Solution of the Fractional Form of Unsteady Squeezing Flow through Porous Medium

We propose two friendly analytical techniques called Adomian decomposition and Picard methods to analyze an unsteady axisymmetric flow of nonconducting, Newtonian fluid. This fluid is assumed to be squeezed between two circular plates passing through porous medium channel with slip and no-slip boundary conditions. A single fractional order nonlinear ordinary differential equation is obtained by means of similarity transformation with the help of the fractional calculus definitions. The resulting fractional boundary value problems are solved by the proposed methods. Convergence of the two methods’ solutions is confirmed by obtaining various approximate solutions and various absolute residuals for different values of the fractional order. Comparison of the results of the two methods for different values of the fractional order confirms that the proposed methods are in a well agreement and therefore they can be used in a simple manner for solving this kind of problems. Finally, graphical study for the longitudinal and normal velocity profiles is obtained for various values of some dimensionless parameters and fractional orders.


Introduction
The importance of the fluid flow through a porous medium goes back a long time in various applications in life such as agriculture, industry, and oil and gas production, where the focus was on estimating and optimizing production.Similarly, another important application is the simulation of ground water pollution, mostly occurring due to leakage of chemicals from tanks and oil pipelines.The objective is to consider ground water as one medium and polluted water as another, so that the spreading in the latter medium and its consequences can be studied.
In recent times, after the introduction of the modified Darcy Law [1], analysis through a porous medium has advantage in an important topic for the research community such as reservoir, petroleum, chemical, civil, environmental, and agricultural and biomedical engineering.Some practical applications in these fields include chemical reactors, filtration, geothermal reservoirs, ground water hydrology, and drainage and recovery of crude oil from pores of reservoir rocks [2][3][4][5][6][7].
The importance of squeezing flow is due to its wide applications in many fields such as chemical, biomechanics, food industries, and mechanical and industrial engineering.Practical applications of squeezing flows in these fields are modeling of lubrication systems, polymer processing, compression and injection molding, and so on.These flows are induced by applying normal stresses or vertical velocities by means of a moving boundary, which can be frequently observed in various hydrodynamical tools and machines.Stefan [8] is one of the first researchers who operated on the development of squeezing flows and an ad hoc asymptotic solution to Newtonian fluids.In [9] Thorpe has established an explicit solution to the squeeze flow considering inertial terms.However, in [10], P. S. Gupta and A. S. Gupta proved that the solution given in [9] fails in satisfying boundary conditions.Verma in [11] and Singh et al. in [12] have presented numerical solutions for squeezing flow between parallel plates.Leider and Byron performed in [13] theoretical analysis of power-law fluid between parallel disks.The optimal homotopy asymptotic method (OHAM) has been used by Qayyum et al. [14] to analyze the unsteady axisymmetric flow of nonconducting Newtonian fluid squeezed between two circular plates with slip and no-slip boundaries.Also, the homotopy perturbation method (HPM) has been developed by Qayyum et al. [15], to model and analyze the unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates passing through a porous medium channel with slip boundary condition.The new iterative and Picard methods had been used by Hemeda and Eladdad in [16] for solving the fractional form of unsteady axisymmetric flow of a nonconducting, Newtonian fluid squeezed between two circular plates with slip and no-slip boundaries.For more studies on squeezing flow through porous medium and also for more theoretical and experimental studies on squeezing flow, you can see [17][18][19][20][21][22][23][24][25].
Fractional calculus becomes one of the important branches of applied mathematics that deals with derivatives and integrals of arbitrary orders.It is the generalized differential and integral calculus of an arbitrary order.Recently, it has the importance of many researchers because of its wide appearance in many applications in fluid mechanics, viscoelasticity, biology, physics, and engineering.Moreover, the ordinary differential operator is a local operator, but the fractional order differential operator is nonlocal.Consequently, attention has been given to the solutions of fractional differential equations for the most important fields of physics, fluid mechanics, and so on.The exact analytical solutions to most nonlinear fractional order problems cannot be found.Therefore, approximation and numerical techniques can be used.Picard method [26][27][28][29] and the Adomian decomposition method [30][31][32][33][34] are two powerful approaches of these techniques which can be used in simple manner and short time to obtain analytical approximations to nonlinear problems and they are particularly valuable as tools for researchers, because they provide immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions to nonlinear differential equations without linearization or discretization.
In the last years, the applications of the proposed methods are extended to the fractional differential equations.Our objective of this article is to prepare and utilize the proposed methods to obtain an analytical solution to the fractional form of an unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates passing through porous medium with slip and no-slip boundary conditions.Validity of the two methods is confirmed by comparing the obtained results.In addition, the effects of various fractional order, constant containing permeability, Reynolds number, and slip parameter on the solution are studied tabularly and graphically.

