MHD fractionalized Jeffrey fluid over an accelerated slipping porous plate

Abstract The primary target of this paper is to obtain the analytic solutions for the incompressible unsteady flow of fractionalized MHD Jeffrey fluid over an accelerating porous plate with linear slip effect is assumed between fluid and the plate. The governing equations of Jeffrey fluid are developed by fractional calculus approach. The velocity distribution and its corresponding shear stress both are obtained in terms of generalized M-function by using Laplace transform technique and considering all initial and boundary conditions. We have also discussed that obtained results of fractionalized MHD Jeffrey fluid for different cases for instance, with and without slip effects, with and without MHD and porosity effects. The influence of the different parameters affected on the flow characteristic is deliberated with the help of graphs. Finally, the analysis among different fluid models exhibits by graphical illustrations.


Introduction
It is investigated from the last few decades, that many researchers are much interested in Non-Newtonian uids. The explanation behind such quickening interest is because of the extensive scope of utilization of non-Newtonian uids. The non-Newtonian uids have several applications in di erent regions for an instant, geophysics and biological sciences, petroleum, and chemical industries. As we know that all non-Newtonian uids have both the properties of viscosity as well as elasticity. Non-Newtonian uids have many examples, such as honey, ketchup, oils, toothpaste and paints, liquid polymers and asphalt are represented some remarkable phenomena. These uids are several interesting applications and also used in our life. Many researchers have been proven that such type of uids has a non-linear relationship between shear rate and shear stress and these are not only signi cant for academia but also important for industries like polymer processing and the making of papers and foods. It is observed that the ow of Newtonian models is explained in a single constitutive equation but non-Newtonian uids ow cannot be represented by a single constitutive equation. In general, the rheological properties of liquids are indicated by their supposed constitutive conditions. Further, it is noticed that uids are satis ed Newtonian law of inner friction that shear stress is proportional to the viscosity of the uid gradient and nonnewtonian uids do not satis es the law of inner fraction. The governing equations of ow the non-Newtonian uids are more complex than Navier-Stokes equations [1][2][3]. In general Non-Newtonian uid is classi ed into three di erent classes namely, integral type, di erential type, and rate type. The Je rey uid model is considered in the present study, this type of uid shows the properties of the ratio of relation to retardation time. It has been proven that the non-Newtonian uid ow in the presence as well as in absence of magnetic eld has many applications in different elds for an instant, the handling of biological uids, plasma, mercury amalgams, liquid metals and alloys, blood and electromagnetic propulsion [4,5].
It is noti ed that Je rey uid is a special kind of non-Newtonian uid. In di erent models of non-Newtonian uids, it is shown that Je rey uid model is one of the signi cant models which de ned the best explanation of the rheological viscoelastic uids, it is used the time derivative rather than convected derivative [6 -11]. It is exhibited that Je rey uid models in nature are de ned in linear viscoelastic uids. As we know that in polymer industries Je rey uids have many applications, dilute polymer is one of them discussed by Farooq et al. [12] and Ara et al. [13]. The important and appropriate role of Je rey uid models is found in biological and uid mechanics due to viscoelastic behavior, it has been successfully used in the blood ow model. Je rey uid is interrelated with Newtonian uid and Maxwell uid as a special case [14].
It is found in physiology, the magnetohydrodynamic eld have many applications such as (i): blood a bio magnetic uid is created in the presence of hemoglobin molecule (ii): magnetic resonance imaging (iii): magnetic particles and magmatic devices are used as a drug carriers [15]. Tripathi et al. [16] investigated the in uence of MHD on nite length of cylindrical tube in Je rey uid ow. Das [17] and Afridi et al. [18] studied that in the existence of heat transform and slip impacts on MHD Jeffrey uid ow. Jamil et al. [19] investigated the in uence of MHD in Oldroyed-B uid ow over a plane. Mohanty et al. [20] have analyzed the in uence of MHD on Je rey uid ow and also discussed the e ects of chemical reaction on MHD ow of Je rey uid. Kashyap et al. [21] studied MHD slip ow of chemically reacting upper convected Maxwell uid through a dilating channel with heat source/sink. Hayat et al. [22] discovered in channel MHD ow of Je rey uid in series solutions. Zheng et al. [23] discussed the inuence of slip on MHD of a generalized oldroyd-B uid ow and determined analytic solution. Sharma et al. [24] studied viscous dissipation and thermal radiation e ects in MHD ow of Je rey nano uid through impermeable surface with heat generation/absorption. Jamil et al. [25] and Yaqing [26] are investigated the in uence of rst order slip condition of fractional Maxwell uid and found exact solutions of velocity eld and shear stress. Further MHD and slip in uence on Je rey uid with fractional calculus have been discussed by various authors in [27 -32].
The main purpose of this present paper is to investigate the in uence of MHD fractionalized Je rey uids over an accelerated slipping porous plate. In this present investigation, classical Je rey model is converted into fractionalized model by using Caputo fractional operator. The governing equations are transformed into fractionalized MHD Je rey uid by taking fractional parameter α. To nd analytic solutions for velocity eld and shear stress, we have used Laplace transform on fractionalized governing equations of Je rey uid. Here we have considered MHD and porous plate. The generalized solutions are determined and express in series form as well as in term of the generalized M−function satisfying all initial and boundary conditions. From the obtained general solutions the related solutions for ordinary Je rey uid can be determined as as special cases of general solutions. The second grade and Newtonian uid solutions can also be analyzed as limiting cases of fractional and ordinary Je rey uids. More precisely, the solutions for without slip condition for fractionalized and ordinary Je rey uid are also determined as a special cases and they are related with obtained pervious results in literature. Finally, the impacts of di erent materials parameters, slip and fractional parameters on the movement of ordinary Je rey uid and fractionalized uid are represented by graphical illustration. The di erence is also highlighted among the fractionalized Je rey uid, ordinary Je rey uid, Newtonian uid with MHD and porous and simple Newtonian uids graphically.

