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Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative
Journal Pre-proof Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative M. Mansha Ghalib, Azhar A. Zafar, M. Bilal Riaz, Z. Hammouch, Khurram Shabbir PII: DOI: Reference:
S0378-4371(19)32178-8 https://doi.org/10.1016/j.physa.2019.123941 PHYSA 123941
To appear in:
Physica A
Received date : 26 September 2019 Revised date : 23 November 2019 Please cite this article as: M.M. Ghalib, A.A. Zafar, M.B. Riaz et al., Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123941. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative
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M. Mansha Ghalib1 , Azhar A. Zafar2 , M. Bilal Riaz3 , Z. Hammouch41 , Khurram Shabbir2 Department of Mathematics and statistics, University of Lahore 54590, Pakistan. 2 Department of Mathematics, GC University, Lahore 54590, Pakistan. 3 Department of Mathematics, University of Management and Technology Lahore 54590, Pakistan. 4∗ FSTE University of Moulay Ismail Errachidia 52000 Morocco. Corresponding Author Email: [email protected]
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Abstract
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This study investigates the unsteady magnetohydrodynamics (MHD) flow of a viscous fluid. The fluid is passing over a vertical plate through porous medium. Additionally conjugate effects of heat and mass transfer with ramped temperatures, slip effect and influence of thermal radiation in the energy equation are taken into account. The dimensionless fractional-order governing equations, in the CaputoFabrizio sense, are solved with the help of Laplace transformation. Moreover, semi analytical technique is used to investigate the velocity field. Some results which present in literature are recovered as limiting cases. Influences of different parameters on the velocity profiles for the case of f (t) = t and f (t) = sin ωt are highlighted. The novelty of the manuscript is the use of the most recent definition of the non integer order derivative operator i.e. Caputo-Fabrizio derivative operator, the use of generalized boundary conditions in terms of general function f (t), from our general results, several particular cases for instance when f (t) is a linear or sinusoidal function could be recovered.
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Keywords Nonsingular-kernel derivative, Unsteady MHD flow, Porous medium, Ramped wall temperature, Closed-form solution. 1
Corresponding Author email:[email protected]
Preprint submitted to Elsevier
December 29, 2019
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1. Introduction
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Heat and mass transfer in case of chemical reaction, evaporation and condensation caused by both temperature difference and concentration difference. The rate of mass transfer depends on the rate of heat transfer. The practical application of heat and mass transfer frequently exist in chemically processed industries such as food processing and polymer production. Free convection flows, force convection flow and mixed convection flow with conjugate effects of heat and mass transfer past a vertical plate have been studied substantially in the literature [3] due to its applications in food processing products and polymer production in various disciplines of engineering and industries, fiber and granular insulation and geothermal system [28], [5], [10]. For most notable articles in this area of research we refer [7], [8], [9], [25], [35] and [36]. Toki et al. [49] have examined the unsteady free convection streams of viscous fluid near a permeable infinite plate with time dependent heating plate. The influence of chemical reactions in two dimensional independent of time free convection stream of an electrically conducting viscous fluid through a permeable medium bounded by vertical surface with slip at the boundary has been examined by Senapatil et al. [43]. The effect of magnetic field has been studied in many artificial flows as well as natural flows. The earth magnetic field is kept by fluid flow in the centre of the earth, solarium magnetic field that generate solarium flickers and sunspot on the surface of the sun and the magnetic field present in galaxies maintain the figure of stars among the clouds of stars [47]. At present, a number of researchers have focused their attention on the application of MHD fluids and heat mass transfer such as geothermal energy extraction,Magneto-hydrodynamic generators and metallurgical processing. The problems relating to heat and mass transfer with Magneto-hydrodynamic flow has got much importance because of their application in the field of different disciplines of engineering. Along these flows, heat and mass exchange with MHD flow has been a subject of interest for many researchers including Hayat et al. [17], Jha and Apere [26] and Fetecau et al. [12]. Besides it, it is observed from the research work that a lot of study on free convection flows over a plate with different boundaries conditions have been done. Many problems have been found also with variable boundary conditions at the wall. The practical application of this thought can be found in the manufacturing of thin film photovoltaic gadgets where ramped wall temperature might be utilized to accomplish an explicit complete of the framework [4]. A large number of researchers have investigated free 2
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convection flow over a vertical wall with variable surface temperature. For some new and important work we refer [24], [40-42], [15], [44], [6]. The fluid which possess slip at boundary, have application in different disciplines of science such as polishing the internal cavities and valves of artificial heart [11]. Analytical solutions of the slip boundary condition problems are not more in literature due to their complexity and also the solutions of combined heat and mass transfer are not very large. Some important analytical solutions having slip boundary conditions are found in [27], [53]. Navier [37] investigated the problems related to slip boundary condition having slip velocity varies linearly to shear stress at boundary. Customarily, the governing equation for slip fluid flow based on slip velocity exhibit by shear stress. The study of slip on the wall leads to an interesting topic having shear stress on solid surface. Taking these into account, Fetecau et al. [13] examined the free convection flow by applying shear stress on fluid over vertical plate by considering both thermal radiation and porosity effect. The classical derivatives are in local nature, i.e., using classical derivatives we can describe changes in a neighborhood of a point but using fractional derivatives we can describe changes in an interval. Namely, fractional derivative is in nonlocal nature. This property makes these derivatives suitable to simulate more physical phenomena such as earthquake vibrations, fluid flow, polymers and etc. For more details, we refer the reader to ”Podluny” book [44]. Since, Fractional calculus is approximately as been around as the standard differential and integral calculus and a list of its applications is very large. However, it is important to mention the fact that fractional derivative generalizations of one dimensional viscoelastic models have been found to be of great utility in modeling the response linear regime [18] and they are in agreement with the second principle of thermodynamics. As fractional order derivative parameter becomes the rheological parameter and controls the dynamics of the fluids. In some more complex classes of fluids, these fractional order parameter serves as sheer thickening and sheer thinning parameters.Furthermore, as it results from the work of Makris et al. [33], a satisfactory agreement of experimental work was achieved when the non-integer order Maxwell model was used in place of the ordinary one. They also proved that the fractional model has a stronger memory of the recent past than the ordinary model. Many researchers are taking keen interest to generalize the problems of classical dynamics to fractional dynamics. However, this generalization is done using different approaches/definitions of fractional derivatives [14], [19-21], [55], [39], [51], [38]. Most recent study on heat and mass transfer of different fluids under different thermal and mechanical conditions with Caputo fractional derivatives can be seen [22-23], [29], [54]. Based on the previous definitions and some 3
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2. Preliminaries
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of the difficulties therein, recently, Caputo and Fabrizio (CF) have introduced a modern definition of the fractional derivatives with an exponential but without singular kernel [2]. The CF derivative problems are found more suitable for the Laplace transform. Nehad and Khan [45] recently applied the CF derivatives to the heat transfer problem of viscoelastic fluid of second grade and obtained the exact solutions via Laplace transform. In present paper, the analytical solution for Newtonian unsteady magneto-hydro dynamic conjugate flow with slip boundary condition over porous medium in the setting of non integer order derivative model has been examined. We consider the vertical plate situated in the (x, z) plane of a cartesian coordinate system oxyz, the domain of the flow is the porous half-space y > 0 and the motion of the wall is a rectilinear translation in its plane with velocity f µ(t) , where f (t) is a piecewise continuous function defined on [0, ∞) with f (0) = 0 and µ is the viscosity. The novelty of the manuscript is the use of the most recent definition of the non integer order derivative operator i.e. Caputo-Fabrizio derivative operator, the use of generalized boundary conditions in terms of general function f (t), from our general results, several particular cases for instance when f (t) is a linear or sinusoidal function could be recovered.
