A comprehensive report on convective flow of fractional (ABC) and (CF) MHD viscous fluid subject to generalized boundary conditions

Chaos, Solitons and Fractals 118 (2019) 274–289

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

A comprehensive report on convective flow of fractional (ABC) and (CF) MHD viscous fluid subject to generalized boundary conditions M.A. Imran b, Maryam Aleem b, M.B. Riaz b, Rizwan Ali b, Ilyas Khan a,∗ a b

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Department of Mathematics, University of Management and Technology Lahore, C-II Johar Town, Lahore 54770, Pakistan

a r t i c l e

i n f o

Article history: Received 1 August 2018 Revised 30 November 2018 Accepted 1 December 2018 Available online 8 December 2018 Keywords: Inclined plat MHD Heat sink Chemical reaction New tonian fluid Comparison study CF and ABC fractional model

a b s t r a c t We have analyzed the magnetohydrodynaimcs (MHD) unsteady free convection flow of incompressible Newtonian fluid passing over an inclined plate through porous medium with variable temperature and concentration at the boundary. Additionally, we have also seen the effects of heat sink and chemical reaction. We have solved dimensionless equations governing the physical problem by Laplace transform method. Firstly, we have found the analytical results for concentration, temperature and velocity fields of classical model. After that we have extended the classical model to some fractional models specifically Caputo–Fabrizio (CF) and Atangana–Baleanu (ABC). Semi analytical results are attained for concentration, temperature and velocity fields for both models and then compared with solutions of classical one. Influence of Fembedded parameters on concentration, temperature and velocity domains can be perceived through MathCad software. As a result, we have observed that both the fractional models (CF) and (ABC) are better in describing the history of the physical problem. Further it is noted that, (ABC) model is wellsuited in stimulating the history functions of temperature, concentration and velocity fields. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Free convection, heat and mass transfer of electrically conducting fluids passing through inclined plate with varying temperature under the influence of magnetic field has drew the attention of astrophysics, geophysical science, metrology, polymer technology and many applied scientific problems, e.g., temperature reduction of boundary layer control in aeromechanics, nuclear reactors and refrigerants. In power technology and thermal physics, MHD (magneto hydrodynamics) problems with heat and mass transfer have tremendous applications like in astrophysics and geophysical sciences hence enforced to analyze the cosmological and solar anatomical structure, propagation of radio waves by the ionosphere, metal forming and continuous fire casting. In applied sciences it is applicable in MHD bearings, MHD pump and electric generators. In stellar structures, theory the mass transfer phenomenon is very common and observable impressions are perceptible on the astrophysical surface. In convective flows the influence of magnetic flux plays a key role in alloys, ionized gases as well as in electrolytes. Experimental and analytical study of ordinary convection for different geometries of the surface is of major concern in recent ∗

Corresponding author. E-mail addresses: [email protected] (M.A. Imran), [email protected] (I. Khan). https://doi.org/10.1016/j.chaos.2018.12.001 0960-0779/© 2018 Elsevier Ltd. All rights reserved.

years. Heat transfer is due to natural convection over an inclined surface have imperative functioning in designing of technical devices and intrinsic atmosphere. Rich [1] was the first who studied the natural convection of fluid over an inclined plate experimentally in 1953. Chen et al. [2] had drawn numerical results of natural convection flow across an inclined surface with variable temperature. They have considered that while plate’s angle from vertical direction steps-up, heat transfer coefficient and rate of friction factor reduces. This is of major concern to applied scientists and chemical engineers. Free or natural convection heat conveyance by an isothermal plate possesses arbitrary angle of inclination had been inquired by [3]. Khaled and Chamkha [6] attained results for hydro-magnetic mass and heat transferal through ordinary convection across bended plate with absorption or heat generation coefficients. Flow of liquid passing across bent plate with changing surface temperature brings about convection in water (H2 O) at 4 °C, this fact had been considered by Palani et al. [7]. Natural convective rate of flow of a chemically reacting Newtonian liquid passing by vertical and inclined plate in the presence of Dufour i.e., diffusion-thermo and Soret i.e., thermal-diffusion increases with heat and mass transfer features were analyzed by Beg et al. [8]. Singh et al. [9] examined analytically the effects of Newtonian heating bearing volumetrical heat generation. Varma et al. [10] had

