- Email: [email protected]
Application of fractal fractional derivative of power law kernel (FFP0Dxα,β) to MHD viscous fluid flow between two plates
Chaos, Solitons and Fractals 134 (2020) 109691
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Application of fractal fractional derivative of power law kernel (F F P 0 Dαx ,β ) to MHD viscous fluid flow between two plates M.A. Imran Department of Mathematics, University of Management and Technology, Lahore, Pakistan
a r t i c l e
i n f o
Article history: Received 31 December 2019 Revised 9 February 2020 Accepted 12 February 2020
Keywords: Fractal fractional derivative Viscous fluid Power law kernel Couette flow
a b s t r a c t In this problem, I have studied the application of newly introduced fractal fractional operators with power law kernel in fluid dynamics. We Considered the MHD viscous fluid flow between two plates such that the upper plate is in motion with constant velocity while the lower plate is at rest. The governing equation developed from the problem can be formulated withe fractal fractional derivative operator with power law kernel. The proposed fractal fractional model can be solved by means of Laplace transform technique and obtained exact solutions. To see the impact of magnetic field M, fractional α as well as fractal parameter β on the fluid velocity field, we plotted some graphs through MathCad software and presented in the graphical section. As a result, we found that for larger values of α and β , a decay in velocity of the fluid was observed. Further, fractal fractional model more slow down the velocity of the model in comparison of fractional only. Therefore, a combined approach of fractal fractional explains the memory of the function better than fractional only. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction Magnetohydrodynamic is a physical-mathematical framework that concerns the dynamics of magnetic fields electrically conducting fluids,e.g. in plasma and liquid metals. The word Magnetohydrodynamics is comprised of the words magneto-meaning magnetic, hydro-meaning water (and liquid) and dynamics referring to the movement. MHD applied to astrophysics, including stars, the interplanetary medium (space and the planets), and possibly within the interstellar medium (space between the stars) and jets. Most astrophysical system are not in local thermal equilibrium and therefore require and additional kinematic treatment to describe all the phenomena within the system. In microfluidics, MHD is studies as a fluid pump for producing a continuous, non-pulsating flow in a complex micro-channel design. A lot of work on MHD has been done by different researchers for example Imran et al. [1] study the exact solutions of Walter’s-B fluid in the presence of radiation and chemical reaction. Khan et al. [2] discussed the coupled heat and mass transfer for an inclined plate using Laplace transform and fractional derivative. Many others applications of MHD in different branches can be seen in [3–10]. Sheikh et al. [11] studied heat and mass transfer of Casson fluid with new modeling of fractional derivatives and found exact solutions with Laplace transform method. Ali et al. [12] discussed
E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2020.109691 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
the thermal radiation effect for different shapes of nanofluids with fractional derivative. Ali et al. [13] investigated the magnetic field influence on blood flow of a Casson fluid with the help of fractional differential operators. Sheikh et al. [14] found the comparison between two solutions obtained with Caputo-Fabrizio and Atangana–Baleanu fractional derivatives. More recent applications of different fractional operators in variety of area can be found in the references [15–27]. The classical laws (Darcy’s, Fourier’s Law and Fick’s law) are not appropriate to the physical model using non-standard derivatives because they are based on Euclidean geometry. Its not possible to apply these derivatives to the media of fractal like porous media and aquifers exhibiting the fractal properties. Also, local derivative is not so common in application to more complicated problems exhibiting fractal behaviors. In (2012) Jan et al. [28] modeled air permeability in hierarchic porous media with fractal derivatives. Kanno et al. [29] gave the representation of random walk in fractal space time derivative. Chen et al. [30,31] investigated the anomalous diffusion modeling by fractal fractional derivative and space time fabric diffusion equation. In [32], (2017) Atangana gave the connection between fractal and fractional derivatives to predict the complex situations in real life problems. This paper has open new horizon of modeling real world problems in complex situations. Further, in (2019), Atangana et al. [33–35] gave the detail properties of newly introduced operators and a numerical sachem associated to solve fractal fractional differential equations. Also introduced the fractal fractional
2
M.A. Imran / Chaos, Solitons and Fractals 134 (2020) 109691
relations of power law kernel, exponential kernel, Dirac delta and generalized Mittage–Leffler kernels respectively. Couette flow in fluid dynamics is a fundamental problem, when fluid motion is caused by the motion of upper plate with constant velocity taking lower plate at rest. MHD effects is also considered in this problem. No doubt, there is a lot of study present in the literature about different motions of Couette flow. But no result was published yet regarding fractal fractional approach. By motivating the above newly introduced fractal fractional operators, we will develop a fractal fractional Couette flow model in fluid dynamics with power law kernel. The exact solutions in series form can be obtained using Laplace transform. To see the impact of fractional α as well as fractal parameter β on the fluid velocity field, we plotted some graphs through MathCad software and presented in the graphical section. Fig. 1. Geometry of the physical model.