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2.1. Caputo-Fabrizio derivatives Definition 1. Let u be a function in H 1 (a; b); b > a; 0 < α < 1 then, the new Caputo-Fabrizio derivative of fractional order α is defined as [2]: Z (t − τ ) M (α) t ∂ CF α Dt u(x, t) = u(x, τ )exp −α dτ, (1) 1 − α 0 ∂τ 1−α
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where M (α) is a normalization function such that M (0) = M (1) = 1.
However, if the function does not belong to H 1 (a; b) then, the new derivative has the form [2]: Z αM (α) t (t − τ ) CF α Dt u(x, t) = (u(x, t) − u(x, τ ))exp −α dτ. (2) 1−α 0 1−α
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The fractional integral associated to the Caputo-Fabrizio fractional derivative is given by [2]: Z t 1−α α CF α It u(x, t) = u(x, t) + u(x, τ )dτ. (3) M (α) M (α) 0 4
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It’s clear that the Caputo-Fabrizio derivative has no singular kernel, since the kernel is based on exponential function.
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2.2. Problem formulation Assume an unsteady free convection viscous fluid flow over a vertical porous plate. We consider x-axis along the plate in vertical direction and fluid is embedded over the plate in the same axis. Initially the temperature and concentration is constant according to the atmosphere. When the time progressed, plate begins to translate along the x-axis with velocity f µ(t) , where f (t) is non negative with f (0) = 0 and its Laplace transformation exist. Moreover, the velocity vector along the plate is V~ = (v(y, t), 0, 0). It is assumed that the slip exist on the wall and velocity of fluid at the bottom is proportional to the shear stress. As time progresses the variation in temperature of the plate is θ∞ + (θω − θ∞ ) tt0 whereas t ≤ t0 and therefore, for t > t0 , is kept the temperature of wall and concentration to be constant. We consider the radiation term is in energy equation, radiative heat flux along x-axis is assumed to be negligible as compare to y-axis. Further, assuming that the fluid is electrically conducting. The set of equations that governs the flow are given by [5], [42] ∂ 2v ν σB◦2 ∂v = ν 2 + gβθ (θ − θ∞ ) + gβc (C − C∞ ) − v − v; y, t > 0, ∂t ∂y k ρ ∂θ ∂ 2 θ ∂qr =k 2 − y, t > 0, ∂t ∂y ∂y
(5)
∂C ∂ 2C =D 2 ∂t ∂y
(6)
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ρCp
(4)
y, t > 0.
The related initial and boundary conditions are v(y, 0) = 0, θ(y, 0) = θ∞ , C(y, 0) = C∞ , ∂v(0, t) f (t) = , ∂y µ
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v(0, t) − L
∀y ≥ 0,
(7)
t C(0, t) = Cω ; t > 0, θ(y, 0) = θ∞ + (θw − θ∞ ) , t0
θ(0, t) = θω ; t > t0 v(∞, t) = θ∞ , C(∞, t) = C∞ ; t > 0. 5
(9)
0 < t < t0 , (8)
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qr = −
4σ ∂θ4 . 3KR ∂y
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Here L is the slip coefficient of velocity . Rosseland approximation for radiation heat flux of optically thick fluid is given by [32], [16], [34] (10)
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It is assumed that the temperature difference within the flow is very small using this fact, we have 3 4 θ4 = 4θ∞ θ − 3θ∞ ,
(11)
by making use of Eq. (8) into Eq. (7) and substituting the result in Eq. (2), we have ∂θ ∂ 2θ = ν (1 + Nr ) 2 ; y, t > 0. ∂t ∂y
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Pr
(12)
The given dimensionless variables are r r t0 ∗ θ − θ∞ C − C∞ ∗ y t ∗ ∗ 1 t0 ∗ ∗ ∗ v =v ,θ = ,C = , y = √ , t = , f (t ) = f (t0 t∗ ),(13) νt0 ν θω − θ∞ Cω − C∞ t0 µ ν used into Eq. (4), (6) and (12), after that dropping the star notations, we have
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∂v ∂ 2v = 2 + Gr θ + Gm C − Kp v − M 2 v, ∂t ∂y
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P ref f
∂θ ∂ 2θ = 2, ∂t ∂y
∂C 1 ∂ 2C = , ∂t Sc ∂y 2
(14)
(15)
(16)
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Pr where P ref f = 1+N is the effective Prandtl number and Sc is the Schmidt number. Effective Prandtl number is the transport parameter regarding the thermal and mass diffusivity. The dimensionless initial and boundary conditions are
v(y, 0) = 0, θ(y, 0) = 0, C(y, 0) = 0; ∀y ≥ 0, v(y, 0) − h 6
∂v |y=0 = f (t),(17) ∂y
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C(0, t) = 1, C(∞, t) = 0, θ(∞, t) = 0, v(∞, t) = 0; t > 0, θ(0, t) = t; 0 < t ≤ 1, θ(0, t) = 1; t > 1.
Dtα v =
∂ 2v + Gr θ + Gm C − Kp v − M 2 v, ∂y 2
(18)
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The corresponding model in terms of fractional order derivative is written as
where CF Dtα is Caputo-Fabrizio time fractional derivative operator [2], [30]. 3. Solution of the problem
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3.1. Temperature Distribution By applying the Laplace transform to Eq. (15) and using initial conditions from Eq. (17))2 we get, the following form, ¯ q) 1 ∂ 2 θ(y, ¯ q) = θ(y, , (19) qP ref f ∂y 2 by applying the corresponding boundary conditions
(20)
√ √ −y ¯ q) = 1 e−y qP ref f − e e−y qP ref f , θ(y, q2 q2
(21)
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we have,
−q ¯ q) = 1 − e , θ(∞, ¯ θ(0, q) = 0, q2
by applying inverse Laplace transform, we have
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where
θ(y, t) = l(y, t) − l(y, t − 1)H(t − 1),
l(y, t) =
y 2 P ref f + t erfc 2
y
p
P ref f √ 2 t
!
−y
r
tP ref f − y2 P ref f 4t e , π
p √ 2 P ref f √ ∂θ(y, t) √ |y=0 = t − t − 1H(t − 1) , ∂y π
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and
(22)
(23)
(24)
is the corresponding heatRtransfer rate also known as Nusselt number, Here erfc(y) = 2 y 1 − erf(y), erf(y) = √2π 0 eη dη and H(t − 1) is the unit step function. 7
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2¯ ¯ q) = 1 ∂ C(y, q) , C(y, qSc ∂y 2
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3.2. For concentration Applying Laplace transform on Eq. (16) and making use of (17)3 , we have
with boundary conditions
¯ q) = 1 , C(∞, ¯ C(0, q) = 0, q we get
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√ ¯ q) = 1 e−y qSc , C(y, q
(25)
(26)
(27)
which upon inverse Laplace transform gives
√ ! y Sc √ , 2 t
C(y, t) = erfc
√ Sc ∂C(y, t) |y=0 = − √ , ∂y πt
(28)
(29)
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is the corresponding mass transfer rate, also known as Sherwood number.
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3.3. Solution for velocity field By taking Laplace transform of Eq. (18), we get q ∂ 2 v¯(y, q) ¯ q) + Gm C(y, ¯ q) − Kp v¯(y, q) − M 2 v¯(y, q),(30) v¯(y, q) = + Gr θ(y, (1 − α)q + α ∂y 2 with boundary conditions
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v¯(0, q) − h
∂¯ v (y, q) |y=0 = F (q), v¯(∞, q) = 0, ∂y
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(31)
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we get 1
q q 1 + h (1−α)q+α + kp + M 2
Gr (1 − e−q ) q2
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
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v¯(y, q) =
qP ref f !