Formulation of the Problem
In this section, the unsteady axisymmetric squeezing flow of incompressible first-grade fluid with density , viscosity , and kinematic viscosity ], squeezed between two circular plates having speed   () and passing through porous medium channel, is considered with a fractional order form.At any time , it is assumed that the distance between the two circular plates is 2ℎ().Also, it is assumed that -axis is the central axis of the channel while -axis is taken as normal to it.Plates move symmetrically with respect to the central axis  = 0 while the flow is axisymmetric about  = 0.The longitudinal and normal velocity components in radial and axial directions are   (, , ) and   (, , ), respectively.For more physical explanation, see [14][15][16].
The basic system of equations describing the motion of the fluid is where where   () = ℎ/ is the plates velocity.The boundary values in (2) are due to slipping at the upper plate when  = ℎ and symmetry at  = 0.If we introduce the dimensionless parameter (1) transforms to The boundary conditions on   and   are The elimination of  between ( 5) and (6) gives where ∇ 2 is the Laplacian operator.
Defining the components of velocity as follows [10]: we see that ( 4) is identically satisfied and (8) becomes where In the above equations  and  are functions of  but, for similarity solution, we consider them as constants.Since   = ℎ/, integrating first equation of (11), we get where  and  are constants.The plates move away from each other symmetrically with respect to  when  > 0 and  > 0. The squeezing flow exists when the plates approach each other when  < 0,  > 0 and ℎ() > 0. From (11) and (12) it follows that  = −.Then (10) becomes Using ( 7) and ( 9), the boundary conditions in cases of no-slip and slip boundaries at the upper plate can be defined in the form: (14b)

Fractional Calculus
In the following, we state some definitions of the fractional calculus, which can be used in this work.
Definition 1.The Riemann-Liouville fractional integral operator of order  > 0 and of a function () ∈   and  ≥ −1 is defined as follows [35]: Also, for    , we can define Definition 2. The fractional order derivative    of () in the Caputo sense is defined as follows [36]: For the Caputo fractional derivative operator,    , we obtain For the Riemann-Liouvill fractional integral and Caputo fractional derivative of order , we have the following relations.

Analysis of the Proposed Methods
In this section, we present the analysis of the proposed methods with their suitable algorithms for the fractional order problems.

Picard Method (PM).
To present this method, let us consider the general fractional order problem with an arbitrary order  > 0 [26][27][28][29]: where    is the time fractional differential operator of order  > 0. According to the fractional integral operators, the fractional order problem (21a) and (21b) takes the equivalent fractional integral form: where and    is the inverse of    .The required solution () for ( 22) which is also the solution for (21a) and (21b) can be obtained as the limit of a sequence of functions  +1 () generated by the recurrence relation: where () = lim →∞   ().

Adomian Decomposition Method (ADM).
To illustrate the idea of this method, let us consider the fractional order problem [30][31][32][33][34]: subject to the initial values: where ,  are linear and nonlinear operators and () is a nonhomogeneous term.The method is based on applying the fractional integral operator    , the inverse of the fractional differential operator    , to both sides of (24a) and (24b) to obtain The ADM suggests that the solution () be decomposed into the infinite series of components: and the nonlinear term  in (24a) is decomposed as where   are called Adomian polynomials.Substituting the decomposition series ( 26) and ( 27) into both sides of ( 25) gives Following the decomposition method, we introduce the recurrence relation as The Adomian polynomial   can be calculated for all forms of nonlinearity according to specific algorithms constructed by Adomian.The general form of the Adomian polynomials formula for   is This formula is easy to compute.Finally, we approximate the solution () by the truncated series:

Applications
In this section, we illustrate the application of the two proposed methods to solve the nonlinear fractional order ordinary differential equation (20) subject to the boundary conditions (14a) and (14b).20) and (14a) and (14b), the initial value fractional order problem

PM. Using (
according to ( 22) and ( 23), becomes Therefore, by Wolframe Mathematica 10 Package, we can obtain the following first few components of the solution: and so on.In the same manner, the rest of the components can be obtained.The 4th-order term solution for (32), in series form, is given by In the special case,  = 4, (35) gives Using the boundary conditions in (14a) and (14b) with the initial conditions in (32), the unknowns  and  for fixed values of  and  in (36) can be easily determined.In case of no-slip boundary, then  = 1.5 and  = −3.0.For  =  = 0.5, the solution is  (38)

ADM.
According to the recurrence relation (29), the initial value fractional order problem (32) gives where and, therefore, the first few components of the solution are as follows: and so on.In the same manner the rest of the components can be obtained.The 4-term solution is In the special case,  = where û is the 4-term approximate solution in (35) or (42) for (32).
It is important to note that, when Re = 0, û will be the exact solution.But, in nonlinear problems, this case does not usually occur.
It is clear from the obtained results that the above considered methods are used simply and accurately without linearization or discretization with their difficulties.Therefore, these methods are powerful methods for solving the nonlinear fractional order differential equations.

