Analysis of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries Using OHAM

Abstract

In this manuscript, An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is studied with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations is reduced to a single fourth order ordinary differential equation. The resulting boundary value problems are solved by optimal homotopy asymptotic method (OHAM) and fourth order explicit Runge-Kutta method (RK4). It is observed that the results obtained from OHAM are in good agreement with numerical results by means of residuals. Furthermore, the effects of various dimensionless parameters on the velocity profiles are investigated graphically.

Academic Editor:María Isabel Herreros

1, Department of Mathematics, National University of Computer & Emerging Sciences, FAST Peshawar Campus, Peshawar 25000, Pakistan

Received 27 May 2014; Revised 11 August 2014; Accepted 12 September 2014; 23 February 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The squeezing of an incompressible viscous fluid between two parallel plates is an essential type of flow that is frequently observed in many hydro dynamical tools and machines. In food industries, squeezing flows have several applications particularly in chemical engineering [1, 2]. Compression and injection molding, polymer processing, and modeling of lubrication system are some practical examples of squeezing flows. The modeling and analysis of squeezing flow has started in nineteenth century and continues to receive significant attention due to its vast applications in biophysical and physical sciences. The initial work in squeezing flows has been done by Stefan [3], who developed an ad hoc asymptotic solution of Newtonian fluids. An explicit solution of the squeeze flow, considering inertial terms, has been established by Thorpe and Shaw [4]. However, P. S. Gupta and A. S. Gupta [5] proved that the solution set up by [4] fails to satisfy the boundary conditions. Considering fluid inertia effects, Elkouh [6] studied the squeeze film between two plane annuli. Verma [7] and Singh et al. [8] have conducted Numerical solutions of the squeezing flow between parallel plates. Leider and Bird [9] performed theoretical analysis for squeezing flow of power-law fluid between parallel disks. Analytic solution for the squeezing flow of viscous fluid between two parallel disks with suction or blowing effect has been proposed by Domairry and Aziz [10]. Islam et al. [11] studied Newtonian squeezing fluid flow in a porous medium channel. Ullah et al. [12] discussed the Newtonian fluid flow with slip boundary condition keeping MHD effect into account. Siddiqui et al. [13] investigated the unsteady squeezing flow of viscous fluid with magnetic field. Apart from the mentioned researchers, other prominent scholars have conducted various theoretical and experimental studies of squeezing flows [14-17].

The difference between fluid and boundary velocity is proportional to the shear stress at the boundary. The dimension of proportionality constant is length, which is known as slip parameter. In fluids with elastic character, slip condition has great importance [18]. It has many applications in medical sciences, for instance, polishing artificial heart valves [19]. There are various situations in which no-slip boundary condition is inappropriate. Some of these situations include polymeric liquids when the weight of molecule is high, flow on multiple interfaces, fluids containing concerted suspensions, and thin film problems. The general boundary condition which shows the fluid slip at the wall was initially proposed by Navier [20]. Recently, Ebaid [21] studied the effect of magnetic field in Newtonian fluid in an asymmetric channel with wall slip conditions.

Most of scientific incidents are modeled by nonlinear partial or ordinary differential equations. In literature, we have variety of perturbation techniques which can solve nonlinear boundary value problems analytically. But the limitations of these techniques are based on the assumption of small parameters. Detailed review of these methods is given by He [22]. In recent times, the ideas of Homotopy and Perturbation have been combined together. Liao [23] and He [24, 25] have done the primary work in this regard. In series of papers, Marinca with various scholars used OHAM to find the approximate solution of nonlinear differential equations arising in heat transfer, steady flow of a fourth-grade fluid, and thin film flow [26-28].

In this work, OHAM is used to analyze an unsteady squeezing fluid flow between two circular nonrotating disks with slip and no-slip boundary conditions. In addition, movement of the circular plates is considered to be symmetric about the axial line and the fluid is considered to be Newtonian, incompressible and viscous. Sections 2 and 3 include the description and mathematical formulation of the problem. Sections 4, 5, and 6 present the basic theory of OHAM and its application in case of no-slip and slip boundaries. Results and discussions are given in Section 7 while conclusions are mentioned in Section 8.