√ √ q 1 + h qSc −y (1−α)q+α +kp +M 2 + F (q) e − (32) q 2 qSc − (1−α)q+α − (kp + M ) √ √ Gr (1 − e−q ) e−y qP ref f Gm e−y qSc − − . q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
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Gm + q
Skin friction
1
Gr (1 − e−q ) q2 !
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
q q qP ref f 1 + h (1−α)q+α + kp + M 2 √ r Gm 1 + h qSc q + F (q) + kp + M 2 − (33) + q q qSc − (1−α)q+α − (kp + M 2 ) (1 − α)q + α p √ qP ref f Gr (1 − e−q ) qSc Gm − − . q q 2 2 q qP ref f − (1−α)q+α − (kp + M ) q qSc − (1−α)q − (kp + M 2 )
4. Limiting Cases
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τ¯(y, q) =
∂¯ v (y, q) |y=0 , ∂y
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In this section we discuss few limiting cases of our general solutions.
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4.1. Solution in the absence of porous effects for ramped and constant wall temperature (kp → 0) As it is clear from Eq.(22) and (27) that the temperature and concentration distribution are not effected by the inverse permeability parameter for the porous medium kp , and the velocity and skin friction with kp → 0 with ramped wall
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temperature. The Eqs. (32) and (33) are given by 1
q q 1 + h (1−α)q+α + M2
Gr (1 − e−q ) q2
! p 1 + h qP ref f + q qP ref f − (1−α)q+α − M2
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v¯(y, q) =
! √ √ q 1 + h qSc −y (1−α)q+α +M 2 + F (q) e − (34) q 2 qSc − (1−α)q+α − M √ √ Gr (1 − e−q ) e−y qP ref f e−y qSc Gm − − , q q q2 qP ref f − (1−α)q+α − M2 q qSc − (1−α)q − M2
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Gm + q
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and
! p 1 + h qP ref f Gr (1 − e−q ) q τ¯(y, q) = + q q q2 qP ref f − (1−α)q+α − M2 + M2 1 + h (1−α)q+α ! √ r 1 + h qSc q Gm + F (q) + M 2 − (35) + q 2 q qSc − (1−α)q+α − M (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q q2 qP ref f − (1−α)q+α − M2 q qSc − (1−α)q − M2
1
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4.2. Solution in the absence of thermal radiation (Nr → 0) In the absence of thermal radiation, the corresponding solution for ramped wall temperature are directly obtained from the general solutions (17)–(22) and (32), (33) by taking Nr → 0 and replacing P ref f by P r i.e. √ ! y Pr √ , (36) θ(y, t) = erfc 2 t
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√ ∂θ(y, t) Pr =−√ , ∂y πt
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(37)
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q q 1 + h (1−α)q+α + kp + M 2
Gr (1 − e−q ) − q2 qP r −
e−y
√
q (1−α)q+α
1
! √ 1 + h qP r + q qP r − (1−α)q+α − (kp + M 2 ) ! √ q +kp +M 2 −y (1−α)q+α − (38) + F (q) e
qP r
Gm − q qSc − − (kp + M 2 )
Gr (1 − e−q ) q2 !
e−y q (1−α)q
√
qSc
− (kp + M 2 )
,
! √ 1 + h qP r + q − (kp + M 2 ) qP r − (1−α)q+α
q q 1 + h (1−α)q+α + kp + M 2 √ r Gm 1 + h qSc q + + F (q) + kp + M 2 − (39) q q qSc − (1−α)q+α − (kp + M 2 ) (1 − α)q + α √ √ Gm Gr (1 − e−q ) qP r qSc − . − q q − (kp + M 2 ) − (kp + M 2 ) q2 qP r − (1−α)q+α q qSc − (1−α)q
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τ¯(y, q) =
Gr (1 − e−q ) q2
√ 1 + h qSc q qSc − (1−α)q+α − (kp + M 2 )
Gm + q
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v¯(y, q) =
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4.3. Solution in the absence of magnetic parameter (M → 0) As it is clear from Eq. (22) and (27) that the temperature and concentration distributions are not effected by the magnetic parameter M , and the velocity and skin friction with M = 0 for ramped temperature are given by ! p 1 + h qP ref f 1 Gr (1 − e−q ) q v¯(y, q) = + q q − kp q2 qP ref f − (1−α)q+α 1 + h (1−α)q+α + kp ! √ √ q Gm 1 + h qSc −y (1−α)q+α +kp + + F (q) e − (40) q q qSc − (1−α)q+α − kp √ √ Gr (1 − e−q ) e−y qP ref f Gm e−y qSc − , − q q q2 qP ref f − (1−α)q+α − kp q qSc − (1−α)q − kp
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! p 1 + h qP ref f Gr (1 − e−q ) q τ¯(y, q) = + q q q2 qP ref f − (1−α)q+α − kp 1 + h (1−α)q+α + kp ! √ r q 1 + h qSc Gm + kp − (41) + F (q) + q q qSc − (1−α)q+α − kp (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q q2 qP ref f − (1−α)q+α q qSc − (1−α)q − kp − kp
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1
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4.4. Solution in the absence of mechanical effects For this situation, we expect that the infinite plate is in static position at each time i.e. the given function f (t) is zero for all values of t. In such a circumstance, the dynamics of the fluid are due the free convection and buoyancy forces. Accordingly, the velocity of the fluid due to the variation of wall temperature is just expressed by their convective parts. 5. Special cases
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5.1. First case :- f (t) = t In first case, the expression of velocity field and skin friction are obtained by substituting the value of f (t) = t in Eqs. (32) and (33), we have ! p 1 + h qP ref f 1 Gr (1 − e−q ) q + v¯(y, q) = q q − (kp + M 2 ) q2 qP ref f − (1−α)q+α 1 + h (1−α)q+α + kp + M 2 ! √ √ q Gm 1 + h qSc 1 −y (1−α)q+α +kp +M 2 + + e − (42) q 2 2 q qSc − (1−α)q+α − (kp + M ) q √ √ e−y qP ref f Gm e−y qSc Gr (1 − e−q ) − , − q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
! p 1 + h qP ref f Gr (1 − e−q ) q τ¯(y, q) = + q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) 1 + h (1−α)q+α + kp + M 2 ! √ r 1 + h qSc 1 Gm q + 2 + kp + M 2 − (43) + q 2 q qSc − (1−α)q+α − (kp + M ) q (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q 2 2 q qP ref f − (1−α)q+α − (kp + M ) q qSc − (1−α)q − (kp + M 2 )
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1
q q 1 + h (1−α)q+α + kp + M 2
qP ref f
! p 1 + h qP ref f + q − (kp + M 2 ) − (1−α)q+α
! √ √ q 1 + h qSc 1 −y (1−α)q+α +kp +M 2 e + − (44) q 2 2 qSc − (1−α)q+α − (kp + M ) (q − 1) √ √ Gr (1 − e−q ) Gm e−y qP ref f e−y qSc − , − q q − (kp + M 2 ) − (kp + M 2 ) q2 qP ref f − (1−α)q+α q qSc − (1−α)q
1
Pr e-
Gm + q
τ¯(y, q) =
Gr (1 − e−q ) q2
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v¯(y, q) =
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5.2. Second case :- f (t) = tet In second case, the expression of velocity field and skin friction are obtained by substituting the value of f (t) = t in Eq. (32) and (33), we have
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
Gr (1 − e−q ) q2 !
q q qP ref f + kp + M 2 1 + h (1−α)q+α √ r Gm 1 + h qSc 1 q + kp + M 2 − (45) + + q 2 2 q qSc − (1−α)q+α − (kp + M ) (q − 1) (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
1
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5.3. Third case :- f (t) = sin(ωt) In third case, the expression of velocity field and skin friction are obtained by substituting the value of f (t) = t in Eq. (32) and (33), we have v¯(y, q) =
q q 1 + h (1−α)q+α + kp + M 2
Gr (1 − e−q ) q2
qP ref f
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
! √ √ q 1 + h qSc ω −y (1−α)q+α +kp +M 2 + e − (46) q qSc − (1−α)q+α − (kp + M 2 ) q2 + ω2 √ √ Gr (1 − e−q ) e−y qP ref f Gm e−y qSc − − , q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
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Gm + q
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1
Gr (1 − e−q ) q2 !