2. Some basic definitions 2.1. Definition We assume that y(x) is continuous in opened interval (a, b), if y is fractal differentiable on (a, b) with order (β ) then, the fractal fractional derivative of y of order (α ) in Riemann–Liouville sense with power law is presented as [32]: FFP
α β 0 Dx , y ( x ) =
1 d (1 − α ) dxβ
a
x
y(t )(x − t )
−α
dt, 0 < α , β ≤ 1, (1)
where
dy(t ) y(x ) − y(t ) = lim . x→t dt β xβ − t β
(2)
Reynolds number is very small and negligible. Also, we have assumed that there is no pressure gradient in the flow direction and neglecting the effect of body force the governing equation for the flow along with initial and boundary conditions are [36]
ρ
∂ V (y, t ) ∂ 2V (y, t ) =μ − σ B2o V (y, t ), y, t > 0, ∂t ∂ y2
and associated initial conditions are
V (y, 0 ) = 0, V (0, t ) = 0, V (L, t ) = Vo,
0 ≤ y ≤ L.
(6)
In order to make the problem free from flow regime, we introduce the dimensionless variables from Eq. (6)
u∗ =
2.2. Definition
(5)
tV 2 V Voy , t ∗ = 0 , y∗ = V0 ν ν
(7)
into Eqs. (5) and (6) we get after ignoring the star notation Usual Laplace transform: Let y(x) be defined for (x ≥ 0). The Laplace transform of y(x) defined by Y(s) or L{y(x)}, is an integral transform presented by the Laplace integral
L {y ( x ) } = Y ( s ) =
∞
0
exp(−sx ) f (x )dx.
(3)
∂ u(y, t ) ∂ 2 u(y, t ) = − Mu(y, t ), ∂t ∂ y2 where M =
(8)
σ B2o ν , is the magnetic field parameter. ρV02
u(y, 0 ) = 0, u(0, t ) = 0, u(1, t ) = 1.
(9)
2.3. Definition
4. Solution of Fractal fractional model of Couette flow
Fractal Laplace transform: Let f be continuous on opened interval I, the fractal-Laplace transform of order α defined as [32]
In this section, we developed fractal fractional model of Eq. (8) we have
F α Lp
{y ( x ) } = Y ( s ) =
∞ 0
exp(−sx )xα −1 f (x )dx.
(4)
Clearly for α → 1 we recovered the usual Laplace transform. We have the Laplace transform of this derivative of Caputo fractional derivative as [8],
L
C
α α −1 . 0 Dx y (x ) = (sL{y (x )} − y (0 ) )s
C
Dtα u(y, t ) =
β t β −1
u(y, 0 ) −α ∂ 2 u(y, t ) − Mu(y, t ) − t (1 − α ) ∂ y2
(10)
By applying the Laplace transform to Eqs. (9) and (10) we have
sα u(y, s ) − u(y, 0 ) = −
3. Application of fractal fractional derivative to Couette flow
β (β )s
−β
∂ 2 u(y, s ) − Mu(y, s ) ∂ y2
u(y, 0 ) α −1 s (1 − α ) (1 − α )
(11)
and Consider a unsteady laminar flow of incompressible Newtonian fluid between two infinite parallel plate which are kept at a distance L apart in oxyz coordinate system as shown in Fig. 1. We assuming that upper plate is moving with constant velocity Vo in the direction of x-axis while lower plate is at rest and y-axis perpendicular to it and there is fluid flow properties contribution in the z-Axis. A transverse Magnetic field of strength Bo is applied to the upper plate perpendicularly by assuming that the magnetic
u(y, 0 ) = 0, u(0, s ) = 0, u(1, s ) =
1 s
The solution of Eq. (11) subject to conditions (12) we get
u(y, s ) =
(12)
√ √ γ γ 1 1 e−y a(s +M) ey a ( s +M ) − . √ √ √ √ s e− a ( sγ +M ) − e a ( sγ +M ) s e− a ( sγ +M ) − e a ( sγ +M ) (13)
M.A. Imran / Chaos, Solitons and Fractals 134 (2020) 109691
Fig. 2. Effects of fractional parameter α on velocity when t = 0.1, M = 0.01, β = 0.1.
n=0
−
Fig. 3. Effects of fractal parameter β on velocity, when t = 0.1, α = 0.1, M = 0.01.
4.3. Solution of classical Couette flow
1 where γ = α + β and a = β ( β ) . or ∞ 1 u(y, s ) = s
3
For α = 1, β = 1 and M=0, we get the result for classical Couette flow
∞ p2 ((2n + 1 − y )) p1 (a(sγ + M )) 2 p1 !
p1 =0
∞ p2 (−(2n + 1 − y )) p2 (a(sγ + M )) 2 p2 !
u(y, t ) =
(14)
∞ ∞ ∞ ∞ (y − 2n − 1 ) p −p −(y + 2n + 1 )q −q t − t . p!(1 − p) q!(1 − q ) n=0 p=0
n=0 q=0
(18)
p2 =0
The inverse Laplace transform of Eq. (14) is given by
5. Numerical results and discussion
1 ∞ ∞ ∞ ( y − 2n − 1 ) p1 a 2 M q1 u(y, t ) = p1 ! q1 ! p
n=0 p1 =0 q1 =0
γ p1
×
( 21 + q1 ) t q1 − 2 − γ p1 (1 + q1 − 2 ) ( p21 )
−
2 ∞ ∞ ∞ (−(y + 2n + 1 )) p2 a 2 Mq2 p2 ! q2 !