2. Description of the Problem

The unsteady axisymmetric squeezing flow of incompressible Newtonian fluid with density [figure omitted; refer to PDF] , viscosity [figure omitted; refer to PDF] and kinematic viscosity [figure omitted; refer to PDF] , squeezed between two circular plates having speed [figure omitted; refer to PDF] is considered. It is assumed that at any time [figure omitted; refer to PDF] , the distance between two circular plates is [figure omitted; refer to PDF] . It is also assumed that [figure omitted; refer to PDF] -axis is the central axis of the channel while [figure omitted; refer to PDF] -axis is taken normal to it. Plates move symmetrically with respect to the central axis [figure omitted; refer to PDF] while the flow is axisymmetric about [figure omitted; refer to PDF] . The longitudinal and normal velocity components in radial and axial directions are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. The geometrical interpretation of the problem is given in Figure 1.

Figure 1: Geometrical interpretation of the problem.

[figure omitted; refer to PDF]

3. Mathematical Formulation

The governing equations of motion are [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the velocity vector, [figure omitted; refer to PDF] is the pressure, [figure omitted; refer to PDF] is the body force, [figure omitted; refer to PDF] is the Cauchy stress tensor, [figure omitted; refer to PDF] is the Rivlin-Ericksen tensor, and [figure omitted; refer to PDF] is the coefficient of viscosity. Now we formulate the unsteady two-dimensional flow. Let us assume that [figure omitted; refer to PDF] and introduce the vorticity function [figure omitted; refer to PDF] and generalized pressure [figure omitted; refer to PDF] as [figure omitted; refer to PDF] Equations (1) are reduced to [figure omitted; refer to PDF] The boundary conditions on [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the velocity of the plates. The boundary conditions (7) are due to symmetry at [figure omitted; refer to PDF] and no-slip at the upper plate when [figure omitted; refer to PDF] . If we introduce the dimensionless parameter [figure omitted; refer to PDF] Equations (4) and (6) transforms to [figure omitted; refer to PDF] The boundary conditions on [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] After eliminating the generalized pressure between (11) and (12), we obtained [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Laplacian operator.

Defining velocity components as [5] [figure omitted; refer to PDF] we see that (10) is identically satisfied and (14) becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Here both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are functions of [figure omitted; refer to PDF] but we consider [figure omitted; refer to PDF] and [figure omitted; refer to PDF] constants for similarity solution. Since [figure omitted; refer to PDF] , Integrate first equation of (17), we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are constants. The plates move away from each other symmetrically with respect to [figure omitted; refer to PDF] when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Also the plates approach to each other and squeezing flow exists with similar velocity profiles when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . From (17) and (18) it follows that [figure omitted; refer to PDF] . Then (16) becomes [figure omitted; refer to PDF] Using (13) and (15) we determine the boundary conditions in case of no-slip and slip at the upper plate as follows: [figure omitted; refer to PDF]

4. Basic Theory of OHAM [26, 29-32]

Let us apply OHAM to the following differential equation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] represents an independent variable, [figure omitted; refer to PDF] is unknown function and [figure omitted; refer to PDF] is known function. [figure omitted; refer to PDF] are boundary, nonlinear, and linear operators, respectively.

According to OHAM, we construct Homotopy [figure omitted; refer to PDF] [figure omitted; refer to PDF] which satisfies [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is an embedding parameter, [figure omitted; refer to PDF] is a nonlinear auxiliary function for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is an unknown function. Clearly, when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , it holds that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

Thus, as [figure omitted; refer to PDF] varies from 0 to 1, the solution [figure omitted; refer to PDF] approaches from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] .

We choose the auxiliary function [figure omitted; refer to PDF] in the form of [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are convergence controlling constants to be determined.

To obtain an approximate solution, we expand [figure omitted; refer to PDF] in a Taylor series about [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Substituting (25) into (23) and equating the coefficients of like powers of [figure omitted; refer to PDF] , we obtain the following equations.