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

studied effects of chemical reaction with MHD free convective flow rate through a porous medium confined by a pitched surface. MHD unsteady Couette flow through two parallel porous plates of myriad length with magnetic field effects and heat transfer was analyzed by Joseph et al. [11]. Ziyauddin [12] acquired exact solutions (closed form) for MHD non-uniform free convective flow of glutinous liquid streaming across inclined surface with fluctuating rates of heat and mass conveyance. Bhuvaneswari [13] examined free convection MHD flow with conjugate heat and mass conveyance. Singh had studied boundary layer flow through wedge with viscid dissipation in porous channel [14]. MHD mass and heat transfer of thermophoresis flow rate over radiating disposed isothermal porous surface bearing heat sink or source [15]. Raju et al. [16– 18] had studied Soret effects due to free convection between inclined animated plates with magnetic flux and as well in the comportment of radiation and chemical reaction [16–18]. Fractional calculus postulating integrals and derivatives of noninteger order is the natural generalization of the classical calculus letting improved posturing and control procedures in various fields of science and technology. The concept of fractional operator and fractional model theory was first introduced in [19]. It is a helpful mechanism to treat complex flow behavior and many mathematicians had applied this technique of fractional dynamics by non-integer order derivatives. Shahid used the definition of Caputo fractional derivatives (CFD) for analyzing the mass and heat transfer flow through oscillating plates [20]. The concept of fractional differential operator was applied to fluid problems by several mathematicians [20–24] but in heat transfer problems Vieru et al. [25] were the first to apply the idea of fractional derivative (FD) and considered free convective flow rate of viscous fluid. Afterwards, many researchers had applied fractional derivatives to attain the exact solutions for fluid flows with different geometries, references are therein [26–33]. Newtonian fluids flowing over an inclined surface have the potential to enhance effectiveness of heat transfer, which could be accomplished by leaving a fluid to flow across a bended surface with influence of magnetic field. The comparison approach for the proposed physical has not been reported yet in the existing literature, hence the need for this research article. The findings therein can greatly benefit respective diligences specifically medical, electronics, food processing and fabricating likewise nuclear reactor engineering sciences.

Fig. 1. Flow geometry.

 and concentration C  . At time t > 0, plate started perature T∞ ∞ to accelerate with a time dependent velocity u (y , t ) in its own plane. Concurrently, the temperature and concentration level is  + T  h (t  ) and also evoked or lowered depending upon time T∞ w  + C  g (t  ). Physical model describing flow is given in Fig. 1. The C∞ w governing equations of the problem are given by [4,5,9]

    ∂ u ∂ 2 u =ν + gβ T  − T  ∞ cos γ + gβ  C  − C  ∞ cos γ  2 ∂t ∂ y 2 σ B0  ν    − u − u , y , t > 0, ρ K  p ρ cp

  ∂T ∂ 2T  =k − S T  − T  ∞ , 2  ∂t ∂y

  ∂C ∂ 2C  =D − Kc C  − C  ∞ ,  ∂t ∂ y 2

An unsteady MHD uniform free convective flow of a radiating, viscid and incompressible fluid passing through an accelerated time dependent Uo f (t ) inclined plate of infinite length with in-

(1)

(2) y , t  > 0,

(3)

with the initial and boundary conditions

t  ≤ 0,

u = 0,

 , T  = T∞

t  > 0 u = uo f  (t  ), 2. Mathematical formulation of the problem

275

y ≥ 0,

 + T  h (t  ), T  = T∞ w

aty = 0,

t > 0, u → 0,

 , C  = C∞

(4)

 + C  g (t  ), C  = C∞ w

(5)  , T  → T∞

 , C  → C∞

as y → ∞.

(6)

Introducing the dimensionless variables, functions and parameters

⎫

uo  u u2 T  − T ∞ C − C∞ ν ν2 ⎪ ⎪ y, u= , t = o t , T = , C= , a = 2 a , K p = 2 K  P , ⎪   ⎪ ν uo ν T w Cw uo uo ⎪ ⎬ 2     μCp σ Bo ν T w−T ∞ C − C ν ν w ∞   M= , Pr = , Gr = g βν , Gc = g β ν , K = K , S = , c c c 3 3 K ρ u2o u2o  u o   u2o   D⎪ ⎪ 2 2 ⎪ ν  16 σ 3 u u u ⎪ o o o   , f (t ) = f   , g(t ) = g  . ⎪ S= S , N = T , h ( t ) = h t t t ⎭ r ∞ 2 3 kkR ν ν ν ρ C p uo y=

constant temperature inserted in a saturated porous medium has been considered in this problem. The x -axis is taken along the plate and y -axis is normal to the plate. Bo is intensity of magnetic field applied in the direction perpendicular to the plate. The angle of inclination to vertical direction is γ . Since the value of Rm (Magnetic Reynolds number) describing the flow is very small consequently induced magnetic field is ignored. Initially we have assumed that plate and the adjoining fluid are at the same tem-