Numerical Results and Discussion
In this work, an unsteady axisymmetric flow of incompressible, nonconducting Newtonian fluid squeezed between two circular plates passing through porous medium channel with slip and no-slip boundary conditions is considered.The obtained nonlinear fractional order problems are solved analytically through PM and ADM using Wolframe Mathematica 10 Package.
In addition to the fractional order , there are three parameters: constant containing permeability , Reynolds number , and slip parameter  in the considered problem.Our discussions of the obtained results are based on different values of  and these parameters.First, in case of no-slip boundary, the 4th-order approximate solutions obtained by the two methods for different values of , , and  are illustrated in Tables 1 and 2. Secondly, Tables 3 and 4 represent the 4th-order absolute residual errors |Re| for different values    13 and 14 for various values of , , and  at  = 1 in case of no-slip and slip boundaries conditions.Finally, the analysis of the absolute residual errors is shown in Tables 15 and 16 for various values of ,while keeping the  other parameters fixed for different order solutions obtained by the two methods in case of no-slip and slip boundaries.
For various values of  and , the results in Tables 1, 2, 5, and 6 indicate that as  → 4,  ADM →  PM and the results in Tables 3, 4, 7, and 8 (for 4th-order solution), 9 and 10 (for 3rd-order solution), and 11 and 12 (for 2nd-order solution) indicate that increasing  and converging it to 4 implies decreasing the absolute residual errors |Re| as  → 1.     7 and 8 indicate the inverse.In addition to the abovementioned, we noted from the results in Tables 13 and 14 that the absolute difference between the ADM and PM solutions are almost equal especially for small values of  for different values of , , and  at  = 1 which means that the two methods are in a good agreement in solving the proposed problem and therefore the ADM is a reliable method in solving the fractional order problems Finally, the results in   boundaries with  = 1 (Figure 1), while in Figure 2 it is by ADM.
The effect of different values of the Reynolds number  and the constant containing permeability  at the fractional order  = 3.7 on velocity profiles of the two methods in case   of no-slip boundary is shown in Figures 3 and 4, while the effect in case of slip boundary for different values of , , and slip parameter  at  = 3.7 are shown in Figures 5, 6,  and 7.In these profiles we varied  as  = 1, 2, 3 for the sake of comparison at  = 3 and  = 3.7 (Figure 3) and varied  as  = 1, 2, 3 for the sake of comparison at  = 3 and  = 3.7 (Figure 4) in case of no-slip boundary.In case of slip boundary, we varied  as  = 1, 2, 3 for the sake of comparison at  = 3,  = 1, and  = 3.7 (Figure 5), varied  as  = 1, 2, 3 for the sake of comparison at  = 3,  = 1, and  = 3.7 (Figure 6), and varied  as  = .75, 1, 1.25 for the sake of comparison at  =  = 3 and  = 3.7 (Figure 7).
According to the abovementioned data we have observed that, in case of no-slip boundary, the normal velocity (NV) increases with increasing  (Figure 3(a)) and the longitudinal velocity (LV) increases near the central axis of the channel (Figure 3(b)).It has been analyzed that the NV monotonically increases while the LV monotonically decreases from  = 0 to  = 1 for fixed value of  at a given time.Almost a similar behavior for the NV and LV is with varying  (Figure 4).In case of slip boundary, we have noted that the NV decreases as  increases (Figure 5(a)) and the LV decreases near the central axis of the channel (Figure 5(b)).Also, almost a similar behavior for the NV and LV is with varying  (Figure 6).Moreover, we have observed that the NV increases with increasing  (Figure 7

Conclusion
In this work, we find the analytical solution and then the similarity solution for the fractional form of an unsteady axisymmetric flow of incompressible, nonconducting Newtonian fluid squeezed between two circular plates passing through porous medium channel using the PM and ADM Mathematical Problems in Engineering in cases of no-slip and slip boundary conditions.The convergence of the ADM is determined through various order approximate solutions.Moreover, the validity of ADM is checked by comparing the analytical ADM and the numerical PM solutions.We observed that the similarity solution occurs when the distance between the plates varies as ( + ) 1/2 and squeezing flow exists when  < 0,  > 0 and ( + ) 1/2 > 0. Some key findings related to the present study are as follows.