p
p
n=0 p2 =0 q2 =0 γ p2
( 22 + q2 ) t q2 − 2 × . γ p2 (1 + q2 − 2 ) ( p22 ) p
(15)
4.1. Fractal Fractional Solution without MHD If we take M=0 in the Eq. (13) and apply the inverse Laplace transform, we have ∞ ∞ ∞ ( y − 2n − 1 ) p1 ( 2 p3 ) p4
u(y, t ) =
p1 ! p4 !β (β ))
p1 =0 p2 =0 p3 =0
−
∞
∞
∞
p1 + p4 2
t ( α +β ) ( 2 − 2 ) (α + β )(− p21 −
− ( 2n + 1 + y ) p2 ( 2 p3 ) p4
p2 =0 p3 =0 p4 =0
p2 ! p4 !β (β )
p2 + p4 2
p1
t − ( α +β ) (
Fractal fractional model of MHD Couette flow in fluid dynamics is developed with power law kernel. Exact solutions are obtained via Laplace transform and expressed in terms of series. We have investigated the fractal and fractional parameters effect on the fluid velocity through some graphs and presented graphically. The α and β variations on velocity field is presented in Figs. 2 and 3. The fluid velocity reduces as we enhance the values of fractional parameters α and β respectively. Decay in the velocity profile is due to the power law kernel because it explains the memory of the function at certain time t. Upon further noticed that for greater values of fractal parameter β and fractional parameter α momentum boundary layer increases. Similar behavior can be observed in Fig. 3
p4
p2 2
+
p4 2
(α + β )( p22 −
p4 2
)
) p4 2
)
.
(16) 4.2. Solution of Fractional Couette Flow For β = 1 and M=0, we get the result for fractional Couette flow 1 4 ∞ ∞ ∞ ((y − 2n − 1 )) p1 (2 p3 ) p4 t (α +1)( 2 − 2 ) p1 p1 ! p4 ! (α + 1 )(− 2 − p =0 p =0 p =0 p
u(y, t ) = −
1
2
3
∞
∞
∞
p2 =0 p3 =0 p4 =0
p
p4 2
)
−(y + 2n + 1 )) p2 (2 p3 ) p4 t −(α +1)( + ) . p2 ! p4 ! (α + 1 )( p22 − p24 ) p2 2
p4 2
(17)
Fig. 4. Effects of magnetic field M on velocity when t = 2.5, β = 0.1, α = 0.1.
4
M.A. Imran / Chaos, Solitons and Fractals 134 (2020) 109691 Table 1 Statistically analysis of combined fractal fractional parameters on fluid velocity. y
u(y, t )α = β = 0.3, t = 0.5
u(y, t )α = β = 0.6, t = 0.5
u(y, t )α = β = 0.8, t = 0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 0.087 0.174 0.263 0.355 0.449 0.548 0.652 0.761 0.877 1
0 0.111 0.222 0.331 0.438 0.542 0.642 0.738 0.83 0.917 1
0 0.12 0.248 0.385 0.52 0.641 0.743 0.828 0.898 0.955 1
Table 2 Statistically comparison between fractional and fractal fractional for different time. Fractional u(y, t )t = 0.5
Fractional u(y, t )t = 1
Fractal fractional u(y, t )t = 1.5
Fractal fractional u(y, t )t = 0.5
Fractal fractional u(y, t )t = 1
Fractal fractional u(y, t )t = 1.5
0 0.104 0.208 0.312 0.414 0.516 0.616 0.715 0.812 0.907 1
0 0.102 0.204 0.306 0.407 0.508 0.608 0.707 0.806 0.903 1
0 0.101 0.202 0.303 0.403 0.504 0.604 0.704 0.803 0.902 1
0 0.084 0.17 0.257 0.347 0.441 0.539 0.643 0.753 0.872 1
0 0.086 0.173 0.262 0.353 0.448 0.546 0.65 0.759 0.876 1
0 0.087 0.175 0.265 0.356 0.451 0.55 0.653 0.762 0.877 1
Fig. 5. Effects of time t on velocity when α = 0.4, β = 0.5, M = 0.001.
Fig. 4 is plotted for altered values of M, the magnetic field parameter. As expected, by maximizing the value of M, abbreviates fluent velocity unusually. 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, hence causing an abbreviation in its velocity. Fig. 5 is plotted to see the influence of time on fluid’s velocity. It is clear from the graph that by increasing the value of time t the fluid’s velocity as well as the boundary layer thickness increases. Fig. 6 is plotted to see the comparison approach between fractional and combined fractal fractional model and found that velocity is smaller for fractal fractional model. Further, as fractional operators involve with kernels (singular or non-singular) used to explain the memory of the function. In the present model one additional parameter β named as fractal parameter introduced by
Fig. 6. Comparison of fractional velocity and fractal fractional velocity.