(7)

Eqs. (1–6) after dropping prime notations we have

1 ∂ u ∂ 2u = − Mu −  u + GrT cos γ + GcC cos γ , ∂t ∂ y2 Kp

y, t > 0, (8)

∂T 1 ∂ 2T = − ST , ∂t Pr ∂ y2

y, t > 0,

(9)

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∂C 1 ∂ 2C = − KcC, ∂ t Sc ∂ y2

y, t > 0,

(10)

with the initial and boundary conditions in dimensionless form are

t≤0:

u = 0,

T = 0,

C = 0,

y ≥ 0,

The inverse Laplace transform of Eq. (24), writing in suitable form

L

  √ −y Sc (q+Kc )  e , C¯ (y, q ) = L−1 qG(q ).

−1

(11) C (y, t ) =

t > 0,

u = f (t ),

T = h(t ),

C = g(t ),

aty = 0,

(12)

T → 0,

C → C,

as y → ∞.

(13)

L−1



=

∂ 2 u¯ (y, q ) 1 − q+M+ u¯ (y, q ) Kp ∂ y2 = −Gr cos γ T¯ (y, q ) − Gc cos γ C¯ (y, q ),

∂ 2 T¯ (y, q ) , ∂ y2

Pr(q + S )T¯ (y, q ) =

∂ C¯ (y, q ) , ∂ y2

(14)

(15)

0

 √  g (t − s )ψ y Sc, s, Kc , 0 ds.

(26)

√

e−y q+a q−b



= ψ (y, t; a, b)



 ebt −y√a+b y e er f c √ − (a + b)t 2 2 t

+ ey

√



a+b

er f c

 y √ + (a + b)t 2 t





.

4.1. Sherwood number The rate of mass transfer from plate to fluid in term of Sherwood number is given by

2

Sc (q + Kc )C¯ (y, q ) =

t



Applying Laplace transform to Eqs. (8)–(13) we get





where L−1 {qG(q )} = g (t ),



t > 0, u → 0,

(25)

q

(16)

 t  ∂ C (y, t ) |y=0 = ScKc g (t − s )er f c Kc s ds ∂y 0 

Sc t g (t − s ) −Kc s + e ds. √ π 0 s

h=−

u ( 0, q ) = F ( q ),

u(y, q ) → 0, asy → ∞

(17)

T ( 0, q ) = H ( q ),

T (y, q ) → 0, asy → ∞,

(18)

(27)

4.2. Solution of temperature field

C¯ (0, q ) = G(q ),

C¯ (y, q ) → 0, asy → ∞.

(19)

3. Preliminaries

Solution of Eq. (15) subjected to (18), same as explained in Section 4

T (y, q ) = H (q )e

−y

√

Pr (q+S )

= qH (q )

e−y

√

The Caputo–Fabrizio time derivative is defined as: CF

Dtα f (y, t ) =

t  α t − α ∂ f (y, τ ) 1 ( ) exp − dτ , 1−α 0 1−α ∂τ 0 < α < 1.

T (y, t ) =

CF



Dtα f (y, t ) =

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

(21)

The Atangana –Baleanu time derivative in Caputo sense (ABC) is defined as: Let f ∈ H1 (a, b), b > a, α ∈ [0, 1], then ABC

Dtα f (y, t ) =

1 1−α





t 0

Eα

 α (t − α )α ∂ f (y, τ ) − dτ , 1−α ∂τ

(22)

The Laplace transform of ABC derivative is

L

ABC

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

(23)

Sc (q+Kc )

.

(29)

4.3. Nusselt number The transfer of heat from moving fluid and solid plate in terms of Nusselt number is

t √ √  ∂ T (y, t ) |y=0 = PrS h (t − s )er f c Ss ds ∂y 0 

t  Pr h (t − s ) −Ss + e ds. √ π 0 s

Nu = −

(30)

By introducing the expressions from Eqs. (24) and (28) into Eq. (14) and it solution subject to conditions (17) is

 √ √ √  Gr cos γ H (q ) e−y Pr q+S − e−y q+λ (1 − Pr )(q − α1 )  √ √  Gc cos γ G(q ) + e−y Sc(q+Kc ) − e−y q+λ . (1 − Sc )(q − α2 )

u¯ (y, q ) = F (q )e−y

Solution of second order differential Eq. (16) satisfying boundary conditions (19)

√

 √  h (t − s )ψ y Pr, s, S, 0 ds.

