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Abstract and Applied Analysis
Volumeย 2012ย (2012), Article IDย 835268, 18 pages
http://dx.doi.org/10.1155/2012/835268
Research Article

Comparison of Different Analytic Solutions to Axisymmetric Squeezing Fluid Flow between Two Infinite Parallel Plates with Slip Boundary Conditions

1Islamia College Peshawar (Chartered University), Khyber Pakhtunkhawa, Peshawar 25120, Pakistan
2Department of Mathematics, CIIT, H-8, Islamabad 44000, Pakistan
3CECOS University of IT and Emerging Sciences, Peshawar 25100, Pakistan

Received 3 August 2011; Revised 6 October 2011; Accepted 6 October 2011

Academic Editor: Ahmetย Yฤฑldฤฑrฤฑm

Copyright ยฉ 2012 Hamid Khan et al. 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.

Abstract

We investigate squeezing flow between two large parallel plates by transforming the basic governing equations of the first grade fluid to an ordinary nonlinear differential equation using the stream functions ๐‘ข๐‘Ÿ(๐‘Ÿ,๐‘ง,๐‘ก)=(1/๐‘Ÿ)(๐œ•๐œ“/๐œ•๐‘ง) and ๐‘ข๐‘ง(๐‘Ÿ,๐‘ง,๐‘ก)=โˆ’(1/๐‘Ÿ)(๐œ•๐œ“/๐œ•๐‘Ÿ) and a transformation ๐œ“(๐‘Ÿ,๐‘ง)=๐‘Ÿ2๐น(๐‘ง). The velocity profiles are investigated through various analytical techniques like Adomian decomposition method, new iterative method, homotopy perturbation, optimal homotopy asymptotic method, and differential transform method.

1. Introduction

The study of squeezing flows has widespread applications in chemical engineering, industrial engineering, mechanical engineering, biomechanics, and food industry. Valves and diarthrodial joints are also the examples for squeeze flows relevant in biology and bioengineering. The first application to squeeze flow problem was made by Stefan in 1874 [1]. The motion of a thin film of lubricant, squeezed flow between two stationary parallel plane surfaces were reported by Tichy and Winner [2] and Wang and Watson [3]. The theoretical and experimental studies of squeezing flows have been conducted by many researchers [4โ€“11]. The mathematical studies of these flows are concerned primarily with the nonlinear partial differential equations which arise from the Navier-Stokes equations. These equations have no general solutions, and only a few exact solutions have been attained by confining some physical aspects of the original problem [12]. To solve these nonlinear differential equations, different perturbation and analytical techniques have been extensively used in fluid mechanics and engineering [13].

In the literature only a few papers deal with the comparison of different analytical methods. In this paper we study the squeezed flow between two large parallel plates with slip boundary conditions. The velocity profile is obtained using various analytical techniques like Adomian decomposition method (ADM), new iterative method (NIM), homotopy perturbation (HPM), optimal homotopy asymptotic method (OHAM), and differential transform method (DTM) [14โ€“23]. The residual of each technique is computed and a comparison is made to assess the efficiency of the above techniques. We select DTM for analyzing the velocity profile under different flow parameters.

Squeezing flows are produced by vertical movements of boundaries or by applying external normal forces. Commonly two types of boundary conditions are employed. For a viscous fluid at a solid wall, it is generally accepted that the fluid velocity matches the velocity of the solid boundary, and it is known as no slip boundary condition. While the no-slip condition is experimentally proven to be accurate for a number of macroscopic flows. Navier [24] proposed a general boundary condition that incorporates the possibility of fluid slip at a solid boundary. He assumed that the velocity๐‘ข๐‘ฅ at a solid surface is proportional to the shear rate at the surface, that is, ๐‘ข๐‘ฅ=๐›ฝ๐œ•๐‘ข๐‘ฅ/๐œ•๐‘ฆ,where ๐›ฝis the slip length or slip coefficient. If๐›ฝ=0, the generally assumed no-slip boundary condition is obtained, and if ๐›ฝ a finite constant, fluid is slip occurs at the wall. Its effect depends upon the length scale of the flow [25โ€“28].

2. Basic Equation

We consider a steady axisymmetric flow where the velocity vector, ฬƒ๐‘ข is represented by๎€บ๐‘ขฬƒ๐‘ข=๐‘Ÿ(๐‘Ÿ,๐‘ง),0,๐‘ข๐‘ง๎€ป(๐‘Ÿ,๐‘ง).(2.1) In the absence of body forces, the Navier-Stokes equations are obtained for the first grade fluid by using equations of continuity and momentum,๎‚๎€ท๎‚๐‘ค๎€ธ+๎‚โˆ‡๎‚€๐œŒโˆ‡โ‹…ฬƒ๐‘ข=0,โˆ’๐œŒฬƒ๐‘ขร—2||||ฬƒ๐‘ข2๎‚๎‚๎‚‹+๐‘=โˆ’๐œ‚โˆ‡ร—๐‘ค,(2.2) where ๐œŒis the constant density, ๐‘ is the pressure, ๐œ‚ is the viscosity, and ๎‚๎‚๐‘ค=โˆ‡ร—ฬƒ๐‘ข is the vorticity vector.