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
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q q qP ref f 1 + h (1−α)q+α + kp + M 2 √ r q 1 + h qSc ω Gm + kp + M 2 − (47) + 2 + q 2 2 q qSc − (1−α)q+α − (kp + M ) q +ω (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q q2 qP ref f − (1−α)q+α q qSc − (1−α)q − (kp + M 2 ) − (kp + M 2 )
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q q qP ref f 1 + h (1−α)q+α + kp + M 2 √ r Gm 1 + h qSc 2qω q + + 2 + kp + M 2 − (49) q q qSc − (1−α)q+α − (kp + M 2 ) q + ω2 (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − . − q q 2 2 q qP ref f − (1−α)q+α − (kp + M ) q qSc − (1−α)q − (kp + M 2 )
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τ¯(y, q) =
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5.4. Forth case :- f (t) = t sin(ωt) In forth case, the expression of velocity field and skin friction are obtained by substituting the value of f (t) = t in Eq. (32) and (33), we have ! p 1 + h qP ref f 1 Gr (1 − e−q ) q + v¯(y, q) = q q − (kp + M 2 ) q2 qP ref f − (1−α)q+α 1 + h (1−α)q+α + kp + M 2 ! √ √ q Gm 1 + h qSc 2qω −y (1−α)q+α +kp +M 2 + e − (48) + 2 q 2 2 q qSc − (1−α)q+α − (kp + M ) q +ω √ √ Gm e−y qSc Gr (1 − e−q ) e−y qP ref f − , − q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
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The Expressions for given by Eq. (32) and (33) are very complicated functions of the transformed parameter so, the inverse Laplace transform with respect to variable is very difficult to obtain by analytical method. For this reason, in this paper, we will use the Stehfest’s algorithm for numerical inversion of Laplace transform with respect to variable t. According with the Stehfest’s algorithm [48], [1], [31], 2m P [46], the solution is approximated by v(y, t) = ln(2) dj v¯(y, j ln(2) ), where m is t t j=1
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Figure 1: Variation of α with time t = 0.1 and other parameters are as h = 0.5,kp = 4,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,M = 0.9,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
the positive integer min(j,m) P im (2i)! dj = (−1)j+m (m−i)!i!(i−1)!(j−i)!(2i−j)! i=[ j+1 ] 2
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also, [r] indicate the integral part of real number r. Nonetheless, we got another guess for v¯(y, s) by Tzou’s calculation for approval of our numerical inverse Laplace N P1 4.7+kπi e4.7 1 4.7 k v(r, t) = t 2 v¯ r, t + Re (−1) v¯ r, t , k=1
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where Re(.) is the real part, i is the imaginary unit and N1 is a natural number [52].
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5.5. Numerical results The physical behaviour of dimensionless velocity with the effect of different values of effective Prandtl number, Grashof number, modified Grashof number, Magnetic parameter, Schmidt number, permeability parameter for the porous medium and α is displayed by the plotting the graphs for dimensionless velocity versus different system parameters by using computer software MATHCAD. Figures 1 to 7 are prepared for f (t) = t whereas in figures 8 to 14 we have f (t) = sin(wt). From the graphical illustrations the useful results are as follows: 15
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Figure 2: Variation of Gr with time t = 0.1 and other parameters are as h = 0.5,kp = 4,N = 0.5, α = 0.5, P ref f = 2, Gm = 0.75,M = 0.9,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 3: Variation of Gm with time t = 0.1 and other parameters are as h = 0.5,kp = 4,N = 0.5, Gr = 2, P ref f = 2, α = 0.5,M = 0.9,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 4: Variation of kp with time t = 0.1 and other parameters are as h = 0.5,α = 0.5,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,M = 0.9,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 5: Variation of M with time t = 0.1 and other parameters are as h = 0.5,α = 0.5,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,kp = 4,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 6: Variation of P ref f with time t = 0.1 and other parameters are as h = 0.5,α = 0.5,N = 0.5, Gr = 2, M = 0.9, Gm = 0.75,kp = 4,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 7: Variation of Sc with time t = 0.1 and other parameters are as h = 0.5,α = 0.5,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,kp = 4,M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 8: Variation of α with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,kp = 4,Sc = 0.5, M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 9: Variation of Gr with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, α = 0.5, P ref f = 2, Gm = 0.75,kp = 4,Sc = 0.5, M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 10: Variation of Gm with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 2, α = 0.5,kp = 4,Sc = 0.5, M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 11: Variation of kp with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.3,α = 0.5,Sc = 0.5, M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 12: Variation of M with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.3,α = 0.5,Sc = 0.5, kp = 0.6 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 13: Variation of P ref f with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, M = 0.9, Gm = 0.75,α = 0.5,Sc = 0.5, kp = 0.6 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 14: Variation of Sc with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 4, Gm = 0.75,α = 0.5,M = 0.9, kp = 0.6 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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1. The dimensionless velocity increases with the increase of Grashof number as well as modified Grashof number and in general the velocity for the case of non slip boundary condition is lesser than the case of slip boundary condition with slip parameter h = 0.65. 2. The velocity is a decreasing function of the parameters Prandtl effective number, Magnetic parameter and α. Moreover, the velocity with the non slip boundary condition i.e. h = 0 is larger than the velocity with non slip boundary condition (at value parameter h = 0.65). 3. By increasing the value of porosity parameter dimensionless velocity decreases and in general the velocity for the case of non slip boundary condition is less then the case of slip boundary condition. 4. With the small increase in Schmidt number the velocity decreases, whereas the influence is non responsive for large values of the parameter Sc . The velocity in non slip boundary condition is larger in comparison to the slip boundary condition.
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Highlights (for review)
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New fractional derivatives of Caputo–Fabrizio fractional derivative is applied to the unsteady magnetohydrodynamics (MHD) flow of a viscous fluid. Closed forms solution are obtained by means of the Laplace transforms. Results obtained illustrated distinct behaviors of fractional order solutions when compared with classical model solutions. A discussion is presented for mentioning how our results are innovative comparing to those already published.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Manuscript : Ms. Ref. No.: PHYSA-193415 Title: ANALYTICAL APPROACH ON THE UNSTEADY MAGNETOHYDRODYNAMIC CONJUGATE VISCOUS FLUID FLOW IN A POROUS MEDIUM WITH NONSINGULAR KERNEL DERIVATIVES Physica A
The corresponding author :
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Prof Zakia Hammouch Moulay Ismail University
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Authors : M. Mansha Ghalib, Azhar A. Zafar, M. Bilal Riaz, Z.Hammouch, Khurram Shabbir
S0378-4371(19)32178-8 https://doi.org/10.1016/j.physa.2019.123941 PHYSA 123941
To appear in:
Physica A
Received date : 26 September 2019 Revised date : 23 November 2019 Please cite this article as: M.M. Ghalib, A.A. Zafar, M.B. Riaz et al., Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123941. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative
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M. Mansha Ghalib1 , Azhar A. Zafar2 , M. Bilal Riaz3 , Z. Hammouch41 , Khurram Shabbir2 Department of Mathematics and statistics, University of Lahore 54590, Pakistan. 2 Department of Mathematics, GC University, Lahore 54590, Pakistan. 3 Department of Mathematics, University of Management and Technology Lahore 54590, Pakistan. 4∗ FSTE University of Moulay Ismail Errachidia 52000 Morocco. Corresponding Author Email: [email protected]
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1
Abstract
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This study investigates the unsteady magnetohydrodynamics (MHD) flow of a viscous fluid. The fluid is passing over a vertical plate through porous medium. Additionally conjugate effects of heat and mass transfer with ramped temperatures, slip effect and influence of thermal radiation in the energy equation are taken into account. The dimensionless fractional-order governing equations, in the CaputoFabrizio sense, are solved with the help of Laplace transformation. Moreover, semi analytical technique is used to investigate the velocity field. Some results which present in literature are recovered as limiting cases. Influences of different parameters on the velocity profiles for the case of f (t) = t and f (t) = sin ωt are highlighted. The novelty of the manuscript is the use of the most recent definition of the non integer order derivative operator i.e. Caputo-Fabrizio derivative operator, the use of generalized boundary conditions in terms of general function f (t), from our general results, several particular cases for instance when f (t) is a linear or sinusoidal function could be recovered.