Fractionalized governing equations
In the following problem we consider a velocity eld and additional stress of the form of where u is the velocity in x-direction. The continuity equation has been satis ed, at t = the rest of uid, constitutive equations of Je rey uid and equation of motion provide that Syy = Syz = Szz = and the meaningful equations where τ = Sxy is the tangential stress, ν = µ/ρ is the kinematic viscosity, µ is the dynamic viscosity, ρ is the density of the uid and K = ϕ k and H = σB o ρ are magnetic and porosity constants, where ϕ is the porosity and κ is the permeability of the porous medium, B is the magnitude of applied magnetic eld and σ is the electrically conductively of uid. It is important to note that by putting λ = , and λ = λ = in above equations we recover the governing equations of second grade and Newtonian uids.
The governing equations of an incompressible MHD Je rey uid with fractional derivatives, performing the same motion in the absence of pressure gradient, are where α is the fractional parameter, and the fractional differential operator so called Caputo fractional operator D α t de ned by [32]

Formulation of the problem & solutions
Consider the incompressible fractionalized MHD Je rey uid models covering the space lying over an accelerating porous plate. It is exhibited that the velocity eld is measured into (y, t) and in the direction of y-axis depict perpendicular to the plate. In starting the velocity distribution of uid movement will be zero. It is considered at time t = + the plate moves with accelerated velocity At p . Due to force stress over the uid, gradually it starts to move above the plate. Its velocity is of the form ( ) while the governing equations are given by Eqs. (4) and (5). The appropriate initial and boundary conditions are The function H(t) is called Heaviside and θ is called slip parameter. If θ = , then considered that no boundary slip condition can be calculated. If slip parameter θ is limiting values, at the wall slip in uence is accrued, it is observed that in uence uid ow depends on length scale. Moreover, the natural conditions are employed u(y, t) and ∂u(y, t) ∂y → as y → ∞ and t > . (8) .

Investigation the solution of Velocity distribution eld
Employing Laplace transform, to Eq. (4) and having in mind initial and boundary conditions in Eqs. (6) and (7), we get subject to boundary condition where u(y, q) is called pre-image of u(y, t) and q is known as transform parameter. Solving Eqs.( ) by utilizing natural conditions in Eq. ( ), we get To avoid the complex computations of residuals and contours integrals, now we will simplify Eq. (12) and represents in-terms of series form we get velocity eld in the form of series Applying discrete inverse Laplace transform to Eq. (13), we obtain velocity eld Equvalently Now represents the above velocity eld in terms of generalized M-function, we get .