Substituting (2.1) into (2.2), we get the following Continuity equation:๐œ•๐‘ข๐‘Ÿ+๐‘ข๐œ•๐‘Ÿ๐‘Ÿ๐‘Ÿ+๐œ•๐‘ข๐‘ง๐œ•๐‘ง=0.(2.3)๐‘Ÿ-component of N.S equation:๐œ•๎‚€๐œŒ๐œ•๐‘Ÿ2๎€ท๐‘ข2๐‘Ÿ+๐‘ข2๐‘ง๎€ธ๎‚+๐‘โˆ’๐œŒ๐‘ข๐‘งฮฉ(๐‘Ÿ,๐‘ง)=โˆ’๐œ‚๐œ•ฮฉ(๐‘Ÿ,๐‘ง)๐œ•๐‘ง.(2.4)๐‘ง-component of N.S equation:๐œ•๎‚€๐œŒ๐œ•๐‘ง2๎€ท๐‘ข2๐‘Ÿ+๐‘ข2๐‘ง๎€ธ๎‚+๐‘โˆ’๐œŒ๐‘ข๐‘ง1ฮฉ(๐‘Ÿ,๐‘ง)=๐œ‚๐‘Ÿ๐œ•(๐‘Ÿฮฉ(๐‘Ÿ,๐‘ง))๐œ•๐‘Ÿ,(2.5) where ฮฉ(๐‘Ÿ,๐‘ง)=๐œ•๐‘ข๐‘Ÿ/๐œ•๐‘งโˆ’๐œ•๐‘ข๐‘ง/๐œ•๐‘Ÿ is the vorticity function.

Now we define a function which is known as generalized pressure,๐œŒฬ‚๐‘=2๎€ท๐‘ข2๐‘Ÿ+๐‘ข2๐‘ง๎€ธ+๐‘.(2.6) Using (2.6), (2.4) and (2.5) take the following form:๐œ•๐œ•๐‘Ÿฬ‚๐‘โˆ’๐œŒ๐‘ข๐‘งฮฉ(๐‘Ÿ,๐‘ง)=โˆ’๐œ‚๐œ•ฮฉ(๐‘Ÿ,๐‘ง),๐œ•๐œ•๐‘ง๐œ•๐‘งฬ‚๐‘โˆ’๐œŒ๐‘ข๐‘ง1ฮฉ(๐‘Ÿ,๐‘ง)=๐œ‚๐‘Ÿ๐œ•(๐‘Ÿฮฉ(๐‘Ÿ,๐‘ง)).๐œ•๐‘Ÿ(2.7) We now introduce the stream functions, ๐‘ข๐‘Ÿ(๐‘Ÿ,๐‘ง,๐‘ก)=(1/๐‘Ÿ)(๐œ•๐œ“/๐œ•๐‘ง),๐‘ข๐‘ง(๐‘Ÿ,๐‘ง,๐‘ก)=(โˆ’1/๐‘Ÿ)ร—(๐œ•๐œ“/๐œ•๐‘Ÿ),and obtain the following results: 1ฮฉ(๐‘Ÿ,๐‘ง)=โˆ’๐‘Ÿ๐ธ2๐œ“,(2.8) where ๐ธ2=๐œ•2/๐œ•๐‘Ÿ2โˆ’(1/๐‘Ÿ)(๐œ•/๐œ•๐‘Ÿ)+๐œ•2/๐œ•๐‘ง2,๐œ•1๐œ•๐‘Ÿฬ‚๐‘โˆ’๐œŒ๐‘Ÿ2๐œ•๐œ“๐ธ๐œ•๐‘Ÿ2๐œ‚๐œ“=โˆ’๐‘Ÿ๐œ•๐ธ๐œ•๐‘ง2๐œ•๐œ“,1๐œ•๐‘งฬ‚๐‘โˆ’๐œŒ๐‘Ÿ2๐œ•๐œ“๐ธ๐œ•๐‘ง2๐œ‚๐œ“=โˆ’๐‘Ÿ๐œ•๐ธ๐œ•๐‘Ÿ2๐œ“.(2.9) Eliminating ฬ‚๐‘, from (2.9), we obtain๐œ•๎€ทโˆ’๐œŒ๐œ“,๐ธ2๐œ“/๐‘Ÿ2๎€ธ=๐œ‚๐œ•(๐‘Ÿ,๐‘ง)๐‘Ÿ๐ธ4๐œ“.(2.10) We consider viscous incompressible fluid, squeezed between two large planar and parallel plates, separated by a distance 2๐‘‘. The plates are moving towards each other with velocity ๐‘ˆ. The surfaces of both plates are covered by special material with slip length (slip coefficient)๐›ฝ. For small values of ๐‘ˆ the gape distance 2๐‘‘ between the plates varies slowly with the time ๐‘ก, so the flow can be taken as quasisteady [29โ€“32]. See Figures 1, 2, 3, and 4.

835268.fig.001
Figure 1: Plots of the residuals, blue: ADM, green: NIM, red: HPM, black: OHAM and dashed: DTM. In this figure, a comparison of the residuals is shown graphically, and it reveals that OHAM is more suitable technique for this problem. One can easily observe that the curve of the OHAM residual is approximately normally distributed while the curves of the residuals of other methods are ๐ฝ-shaped and highly skewed.
835268.fig.002
Figure 2: ๐‘ง-component ๐‘ข๐‘ง(๐‘Ÿ,๐œƒ,๐‘ง)=(โˆ’1/๐‘Ÿ)(๐œ•๐œ“/๐œ•๐‘Ÿ) of the velocity profile given by OHAM, when ๐‘…=1,๐›พ=1.
835268.fig.003
Figure 3: ๐‘Ÿ-component ๐‘ข๐‘Ÿ(๐‘Ÿ,๐œƒ,๐‘ง)=(1/๐‘Ÿ)(๐œ•๐œ“/๐œ•๐‘ง)of the velocity profile given by OHAM, when ๐‘…=1, ๐›พ=1.
835268.fig.004
Figure 4: Vorticity function ฮฉ(๐‘Ÿ,๐‘ง)=๐œ•๐‘ข๐‘Ÿ/๐œ•๐‘งโˆ’๐œ•๐‘ข๐‘ง/๐œ•๐‘Ÿ where the velocity components have been obtained by OHAM, for ๐‘…=1, ๐›พ=1.