giving relation between fractional and fractal derivative. When we used them together to their effect on the fluid flow, can explain memory better than fractional only. To see statistically approach of combined fractal fractional parameters for small time table one drawn. In the Table 1 interesting results were found that overall fluid behavior can enhanced for greater values of α and β respectively. Table 2 is drawn to see comparison for small and large time between the fractional and fractal fractional and found that fractal fractional model has smaller velocity than fractional one. 6. Conclusions This paper deals with the fractal and fractional derivative application in Couette flow with power law kernel. Laplace transform
M.A. Imran / Chaos, Solitons and Fractals 134 (2020) 109691
used to obtain the exact solutions for the Couette flow in terms of series. The main outcomes of the present study are: • Fluid velocity field decay with the larger values of fractional α and fractal β parameters respectively. • Velocity is decreasing function for small values of time t. • For same values of fractal and fractional parameter by taking large values of time fluid velocity can be enhanced. • By increasing magnetic field parameter M, velocity decreases due to drag force opposing the flow. • Fractal fractional model exhibits better memory in comparison of fractional model. Declaration of Competing Interest I hereby certify that the information on this manuscript is true and further authors declare that there is no conflict of interest for the paper titled given below. CRediT authorship contribution statement M.A. Imran: Methodology, Software, Writing - original draft, Visualization, Investigation, Validation, Writing - review & editing. Acknowledgments Author is greatly obliged to the reviewers for their positive remarks and suggestions to improve the present form of the paper, my wife to support me and also thankful to the University of Management and Technology Lahore, Pakistan for facilitating and supporting the research work. References [1] Imran MA, Aleem M, Riaz MB. Exact analysis of MHD walters’-b fluid flow with non-singular fractional derivatives of Caputo–Fabrizio in the presence of radiation and chemical reaction. J Polym Sci Eng 2018;1:599. [2] Khan I, Ahmad M, Shah NA. Effects of non-integer order time fractional derivative on coupled heat and mass transfer of MHD viscous fluid over an infinite inclined plane with heat absorption. Int J Innov Res Sci Eng Technol 2017:6. [3] Nazar M, Ahmad M, Imran MA. Double convection of heat and mass transfer flow of MHD generalized second grade fluid over an exponentially accelerated infinite vertical plate with heat absorption. J Math Anal 2017;8:1–10. [4] Shah NA, Khan I, Aleem M, Imran MA. Influence of magnetic field on double convection problem of fractional viscous fluid over an exponentially moving vertical plate: new trends of Caputo time-fractional derivative model. Adv Mech Eng 2019;11(7):1–11. [5] Hussanan A, Khan I, Hashim H, Salleh MZ. Unsteady MHD flow of some nanofluids past an accekerated vertical plate embedded in porous medium. J Teknologi 2016;78(2). 121-6 [6] Aleem M, Imran MA, Shaheen A, Khan I. MHD influence on different water based nanofluids (TiO2 , Al2 O3 , CuO) in porous medium with chemical reaction and newtonian heating. Chaos Solitons Fractals 2020;130:109437. [7] Imran MA, Shah NA, Aleem M, Khan I. Heat transfer analysis of fractional second-grade fluid subject to newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: a comparison. Eur Phys J Plus 2017;132:340. [8] Imran MA. Fractional mechanism with power law (singular) and exponential (non-singular) kernels and its applications in bio heat transfer model. Int. J. Heat Technol. 2019;37:846–52. [9] Imran MA, Shah NA, Khan I, Aleem M. Applications of non-integer Caputo time fractional derivatives to natural convection flow subject to arbitrary velocity and newtonian heating. Neural Comput Appl 2018;30(5):1589–99.
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[10] Imran MA, Shah NA, Aleem M, Khan I. Heat transfer analysis of fractional second-grade fluid subject to newtonian heating with Caputo and Caputo–Fabrizio fractional derivatives: a comparison. Eur Phys J Plus 2017;132(340):1–19. [11] Sheikha NA, Ching DLC, Khan I, Kumar D, S Nisar K. A new model of fractional casson fluid based on generalized fick’s and fourier’s laws together with heat and mass transfer. Alexandria Engineering Journal 2019. https://doi.org/ 10.1016/j.aej.2019.12.023 [12] Ali F., Gohar M., Khan I., Sheikh N.A.. S. a. a. jan thermal radiation and magnetic field effects on different channel flows of CNTs Brinkman-type nanofluids with water, kerosene and engine-oil. International Journal of Computational Analysis, Vol (1), No. (2), pp. 01–17. [13] Ali F, Sheikh NA, Khan I, Saqib M. Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: a fractional model. J Magn Magn Mater 2017;423:327–36. [14] Sheikh NA, Ali F, Saqib M, Khan I, Alam SA, SaidAlghamdi SAM. Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results Phys 2017;7:789–800. [15] Imran MA, Khan I, Ahmad M, Shah NA, Nazar M. Heat and mass transport of differential type fluid with non-integer order time-fractional Caputo derivatives. J Mol Liq 2017;229:67–75. [16] Atangana A, Jain S. Models of fluid flowing in non-conventional media: new numerical analysis. Discret Contin Dyn Syst-S 2019:757–63. [17] Goufo EFD. A Atangana dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discret Contin Dyn Syst-S 2019:104–14. [18] Altaf KM, Atangana A. Dynamics of ebola disease in the framework of different fractional derivatives. Entropy 2019;21(3):303. [19] Owolabi KM, Atangana A. On the formulation of Adams–Bashforth scheme with Atangana–Baleanu–Caputo fractional derivative to model chaotic problems. Chaos 2019;29(2):23111. [20] Atangana A, Mekkaoui T. Capturing complexities with composite operator and differential operators with non-singular kernel. Chaos 2019;29(2):23103. [21] Bonyah E, Atangana A, Elsadany AA. A fractional model for predator-prey with omnivore. Chaos 2019;29(1):13136. [22] Gómez-Aguilar JF, Atangana A. Time-fractional variable-order telegraph equation involving operators with mittag-leffler kernel. J Electromagn Waves Appl 2019;33(2):165–77. [23] Fahd J, Abdeljawad TA. A modified laplace transform for certain generalized fractional operators. Results Nonlinear Anal 2018;2:88–98. [24] Singh J, Kumar D, Hammouch Z, Atangana A. A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Appl Math Comput 2018;316:504–15. [25] Hammouch Z, Mekkaoui T. Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system. Complex Intell Syst 2018;4:251. [26] Mekkaoui T, Hammouch Z, Belgacem F, ElAbbassi A. Fractional calculus with power law: The cradle of our ancestors. Fract Dyn 2015. [27] Yang XJ. Advanced local fractional calculus and its applications. New York, NY, USA: World Science; 2012. [28] Fan J, He J. Fractal derivative model for air permeability in hierarchic porous media. Abstr Appl Anal 2012. [29] Kanno R. Representation of random walk in fractal space-time. Physica A 1998;248. 165-75 [30] Chen W, Sun HG, Zhang X, Korosak D. Anomalous diffusion modeling by fractal and fractional derivatives. Comput Math Appl 2010;59(5). 1754-8 [31] Chen W. Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractal 2006;28. 923-9 [32] Atangana A. Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals 2017;0 0 0:1–11. [33] Atangana A, Khan MA. Validity of fractal derivative to capturing chaotic attractors. Chaos Solitons Fractals 2019;126:50–9. [34] Atangana A, Shafiq A. Differential and integral operators with constant fractional order and variable fractional dimension. Chaos Solitons Fractals 2019;127:226–43. [35] Atangana A, Qureshi S. Modeling attractors of chaotic dynamical systems with fractal-fractional operators. Chaos Solitons Fractals 2019;123:320–37. [36] Barhoi A, Hazarika GC, Dutta P. Numerical study of fuzzified boundary value problem for Couette type flow of fluid mechanics. Ann Pure Appl Math 2018;16(2):373–84.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Application of fractal fractional derivative of power law kernel (F F P 0 Dαx ,β ) to MHD viscous fluid flow between two plates M.A. Imran Department of Mathematics, University of Management and Technology, Lahore, Pakistan
a r t i c l e
i n f o
Article history: Received 31 December 2019 Revised 9 February 2020 Accepted 12 February 2020
Keywords: Fractal fractional derivative Viscous fluid Power law kernel Couette flow
a b s t r a c t In this problem, I have studied the application of newly introduced fractal fractional operators with power law kernel in fluid dynamics. We Considered the MHD viscous fluid flow between two plates such that the upper plate is in motion with constant velocity while the lower plate is at rest. The governing equation developed from the problem can be formulated withe fractal fractional derivative operator with power law kernel. The proposed fractal fractional model can be solved by means of Laplace transform technique and obtained exact solutions. To see the impact of magnetic field M, fractional α as well as fractal parameter β on the fluid velocity field, we plotted some graphs through MathCad software and presented in the graphical section. As a result, we found that for larger values of α and β , a decay in velocity of the fluid was observed. Further, fractal fractional model more slow down the velocity of the model in comparison of fractional only. Therefore, a combined approach of fractal fractional explains the memory of the function better than fractional only. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction Magnetohydrodynamic is a physical-mathematical framework that concerns the dynamics of magnetic fields electrically conducting fluids,e.g. in plasma and liquid metals. The word Magnetohydrodynamics is comprised of the words magneto-meaning magnetic, hydro-meaning water (and liquid) and dynamics referring to the movement. MHD applied to astrophysics, including stars, the interplanetary medium (space and the planets), and possibly within the interstellar medium (space between the stars) and jets. Most astrophysical system are not in local thermal equilibrium and therefore require and additional kinematic treatment to describe all the phenomena within the system. In microfluidics, MHD is studies as a fluid pump for producing a continuous, non-pulsating flow in a complex micro-channel design. A lot of work on MHD has been done by different researchers for example Imran et al. [1] study the exact solutions of Walter’s-B fluid in the presence of radiation and chemical reaction. Khan et al. [2] discussed the coupled heat and mass transfer for an inclined plate using Laplace transform and fractional derivative. Many others applications of MHD in different branches can be seen in [3–10]. Sheikh et al. [11] studied heat and mass transfer of Casson fluid with new modeling of fractional derivatives and found exact solutions with Laplace transform method. Ali et al. [12] discussed
E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2020.109691 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