(28)

4.4. Velocity field

4. Solution of concentration field

C¯ (y, q ) = G(q )e−y

0

t

,

(20)

The Laplace transform of Caputo–Fabrizio time derivative is

L



Pr (q+S )

q

(24)

√

q+λ

+

(31)

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

277

Fig. 2. Concentration profiles of classical, CF and ABC model for altered values of Sc.

Fig. 3. Concentration profiles of classical, CF and ABC model for altered values of t.

By applying inverse Laplace transform to Eq. (31) we have

u(y, t ) =





Gr cos γ h(t − s ) 1 − Pr 0 0 [φ (y, s; α1 , S, Pr) − φ (y, s; α1 , λ, 1)]ds

Gc cos γ t + g(t −s )[φ (y, s; α2 , Kc, Sc ) 1 − Sc 0 − φ (y, s; α2 , λ, 1)]ds, t

f (t − s )φ (y, s; λ )ds +

t

5. Concentration field with Caputo–Fabrizio fractional derivative Solving Eq. (14) by using (21)

(32)

where

λ=M+

1 , Kp

α1 =

S Pr −λ , 1 − Pr

α2 =

ScKc − λ , 1 − Sc

Pr = Sc = 1.

 Sc

q q (1 − α ) + α



+ Kc C¯ (y, q ) =

∂ 2C¯ (y, q ) ∂ y2

(33)

278

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Fig. 4. Concentration profiles of classical, CF and ABC model for altered values of α.

Solution of the second order differential (33) satisfying conditions (19)

1 −y e C¯ (y, q ) = qG(q ) q

√



Sc (a+Kc )

q+c q+b

,

(34)

where a = b = aα and c = Eq. (34) can be written equivalently as 1 1−α ,

C¯ (y, q ) =

ψ1 (q ) . ψ2 (y, q ) .

Fig. 5. Concentration profiles of classical, CF and ABC model for altered values of Kc .

The inverse Laplace transform of Eq. (35) is obtained as

C (y, t ) =



t 0

ψ1 (t − τ ) ψ2 (y, τ )dτ ,

where

ψ1 (t ) = g (t ), ψ2 (y, t ) = e

bKc a+Kc .

(35)

2

√

π



−y

∞

o

0

t

√

Sc (a+Kc )



−

y Sc (a + Kc )(c − b)

  1  y(Sc(a+Kc ))  √ e bt− 4u −u × I1 2 (c − b)ut dtdu. t

(36)

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

279

Fig. 7. Temperature profiles of classical, CF and ABC Model for altered values of S. Fig. 6. Temperature profiles of classical, CF and ABC model for altered values of Pr.

Eq. (37) can be written equivalently as For α → 1 we recover the solution for ordinary concentration as in Eq. (26).

T (y, q ) =

5.1. Temperature field with Caputo–Fabrizio fractional model

T (y, t ) =

Solving Eq. (15) by using (21) subject to conditions (18) we get

1 −y T¯ (y, q ) = qH (q ) e q where a =

1 1−α ,

√



Pr(a+S )

q+c1 q+b

b = aα and c1 =

, bS a+S .

(37)

ψ1 (q ) . ψ2 (y, q ) .

(38)

The inverse Laplace transform of Eq. (38) is obtained as



t 0

ψ1 (t − τ ) ψ2 (y, τ )dτ ,

where ψ1 (t ) = g (t ) and

ψ2 (y, t ) = e−y

o

∞



t 0

√

Pr (a+S )

(39)



−

y Pr (a + S )(c1 − b) √ 2 π





 y (Pr (a+S ) ) 1 √ e(bt− 4u −u ) × I1 2 (c1 − b)ut dtdu. t

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M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

Fig. 8. Temperature profiles of classical, CF and ABC model for altered values of α .

Fig. 9. Temperature profiles of classical, CF and ABC model for altered values of t.

For α → 1 we recover the solution for ordinary temperature as in Eq. (29). 5.2. Velocity field with Caputo–Fabrizio fractional model The solution of Eq. (13) using (21) subject to conditions (17)

u¯ (y, q ) = F (q )e



e

−y



qao q+α ao

+λ

−y

−e



−y

qao q+α ao



+λ

Prqao q+α ao

+

+Pr S



+

Gc cos γ G(q )(q + a5 ) −y e a3 ( q + a4 )



qao q+α ao

+λ

−e

−y



Scqao q+α ao

+ScKc

 .