835268.fig.005

The boundary conditions are as follows:

โ€ƒโ€ƒ (i)At๐‘ง=๐‘‘,๐‘ข๐‘Ÿ=๐›ฝ๐œ•๐‘ข๐‘Ÿ๐œ•๐‘ง,๐‘ข๐‘ง=โˆ’๐‘ˆ.(ii)At๐‘ง=0,๐‘ข๐‘ง=0,๐œ•๐‘ข๐‘Ÿ๐œ•๐‘ง=0.(2.11) Now we use the transformation ๐œ“(๐‘Ÿ,๐‘ง)=๐‘Ÿ2๐‘“(๐‘ง).(2.12)

By virtue of ๐ธ2=๐œ•2/๐œ•๐‘Ÿ2โˆ’(1/๐‘Ÿ)(๐œ•/๐œ•๐‘Ÿ)+๐œ•2/๐œ•๐‘ง2 and (2.12), the compatibility equation (2.10) and the boundary conditions equation (2.11) become๐น๐‘–๐‘ฃ๐œŒ(๐‘ง)+2๐œ‚๐น(๐‘ง)๐นโ€ฒ๎…ž๎…ž(๐‘ง)=0,๐น(0)=0,๐น๎…ž๎…ž๐‘ˆ(0)=0,๐น(๐‘‘)=2,๐น๎…ž(๐‘‘)=๐›ฝ๐น๎…ž๎…ž(๐‘‘).(2.13) Now by introducing dimensionless parameters๐นโˆ—=๐น๐‘ˆ/2,๐‘งโˆ—=๐‘ง๐‘‘๐›ฝ,๐›พ=๐‘‘,๐‘…=๐œŒ๐‘‘๐‘ˆ๐œ‚,(2.14) and dropping โ€œ*โ€ for simplicity, the boundary value problem (2.13) become,๐‘‘4๐น๐‘‘๐‘ง4๐‘‘+๐‘…๐น3๐น๐‘‘๐‘ง3=0,(2.15) with boundary conditions๐น(0)=0,๐น๎…ž๎…ž(0)=0,๐น(1)=1,๐น๎…ž(1)=๐›พ๐น๎…ž๎…ž(1).(2.16)

3. Analytical Techniques

In this section, we give the basic idea of various analytical techniques and evaluate the velocity profile of our problem by considering ๐‘…=1,๐›พ=1. The residuals of all the techniques are computed and the results are displayed in Table 2.

3.1. Adomian Decomposition Method

According to [13, 14], we consider the differential equation๐ฟ(๐น(๐‘ง))+๐‘€(๐น(๐‘ง))+๐‘(๐น(๐‘ง))=๐‘”(๐‘ง),(3.1) where ๐ฟis the operator of the highest order derivative with respect to๐‘ง, ๐ฟ=๐‘‘4/๐‘‘๐‘ง4, ๐‘…is the reminder of the linear term, and the nonlinear term is represented by๐‘(๐น(๐‘ง)). Operating ๐ฟโˆ’1 on both sides of (3.1) we get the following: ๐น(๐‘ง)=๐›ผ0+๐›ผ1๐‘ง+๐›ผ2๐‘ง22!+๐›ผ3๐‘ง33!+๐ฟโˆ’1(๐‘”(๐‘ง))โˆ’๐ฟโˆ’1๐‘€(๐น(๐‘ง))โˆ’๐ฟโˆ’1๐‘(๐น(๐‘ง)),(3.2) where๐›ผ๐‘–:๐‘–=0,1,2,3, the constants can be determined by using initial or boundary conditions.

The unknown function ๐น(๐‘ง)can be expressed by an infinite series of the form๐น(๐‘ง)=โˆž๎“๐‘›=0๐น๐‘›(๐‘ง),(3.3) where๐น0(๐‘ง)=๐›ผ0+๐›ผ1๐‘ง+๐›ผ2๐‘ง2/2!+๐›ผ3๐‘ง3/3!+๐ฟโˆ’1(๐‘”(๐‘ง)) and ๐น๐‘›+1=โˆ’๐ฟโˆ’1๐‘€(๐น๐‘›(๐‘ง))โˆ’๐ฟโˆ’1(๐ด๐‘›), ๐‘›=0,1,2,โ€ฆ. The nonlinear term ๐‘(๐น(๐‘ง)) is decomposed by an infinite series of polynomial given by๐‘(๐น(๐‘ง))=โˆž๎“๐‘›=0๐ด๐‘›,(3.4) where ๐ด๐‘› are the so-called Adomian polynomials that can be determined by the formula๐ด๐‘›=1๐‘‘๐‘›!๐‘›๐‘‘๐œ†๐‘›๎ƒฌ๐‘๎ƒฉ๐‘›๎“๐‘–=0๐œ†๐‘–๐น๐‘–๎ƒช๎ƒญ๐œ†=0.(3.5)

It has been observed that these polynomials can be constructed for a wide class of nonlinear functions.