the thermal radiation effect for different shapes of nanofluids with fractional derivative. Ali et al. [13] investigated the magnetic field influence on blood flow of a Casson fluid with the help of fractional differential operators. Sheikh et al. [14] found the comparison between two solutions obtained with Caputo-Fabrizio and Atangana–Baleanu fractional derivatives. More recent applications of different fractional operators in variety of area can be found in the references [15–27]. The classical laws (Darcy’s, Fourier’s Law and Fick’s law) are not appropriate to the physical model using non-standard derivatives because they are based on Euclidean geometry. Its not possible to apply these derivatives to the media of fractal like porous media and aquifers exhibiting the fractal properties. Also, local derivative is not so common in application to more complicated problems exhibiting fractal behaviors. In (2012) Jan et al. [28] modeled air permeability in hierarchic porous media with fractal derivatives. Kanno et al. [29] gave the representation of random walk in fractal space time derivative. Chen et al. [30,31] investigated the anomalous diffusion modeling by fractal fractional derivative and space time fabric diffusion equation. In [32], (2017) Atangana gave the connection between fractal and fractional derivatives to predict the complex situations in real life problems. This paper has open new horizon of modeling real world problems in complex situations. Further, in (2019), Atangana et al. [33–35] gave the detail properties of newly introduced operators and a numerical sachem associated to solve fractal fractional differential equations. Also introduced the fractal fractional
2
M.A. Imran / Chaos, Solitons and Fractals 134 (2020) 109691
relations of power law kernel, exponential kernel, Dirac delta and generalized Mittage–Leffler kernels respectively. Couette flow in fluid dynamics is a fundamental problem, when fluid motion is caused by the motion of upper plate with constant velocity taking lower plate at rest. MHD effects is also considered in this problem. No doubt, there is a lot of study present in the literature about different motions of Couette flow. But no result was published yet regarding fractal fractional approach. By motivating the above newly introduced fractal fractional operators, we will develop a fractal fractional Couette flow model in fluid dynamics with power law kernel. The exact solutions in series form can be obtained using Laplace transform. To see the impact of fractional α as well as fractal parameter β on the fluid velocity field, we plotted some graphs through MathCad software and presented in the graphical section. Fig. 1. Geometry of the physical model.
2. Some basic definitions 2.1. Definition We assume that y(x) is continuous in opened interval (a, b), if y is fractal differentiable on (a, b) with order (β ) then, the fractal fractional derivative of y of order (α ) in Riemann–Liouville sense with power law is presented as [32]: FFP
α β 0 Dx , y ( x ) =
1 d (1 − α ) dxβ
a
x
y(t )(x − t )
−α
dt, 0 < α , β ≤ 1, (1)
where
dy(t ) y(x ) − y(t ) = lim . x→t dt β xβ − t β
(2)
Reynolds number is very small and negligible. Also, we have assumed that there is no pressure gradient in the flow direction and neglecting the effect of body force the governing equation for the flow along with initial and boundary conditions are [36]
ρ
∂ V (y, t ) ∂ 2V (y, t ) =μ − σ B2o V (y, t ), y, t > 0, ∂t ∂ y2
and associated initial conditions are
V (y, 0 ) = 0, V (0, t ) = 0, V (L, t ) = Vo,
0 ≤ y ≤ L.
(6)
In order to make the problem free from flow regime, we introduce the dimensionless variables from Eq. (6)
u∗ =
2.2. Definition
(5)
tV 2 V Voy , t ∗ = 0 , y∗ = V0 ν ν
(7)
into Eqs. (5) and (6) we get after ignoring the star notation Usual Laplace transform: Let y(x) be defined for (x ≥ 0). The Laplace transform of y(x) defined by Y(s) or L{y(x)}, is an integral transform presented by the Laplace integral
L {y ( x ) } = Y ( s ) =
∞
0
exp(−sx ) f (x )dx.
(3)
∂ u(y, t ) ∂ 2 u(y, t ) = − Mu(y, t ), ∂t ∂ y2 where M =
(8)
σ B2o ν , is the magnetic field parameter. ρV02
u(y, 0 ) = 0, u(0, t ) = 0, u(1, t ) = 1.
(9)
2.3. Definition
4. Solution of Fractal fractional model of Couette flow
Fractal Laplace transform: Let f be continuous on opened interval I, the fractal-Laplace transform of order α defined as [32]
In this section, we developed fractal fractional model of Eq. (8) we have
F α Lp
{y ( x ) } = Y ( s ) =
∞ 0
exp(−sx )xα −1 f (x )dx.
(4)
Clearly for α → 1 we recovered the usual Laplace transform. We have the Laplace transform of this derivative of Caputo fractional derivative as [8],
L
C
α α −1 . 0 Dx y (x ) = (sL{y (x )} − y (0 ) )s
C
Dtα u(y, t ) =
β t β −1
u(y, 0 ) −α ∂ 2 u(y, t ) − Mu(y, t ) − t (1 − α ) ∂ y2
(10)
By applying the Laplace transform to Eqs. (9) and (10) we have
sα u(y, s ) − u(y, 0 ) = −
3. Application of fractal fractional derivative to Couette flow
β (β )s
−β
∂ 2 u(y, s ) − Mu(y, s ) ∂ y2
u(y, 0 ) α −1 s (1 − α ) (1 − α )
(11)
and Consider a unsteady laminar flow of incompressible Newtonian fluid between two infinite parallel plate which are kept at a distance L apart in oxyz coordinate system as shown in Fig. 1. We assuming that upper plate is moving with constant velocity Vo in the direction of x-axis while lower plate is at rest and y-axis perpendicular to it and there is fluid flow properties contribution in the z-Axis. A transverse Magnetic field of strength Bo is applied to the upper plate perpendicularly by assuming that the magnetic
u(y, 0 ) = 0, u(0, s ) = 0, u(1, s ) =
1 s
The solution of Eq. (11) subject to conditions (12) we get
u(y, s ) =
(12)
√ √ γ γ 1 1 e−y a(s +M) ey a ( s +M ) − . √ √ √ √ s e− a ( sγ +M ) − e a ( sγ +M ) s e− a ( sγ +M ) − e a ( sγ +M ) (13)
M.A. Imran / Chaos, Solitons and Fractals 134 (2020) 109691
Fig. 2. Effects of fractional parameter α on velocity when t = 0.1, M = 0.01, β = 0.1.