The zeroth-order problem is [figure omitted; refer to PDF] First-order problem is [figure omitted; refer to PDF] Second-order problem is [figure omitted; refer to PDF] The general equations for [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF] where the coefficient of [figure omitted; refer to PDF] in the expansion of [figure omitted; refer to PDF] about [figure omitted; refer to PDF] is [figure omitted; refer to PDF] . [figure omitted; refer to PDF]

It is noted that the convergence of the series (25) depends upon [figure omitted; refer to PDF] . For convergence at [figure omitted; refer to PDF] , the [figure omitted; refer to PDF] th order approximation [figure omitted; refer to PDF] is [figure omitted; refer to PDF] Substituting (31) in (22), the expression for residual is [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] will be the exact solution but usually this does not happen in nonlinear problems.

There are various methods to find the optimal values of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . We apply the method of least square and Galerkin's method in the following manner:

In method of least square [figure omitted; refer to PDF] Minimizing [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] In Galerkin's method, we solve the following system for [figure omitted; refer to PDF] : [figure omitted; refer to PDF] To find appropriate [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in the domain of the problem. Approximate solution of order [figure omitted; refer to PDF] is well-determined with these known constants.

5. Application of OHAM in Case of No-Slip Boundary

Using (19) and (20) various order problems are as follows:

Zeroth-order problem [figure omitted; refer to PDF] First-order problem [figure omitted; refer to PDF] Second-order problem [figure omitted; refer to PDF] Third-order problem [figure omitted; refer to PDF] Fourth-order problem [figure omitted; refer to PDF] By considering fourth-order solution, we have [figure omitted; refer to PDF] The residual of the problem is [figure omitted; refer to PDF] We apply Galerkin's method to find constant [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Solving (43) and keeping [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] Using above value of [figure omitted; refer to PDF] , the approximate solution is [figure omitted; refer to PDF]

6. Application of OHAM in Case of Slip Boundary

Using (19) and (21) different order problems are as follows.

Zeroth-order problem [figure omitted; refer to PDF] First-order problem [figure omitted; refer to PDF] Second-order problem [figure omitted; refer to PDF] Third-order problem [figure omitted; refer to PDF] Fourth-order problem [figure omitted; refer to PDF] By considering fourth-order solution, we have [figure omitted; refer to PDF] The residual of the problem is [figure omitted; refer to PDF] We apply Galerkin's method to find constant [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Solving (53) and taking [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] Using above value of [figure omitted; refer to PDF] , the approximate solution is [figure omitted; refer to PDF]

7. Results and Discussions

In this article we considered the unsteady axisymmetric flow of nonconducting, incompressible Newtonian fluid between two circular plates. The resulting nonlinear boundary value problems are solved with OHAM and fourth-order Runge-Kutta method using Mathematica 7.0.

Tables 1, 3, and 5 reflect OHAM solutions along with residuals in case of no-sip and slip boundaries for various values of Reynolds number [figure omitted; refer to PDF] and slip parameter [figure omitted; refer to PDF] . Also, Tables 2, 4, and 6 represent RK4 solutions along with residuals in case of no-slip and slip boundaries for various values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . All the tables demonstrate that results obtained using OHAM are in agreement with RK4 by means of residuals. In addition to above mentioned tables, Table 7 shows the comparison of solutions obtained from OHAM and RK4 for various values of Reynolds number [figure omitted; refer to PDF] .