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Keywords Nonsingular-kernel derivative, Unsteady MHD flow, Porous medium, Ramped wall temperature, Closed-form solution. 1
Corresponding Author email:[email protected]
Preprint submitted to Elsevier
December 29, 2019
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1. Introduction
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Heat and mass transfer in case of chemical reaction, evaporation and condensation caused by both temperature difference and concentration difference. The rate of mass transfer depends on the rate of heat transfer. The practical application of heat and mass transfer frequently exist in chemically processed industries such as food processing and polymer production. Free convection flows, force convection flow and mixed convection flow with conjugate effects of heat and mass transfer past a vertical plate have been studied substantially in the literature [3] due to its applications in food processing products and polymer production in various disciplines of engineering and industries, fiber and granular insulation and geothermal system [28], [5], [10]. For most notable articles in this area of research we refer [7], [8], [9], [25], [35] and [36]. Toki et al. [49] have examined the unsteady free convection streams of viscous fluid near a permeable infinite plate with time dependent heating plate. The influence of chemical reactions in two dimensional independent of time free convection stream of an electrically conducting viscous fluid through a permeable medium bounded by vertical surface with slip at the boundary has been examined by Senapatil et al. [43]. The effect of magnetic field has been studied in many artificial flows as well as natural flows. The earth magnetic field is kept by fluid flow in the centre of the earth, solarium magnetic field that generate solarium flickers and sunspot on the surface of the sun and the magnetic field present in galaxies maintain the figure of stars among the clouds of stars [47]. At present, a number of researchers have focused their attention on the application of MHD fluids and heat mass transfer such as geothermal energy extraction,Magneto-hydrodynamic generators and metallurgical processing. The problems relating to heat and mass transfer with Magneto-hydrodynamic flow has got much importance because of their application in the field of different disciplines of engineering. Along these flows, heat and mass exchange with MHD flow has been a subject of interest for many researchers including Hayat et al. [17], Jha and Apere [26] and Fetecau et al. [12]. Besides it, it is observed from the research work that a lot of study on free convection flows over a plate with different boundaries conditions have been done. Many problems have been found also with variable boundary conditions at the wall. The practical application of this thought can be found in the manufacturing of thin film photovoltaic gadgets where ramped wall temperature might be utilized to accomplish an explicit complete of the framework [4]. A large number of researchers have investigated free 2
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convection flow over a vertical wall with variable surface temperature. For some new and important work we refer [24], [40-42], [15], [44], [6]. The fluid which possess slip at boundary, have application in different disciplines of science such as polishing the internal cavities and valves of artificial heart [11]. Analytical solutions of the slip boundary condition problems are not more in literature due to their complexity and also the solutions of combined heat and mass transfer are not very large. Some important analytical solutions having slip boundary conditions are found in [27], [53]. Navier [37] investigated the problems related to slip boundary condition having slip velocity varies linearly to shear stress at boundary. Customarily, the governing equation for slip fluid flow based on slip velocity exhibit by shear stress. The study of slip on the wall leads to an interesting topic having shear stress on solid surface. Taking these into account, Fetecau et al. [13] examined the free convection flow by applying shear stress on fluid over vertical plate by considering both thermal radiation and porosity effect. The classical derivatives are in local nature, i.e., using classical derivatives we can describe changes in a neighborhood of a point but using fractional derivatives we can describe changes in an interval. Namely, fractional derivative is in nonlocal nature. This property makes these derivatives suitable to simulate more physical phenomena such as earthquake vibrations, fluid flow, polymers and etc. For more details, we refer the reader to ”Podluny” book [44]. Since, Fractional calculus is approximately as been around as the standard differential and integral calculus and a list of its applications is very large. However, it is important to mention the fact that fractional derivative generalizations of one dimensional viscoelastic models have been found to be of great utility in modeling the response linear regime [18] and they are in agreement with the second principle of thermodynamics. As fractional order derivative parameter becomes the rheological parameter and controls the dynamics of the fluids. In some more complex classes of fluids, these fractional order parameter serves as sheer thickening and sheer thinning parameters.Furthermore, as it results from the work of Makris et al. [33], a satisfactory agreement of experimental work was achieved when the non-integer order Maxwell model was used in place of the ordinary one. They also proved that the fractional model has a stronger memory of the recent past than the ordinary model. Many researchers are taking keen interest to generalize the problems of classical dynamics to fractional dynamics. However, this generalization is done using different approaches/definitions of fractional derivatives [14], [19-21], [55], [39], [51], [38]. Most recent study on heat and mass transfer of different fluids under different thermal and mechanical conditions with Caputo fractional derivatives can be seen [22-23], [29], [54]. Based on the previous definitions and some 3
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2. Preliminaries
Pr e-
p ro
of
of the difficulties therein, recently, Caputo and Fabrizio (CF) have introduced a modern definition of the fractional derivatives with an exponential but without singular kernel [2]. The CF derivative problems are found more suitable for the Laplace transform. Nehad and Khan [45] recently applied the CF derivatives to the heat transfer problem of viscoelastic fluid of second grade and obtained the exact solutions via Laplace transform. In present paper, the analytical solution for Newtonian unsteady magneto-hydro dynamic conjugate flow with slip boundary condition over porous medium in the setting of non integer order derivative model has been examined. We consider the vertical plate situated in the (x, z) plane of a cartesian coordinate system oxyz, the domain of the flow is the porous half-space y > 0 and the motion of the wall is a rectilinear translation in its plane with velocity f µ(t) , where f (t) is a piecewise continuous function defined on [0, ∞) with f (0) = 0 and µ is the viscosity. The novelty of the manuscript is the use of the most recent definition of the non integer order derivative operator i.e. Caputo-Fabrizio derivative operator, the use of generalized boundary conditions in terms of general function f (t), from our general results, several particular cases for instance when f (t) is a linear or sinusoidal function could be recovered.
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2.1. Caputo-Fabrizio derivatives Definition 1. Let u be a function in H 1 (a; b); b > a; 0 < α < 1 then, the new Caputo-Fabrizio derivative of fractional order α is defined as [2]: Z (t − τ ) M (α) t ∂ CF α Dt u(x, t) = u(x, τ )exp −α dτ, (1) 1 − α 0 ∂τ 1−α
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where M (α) is a normalization function such that M (0) = M (1) = 1.