. Investigation of shear stress
Using the Laplace transform technique, to Eq. (5), by using initial condition to Eq. ( ) , we get where τ(y, q) is known as pre-image of τ(y, t). Now simplify Eq.(17) by using Eq. ( ), we get Rewrite above equation in terms of series form, we obtain Applying discrete inverse Laplace transform to Eq. (19), we get results in terms of series Express Eq. (20) in the form of generalized M-function, we get the required shear stress

. Fractionalized Je rey fluid in porous plate without MHD
Making H −→ into Eqs. ( ) and ( ), we obtain following velocity eld

MHD fractionalized Je rey fluid without porous e ects
Making K −→ into Eqs. ( ) and ( ), we obtain following velocity eld and corresponding shear stress for the above mentioned case.

. Fractionalized Je rey fluid without MHD and without porous e ects
Putting K, H −→ into Eqs. ( ) and ( ) or using K −→ into Eqs. ( ) and ( ) or making H −→ into Eqs. ( ) and ( ), we get following velocity eld and corresponding shear stress fractionalized Je rey uid without MHD and without porus e ects.
for MHD Newtonian uid on porous plate.

. MHD Newtonian fluid without porous e ect
Making K −→ into Eqs. ( ) and ( ) , we obtain following velocity eld and corresponding shear stress . .

MHD fractionalized Je rey fluid without slip e ects on porous plate
Letting θ −→ into Eqs. ( ) and ( ) , we get following velocity eld and corresponding shear stress for mentioned case

Numerical results and conclusions
In this paper, we have used fractional calculus because it is proved that for practical purpose it is very useful in engineering and sciences, fractional approach is help full when it comes to generalization of complex dynamics of uid motion. It shown in various eld like electromagnetism, electrochemistry nance, signal process and biochemistry, the important application subject to magnetic eld is the ow of electrical conducting uid, MHD deals with the dynamics of electrically conducting uids. In addition, the Magnetohydrodynamics (MHD) stream problem has increased signi cant premium in view of its extensive engineering and medical applications. The standards of MHD are utilized in the design of heat exchangers, pumps, ow meters, radar systems, power generation and so on. The signi cant of Je rey uid is that the result is limited as for polymer industry is concerned.
The object of this paper is to determine the exact solutions unsteady ow of MHD fractionalized Je rey uid over an accelerating porous plate under the consideration of rst order slip boundary condition. The analytic solution is obtained for the velocity eld and shear stress in terms of generalized M-function and by employing Laplace transform techniques. At time t=0 the uid velocity eld is zero, but at t = + the uid is immediately moved with velocity At p in its plate with assuming rst order linear slip between uid and the plate. Here the velocity distribution and shear stress of uid are determined against the physical parameters, such as, MHD parameter H, porosity K, slip impact θ, relaxation and retardation time parameters λ and λ and fractional parameter α. The obtained results for velocity uid ow and its corresponding shear stress are exhibited through limiting cases by taking α = , H = , K = , M = , λ = λ = and θ = . Here the results of fractionalized MHD Je rey uid are minimized into the ordinary Je rey uid, without MHD e ect, without porosity e ect, Newtonian uid and with and without slip e ects. Furthermore, in order to discuss some physical aspect of the determined solutions, we have made graphs of velocity and shear stress against vertical height y. In all sketched diagram the values of physical parameters are taken as common, for instance, A = , ν = .
The impacts of time are vital for us to discuss here, Figs. 2 is represented that the velocity distribution and shear stress at di erent values in t, it is noted from these diagram, the velocity eld and shear stress (in absolute Finally, for analysis for the velocity eld and shear stress to four di erent uid models (fractionalized Je rey, ordinary Je rey, MHD Newtonian uid with porosity, simple Newtonian uids) for three di erent values of slip e ect parameter θ = . , . and 0. These uid models are disclosed together in sketched diagrams 10-12 for same values of t and material parameters, it represented clearly from plotted Figs. 10, 11 and 12 that as anticipated the simple Newtonian uid is more rapidly grow and fractionalized Je rey uid is slowly grow. It is clearly seen that the ordinary and fractionalized Je rey uids at di erent values of θ, represents the ordinary Je rey is moved faster than fractionalized Je rey uid. It can also be described from these gures that for highest values of slip parameter the four uid models are decelerated as expected. In the making of all graphs, we used SI units and all graphs are prepared on Mathcad software and then using paint and Inkscape, we obtained the nal form of the graphs.         , µ = , λ = , λ = , H = . , K = , α = . , θ = , p = and t = s. , µ = , λ = , λ = , H = . , K = , α = . , θ = . , p = and t = s. partment of Mathematics, NED University of Engineering & Technology, Karachi-75270, Pakistan and also would like to express their gratitude to Higher Education Commission of Pakistan for assisting and facilitating this research work.