Table 1: OHAM solutions along with residuals for various [figure omitted; refer to PDF] in case of no-slip boundary.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Solution	Residual	Solution	Residual	Solution	Residual
0.0	0.	0.	0.	0.	0.	0.
0.1	0.150158	5.37334 × 10-11	0.151534	-6.69283 × 10-9	0.152999	-6.23412 × 10-8
0.2	0.297237	3.14675 × 10-10	0.299827	1.66025 × 10-9	0.302582	6.58388 × 10-8
0.3	0.438170	8.51847 × 10-10	0.441661	2.33504 × 10-8	0.445373	3.42092 × 10-7
0.4	0.569900	1.51431 × 10-9	0.573869	3.93945 × 10-8	0.578082	4.93942 × 10-7
0.5	0.689397	1.90824 × 10-9	0.693354	2.81614 × 10-8	0.697548	2.42092 × 10-7
0.6	0.793661	1.33301 × 10-9	0.797122	-3.19654 × 10-8	0.800780	-6.53046 × 10-7
0.7	0.879734	-1.59255 × 10-9	0.882300	-2.21911 × 10-7	0.885004	-3.23415 × 10-6
0.8	0.944705	-1.02814 × 10-8	0.946167	-8.84707 × 10-7	0.947702	-1.2206 × 10-5
0.9	0.985722	-3.446 × 10-8	0.986180	-3.21268 × 10-6	0.986659	-4.41008 × 10-5
1.0	1.	-1.03509 × 10-7	1.	-1.11826 × 10-5	1.	-1.54022 × 10-4

Table 2: RK4 solutions along with residuals for various [figure omitted; refer to PDF] in case of no-slip boundary.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Solution	Residual	Solution	Residual	Solution	Residual
0.0	0.	2.88535 × 10-7	0.	9.66906 × 10-5	0.	5.30549 × 10-4
0.1	0.150158	3.88278 × 10-8	0.151534	5.71393 × 10-6	0.152999	3.04600 × 10-5
0.2	0.297237	-9.97789 × 10-9	0.299827	-1.41434 × 10-6	0.302582	-7.52772 × 10-6
0.3	0.438170	2.93767 × 10-9	0.441661	3.95997 × 10-7	0.445373	2.10296 × 10-6
0.4	0.569900	-7.51436 × 10-10	0.573869	-1.15203 × 10-7	0.578082	-6.18081 × 10-7
0.5	0.689397	-4.71705 × 10-10	0.693354	-9.6761 × 10-9	0.697548	-2.95916 × 10-8
0.6	0.793661	2.23004 × 10-9	0.797122	1.4593 × 10-7	0.800780	7.14133 × 10-7
0.7	0.879734	-7.18245 × 10-9	0.882300	-4.86272 × 10-7	0.885004	-2.39245 × 10-6
0.8	0.944705	2.68744 × 10-8	0.946167	1.77796 × 10-6	0.947702	8.70664 × 10-6
0.9	0.985722	-1.11599 × 10-7	0.986180	-7.28952 × 10-6	0.986659	3.56157 × 10-5
1.0	1.	-2.71683 × 10-6	1.	-1.47695 × 10-4	1.	6.90088 × 10-4

Table 3: OHAM solutions along with residuals for various values of [figure omitted; refer to PDF] in case of slip boundary.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Solution	Residual	Solution	Residual	Solution	Residual
0.0	0.	0.	0.	0.	0.	0.
0.1	0.072692	3.18345 × 10-8	0.071220	3.15621 × 10-7	0.069593	1.74626 × 10-6
0.2	0.147091	-1.72789 × 10-8	0.144268	-9.14579 × 10-8	0.141147	-1.13148 × 10-7
0.3	0.224893	-1.96995 × 10-7	0.220953	-1.66461 × 10-6	0.216599	-7.79061 × 10-6
0.4	0.307773	-4.94482 × 10-7	0.303048	-4.29173 × 10-6	0.297832	-2.07401 × 10-5
0.5	0.397370	-7.99437 × 10-7	0.392277	-6.99917 × 10-6	0.386657	-3.41643 × 10-5
0.6	0.495283	-8.5784 × 10-7	0.490289	-7.5499 × 10-6	0.484784	-3.70689 × 10-5
0.7	0.603056	-2.1053 × 10-7	0.598654	-1.91141 × 10-6	0.593805	-9.6958 × 10-6
0.8	0.722172	1.87065 × 10-6	0.718841	1.62936 × 10-5	0.715176	7.91082 × 10-5
0.9	0.854042	6.40576 × 10-6	0.852211	5.59717 × 10-5	0.850196	2.72708 × 10-4
1.0	1.	1.46076 × 10-5	1.	1.27671 × 10-4	1.	6.22242 × 10-4