The solutionโˆ‘๐น(๐‘ง)=โˆž๐‘›=0๐น๐‘›(๐‘ง) is approximated by the truncated series of order๐พ, that is,๎‚๐น(๐‘ง)=๐พ๎“๐‘›=0๐น๐‘›(๐‘ง).(3.6) In our case๐‘”(๐‘ง)=0, ๐‘€(๐น(๐‘ง))=0.

Now appling ADM on (2.15) and (2.16), for ๐‘›=0,1,โ€ฆ,5.

We obtain ๐น0๐‘ง(๐‘ง)=๐ด36๐น+๐ต๐‘ง1๎‚ต1(๐‘ง)=โˆ’120๐ด๐ต๐‘ง5+๐ด2๐‘ง7๎‚ถ๐น50402(๐‘ง)=๐ด๐ต2๐‘ง7+๐ด16802๐ต๐‘ง9+๐ด226803๐‘ง111108800โ‹ฎ.(3.7)

Considering the Adomian 5th-order solution,๎‚๐น(๐‘ง)=๐น0(๐‘ง)+๐น1(๐‘ง)+โ‹…โ‹…โ‹…+๐น5๎€ท๐‘ง(๐‘ง)+๐‘‚20๎€ธ.(3.8)

The boundary conditions at ๐‘ง=1 are used to get the following values of ๐ดand ๐ต.๐ด=1.74911,๐ต=0.71896.(3.9)

By substituting these values our solution is๎‚๐น(๐‘ง)=0.718965๐‘ง+0.291518๐‘ง3โˆ’0.0104795๐‘ง5โˆ’0.0000688452๐‘ง7+0.0000701133๐‘ง9โˆ’3.49247ร—10โˆ’6๐‘ง11โˆ’4.73949ร—10โˆ’7๐‘ง13+5.95015ร—10โˆ’8๐‘ง15โˆ’8.90909ร—10โˆ’10๐‘ง17โˆ’1.65991ร—10โˆ’9๐‘ง19๎€ท๐‘ง+๐‘‚20๎€ธ.(3.10)

3.2. New Iterative Method

The basic idea of new iterative method [15, 16]. Consider the following nonlinear general differential equation:๐ฟ(๐น(๐‘ง))+๐‘€(๐น(๐‘ง))+๐‘(๐น(๐‘ง))=๐‘”(๐‘ง),(3.11) where ๐ฟis the operator of the highest order derivative with respect to๐‘ง, ๐ฟ=๐‘‘4/๐‘‘๐‘ง4, ๐‘€ is the reminder of the linear term, and the nonlinear term is represented by๐‘(๐น(๐‘ง)). Operating ๐ฟโˆ’1 on both sides of (3.11) we get,๐น(๐‘ง)=๐›ผ0+๐›ผ1๐‘ง+๐›ผ2๐‘ง22!+๐›ผ3๐‘ง33!+๐ฟโˆ’1(๐‘”(๐‘ง))โˆ’๐ฟโˆ’1๐‘€(๐น(๐‘ง))โˆ’๐ฟโˆ’1๐‘(๐น(๐‘ง)),(3.12) where ๐›ผ๐‘–:๐‘–=0,1,2,3, are constants to be determined by using initial or boundary conditions.

The unknown function ๐น(๐‘ง)can be expressed by an infinite series of the form๐น(๐‘ง)=โˆž๎“๐‘›=0๐น๐‘›(๐‘ง),(3.13) where ๐น0(๐‘ง)=๐›ผ0+๐›ผ1๐‘ง+๐›ผ2๐‘ง2/2!+๐›ผ3๐‘ง3/3!+๐ฟโˆ’1๐‘”(๐‘ง) and ๐น๐‘›+1=โˆ’๐ฟโˆ’1๐‘€(๐น๐‘›(๐‘ง))โˆ’๐ฟโˆ’1(๐บ๐‘›), ๐‘›=0,1,2,โ€ฆ The nonlinear term ๐‘(๐น(๐‘ง)) is decomposed by an infinite series of polynomials given by๐‘(๐น(๐‘ง))=โˆž๎“๐‘›=0๐บ๐‘›,(3.14) where ๐บ๐‘›โˆ‘=๐‘(๐‘›๐‘˜=0๐น๐‘˜โˆ‘)โˆ’๐‘(๐‘›โˆ’1๐‘˜=0๐น๐‘˜) and ๐บ0=๐‘(๐น0(๐‘ง)).

The solutionโˆ‘๐น(๐‘ง)=โˆž๐‘›=0๐น๐‘›(๐‘ง), is approximated by the truncated series of order ๐พ, that is,๎‚๐น(๐‘ง)=๐พ๎“๐‘›=0๐น๐‘›(๐‘ง).(3.15)

In our case ๐‘”(๐‘ง)=0 and ๐‘€(๐น(๐‘ง))=0.

Now using NIM on (2.15) and (2.16), for ๐‘›=0,1,โ€ฆ,5.