However, if the function does not belong to H 1 (a; b) then, the new derivative has the form [2]: Z αM (α) t (t − τ ) CF α Dt u(x, t) = (u(x, t) − u(x, τ ))exp −α dτ. (2) 1−α 0 1−α
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The fractional integral associated to the Caputo-Fabrizio fractional derivative is given by [2]: Z t 1−α α CF α It u(x, t) = u(x, t) + u(x, τ )dτ. (3) M (α) M (α) 0 4
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It’s clear that the Caputo-Fabrizio derivative has no singular kernel, since the kernel is based on exponential function.
Pr e-
p ro
2.2. Problem formulation Assume an unsteady free convection viscous fluid flow over a vertical porous plate. We consider x-axis along the plate in vertical direction and fluid is embedded over the plate in the same axis. Initially the temperature and concentration is constant according to the atmosphere. When the time progressed, plate begins to translate along the x-axis with velocity f µ(t) , where f (t) is non negative with f (0) = 0 and its Laplace transformation exist. Moreover, the velocity vector along the plate is V~ = (v(y, t), 0, 0). It is assumed that the slip exist on the wall and velocity of fluid at the bottom is proportional to the shear stress. As time progresses the variation in temperature of the plate is θ∞ + (θω − θ∞ ) tt0 whereas t ≤ t0 and therefore, for t > t0 , is kept the temperature of wall and concentration to be constant. We consider the radiation term is in energy equation, radiative heat flux along x-axis is assumed to be negligible as compare to y-axis. Further, assuming that the fluid is electrically conducting. The set of equations that governs the flow are given by [5], [42] ∂ 2v ν σB◦2 ∂v = ν 2 + gβθ (θ − θ∞ ) + gβc (C − C∞ ) − v − v; y, t > 0, ∂t ∂y k ρ ∂θ ∂ 2 θ ∂qr =k 2 − y, t > 0, ∂t ∂y ∂y
(5)
∂C ∂ 2C =D 2 ∂t ∂y
(6)
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ρCp
(4)
y, t > 0.
The related initial and boundary conditions are v(y, 0) = 0, θ(y, 0) = θ∞ , C(y, 0) = C∞ , ∂v(0, t) f (t) = , ∂y µ
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v(0, t) − L
∀y ≥ 0,
(7)
t C(0, t) = Cω ; t > 0, θ(y, 0) = θ∞ + (θw − θ∞ ) , t0
θ(0, t) = θω ; t > t0 v(∞, t) = θ∞ , C(∞, t) = C∞ ; t > 0. 5
(9)
0 < t < t0 , (8)
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qr = −
4σ ∂θ4 . 3KR ∂y
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Here L is the slip coefficient of velocity . Rosseland approximation for radiation heat flux of optically thick fluid is given by [32], [16], [34] (10)
p ro
It is assumed that the temperature difference within the flow is very small using this fact, we have 3 4 θ4 = 4θ∞ θ − 3θ∞ ,
(11)
by making use of Eq. (8) into Eq. (7) and substituting the result in Eq. (2), we have ∂θ ∂ 2θ = ν (1 + Nr ) 2 ; y, t > 0. ∂t ∂y
Pr e-
Pr
(12)
The given dimensionless variables are r r t0 ∗ θ − θ∞ C − C∞ ∗ y t ∗ ∗ 1 t0 ∗ ∗ ∗ v =v ,θ = ,C = , y = √ , t = , f (t ) = f (t0 t∗ ),(13) νt0 ν θω − θ∞ Cω − C∞ t0 µ ν used into Eq. (4), (6) and (12), after that dropping the star notations, we have
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∂v ∂ 2v = 2 + Gr θ + Gm C − Kp v − M 2 v, ∂t ∂y
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P ref f
∂θ ∂ 2θ = 2, ∂t ∂y
∂C 1 ∂ 2C = , ∂t Sc ∂y 2
(14)
(15)
(16)
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Pr where P ref f = 1+N is the effective Prandtl number and Sc is the Schmidt number. Effective Prandtl number is the transport parameter regarding the thermal and mass diffusivity. The dimensionless initial and boundary conditions are
v(y, 0) = 0, θ(y, 0) = 0, C(y, 0) = 0; ∀y ≥ 0, v(y, 0) − h 6
∂v |y=0 = f (t),(17) ∂y
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C(0, t) = 1, C(∞, t) = 0, θ(∞, t) = 0, v(∞, t) = 0; t > 0, θ(0, t) = t; 0 < t ≤ 1, θ(0, t) = 1; t > 1.
Dtα v =
∂ 2v + Gr θ + Gm C − Kp v − M 2 v, ∂y 2
(18)
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CF
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The corresponding model in terms of fractional order derivative is written as
where CF Dtα is Caputo-Fabrizio time fractional derivative operator [2], [30]. 3. Solution of the problem
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3.1. Temperature Distribution By applying the Laplace transform to Eq. (15) and using initial conditions from Eq. (17))2 we get, the following form, ¯ q) 1 ∂ 2 θ(y, ¯ q) = θ(y, , (19) qP ref f ∂y 2 by applying the corresponding boundary conditions
(20)
√ √ −y ¯ q) = 1 e−y qP ref f − e e−y qP ref f , θ(y, q2 q2
(21)
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we have,
−q ¯ q) = 1 − e , θ(∞, ¯ θ(0, q) = 0, q2
by applying inverse Laplace transform, we have
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where
θ(y, t) = l(y, t) − l(y, t − 1)H(t − 1),
l(y, t) =
y 2 P ref f + t erfc 2
y
p
P ref f √ 2 t
!
−y
r
tP ref f − y2 P ref f 4t e , π
p √ 2 P ref f √ ∂θ(y, t) √ |y=0 = t − t − 1H(t − 1) , ∂y π
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and
(22)
(23)
(24)
is the corresponding heatRtransfer rate also known as Nusselt number, Here erfc(y) = 2 y 1 − erf(y), erf(y) = √2π 0 eη dη and H(t − 1) is the unit step function. 7
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p ro
2¯ ¯ q) = 1 ∂ C(y, q) , C(y, qSc ∂y 2
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3.2. For concentration Applying Laplace transform on Eq. (16) and making use of (17)3 , we have
with boundary conditions
¯ q) = 1 , C(∞, ¯ C(0, q) = 0, q we get
Pr e-
√ ¯ q) = 1 e−y qSc , C(y, q
(25)
(26)
(27)
which upon inverse Laplace transform gives
√ ! y Sc √ , 2 t
C(y, t) = erfc
√ Sc ∂C(y, t) |y=0 = − √ , ∂y πt
(28)
(29)
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is the corresponding mass transfer rate, also known as Sherwood number.
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3.3. Solution for velocity field By taking Laplace transform of Eq. (18), we get q ∂ 2 v¯(y, q) ¯ q) + Gm C(y, ¯ q) − Kp v¯(y, q) − M 2 v¯(y, q),(30) v¯(y, q) = + Gr θ(y, (1 − α)q + α ∂y 2 with boundary conditions
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v¯(0, q) − h
∂¯ v (y, q) |y=0 = F (q), v¯(∞, q) = 0, ∂y
8
(31)
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we get 1
q q 1 + h (1−α)q+α + kp + M 2
Gr (1 − e−q ) q2
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
of
v¯(y, q) =
qP ref f !
√ √ q 1 + h qSc −y (1−α)q+α +kp +M 2 + F (q) e − (32) q 2 qSc − (1−α)q+α − (kp + M ) √ √ Gr (1 − e−q ) e−y qP ref f Gm e−y qSc − − . q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
p ro
Gm + q
Skin friction
1
Gr (1 − e−q ) q2 !