Table 4: RK4 solutions along with residuals for various values of [figure omitted; refer to PDF] in case of slip boundary.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Solution	Residual	Solution	Residual	Solution	Residual
0.0	0.	4.12179 × 10-6	0.	7.56765 × 10-6	0.	3.62897 × 10-6
0.1	0.072692	1.74999 × 10-7	0.071220	2.16125 × 10-7	0.069593	-3.49994 × 10-7
0.2	0.147091	-4.21503 × 10-8	0.144268	-4.97423 × 10-8	0.141147	9.53266 × 10-8
0.3	0.224893	1.12883 × 10-8	0.220953	1.22294 × 10-8	0.216598	-3.0735 × 10-8
0.4	0.307773	-3.52272 × 10-9	0.303048	-4.30187 × 10-9	0.297832	7.2493 × 10-9
0.5	0.397370	7.80684 × 10-10	0.392276	3.06668 × 10-9	0.386657	8.46409 × 10-9
0.6	0.495283	1.10391 × 10-9	0.490289	-5.28586 × 10-9	0.484783	-3.36828 × 10-8
0.7	0.603056	-4.35033 × 10-9	0.598654	1.54159 × 10-8	0.593805	1.06816 × 10-7
0.8	0.722172	1.44198 × 10-8	0.718841	-6.05873 × 10-8	0.715175	-3.98519 × 10-7
0.9	0.854042	-5.53933 × 10-8	0.852211	2.59233 × 10-7	0.850196	1.65518 × 10-6
1.0	1.	-1.97434 × 10-7	1.	7.96168 × 10-6	1.	3.89126 × 10-5

Table 5: OHAM solutions along with residuals for various values of [figure omitted; refer to PDF] in case of slip boundary.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Solution	Residual	Solution	Residual	Solution	Residual
0.0	0.	0.	0.	0.	0.	0.
0.1	-0.00880011	-1.07933 × 10-6	0.0336596	-7.54858 × 10-8	0.0522402	-7.74445 × 10-9
0.2	-0.010881	-2.2445 × 10-6	0.0714091	-1.73419 × 10-7	0.1074223	-2.51485 × 10-8
0.3	0.000449012	-3.45224 × 10-6	0.117324	-3.00407 × 10-7	0.168479	-5.68899 × 10-8
0.4	0.0318278	-4.41058 × 10-6	0.175449	-4.32397 × 10-7	0.238322	-9.8056 × 10-8
0.5	0.0898133	-4.47787 × 10-6	0.249785	-5.00627 × 10-7	0.319832	-1.29675 × 10-7
0.6	0.18086	-2.60114 × 10-6	0.344278	-3.77978 × 10-7	0.415852	-1.13733 × 10-7
0.7	0.311298	2.65136 × 10-6	0.462801	1.30706 × 10-7	0.529175	1.20785 × 10-8
0.8	0.487312	1.28583 × 10-5	0.609145	1.28166 × 10-6	0.662538	3.36724 × 10-7
0.9	0.714929	2.91822 × 10-5	0.787012	3.34515 × 10-6	0.818613	9.67763 × 10-7
1.0	1.	5.14019 × 10-5	1.	6.49074 × 10-6	1.	2.00176 × 10-6