We obtain the following: ๐น0๐‘ง(๐‘ง)=๐ด36๐น+๐ต๐‘ง1๎‚ต1(๐‘ง)=โˆ’120๐ด๐ต๐‘ง5+๐ด2๐‘ง7๎‚ถ๐น50402(๐‘ง)=โˆ’๐ด๐ต2๐‘ง7+๐ด16802๐ต๐‘ง9+๐ด226803๐‘ง11โˆ’๐ด11088002๐ต2๐‘ง11โˆ’๐ด19008003๐ต๐‘ง13โˆ’๐ด384384004๐‘ง153962649600โ‹ฎ.(3.16) Considering the NIM 5th-order solution,๎‚๐น(๐‘ง)=๐น0(๐‘ง)+๐น1(๐‘ง)+๐น3(๐‘ง)+๐น4(๐‘ง)+๐น5๎€ท๐‘ง(๐‘ง)+๐‘‚20๎€ธ,(3.17) we have the following:๎‚๐น(๐‘ง)=๐ต๐‘ง+๐ด๐‘ง36โˆ’1120๐ด๐ต๐‘ง5+๎‚ตโˆ’๐ด2+5040๐ด๐ต2๎‚ถ๐‘ง16807+๎‚ต๐ด2๐ตโˆ’22680๐ด๐ต3๎‚ถ๐‘ง241929+๎‚ต๐ด3โˆ’1108800241๐ด2๐ต2+39916800๐ด๐ต4๎‚ถ๐‘ง38016011+๎‚ตโˆ’71๐ด3๐ต+239500800131๐ด2๐ต3โˆ’207567360๐ด๐ต5๎‚ถ๐‘ง658944013+๎‚ตโˆ’1051๐ด4+2179457280001357๐ด3๐ต2โˆ’251475840004759๐ด2๐ต4๎‚ถ๐‘ง8717829120015+๎‚ต2243๐ด4๐ตโˆ’1140023808000359๐ด3๐ต3+50523782400179๐ด2๐ต5๎‚ถ๐‘ง13028843520017+๎‚ต21919๐ด5โˆ’84475764172800071999๐ด4๐ต2+167094918144000104977๐ด3๐ต4โˆ’30035827261440031๐ด2๐ต6๎‚ถ๐‘ง64981357056019๎€ท๐‘ง+๐‘‚20๎€ธ.(3.18) Using the boundary conditions at ๐‘ง=1, we get the following value of ๐ด and ๐ต๐ด=1.74911,๐ต=0.71897.(3.19) The approximate solution is as follows:๎‚๐น(๐‘ง)=0.718965๐‘ง+0.291518๐‘ง3โˆ’0.0104795๐‘ง5โˆ’0.0000688448๐‘ง7+0.0000701132๐‘ง9โˆ’3.49247ร—10โˆ’6๐‘ง11โˆ’4.73948ร—10โˆ’7๐‘ง13+5.95015ร—10โˆ’8๐‘ง15โˆ’8.34647ร—10โˆ’11๐‘ง17โˆ’1.18034ร—10โˆ’9๐‘ง19๎€ท๐‘ง+๐‘‚20๎€ธ.(3.20)

3.3. HPM

To illustrate the basic idea of homotopy perturbation method [17โ€“21], we consider the following nonlinear differential equation:๐ด(๐น)=๐‘“(๐‘Ÿ).(3.21)๐ด is a general differential operator, the operator ๐ด can usually be divided into two parts ๐ฟand ๐‘, where ๐ฟ is linear, and ๐‘ is nonlinear:๐ด=๐ฟ+๐‘.(3.22) so๐ฟ(๐น)+๐‘(๐น)โˆ’๐‘“(๐‘Ÿ)=0,๐‘Ÿโˆˆฮฉ.(3.23)๐‘“(๐‘Ÿ)is a known analytic function.

With the boundary condition ๐ต(๐น,๐œ•๐น/๐œ•๐‘›)=0,๐‘Ÿโˆˆฮ“.

๐ต is a boundary operator, and ฮ“is the boundary of the domain ฮฉ.

Now we construct the following homotopy:๐ป๎€บ๐ฟ๎€ท๐น(๐‘ฃ,๐‘)=(1โˆ’๐‘)(๐‘ฃ)โˆ’๐ฟ0[]๎€ธ๎€ป+๐‘๐ด๐‘ฃโˆ’๐‘“(๐‘Ÿ),(3.24) where ๐‘โˆˆ[0,1]is an embedding parameter and ๐น0 is the first approximation that satisfied the boundary condition. To get an approximate solution, we expand ๐น(๐‘Ÿ,๐‘)in Taylorโ€™s series about๐‘ in the following manner:๐น(๐‘Ÿ)=๐‘ฃ0(๐‘Ÿ)+โˆž๎“๐‘š=1๐‘ฃ๐‘š(๐‘Ÿ)๐‘๐‘š.(3.25) Plugging (3.25) into (3.24) and then equating the coefficient of like powers of๐‘, we get the following problems which are directly integrable.