n=0
−
Fig. 3. Effects of fractal parameter β on velocity, when t = 0.1, α = 0.1, M = 0.01.
4.3. Solution of classical Couette flow
1 where γ = α + β and a = β ( β ) . or ∞ 1 u(y, s ) = s
3
For α = 1, β = 1 and M=0, we get the result for classical Couette flow
∞ p2 ((2n + 1 − y )) p1 (a(sγ + M )) 2 p1 !
p1 =0
∞ p2 (−(2n + 1 − y )) p2 (a(sγ + M )) 2 p2 !
u(y, t ) =
(14)
∞ ∞ ∞ ∞ (y − 2n − 1 ) p −p −(y + 2n + 1 )q −q t − t . p!(1 − p) q!(1 − q ) n=0 p=0
n=0 q=0
(18)
p2 =0
The inverse Laplace transform of Eq. (14) is given by
5. Numerical results and discussion
1 ∞ ∞ ∞ ( y − 2n − 1 ) p1 a 2 M q1 u(y, t ) = p1 ! q1 ! p
n=0 p1 =0 q1 =0
γ p1
×
( 21 + q1 ) t q1 − 2 − γ p1 (1 + q1 − 2 ) ( p21 )
−
2 ∞ ∞ ∞ (−(y + 2n + 1 )) p2 a 2 Mq2 p2 ! q2 !
p
p
n=0 p2 =0 q2 =0 γ p2
( 22 + q2 ) t q2 − 2 × . γ p2 (1 + q2 − 2 ) ( p22 ) p
(15)
4.1. Fractal Fractional Solution without MHD If we take M=0 in the Eq. (13) and apply the inverse Laplace transform, we have ∞ ∞ ∞ ( y − 2n − 1 ) p1 ( 2 p3 ) p4
u(y, t ) =
p1 ! p4 !β (β ))
p1 =0 p2 =0 p3 =0
−
∞
∞
∞
p1 + p4 2
t ( α +β ) ( 2 − 2 ) (α + β )(− p21 −
− ( 2n + 1 + y ) p2 ( 2 p3 ) p4
p2 =0 p3 =0 p4 =0
p2 ! p4 !β (β )
p2 + p4 2
p1
t − ( α +β ) (
Fractal fractional model of MHD Couette flow in fluid dynamics is developed with power law kernel. Exact solutions are obtained via Laplace transform and expressed in terms of series. We have investigated the fractal and fractional parameters effect on the fluid velocity through some graphs and presented graphically. The α and β variations on velocity field is presented in Figs. 2 and 3. The fluid velocity reduces as we enhance the values of fractional parameters α and β respectively. Decay in the velocity profile is due to the power law kernel because it explains the memory of the function at certain time t. Upon further noticed that for greater values of fractal parameter β and fractional parameter α momentum boundary layer increases. Similar behavior can be observed in Fig. 3
p4
p2 2
+
p4 2
(α + β )( p22 −
p4 2
)
) p4 2
)
.
(16) 4.2. Solution of Fractional Couette Flow For β = 1 and M=0, we get the result for fractional Couette flow 1 4 ∞ ∞ ∞ ((y − 2n − 1 )) p1 (2 p3 ) p4 t (α +1)( 2 − 2 ) p1 p1 ! p4 ! (α + 1 )(− 2 − p =0 p =0 p =0 p
u(y, t ) = −
1
2
3
∞
∞
∞
p2 =0 p3 =0 p4 =0
p
p4 2
)
−(y + 2n + 1 )) p2 (2 p3 ) p4 t −(α +1)( + ) . p2 ! p4 ! (α + 1 )( p22 − p24 ) p2 2
p4 2
(17)
Fig. 4. Effects of magnetic field M on velocity when t = 2.5, β = 0.1, α = 0.1.