Table 6: RK4 solutions along with residuals for various values of [figure omitted; refer to PDF] in case of slip boundary.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Solution	Residual	Solution	Residual	Solution	Residual
0.0	0.	2.04594 × 10-5	0.	7.34794 × 10-6	0.	3.56291 × 10-6
0.1	-0.00880012	7.72953 × 10-7	0.0336596	3.21066 × 10-7	0.0522402	1.63598 × 10-7
0.2	-0.0108811	-1.8406 × 10-7	0.0714091	-7.75305 × 10-8	0.1074223	-3.96814 × 10-8
0.3	0.000448983	4.82747 × 10-8	0.117324	2.08509 × 10-8	0.168479	1.0753 × 10-8
0.4	0.0318277	-1.54235 × 10-8	0.175449	-6.43999 × 10-9	0.238322	-3.28063 × 10-9
0.5	0.0898132	5.14031 × 10-9	0.249785	1.21029 × 10-9	0.319832	4.50154 × 10-10
0.6	0.18086	-6.51866 × 10-10	0.344278	2.64494 × 10-9	0.415852	1.84079 × 10-9
0.7	0.311298	-1.86837 × 10-9	0.462801	-9.88382 × 10-9	0.529175	-6.51979 × 10-9
0.8	0.487312	-1.33003 × 10-9	0.609145	3.37336 × 10-8	0.662538	2.27322 × 10-8
0.9	0.714929	2.66400 × 10-8	0.787012	-1.32232 × 10-7	0.818613	-9.05714 × 10-8
1.0	1.	5.55277 × 10-6	1.	-1.1671 × 10-6	1.	-1.1767 × 10-6

Table 7: Comparison of OHAM and RK4 solutions for various [figure omitted; refer to PDF] in case of slip and no-slip boundary.

[figure omitted; refer to PDF]	In case of no-slip boundary	In case of slip boundary
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.0	0.	0.	0.	0.	0.	0.
0.1	2.33147 × 10-15	1.62249 × 10-10	2.41725 × 10-9	8.44379 × 10-10	7.90896 × 10-9	4.1479 × 10-8
0.2	9.49241 × 10-14	3.14228 × 10-10	4.67809 × 10-9	1.6672 × 10-9	1.55832 × 10-8	8.15527 × 10-8
0.3	3.82361 × 10-13	4.46143 × 10-10	6.63065 × 10-9	2.44889 × 10-9	2.28102 × 10-8	1.18952 × 10-7
0.4	9.13492 × 10-13	5.47932 × 10-10	8.11705 × 10-9	3.1663 × 10-9	2.93539 × 10-8	1.52339 × 10-7
0.5	1.65451 × 10-12	6.07476 × 10-10	8.95202 × 10-9	3.77514 × 10-9	3.48026 × 10-8	1.79574 × 10-7
0.6	2.42317 × 10-12	6.09434 × 10-10	8.91445 × 10-9	4.18241 × 10-9	3.8327 × 10-8	1.96532 × 10-7
0.7	2.86218 × 10-12	5.36833 × 10-10	7.77928 × 10-9	4.22088 × 10-9	3.84569 × 10-8	1.96009 × 10-7
0.8	2.52039 × 10-12	3.78563 × 10-10	5.4278 × 10-9	3.65208 × 10-9	3.31085 × 10-8	1.67861 × 10-7
0.9	1.20232 × 10-12	1.54202 × 10-10	2.18716 × 10-9	2.24454 × 10-9	2.02732 × 10-8	1.02385 × 10-7
1.0	1.26281 × 10-16	2.53545 × 10-16	2.93337 × 10-17	1.7436 × 10-14	4.92715 × 10-14	3.8179 × 10-14

Furthermore, Figures 2, 3, and 4 indicate the OHAM residuals in case of no-slip and slip boundaries for various values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Figure 2: OHAM residuals at various values of [figure omitted; refer to PDF] in case of no-slip boundary.

[figure omitted; refer to PDF]

Figure 3: OHAM residuals at various values of [figure omitted; refer to PDF] in case of slip boundary.

[figure omitted; refer to PDF]

Figure 4: OHAM residuals at various values of [figure omitted; refer to PDF] in case of slip boundary.

[figure omitted; refer to PDF]

The effect of Reynolds number [figure omitted; refer to PDF] on velocity profiles in case of no-slip boundary is shown in Figure 5. In these profiles we varied [figure omitted; refer to PDF] as [figure omitted; refer to PDF] and observed that the normal velocity is increased with the increase of Reynolds number (Figure 5(a)). It is also noted that the normal velocity monotonically increases from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] for fixed positive value of [figure omitted; refer to PDF] at a given time. Figure 5(b) describes the impact of [figure omitted; refer to PDF] on the longitudinal velocity in case of no-slip boundary. It is experienced that this component of velocity deceases near the wall but increases near the central axis of the channel.