Zeroth-order problem:๐น0(๐‘–๐‘ฃ)๐น(๐‘ง)=0,0(0)=0,๐นโ€ฒ๎…ž0(0)=0,๐น0(1)=1,๐น๎…ž0(1)=๐น0๎…ž๎…ž(1).(3.26)

First-order problem:๐น1๐‘–๐‘ฃ(๐‘ง)=โˆ’๐น0(๐‘ง)๐น0๎…ž๎…ž๎…ž๐น(๐‘ง),1(0)=0,๐น1๎…ž๎…ž(0)=0,๐น1(1)=0,๐น๎…ž1(1)=๐น1๎…ž๎…ž(1).(3.27)

Second-order problem:๐น2๐‘–๐‘ฃ(๐‘ง)=โˆ’๐น1(๐‘ง)๐น0๎…ž๎…ž๎…ž(๐‘ง)โˆ’๐น0(๐‘ง)๐น1๎…ž๎…ž๎…ž๐น(๐‘ง),2(0)=0,๐น2๎…ž๎…ž(0)=0,๐น2(1)=0,๐น๎…ž2(1)=๐น2๎…ž๎…ž(1)โ‹ฎ.(3.28) We consider the following 5th-order solution,๎‚๐น(๐‘ง)=๐น0(๐‘ง)+๐น1(๐‘ง)+๐น3(๐‘ง)+๐น4(๐‘ง)+๐น5๎€ท๐‘ง(๐‘ง)+๐‘‚20๎€ธ,๎‚๐น(๐‘ง)=2379640217780939164049๐‘ง+3309814671645081600000723651501050808628889๐‘ง3โˆ’248236100373381120000043087465806546383๐‘ง5โˆ’41115710206771200001652256496336207๐‘ง7+23984164287283200000121738793951๐‘ง9โˆ’173581664256000031017787๐‘ง11โˆ’8879270400000430067207๐‘ง13+90923728896000074925133๐‘ง15+122397327360000030859๐‘ง17โˆ’1542948126720016191589๐‘ง19๎€ท๐‘ง18688959184896000+๐‘‚20๎€ธ.(3.29)

3.4. OHAM

According to [22โ€“24], we consider the following differential equation:๐ด(๐น(๐‘ง))=๐‘“(๐‘Ÿ),๐‘Ÿโˆˆฮฉ,(3.30) with the boundary conditions๐ต๎‚ต๐น,๐œ•๐น(๐‘ง)๎‚ถ๐œ•๐‘›=0,๐‘Ÿโˆˆฮ“,(3.31) where ๐ดis a differential operator, ๐ตis a boundary operator, and ๐‘“(๐‘Ÿ)is a known function of ๐‘Ÿ: ๐‘Ÿโˆˆฮฉ. The operator ๐ด can be written as ๐ด=๐ฟ+๐‘, where๐ฟ is linear and ๐‘is a nonlinear operator. In OHAM we first construct a homotopy equation,[](1โˆ’๐‘)๐ฟ(๐น(๐‘Ÿ,๐‘))โˆ’โ„Ž(๐‘)๐ฟ(๐น(๐‘Ÿ,๐‘))+๐‘(๐น(๐‘Ÿ,๐‘))โˆ’๐‘“(๐‘Ÿ)=0,(3.32) where ๐‘โˆˆ[0,1]is an embedding parameter, โ„Ž(๐‘)is a nonzero auxiliary function for ๐‘โ‰ 0 andโ„Ž(0)=0๐ฟ(๐น(๐‘ง))=0,for๐‘=0๐ด(๐น(๐‘ง))=๐‘“(๐‘Ÿ),for๐‘=1.(3.33) The solution๐น(๐‘Ÿ,0)=๐‘ฃ0(๐‘Ÿ) of ๐ฟ(๐น(๐‘ง))=0 traces the solution curve ๐‘ฃ(๐‘Ÿ) continuously as ๐‘ approaches to 1, where ๐‘ฃ0 is the solution of the zeroth-order problem that will come in the next few lines. We next choose the auxiliary function โ„Ž(๐‘) in the following form:โ„Ž(๐‘)=๐‘š๎“๐‘–=1๐‘๐‘–๐ถ๐‘–,(3.34) where ๐ถ1,๐ถ2,โ€ฆare the convergence controlling constants which are to be determined. To get an approximate solution, we expand ๐น(๐‘Ÿ,๐‘)in Taylorโ€™s series about๐‘ in the following manner:๐น๎€ท๐‘Ÿ,๐‘,๐ถ๐‘–๎€ธ=๐‘ฃ0(๐‘Ÿ)+โˆž๎“๐‘š=1๎€ท๐‘Ÿ,๐ถ1,๐ถ1,โ€ฆ,๐ถ๐‘š๎€ธ๐‘๐‘š.(3.35) Now after substituting the auxiliary function โ„Ž(๐‘) and ๐น(๐‘Ÿ,๐‘,๐ถ๐‘–)in homotopy equation we compare the coefficient of like powers of๐‘, to obtain the following linear equations.

Zeroth-order problem:๐ฟ๐‘ฃ0๎‚ต๐‘ฃ=0,๐ต0,๐œ•๐‘ฃ0๎‚ถ๐œ•๐‘›=0.(3.36)

First-order problem:๐ฟ๐‘ฃ1โˆ’๎€ท1+๐ถ1๐‘ฃ0๎€ธ๐ฟ+๐ถ1๐‘“(๐‘Ÿ)=๐ถ1๐‘0๐‘ฃ0๎‚ต๐‘ฃ,๐ต1,๐œ•๐‘ฃ1๎‚ถ๐œ•๐‘›=0.(3.37)

Second-order problem:๐ฟ๐‘ฃ2โˆ’๎€ท1+๐ถ1๎€ธ๐ฟ๐‘ฃ1โˆ’๐ถ2๐ฟ๐‘ฃ0โˆ’๐ถ2๐‘“(๐‘Ÿ)=๐ถ2๐‘0๐‘ฃ0+๐ถ1๐‘1๐‘ฃ1๎‚ต๐‘ฃ,๐ต2,๐œ•๐‘ฃ2๎‚ถ๐œ•๐‘›=0,(3.38) and so on.