(
Slip at the wall) .

Table 9 :
Third-order absolute residual errors |Re| for different values of  and , at  = 0.3,  = 0.5,  = 1 by the two methods.

Figure 1 :
Figure 1: Residuals for various values of  at  = .5, = .7,and  = 1 by PM: (a) in case of no-slip boundary and (b) in case of slip boundary.

Figure 2 :Figure 3 :
Figure 2: Residuals for various values of  at  = .5, = .7,and  = 1 by ADM: (a) in case of no-slip boundary and (b) in case of slip boundary.
(a)) and the LV increases near the wall and decreases near the central axis of the channel (Figure 7(b)).

Figure 4 :Figure 5 :
Figure 4: Velocity profiles for various values of  at  = 3.7 and  = 3 by ADM in case of no-slip boundary: (a) normal velocity and (b) longitudinal velocity.
No-Slip Boundary.(i) The NV increases with the increase in .(ii) The LV increases near the central axis of the channel with the increase in .(iii) The NV increases monotonically while the LV decreases monotonically from  = 0 to  = 1 for fixed value of  at a given time.(iv) For the variation in , we have noted almost a similar behavior.In Case of Slip Boundary.(i) The NV decreases with the increase in  with fixed  and .(ii) The LV decreases near the central axis of the channel with the increase in  with fixed  and .(iii) For the variation in , we have observed almost a similar behavior for fixed  and .(iv) The NV increases with the increase in  with fixed  and .(v) The LV increases near the wall and decreases near the central axis of the channel with fixed  and .(vi)  and  have similar effects on the NV and LV, while ,  have opposite effects on the NV and LV components.

Table 1 :
Solutions for different values of  and  at  = 0.5,  = 0.3, in case of no-slip boundary, by the two methods.

Table 3 :
Fourth-order absolute residual errors |Re| for different values of ,  at  = 0.3,  = 0.5 in case of no-slip boundary, by the two methods.

Table 4 :
Fourth-order absolute residual errors |Re| for different values of ,  at  = 0.3,  = 0.7 in case of no-slip boundary, by the two methods.

Table 5 :
Solutions for different values of  and  at  = 0.5,  = 0.3, and  = 1, in case of slip boundary, by the two methods.of,,and .Thirdly, Tables5, 6, 7, and 8 are in case of slip boundary at  = 1.Fourthly, Tables 9 and 10 and 11 and 12 represent the 3rd-order and 2nd-order absolute residual errors |Re| for different values of , , and  in case of no-slip and slip boundaries with  = 1 for the two methods.Fifthly, the absolute difference between the ADM and PM solutions is illustrated in Tables

Table 6 :
Solutions for different values of  and  at  = 0.3,  = 0.5, and  = 1, in case of slip boundary, by the two methods.

Table 7 :
Fourth-order absolute residual errors |Re| for different values of  and , at  = 0.3,  = 0.5, and  = 1, in case of slip boundary, by the two methods.

Table 8 :
Fourth-order absolute residual errors |Re| for different values of  and , at  = 0.3,  = 0.7, and  = 1 in case of slip boundary, by the two methods.

Table 10 :
Third-order absolute residual errors |Re| for different values of  and , at  = 0.3,  = 0.7, and  = 1 by the two methods.

Table 11 :
Second-order absolute residual errors |Re| for different values of  and , at  = 0.3,  = 0.5, and  = 1 by the two methods.Also, increasing the order of solution implies decreasing |Re|.Moreover, the results in Tables3 and 4indicate that |Re| obtained by PM are less than |Re| that obtained by ADM in case of no-slip boundary, while, in case of slip boundary, the results in Tables

Table 12 :
Second-order absolute residual errors |Re| for different values of  and , at  = 0.3,  = 0.7, and  = 1 by the two methods.

Table 13 :
Absolute difference of ADM and PM solutions; | ADM −  PM | for various values of  and , at  = 0.3,  = 0.5, and  = 1.

Table 14 :
Absolute difference of ADM and PM solutions; | ADM −  PM | for various values of  and , at  = 0.3,  = 0.7, and  = 1.Tables 9 and 10 and 11 and 12 indicate that the 3rd-order and 2nd-order absolute residual errors |Re|, respectively, are equal for the two methods in case of no-slip and slip boundaries.Figures1 and 2show the residual errors Re for various values of , , and  by PM in cases of no-slip and slip

Table 15 :
Average absolute residual errors for fourth-order solutions for various values of , at  = 0.3,  = 0.7, and  = 1 by the two methods.

Table 16 :
Average absolute residual errors for third-order and second-order solutions for various values of , at  = 0.3,  = 0.7, and  = 1 by the two methods.