(40)

Gr cos γ H (q )(q + a5 ) a1 ( q + a2 )



For α → 1 we recover the solution for ordinary velocity as in Eq. (32).

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

Fig. 10. Velocity profiles of classical, CF and ABC model for altered values of α .

Solving Eq. (16) by using (23) subject to conditions (19)



qα Sc α + Kc C¯ (y, q ) = q (1 − α ) + α

∂ 2C¯ (y, q ) ∂ y2

6.1. Temperature field with Atangana–Beleanu fractional model

(41)

The solution of the second order differential Eq. (41) satisfying boundary conditions (19)  

C¯ (y, q ) = G(q )e

−y Sc

qα qα (1−α )+α

+Kc

Fig. 11. Velocity profiles of classical, CF and ABC model for altered values of γ .

For α → 1 we recover the solution for ordinary concentration as in Eq. (26).

6. Concentration field with Atangana–Baleanu fractional derivative



281



(42)

Solving Eq. (15) by using (23) subject to conditions (18) we get −y T¯ (y, q ) = H (q )e

√  α Pr qα (1q−α )+α +S

.

(43)

For α → 1 we recover the solution for ordinary concentration as in Eq. (29).

282

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Fig. 13. Velocity profiles of classical, CF and ABC model for altered values of Gc.

Fig. 12. Velocity profiles of classical, CF and ABC model for altered values of Gr.

6.2. Velocity field with Atangana–Beleanu fractional model The solution of Eq. (14) by using (23) subject to conditions (17)

u¯ (y, q ) = F (q )e



e

 −y

qα a0 qα +α ao

+λ

−y



−e

qα ao qα +α ao

−y



+λ

Pr qα ao qα +α ao

+

Gr cos γ H (q )(qα + a5 ) a1 ( qα + a2 )

+Pr S





+

Gc cos γ G(q )(qα + a5 ) −y e a3 ( qα + a4 )



qα ao qα +α ao

+λ

−e

−y



Scqα ao qα +α ao

+ScKc

 , (44)

where

λ=M+

1 (Pr S − λ )α ao , a1 = Pr ao + Pr S − ao − λ, a2 = , Kp a1

a3 = Scao + KcSc − ao − λ a4 =

(ScKc − λ )α ao a3

, ao =

1 , 1−α

α = 1.

The inverse Laplace transform of Eqs. (40)–(44) will be found numerically by applying Stehfest’s and Tzou’s algorithms [34,35].

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

Fig. 14. Velocity profiles of classical, CF and ABC model for altered values of Kc .

283

Fig. 15. Velocity profiles of classical, CF and ABC model for altered values of Kp .

For α → 1 we recover the solution for ordinary velocity as in Eq. (32). 7. Numerical results and discussion The current study aims to mention the effect of viscous, radiating and incompressible fluid passing through an accelerated inclined flat plate of infinite length with changeable temperature in a porous medium. The Newtonian fluid model is solved by Laplace transform method satisfying all levied initial conditions (initial and boundary) to attain analytical solutions. Numerical results for ve-

locity, concentration and temperature fields are obtained by applying three different differential operators; i.e. the Classical model, (CF) and (ABC) model and results are shown graphically for several flow parameters like the Prandtl number Pr, the magnetic field parameter M, the permeability of porous medium Kp , the chemical reaction parameter Kc , the constant heat sink S, the mass Grashof number Gm, the thermal Grashof number Gr, the inclination angle from the vertical directionγ , the Schmidt number Sc, the fractional parameter α and time t.

284

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Fig. 16. Velocity profiles of classical, CF and ABC model for altered values of M. Fig. 17. Velocity profiles of classical, CF and ABC model for altered values of S.

Fig. 2 is graphically plotted to perceive the impact of Schmidt number on the concentration. Physically it measures the ratio between the viscous diffusion rate and molecular diffusion rate. As value of Sc increases the concentration decreases due to the fact that for larger values of Sc molecular diffusivity drops leading to decrease the boundary layer thickness of concentration neat the plate. It can be seen that the classical model has high concentration than (CF) and (ABC) model in comparison sense whereas in terms of memory effects, we can say that (ABC) model depicts more memory effects than (CF) and classical model.