If the series is convergent at๐‘=1 for suitable auxiliary constants ๐ถ1,๐ถ2,โ€ฆ, then๎€ท๐‘ฃ(๐‘Ÿ)=๐น๐‘Ÿ,๐ถ๐‘–๎€ธ=๐‘ฃ0(๐‘Ÿ)+โˆž๎“๐‘š=1๐‘ฃ๐‘š๎€ท๐‘Ÿ,๐ถ1,๐ถ1,โ€ฆ,๐ถ๐‘š๎€ธ.(3.39)

The result of the mth-order approximations are given byฬ†๐‘ฃ(๐‘Ÿ)=๐‘ฃ0(๐‘Ÿ)+๐‘š๎“๐‘–=1๐‘ฃ๐‘–๎€ท๐‘Ÿ,๐ถ1,๐ถ1,โ€ฆ,๐ถ๐‘–๎€ธ.(3.40)

Residual of the solution is๐‘…๎€ท๐‘Ÿ,๐ถ1,๐ถ2,โ€ฆ,๐ถ๐‘š๎€ธฬ†=๐ด๐‘ฃโˆ’๐‘“(๐‘Ÿ).(3.41) If๐‘…=0, ฬ†๐‘ฃ will be the exact solution, but it does not happen specially in nonlinear problems. To find the optimal values of๐ถ๐‘–, many methods can be applied. We follow the method of least squares. According to the method of least squares, we first construct the functional๐ฝ๎€ท๐‘Ÿ,๐ถ1,๐ถ2,โ€ฆ,๐ถ๐‘š๎€ธ=๎€œ๐‘๐‘Ž๐‘…2๐‘‘๐‘Ÿ,(3.42) and then minimizing it, we have๐œ•๐ฝ๐œ•๐ถ1=๐œ•๐ฝ๐œ•๐ถ2=โ‹ฏ=๐œ•๐ฝ๐œ•๐ถ๐‘š=0,(3.43) where ๐‘Žand ๐‘are in the domain of the problem. With these constants known, the approximate solution (of order๐‘š) is well determined.

Now applying OHAM to (2.15) and (2.16), we obtain the following problems which are directly integrable.

Zeroth-order problem:๐น0(๐‘–๐‘ฃ)๐น(๐‘ง)=0,0(0)=0,๐นโ€ฒ๎…ž0(0)=0,๐น0(1)=1,๐น๎…ž0(1)=๐น0๎…ž๎…ž(1).(3.44)

First-order problem:๐น1๐‘–๐‘ฃ(๐‘ง)=๐ถ1๐น0(๐‘ง)๐น0๎…ž๎…ž๎…ž(๐‘ง)+๐น0๐‘–๐‘ฃ(๐‘ง)+๐ถ1๐น0๐‘–๐‘ฃ๐น(๐‘ง),1(0)=0,๐น1๎…ž๎…ž(0)=0,๐น1(1)=0,๐น๎…ž1(1)=๐น1๎…ž๎…ž(1).(3.45)

Second-order problem:๐น2๐‘–๐‘ฃ(๐‘ง)=๐ถ1๐น1(๐‘ง)๐น0๎…ž๎…ž๎…ž(๐‘ง)+๐ถ1๐น0(๐‘ง)๐น1๎…ž๎…ž๎…ž(๐‘ง)+๐น1๐‘–๐‘ฃ(๐‘ง)+๐ถ1๐น1๐‘–๐‘ฃ๐น(๐‘ง),2(0)=0,๐น2๎…ž๎…ž(0)=0,๐น2(1)=0,๐น๎…ž2(1)=๐น2๎…ž๎…ž(1)โ‹ฎ.(3.46) Considering the OHAM 5th-order solution and using the method of least squares, we obtain ๐ถ1=โˆ’0.93281. Hence the solution is ๎‚๐น(๐‘ง)=0.718965๐‘ง+0.291517๐‘ง3โˆ’0.0104795๐‘ง5โˆ’0.0000688407๐‘ง7+0.0000700828๐‘ง9โˆ’3.4536ร—10โˆ’6๐‘ง11โˆ’4.87073ร—10โˆ’7๐‘ง13+5.80616ร—10โˆ’8๐‘ง15+3.31269ร—10โˆ’9๐‘ง17โˆ’5.61766ร—10โˆ’10๐‘ง19๎€ท๐‘ง+๐‘‚20๎€ธ.(3.47)

3.5. NDSolve

NDSolve is a mathematica code, utilized for solution of ordinary and partial differential equations. This code is also used for differential-algebraic equations and system of ordinary differential equation. NDSolve gives solution on discrete points rather than for the function ๐น itself. List interpolation is used for the construction of approximating polynomial.

Apply NDSolve to (2.15) and (2.16), the following approximate solution is obtained:๐น(๐‘ง)=0.718965๐‘งโˆ’2.5833ร—10โˆ’6๐‘ง2+0.291529๐‘ง3โˆ’0.000025575๐‘ง4โˆ’0.0104737๐‘ง5+0.0000970056๐‘ง6โˆ’0.000294814๐‘ง7+0.000231011๐‘ง8โˆ’0.0000378459๐‘ง9+0.0000122062๐‘ง10.(3.48)

3.6. DTM

According to [33, 34], the basic idea of differential transforms method (DTM) starts from the following definition.