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
q q qP ref f 1 + h (1−α)q+α + kp + M 2 √ r Gm 1 + h qSc q + F (q) + kp + M 2 − (33) + q q qSc − (1−α)q+α − (kp + M 2 ) (1 − α)q + α p √ qP ref f Gr (1 − e−q ) qSc Gm − − . q q 2 2 q qP ref f − (1−α)q+α − (kp + M ) q qSc − (1−α)q − (kp + M 2 )
4. Limiting Cases
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τ¯(y, q) =
∂¯ v (y, q) |y=0 , ∂y
Pr e-
τ¯(y, q) = −
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In this section we discuss few limiting cases of our general solutions.
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4.1. Solution in the absence of porous effects for ramped and constant wall temperature (kp → 0) As it is clear from Eq.(22) and (27) that the temperature and concentration distribution are not effected by the inverse permeability parameter for the porous medium kp , and the velocity and skin friction with kp → 0 with ramped wall
9
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temperature. The Eqs. (32) and (33) are given by 1
q q 1 + h (1−α)q+α + M2
Gr (1 − e−q ) q2
! p 1 + h qP ref f + q qP ref f − (1−α)q+α − M2
of
v¯(y, q) =
! √ √ q 1 + h qSc −y (1−α)q+α +M 2 + F (q) e − (34) q 2 qSc − (1−α)q+α − M √ √ Gr (1 − e−q ) e−y qP ref f e−y qSc Gm − − , q q q2 qP ref f − (1−α)q+α − M2 q qSc − (1−α)q − M2
p ro
Gm + q
Pr e-
and
! p 1 + h qP ref f Gr (1 − e−q ) q τ¯(y, q) = + q q q2 qP ref f − (1−α)q+α − M2 + M2 1 + h (1−α)q+α ! √ r 1 + h qSc q Gm + F (q) + M 2 − (35) + q 2 q qSc − (1−α)q+α − M (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q q2 qP ref f − (1−α)q+α − M2 q qSc − (1−α)q − M2
1
urn
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4.2. Solution in the absence of thermal radiation (Nr → 0) In the absence of thermal radiation, the corresponding solution for ramped wall temperature are directly obtained from the general solutions (17)–(22) and (32), (33) by taking Nr → 0 and replacing P ref f by P r i.e. √ ! y Pr √ , (36) θ(y, t) = erfc 2 t
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√ ∂θ(y, t) Pr =−√ , ∂y πt
10
(37)
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q q 1 + h (1−α)q+α + kp + M 2
Gr (1 − e−q ) − q2 qP r −
e−y
√
q (1−α)q+α
1
! √ 1 + h qP r + q qP r − (1−α)q+α − (kp + M 2 ) ! √ q +kp +M 2 −y (1−α)q+α − (38) + F (q) e
qP r
Gm − q qSc − − (kp + M 2 )
Gr (1 − e−q ) q2 !
e−y q (1−α)q
√
qSc
− (kp + M 2 )
,
! √ 1 + h qP r + q − (kp + M 2 ) qP r − (1−α)q+α
q q 1 + h (1−α)q+α + kp + M 2 √ r Gm 1 + h qSc q + + F (q) + kp + M 2 − (39) q q qSc − (1−α)q+α − (kp + M 2 ) (1 − α)q + α √ √ Gm Gr (1 − e−q ) qP r qSc − . − q q − (kp + M 2 ) − (kp + M 2 ) q2 qP r − (1−α)q+α q qSc − (1−α)q
Pr e-
τ¯(y, q) =
Gr (1 − e−q ) q2
√ 1 + h qSc q qSc − (1−α)q+α − (kp + M 2 )
Gm + q
of
1
p ro
v¯(y, q) =
Jo
urn
al
4.3. Solution in the absence of magnetic parameter (M → 0) As it is clear from Eq. (22) and (27) that the temperature and concentration distributions are not effected by the magnetic parameter M , and the velocity and skin friction with M = 0 for ramped temperature are given by ! p 1 + h qP ref f 1 Gr (1 − e−q ) q v¯(y, q) = + q q − kp q2 qP ref f − (1−α)q+α 1 + h (1−α)q+α + kp ! √ √ q Gm 1 + h qSc −y (1−α)q+α +kp + + F (q) e − (40) q q qSc − (1−α)q+α − kp √ √ Gr (1 − e−q ) e−y qP ref f Gm e−y qSc − , − q q q2 qP ref f − (1−α)q+α − kp q qSc − (1−α)q − kp
11
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! p 1 + h qP ref f Gr (1 − e−q ) q τ¯(y, q) = + q q q2 qP ref f − (1−α)q+α − kp 1 + h (1−α)q+α + kp ! √ r q 1 + h qSc Gm + kp − (41) + F (q) + q q qSc − (1−α)q+α − kp (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q q2 qP ref f − (1−α)q+α q qSc − (1−α)q − kp − kp
of
1
p ro
Pr e-
4.4. Solution in the absence of mechanical effects For this situation, we expect that the infinite plate is in static position at each time i.e. the given function f (t) is zero for all values of t. In such a circumstance, the dynamics of the fluid are due the free convection and buoyancy forces. Accordingly, the velocity of the fluid due to the variation of wall temperature is just expressed by their convective parts. 5. Special cases
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5.1. First case :- f (t) = t In first case, the expression of velocity field and skin friction are obtained by substituting the value of f (t) = t in Eqs. (32) and (33), we have ! p 1 + h qP ref f 1 Gr (1 − e−q ) q + v¯(y, q) = q q − (kp + M 2 ) q2 qP ref f − (1−α)q+α 1 + h (1−α)q+α + kp + M 2 ! √ √ q Gm 1 + h qSc 1 −y (1−α)q+α +kp +M 2 + + e − (42) q 2 2 q qSc − (1−α)q+α − (kp + M ) q √ √ e−y qP ref f Gm e−y qSc Gr (1 − e−q ) − , − q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
! p 1 + h qP ref f Gr (1 − e−q ) q τ¯(y, q) = + q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) 1 + h (1−α)q+α + kp + M 2 ! √ r 1 + h qSc 1 Gm q + 2 + kp + M 2 − (43) + q 2 q qSc − (1−α)q+α − (kp + M ) q (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q 2 2 q qP ref f − (1−α)q+α − (kp + M ) q qSc − (1−α)q − (kp + M 2 )
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1
12
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1
q q 1 + h (1−α)q+α + kp + M 2
qP ref f
! p 1 + h qP ref f + q − (kp + M 2 ) − (1−α)q+α
! √ √ q 1 + h qSc 1 −y (1−α)q+α +kp +M 2 e + − (44) q 2 2 qSc − (1−α)q+α − (kp + M ) (q − 1) √ √ Gr (1 − e−q ) Gm e−y qP ref f e−y qSc − , − q q − (kp + M 2 ) − (kp + M 2 ) q2 qP ref f − (1−α)q+α q qSc − (1−α)q
1
Pr e-
Gm + q
τ¯(y, q) =
Gr (1 − e−q ) q2
p ro
v¯(y, q) =
of
5.2. Second case :- f (t) = tet In second case, the expression of velocity field and skin friction are obtained by substituting the value of f (t) = t in Eq. (32) and (33), we have
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
Gr (1 − e−q ) q2 !
q q qP ref f + kp + M 2 1 + h (1−α)q+α √ r Gm 1 + h qSc 1 q + kp + M 2 − (45) + + q 2 2 q qSc − (1−α)q+α − (kp + M ) (q − 1) (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
1
urn
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5.3. Third case :- f (t) = sin(ωt) In third case, the expression of velocity field and skin friction are obtained by substituting the value of f (t) = t in Eq. (32) and (33), we have v¯(y, q) =
q q 1 + h (1−α)q+α + kp + M 2
Gr (1 − e−q ) q2
qP ref f
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
! √ √ q 1 + h qSc ω −y (1−α)q+α +kp +M 2 + e − (46) q qSc − (1−α)q+α − (kp + M 2 ) q2 + ω2 √ √ Gr (1 − e−q ) e−y qP ref f Gm e−y qSc − − , q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
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Gm + q
13
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1
Gr (1 − e−q ) q2 !