It can be seen from Fig. 3 that fluid’s concentration decreases over time. However, the classical model depicts higher concentration than (CF) and (ABC) models. Fig. 4 is plotted to see the impact of fractional parameter α on concentration field. Concentration decreases by increasing the value ofα . As α increases, concentration near the plate reduces as well as the diffusion boundary layer thickness. For α → 1 both fractional models (CF) and (ABC) collapse back to the classical formulation. Fig. 5 is plotted to see

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

285

Fig. 18. Velocity profiles of classical, CF and ABC model for altered values of S. Fig. 19. Velocity profiles of classical, CF and ABC model for altered values of Sc.

the effect of chemical reaction Kc on concentration profile. By increasing the value of Kc , fluid’s concentration decreases. Fig. 6 is plotted to show impact of Pr on temperature. It is found that temperature and thermal boundary layer decreases by increasing the value of Pr. Physically it happens for small value of Pr thermal conductivity increases which allows heat to defuse away from the plate rapidly for higher values of Pr. As Pr increases, temperature obtained by (ABC) model rapidly decays to zero asymptotically as compared to (CF) and Classical model. The influence of heat sink constant S on temperature field can be seen

in Fig. 7. By increasing the value of S, temperature and thermal boundary layer thickness decreases. Fig. 8 is plotted to see the influence of fractional parameter α on temperature fields. Temperature of fluid obtained by three models increases as the value of α enhances away from the plate. Further, it is found that thermal boundary layer decreases as a whole. By raising time parameter the fluid temperature increases as shown in Fig. 9. Fig. 10 is plotted to see the effect of fractional parameter α on fluid’s velocity. The effect of α is significantly near the verti-

286

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

Fig. 21. Velocity profiles of our models compared with the velocity profiles of [36].

Fig. 20. Velocity profiles of classical, CF and ABC model for altered values of t.

cal plate. For larger α , velocity near the plate reduces as well as momentum boundary layer thickness. We have observed that as α → 1, both the fractional models (ABC) and (CF) collapse back to the classical formulation. Fig. 11 is plotted to describe the effect of inclination angle γ on fluid’s velocities. It is pertinent to mention that by increasing the value of γ the boundary layer thickness increases and the velocity becomes greater when fluid is moving away from the plate. Fig. 12 is plotted to get the impact of Gr (the thermal Grashof number). Gr is the ratio of buoyancy forces to viscid forces on motion

of fluid which stimulates free or innate convection. For large values of Gr buoyancy force increases and consequently accelerates the flow. Similar behavior can be seen in Fig. 13 Gc . Fig. 14 is plotted for altered values of Kc that is chemical reaction parameter. Fluid’s velocity drops down by raising the value of chemical reaction constant Kc . Physically, the positive values of reaction are termed as destructive while negative reaction labeled as productive or structural. That is why the fluid velocity decreased as Kc increases. Fig. 15 is plotted to see the impact of porous medium permeability on the fluid’s velocity. As Kp increases the fluid velocity decreases near the plate but increases when it outflows from the plate. Impact of magnetic field parameter M is discussed in Fig. 16. As usual, by raising the value of magnetic field, the fluent velocity decreases due to the fact that magnetic flux brings forth resistive forces which slow down the flow of fluid and hence velocity decreases. By increasing the value of Pr the fluid’s velocity steps-down due to the reason that fluent with greater values of Pr have more viscosity and less thermic conductivity, forming the fluid denser to decrement fluid’s velocity as shown in Fig. 17. In Fig. 18 the effects heat sink S can be seen. By increasing the value of S the fluid’s velocity decreases. Velocity decreases by increasing the value of Sc as displayed in Fig. 19. Physically Sc characterizes fluid’s motion as it transmits the relative thickness of hydro dynamic boundary layer as well as mass transfer boundary layer. Fig. 20 is plotted to see the time influence on both fractional and ordinary models and observed that for small time velocity of (ABC) fractional model is greater than (CF) and classical model. For large time the velocity of (ABC) and (CF) increases and have maximum value near the plate. Tables 1–3 are made to check the effects of α on dimensionless temperature filed, velocity domain and concentration filed for classical, (CF) and (ABC) models. By increasing α temperature, concentration and velocity decreases. As α → 1, fractional models collapse back to the classical one. Figs. 21 and 22 (a and b) are drawn to see the validity of our obtained results for velocity, concentration and temperature profiles. It was compared with Nehad et al. [36]. It can be seen from these graphs that by ignoring the effects of chemical reaction parameter Kc , heat sink S, magnetic field M, and taking α →1 our results are identical to those obtained by [36].