If ๐น(๐‘ง) is a given function, its differential transform is defined as follows:๐น1(๐‘Ÿ)=๐‘‘๐‘Ÿ!๐‘Ÿ๐น(๐‘ง)๐‘‘๐‘ง๐‘Ÿ||||๐‘ง=0.(3.49) The inverse transform of ๐น(๐‘Ÿ) is defined by๐น(๐‘ง)=โˆž๎“๐‘Ÿ=0๐‘ง๐‘Ÿ๐น(๐‘Ÿ).(3.50) In actual application, the function ๐น(๐‘ง) is expressed by a finite series๐น(๐‘ง)=๐‘๎“๐‘Ÿ=0๐‘ง๐‘Ÿ๐น(๐‘Ÿ).(3.51) Equation (3.51) implies that โˆ‘๐น(๐‘ง)=โˆž๐‘Ÿ=๐‘+1๐‘ง๐‘Ÿ๐น(๐‘Ÿ). is negligibly small.

The fundamental operations of the DTM are given in Table 1.

tab1
Table 1
tab2
Table 2: Comparison of residuals. In this table residuals of all the techniques at the different domain points are displayed. We observe that OHAM provides more stable solution.
3.6.1. Analysis of the Method

Consider a fourth-order boundary value problem๐น(4)(๐‘ง)=๐บ(๐‘ง,๐น),0<๐‘ง<๐ป,(3.52) with the boundary conditions:๐น(0)=๐›ผ0,๐น(๐ป)=๐›ผ1,๐น(1)(๐ป)=๐›ผ2,๐น(2)(0)=๐›ผ3,(3.53) where ๐›ผ๐‘–:๐‘–=0,1,2,3 are given values.

The differential transform of (3.52) is as follows:๐น(๐‘Ÿ+4)=๐บ(๐‘Ÿ)โˆ4๐‘–=1(๐‘Ÿ+๐‘–),(3.54) where ๐บ(๐‘Ÿ) is the differential transform of๐บ(๐‘ง,๐น).

The transformed boundary conditions (3.53) are given by๐น(0)=๐›ผ0,๐‘๎“๐‘Ÿ=0๐ป๐‘Ÿ๐น(๐‘Ÿ)=๐›ผ1,๐‘๎“๐‘Ÿ=0๐‘Ÿ๐ป๐‘Ÿ๐น(๐‘Ÿ)=๐›ผ2,๐›ผ๐น(2)=32.(3.55) Using (3.54) and (3.55) values of ๐น(๐‘–)โˆถ๐‘–=4,5,โ€ฆare obtained which give the following series solution up to ๐‘‚(๐‘ง๐‘+1),๐น(๐‘ง)=๐‘๎“๐‘Ÿ=0๐‘ง๐‘Ÿ๎€ท๐‘ง๐น(๐‘Ÿ)+๐‘‚๐‘+1๎€ธ,(3.56) applying DTM on (2.15) and (2.16), the transformed boundary conditions and differential are as follows: ๐น(0)=0,๐น(1)=๐‘Ž,๐น(2)=0,๐น(3)=๐‘,๐น(๐‘Ÿ+4)=๐‘Ÿ!๎ƒฏ(๐‘Ÿ+4)โˆ’๐‘…๐‘Ÿ๎“๐‘˜=0(๐‘˜+1)(๐‘˜+2)(๐‘˜+3)๐น(๐‘˜+3)๎ƒฐ.๐น(๐‘Ÿโˆ’๐‘˜)(3.57) Using (3.57), we obtain the following values of ๐น(๐‘–)โˆถ๐‘–=1,2,3โ€ฆ,19.

For these values, the unknowns ๐‘Žand ๐‘ are determined by the following system:19๎“๐‘Ÿ=0๎‚๐น(๐‘Ÿ)=1,19๎“๐‘Ÿ=0๐‘Ÿ๎‚๐น(๐‘Ÿ)=19๎“๐‘Ÿ=0๎‚๐‘Ÿ(๐‘Ÿโˆ’1)๐น(๐‘Ÿ).(3.58) We find, ๐‘Ž=1.53266and๐‘=โˆ’0.570353.

Now using the inverse differential transform, the following approximate solution of ๐‘‚(๐‘ง20)is obtained:๎‚๐น(๐‘ง)=1.53266๐‘งโˆ’0.570353๐‘ง3+0.043708๐‘ง5โˆ’0.00710856๐‘ง7+0.00130068๐‘ง9โˆ’0.000251907๐‘ง11+0.0000492109๐‘ง13โˆ’9.6216ร—10โˆ’6๐‘ง15+1.86893ร—10โˆ’6๐‘17โˆ’3.60367ร—10โˆ’7๐‘ง19๎€ท๐‘ง+๐‘‚20๎€ธ.(3.59)

4. Conclusion

In this paper we have used Adomianโ€™s decomposition method, new iterative method, homotopy perturbation method, optimal homotopy asymptotic method, and differential transform method to an axisymmetric squeezing flow problem. Though all the methods are based on Taylorโ€™s series expansion, they produce different results because each one has its own environment. Homotopy methods combine homotopy from topology and perturbation method. It has been observed that homotopy perturbation method may suffer convergence in many problems while OHAM controls the convergence region by employing the auxiliary function. Differential transform method, Adomian decomposition method, and new iterative method are the straightforward application of Taylorโ€™s series, and they suffer from divergence in general and particularly in initial value problems. Besides all these facts, ADM, HPM, and DTM can lead easily to closed-form solutions of many problems.

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