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
of
q q qP ref f 1 + h (1−α)q+α + kp + M 2 √ r q 1 + h qSc ω Gm + kp + M 2 − (47) + 2 + q 2 2 q qSc − (1−α)q+α − (kp + M ) q +ω (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − − . q q q2 qP ref f − (1−α)q+α q qSc − (1−α)q − (kp + M 2 ) − (kp + M 2 )
p ro
τ¯(y, q) =
1
Gr (1 − e−q ) q2 !
! p 1 + h qP ref f + q − (1−α)q+α − (kp + M 2 )
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q q qP ref f 1 + h (1−α)q+α + kp + M 2 √ r Gm 1 + h qSc 2qω q + + 2 + kp + M 2 − (49) q q qSc − (1−α)q+α − (kp + M 2 ) q + ω2 (1 − α)q + α p √ qP ref f Gr (1 − e−q ) Gm qSc − . − q q 2 2 q qP ref f − (1−α)q+α − (kp + M ) q qSc − (1−α)q − (kp + M 2 )
urn
τ¯(y, q) =
Pr e-
5.4. Forth case :- f (t) = t sin(ωt) In forth case, the expression of velocity field and skin friction are obtained by substituting the value of f (t) = t in Eq. (32) and (33), we have ! p 1 + h qP ref f 1 Gr (1 − e−q ) q + v¯(y, q) = q q − (kp + M 2 ) q2 qP ref f − (1−α)q+α 1 + h (1−α)q+α + kp + M 2 ! √ √ q Gm 1 + h qSc 2qω −y (1−α)q+α +kp +M 2 + e − (48) + 2 q 2 2 q qSc − (1−α)q+α − (kp + M ) q +ω √ √ Gm e−y qSc Gr (1 − e−q ) e−y qP ref f − , − q q q2 qP ref f − (1−α)q+α − (kp + M 2 ) q qSc − (1−α)q − (kp + M 2 )
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The Expressions for given by Eq. (32) and (33) are very complicated functions of the transformed parameter so, the inverse Laplace transform with respect to variable is very difficult to obtain by analytical method. For this reason, in this paper, we will use the Stehfest’s algorithm for numerical inversion of Laplace transform with respect to variable t. According with the Stehfest’s algorithm [48], [1], [31], 2m P [46], the solution is approximated by v(y, t) = ln(2) dj v¯(y, j ln(2) ), where m is t t j=1
14
Pr e-
p ro
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Figure 1: Variation of α with time t = 0.1 and other parameters are as h = 0.5,kp = 4,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,M = 0.9,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
the positive integer min(j,m) P im (2i)! dj = (−1)j+m (m−i)!i!(i−1)!(j−i)!(2i−j)! i=[ j+1 ] 2
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also, [r] indicate the integral part of real number r. Nonetheless, we got another guess for v¯(y, s) by Tzou’s calculation for approval of our numerical inverse Laplace N P1 4.7+kπi e4.7 1 4.7 k v(r, t) = t 2 v¯ r, t + Re (−1) v¯ r, t , k=1
urn
where Re(.) is the real part, i is the imaginary unit and N1 is a natural number [52].
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5.5. Numerical results The physical behaviour of dimensionless velocity with the effect of different values of effective Prandtl number, Grashof number, modified Grashof number, Magnetic parameter, Schmidt number, permeability parameter for the porous medium and α is displayed by the plotting the graphs for dimensionless velocity versus different system parameters by using computer software MATHCAD. Figures 1 to 7 are prepared for f (t) = t whereas in figures 8 to 14 we have f (t) = sin(wt). From the graphical illustrations the useful results are as follows: 15
Pr e-
p ro
of
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Figure 2: Variation of Gr with time t = 0.1 and other parameters are as h = 0.5,kp = 4,N = 0.5, α = 0.5, P ref f = 2, Gm = 0.75,M = 0.9,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 3: Variation of Gm with time t = 0.1 and other parameters are as h = 0.5,kp = 4,N = 0.5, Gr = 2, P ref f = 2, α = 0.5,M = 0.9,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
16
Pr e-
p ro
of
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Figure 4: Variation of kp with time t = 0.1 and other parameters are as h = 0.5,α = 0.5,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,M = 0.9,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 5: Variation of M with time t = 0.1 and other parameters are as h = 0.5,α = 0.5,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,kp = 4,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 6: Variation of P ref f with time t = 0.1 and other parameters are as h = 0.5,α = 0.5,N = 0.5, Gr = 2, M = 0.9, Gm = 0.75,kp = 4,Sc = 0.5 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 7: Variation of Sc with time t = 0.1 and other parameters are as h = 0.5,α = 0.5,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,kp = 4,M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.65.
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Figure 8: Variation of α with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.75,kp = 4,Sc = 0.5, M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 9: Variation of Gr with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, α = 0.5, P ref f = 2, Gm = 0.75,kp = 4,Sc = 0.5, M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 10: Variation of Gm with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 2, α = 0.5,kp = 4,Sc = 0.5, M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 11: Variation of kp with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.3,α = 0.5,Sc = 0.5, M = 0.9 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 12: Variation of M with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 2, Gm = 0.3,α = 0.5,Sc = 0.5, kp = 0.6 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 13: Variation of P ref f with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, M = 0.9, Gm = 0.75,α = 0.5,Sc = 0.5, kp = 0.6 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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Figure 14: Variation of Sc with time t = 0.1 and other parameters are as h = 0.5,ω = 0.2,N = 0.5, Gr = 2, P ref f = 4, Gm = 0.75,α = 0.5,M = 0.9, kp = 0.6 and slip and non-slip effect is also observed for slip parameter h = 0 and h = 0.2.
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1. The dimensionless velocity increases with the increase of Grashof number as well as modified Grashof number and in general the velocity for the case of non slip boundary condition is lesser than the case of slip boundary condition with slip parameter h = 0.65. 2. The velocity is a decreasing function of the parameters Prandtl effective number, Magnetic parameter and α. Moreover, the velocity with the non slip boundary condition i.e. h = 0 is larger than the velocity with non slip boundary condition (at value parameter h = 0.65). 3. By increasing the value of porosity parameter dimensionless velocity decreases and in general the velocity for the case of non slip boundary condition is less then the case of slip boundary condition. 4. With the small increase in Schmidt number the velocity decreases, whereas the influence is non responsive for large values of the parameter Sc . The velocity in non slip boundary condition is larger in comparison to the slip boundary condition.
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Highlights (for review)
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New fractional derivatives of Caputo–Fabrizio fractional derivative is applied to the unsteady magnetohydrodynamics (MHD) flow of a viscous fluid. Closed forms solution are obtained by means of the Laplace transforms. Results obtained illustrated distinct behaviors of fractional order solutions when compared with classical model solutions. A discussion is presented for mentioning how our results are innovative comparing to those already published.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Manuscript : Ms. Ref. No.: PHYSA-193415 Title: ANALYTICAL APPROACH ON THE UNSTEADY MAGNETOHYDRODYNAMIC CONJUGATE VISCOUS FLUID FLOW IN A POROUS MEDIUM WITH NONSINGULAR KERNEL DERIVATIVES Physica A
The corresponding author :
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Prof Zakia Hammouch Moulay Ismail University
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Authors : M. Mansha Ghalib, Azhar A. Zafar, M. Bilal Riaz, Z.Hammouch, Khurram Shabbir