Velocity profiles for various values of [figure omitted; refer to PDF] in case of no-slip boundary.

(a) The effect of [figure omitted; refer to PDF] on the Normal velocity profiles

[figure omitted; refer to PDF]

(b) The effect of [figure omitted; refer to PDF] on the longitudinal velocity profiles

[figure omitted; refer to PDF]

The effect of Reynolds number [figure omitted; refer to PDF] on velocity profiles in case of slip boundary is depicted in Figure 6. In these profiles, we fixed slip parameter [figure omitted; refer to PDF] and varied Reynolds number [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . It is noted that the normal velocity decreases as the Reynolds number increases (Figure 6(a)). It is also observed that longitudinal velocity decreases near the central axis of the channel but increases near the walls when [figure omitted; refer to PDF] increases (Figure 6(b)).

Velocity profiles for various values of [figure omitted; refer to PDF] fixing [figure omitted; refer to PDF] in case of slip boundary.

(a) The effect of [figure omitted; refer to PDF] on the Normal velocity profiles

[figure omitted; refer to PDF]

(b) The effect of [figure omitted; refer to PDF] on the longitudinal velocity profiles

[figure omitted; refer to PDF]

Figure 7 demonstrates the effect of slip parameter [figure omitted; refer to PDF] on the velocity profiles. After fixing Reynolds number [figure omitted; refer to PDF] we varied [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . We find that normal velocity increases as [figure omitted; refer to PDF] increases. It is also noted that longitudinal velocity decreases near the walls but increases near central axis of the channel.

Velocity profiles for various values of [figure omitted; refer to PDF] fixing [figure omitted; refer to PDF] in case of slip boundary.

(a) The effect of [figure omitted; refer to PDF] on the Normal velocity profiles

[figure omitted; refer to PDF]

(b) The effect of [figure omitted; refer to PDF] on the longitudinal velocity profiles

[figure omitted; refer to PDF]

8. Conclusions

In this article, we find the similarity solution for unsteady axisymmetric squeezing flow of incompressible Newtonian fluid between two circular plates. We observed that the similarity solution exists only when distance between the plates varies as [figure omitted; refer to PDF] , and squeezing flow occurs when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The key findings of the present analysis are as follows:

In case of no-slip at boundary;

(i) It has been found that increase in Reynolds number [figure omitted; refer to PDF] increases the normal velocity.

(ii) It has been observed that normal velocity increases monotonically from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] for fixed positive value of [figure omitted; refer to PDF] at a given time.

(iii): It has been seen that longitudinal velocity deceases near the walls and increases near the central axis of the channel.

In case of slip at boundary;

(i) It has been noted that after fixing slip parameter [figure omitted; refer to PDF] and varying the Reynolds number [figure omitted; refer to PDF] , the normal velocity profile decreases with the increase in [figure omitted; refer to PDF] . Also the longitudinal velocity increases near the walls but decreases near the central axis of the channel.

(ii) It has been examined that for a fixed Reynolds number [figure omitted; refer to PDF] when we vary slip parameter [figure omitted; refer to PDF] , the normal velocity increases with the increase in [figure omitted; refer to PDF] . Also the longitudinal velocity decreases near the walls and increases near the central axis of the channel.

(iii): It has been investigated that Reynolds number [figure omitted; refer to PDF] and slip parameter [figure omitted; refer to PDF] have opposite effects on the normal and longitudinal velocity components.

In case of slip versus no-slip boundary;

(i) It has been observed that Reynolds number [figure omitted; refer to PDF] has opposite behavior on the normal velocity in case of slip and no-slip boundaries.

(ii) It has been also noticed that Reynolds number [figure omitted; refer to PDF] has opposite effect on the longitudinal velocity near the central axis of the channel, while near the wall longitudinal velocity increases in case of slip boundary and decease in no-slip boundary. This is in conformance to [33].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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