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

287

Table 1 (a and b) Effect of fractional parameter α on dimensionless temperature on Classical, (CF) and (ABC) model when Pr = 0.71, t = 12 , S = 0.1. (a) y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T(y, t) Classical model

T(y, t)

T(y, t)

CF

AB

T(y, t) Classical model

T(y, t)

T(y, t)

CF

AB

1 0.973 0.946 0.920 0.895 0.871 0.847 0.824 0.801 0.779 0.757

1 0.930 0.889 0.798 0.799 0.755 0.732 0.712 0.699 0.551 0.515

1 0.918 0.838 0.778 0.708 0.651 0.589 0.545 0.450 0.451 0.432

1 0.973 0.946 0.920 0.895 0.871 0847 0.824 0.801 0.779 0.757

1 0.958 0.918 0.879 0.841 0.804 0.768 0.734 0.701 0.669 0.638

T(y, t) Classical model

T(y, t)

T(y, t)

CF

AB

T(y, t) Classical model

1 0.973 0.946 0.920 0.895 0.871 0.847 0.824 0.801 0.779 0.757

1 0.971 0.943 0.915 0.888 0.861 0836 0.810 0.786 0.762 0.739

1 0.973 0.946 0920 0.895 0.871 0.847 0.824 0.801 0.779 0.757

1 0.972 0.945 0.919 0.893 0.868 0.843 0.819 0.796 0.773 0.751

α = 0.1

T(y, t) Classical model

T(y, t)

T(y, t)

CF

AB

1 0.921 0.848 0.781 0.719 0.662 0.609 0.561 0.516 0.476 0.438

1 0.973 0.946 0.92 0.895 0.871 0.847 0.824 0.801 0.779 0.757

1 0.968 0.937 0.906 0.876 0.847 0.819 0.791 0.764 0.737 0.712

T(y, t)

T(y, t)

T(y, t)

AB

T(y, t) Classical model

T(y, t)

CF

CF

AB

1 0.973 0.946 0.920 0.895 0.871 0.847 0.824 0.801 0.779 0.757

1 0.973 0.946 0.920 0.895 0.871 0.847 0.826 0.801 0.779 0.757

α = 0.2

α = 0.4 1 0.936 0.877 0.821 0.768 0.719 0.673 0.630 0.589 0.551 0.516

(b) y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

α = 0.6 1 0.954 0.910 0.868 0.828 0.789 0.752 0.717 0.683 0.651 0.620

α = 0.8 1 0.972 0.945 0.919 0.893 0.868 0.843 0.819 0.796 0.773 0.751

α = 0.99 1 0.973 0.946 0.920 0.895 0.870 0.84 0.822 0.799 0.777 0.755

Table 2 (a and b) Effect of fractional parameter α on dimensionless concentration on classical, (CF) and (ABC) model when Kc = 0.5, t = 3, Sc = 0.22. (a) y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

C(y, t) Classical model

C(y, t)

C(y, t)

CF

AB

1 0.967 0.935 00.903 0.873 0.844 0.816 0.788 0.762 0.736 0.711

1 0.945 0.897 0.845 0.799 0.765 0.719 0.685 0.652 0.602 0.589

α =0.1 1 0.945 0.897 0.845 0.799 0.765 0.719 0.685 0.652 0.602 0.589

C(y, t) Classical model

C(y, t)

C(y, t)

CF

AB

C(y, t) Classical model

C(y, t)

C(y, t)

CF

AB

1 0.967 0.935 0.903 0.873 0.844 0.816 0.788 0.762 0.736 0.711

1 0.952 0.905 0.861 0.820 0.780 0.742 0.706 0.671 0.638 0.607

1 0.945 0.894 0.845 0.799 0.755 0.714 0.675 0.638 0.604 0.571

1 0.967 0.935 0.903 0.873 0.844 0.816 0.788 0.762 0.736 0.711

1 0.959 0.919 0.880 0.843 0.808 0.774 0.741 0.709 0.679 0.650

C(y, t) Classical model

C(y, t)

C(y, t)

CF

AB

1 0.967 0.935 0.903 0.873 0.844 0.816 0.788 0.762 0.736 0.711

1 0.967 0.934 0.903 0.873 0.844 0.816 0.788 0.762 0.736 0.711

α =0.2

α =0.4 1 0.949 0.901 0.855 0.812 0.771 0.732 0.694 0.659 0.626 0.594

(b) y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

C(y, t) Classical model

C(y, t)

C(y, t)

CF

AB

1 0.967 0.935 0.903 0.873 0.844 0.816 0.788 0.762 0.736 0.711

1 0.964 0.928 0.894 0.861 0.829 0.798 0.768 0.739 0.711 0.683

α = 0.6 1 0.955 0.913 0.872 0.833 0.795 0.759 0.725 0.692 0.661 0.631

C(y, t) Classical model

C(y, t)

C(y, t)

