value of N = 0, there is no mass transfer and buoyancy force is mainly due to the thermal diffusivity. By larger the values of N the fluid velocity as well as the boundary layer thickness increases. Physical impact of Prandtl number Pr on fluid’s velocity by taking the other flow parameters constant can be seen in Figure 11. It is pertinent to mention that as value of Pr steps up, the fluid’s velocity and the boundary layer thickness decreases. It is due to the reason that increasing values of Pr lead to increase in viscosity of fluid and decrease in thermal boundary layer resulted in slower motion of the fluid. 12

Figure 12 is plotted to see the impact of Sc, the Schmidt number on fluid’s velocity by fixing other parametric values. It is noticed that an upsurge in the value of Schmidt number Sc leads to decrement in the velocity field. Since for greater values of Sc molecular diffusivity declines and it tends to decrease the thickness of the boundary layer. Figure 13 is plotted to see the influence of time on fluid’s velocity. It is clear from the graph that as time t increases the fluid’s velocity as well as the boundary layer thickness decreases. Figure 14 and 15 are plotted to see the validity of inversion algorithms applied to find the numerical solutions of velocity field obtained via Caputo model and Caputo-Fabrizio model. Overlapping curves clearly indicates the validation of inversion algorithms. The numerical results for Nusselt number, Sherwood number and skin friction under the influence of fractional parameters are given in table 1 for Caputo and Caputo-Fabrizio fractional models. The rates of heat and mass transfer and skin friction in case of Caputo model is more than the Caputo-Fabrizio.

Figure 2: Profiles of the concentration field C(y, t) for altered values of γ

13

Figure 3: Profiles of the concentration field C(y, t) for altered values of Sc

Figure 4: Profiles of the concentration field C(y, t) for altered values of t

14

Figure 5: Profiles of the temperature field θ(y, t) for altered values of β

Figure 6: Profiles of the temperature field θ(y, t) for altered values of Pr

15

Figure 7: Profiles of the temperature field θ(y, t) for altered values of t

Figure 8: Profiles of the velocity field u(y, t) for altered values of fractional parameters

16

Figure 9: Profiles of the velocity field u(y, t) for altered values of M

Figure 10: Profiles of the velocity field u(y, t) for altered values of N

17

Figure 11: Profiles of the velocity field u(y, t) for altered values of Pr

Figure 12: Profiles of the velocity field u(y, t) for altered values of Sc

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Figure 13: Profiles of the velocity field u(y, t) for altered values of t

Figure 14: Validity of inversion algorithms, Profiles of the velocity field u(y, t) with Caputo-Fabrizio model

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Figure 15: Validity of inversion algorithms, Profiles of the velocity field u(y, t) with Caputo model

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4.6

Conclusion

In this paper, transient Newtonian fluid model has been solved analytically by applying Laplace transform technique. Results for velocity, concentration and temperature fields are obtained analytically and displayed graphically. Numerical inversion Laplace transform techniques namely; Tzou’s and Stehfest’s are applied to find inverse Laplace transform for velocity field. Some key findings of this study are outlined as: 1. Temperature is a decreasing function of fractional parameter β and Prandtl number Pr whereas it increases by increasing the values of time t. 2. Concentration increases by increasing the values of fractional parameter γ and time t while reduces by increasing the values of Sc as well as boundary layer thickness while it is an increasing function of time. 3. Velocity is an increasing function of N and time t whereas it decreases with increasing the value Prandtl number Pr, Schimdt number Sc and magnetic parameter M. Velocity shows the dual behavior for smaller and larger time. 4. The fluid velocity can be enhanced with Caputo-Fabrizio fractional model rather than Caputo. The rates of heat and mass transfer and skin friction in case of Caputo model is more than the Caputo-Fabrizio.

Availability of data and material The data and material is available in the main paper file.

Competing interests The authors declare that there is no competing interests.

Funding University of Management and Technology Lahore, Pakistan.

Authors’ contributions All the authors contributed equally.

Acknowledgments The authors are highly thankful to the reviewers for their fruitful suggestions to improve the manuscript. The corresponding author would like to acknowledge Ton Duc Thang University, Ho Chi Minh City, 700000 Viet Nam for the financial support.

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5

Appendix L−1

n

e−aq qc

b

o

= tc−1 φ(c, −b; −at−b ); φ(x, y, z) =

References

P∞

zn 0 Γ(n+1)Γ(x+ny) ,

0 < b < 1.

(A1)

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