4
M.A. Imran / Chaos, Solitons and Fractals 134 (2020) 109691 Table 1 Statistically analysis of combined fractal fractional parameters on fluid velocity. y
u(y, t )α = β = 0.3, t = 0.5
u(y, t )α = β = 0.6, t = 0.5
u(y, t )α = β = 0.8, t = 0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 0.087 0.174 0.263 0.355 0.449 0.548 0.652 0.761 0.877 1
0 0.111 0.222 0.331 0.438 0.542 0.642 0.738 0.83 0.917 1
0 0.12 0.248 0.385 0.52 0.641 0.743 0.828 0.898 0.955 1
Table 2 Statistically comparison between fractional and fractal fractional for different time. Fractional u(y, t )t = 0.5
Fractional u(y, t )t = 1
Fractal fractional u(y, t )t = 1.5
Fractal fractional u(y, t )t = 0.5
Fractal fractional u(y, t )t = 1
Fractal fractional u(y, t )t = 1.5
0 0.104 0.208 0.312 0.414 0.516 0.616 0.715 0.812 0.907 1
0 0.102 0.204 0.306 0.407 0.508 0.608 0.707 0.806 0.903 1
0 0.101 0.202 0.303 0.403 0.504 0.604 0.704 0.803 0.902 1
0 0.084 0.17 0.257 0.347 0.441 0.539 0.643 0.753 0.872 1
0 0.086 0.173 0.262 0.353 0.448 0.546 0.65 0.759 0.876 1
0 0.087 0.175 0.265 0.356 0.451 0.55 0.653 0.762 0.877 1
Fig. 5. Effects of time t on velocity when α = 0.4, β = 0.5, M = 0.001.
Fig. 4 is plotted for altered values of M, the magnetic field parameter. As expected, by maximizing the value of M, abbreviates fluent velocity unusually. 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, hence causing an abbreviation in its velocity. Fig. 5 is plotted to see the influence of time on fluid’s velocity. It is clear from the graph that by increasing the value of time t the fluid’s velocity as well as the boundary layer thickness increases. Fig. 6 is plotted to see the comparison approach between fractional and combined fractal fractional model and found that velocity is smaller for fractal fractional model. Further, as fractional operators involve with kernels (singular or non-singular) used to explain the memory of the function. In the present model one additional parameter β named as fractal parameter introduced by
Fig. 6. Comparison of fractional velocity and fractal fractional velocity.
giving relation between fractional and fractal derivative. When we used them together to their effect on the fluid flow, can explain memory better than fractional only. To see statistically approach of combined fractal fractional parameters for small time table one drawn. In the Table 1 interesting results were found that overall fluid behavior can enhanced for greater values of α and β respectively. Table 2 is drawn to see comparison for small and large time between the fractional and fractal fractional and found that fractal fractional model has smaller velocity than fractional one. 6. Conclusions This paper deals with the fractal and fractional derivative application in Couette flow with power law kernel. Laplace transform
M.A. Imran / Chaos, Solitons and Fractals 134 (2020) 109691
used to obtain the exact solutions for the Couette flow in terms of series. The main outcomes of the present study are: • Fluid velocity field decay with the larger values of fractional α and fractal β parameters respectively. • Velocity is decreasing function for small values of time t. • For same values of fractal and fractional parameter by taking large values of time fluid velocity can be enhanced. • By increasing magnetic field parameter M, velocity decreases due to drag force opposing the flow. • Fractal fractional model exhibits better memory in comparison of fractional model. Declaration of Competing Interest I hereby certify that the information on this manuscript is true and further authors declare that there is no conflict of interest for the paper titled given below. CRediT authorship contribution statement M.A. Imran: Methodology, Software, Writing - original draft, Visualization, Investigation, Validation, Writing - review & editing. Acknowledgments Author is greatly obliged to the reviewers for their positive remarks and suggestions to improve the present form of the paper, my wife to support me and also thankful to the University of Management and Technology Lahore, Pakistan for facilitating and supporting the research work. References [1] Imran MA, Aleem M, Riaz MB. Exact analysis of MHD walters’-b fluid flow with non-singular fractional derivatives of Caputo–Fabrizio in the presence of radiation and chemical reaction. J Polym Sci Eng 2018;1:599. [2] Khan I, Ahmad M, Shah NA. Effects of non-integer order time fractional derivative on coupled heat and mass transfer of MHD viscous fluid over an infinite inclined plane with heat absorption. Int J Innov Res Sci Eng Technol 2017:6. [3] Nazar M, Ahmad M, Imran MA. Double convection of heat and mass transfer flow of MHD generalized second grade fluid over an exponentially accelerated infinite vertical plate with heat absorption. J Math Anal 2017;8:1–10. [4] Shah NA, Khan I, Aleem M, Imran MA. Influence of magnetic field on double convection problem of fractional viscous fluid over an exponentially moving vertical plate: new trends of Caputo time-fractional derivative model. Adv Mech Eng 2019;11(7):1–11. [5] Hussanan A, Khan I, Hashim H, Salleh MZ. Unsteady MHD flow of some nanofluids past an accekerated vertical plate embedded in porous medium. J Teknologi 2016;78(2). 121-6 [6] Aleem M, Imran MA, Shaheen A, Khan I. MHD influence on different water based nanofluids (TiO2 , Al2 O3 , CuO) in porous medium with chemical reaction and newtonian heating. Chaos Solitons Fractals 2020;130:109437. [7] Imran MA, Shah NA, Aleem M, Khan I. Heat transfer analysis of fractional second-grade fluid subject to newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: a comparison. Eur Phys J Plus 2017;132:340. [8] Imran MA. Fractional mechanism with power law (singular) and exponential (non-singular) kernels and its applications in bio heat transfer model. Int. J. Heat Technol. 2019;37:846–52. [9] Imran MA, Shah NA, Khan I, Aleem M. Applications of non-integer Caputo time fractional derivatives to natural convection flow subject to arbitrary velocity and newtonian heating. Neural Comput Appl 2018;30(5):1589–99.
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