CF

AB

1 0.967 0.935 0.903 0.873 0.844 0.816 0.788 0.762 0.736 0.711

1 0.966 0.933 0.901 0.870 0.840 0.810 0.782 0.755 0.728 0.703

α = 0.8 1 0.962 0.925 0.890 0.856 0.823 0.791 0.761 0.731 0.703 0.675

α = 0.99 1 0.967 0.934 0.903 0.873 0.843 0.815 0.787 0.761 0.735 0.710

288

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289 Table 3 (a and b): Effect of fractional parameter α on dimensionless velocity on classical, (CF) and (ABC) model when Pr = 0.71, M = 0.2, Kc = 0.5, Gr = Gc = 2, K p = 0.2, t = 20, S = 0.2. (a) y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

u(y, t) Classical model

u(y, t)

u(y, t)

u(y, t) Classical model

u(y, t)

u(y, t)

CF

AB

CF

AB

1 0.865 0.756 0.668 0.596 0.538 0.489 0.448 0.414 0.386 0.361

1 0.842 0.715 0.599 0.554 0.489 0.480 0.395 0.312 0.285 0.256

1 0.842 0.715 0.599 0.554 0.489 0.480 0.395 0.312 0.285 0.256

1 0.865 0.756 0.668 0.591 0.531 0.481 0.440 0.405 0.375 0.349

1 0.864 0.754 0.664 0.591 0.531 0.481 0.440 0.405 0.375 0.349

u(y, t) Classical model

u(y, t)

u(y, t)

CF

AB

u(y, t) Classical model

1 0.865 0.756 0.668 0.596 0.538 0.489 0.448 0.414 0.386 0.361

1 0.865 0.756 0.668 0.596 0.537 0.489 0.448 0.414 0.385 0.360

1 0.865 0.756 0.668 0.596 0.538 0.489 0.448 0.414 0.386 0.361

1 0.865 0.756 0.668 0.596 0.538 0.489 0.448 0.414 0.385 0.361

α = 0.1

u(y, t) Classical model

u(y, t)

u(y, t)

CF

AB

1 0.839 0.710 0.607 0.523 0.455 0.399 0.353 0.315 0.282 0.255

1 0.865 0.756 0.668 0.596 0.538 0.489 0.448 0.414 0.386 0.361

1 0.865 0.756 0.668 0.596 0.537 0.488 0.447 0.413 0.384 0.359

u(y, t)

u(y, t)

u(y, t)

AB

u(y, t) Classical model

u(y, t)

CF

CF

AB

1 0.865 0.756 0.668 0.596 0.538 0.489 0.448 0.414 0.386 0.361

1 0.865 0.756 0.668 0.596 0.538 0.489 0.448 0.414 0.386 0.361

α = 0.2

α = 0.4 1 0.850 0.730 0.632 0.553 0.488 0.434 0.390 0.353 0.321 0.294

(b) y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

α = 0.6 1 0.860 0.747 0.655 0.580 0.519 0.468 0.425 0.390 0.359 0.333

α = 0.8 1 0.864 0.754 0.665 0.593 0.533 0.484 0.443 0.409 0.379 0.354

α = 0.99 1 0.865 0.756 0.668 0.596 0.537 0.489 0.448 0.414 0.385 0.361

Fig. 22. (a) Concentration profiles of our models compared with the concentration profile of [36]. (b) Temperature profiles of our models compared with the temperature profile of [36].

8. Conclusion In the present study the comparison between ordinary fluid model, fractional models i.e. (CF) and (ABC) is drawn graphically as well as analytically to observe that how the non-integer order time fractional parameter α controls the fluid flow. Caputo– Fabrizio fractional derivative operator has non singular local exponential kernel which describes velocity, concentration and temperature filed at previous stages. Atangana– Baleanu fractional derivative operator has Mittag Leffler kernel which is non singular as well as non local and it can describe the full memory effect for a given system. Since Mittag Leffler function is the generalization

of exponential function it can describe the power and exponential decay at the same time. It is noted the fractional fluid model with (ABC) has shown more memory effect on concentration, temperature and velocity fields than CF and ordinary models. The key findings of our work are: (i) Temperature, concentration and velocities of the physical model studied with fractional derivatives of (CF) and (ABC) are found to be decrease near the plate for large values of fractional parameter α . (ii) Thermal diffusion and momentum boundary layer thickness are also decrease and the fractional model ultimately collapse back to the classical formulation when α → 1.

M.A. Imran, M. Aleem and M.B. Riaz et al. / Chaos, Solitons and Fractals 118 (2019) 274–289

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