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 š‘¢š‘Ÿ(š‘Ÿ,š‘§,š‘”)=(1/š‘Ÿ)(šœ•šœ“/šœ•š‘§) and š‘¢š‘§(š‘Ÿ,š‘§,š‘”)=āˆ’(1/š‘Ÿ)(šœ•šœ“/šœ•š‘Ÿ) and a transformation šœ“(š‘Ÿ,š‘§)=š‘Ÿ2š¹(š‘§). 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 [4ā€“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) [14ā€“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 velocityš‘¢š‘„ at a solid surface is proportional to the shear rate at the surface, that is, š‘¢š‘„=š›½šœ•š‘¢š‘„/šœ•š‘¦,where š›½is the slip length or slip coefficient. Ifš›½=0, the generally assumed no-slip boundary condition is obtained, and if š›½ a finite constant, fluid is slip occurs at the wall. Its effect depends upon the length scale of the flow [25ā€“28].

2. Basic Equation

We consider a steady axisymmetric flow where the velocity vector, Ģƒš‘¢ is represented byī€ŗš‘¢Ģƒš‘¢=š‘Ÿ(š‘Ÿ,š‘§),0,š‘¢š‘§ī€»(š‘Ÿ,š‘§).(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,ī‚ī€·ī‚š‘¤ī€ø+ī‚āˆ‡ī‚€šœŒāˆ‡ā‹…Ģƒš‘¢=0,āˆ’šœŒĢƒš‘¢Ć—2||||Ģƒš‘¢2ī‚ī‚ī‚‹+š‘=āˆ’šœ‚āˆ‡Ć—š‘¤,(2.2) where šœŒis the constant density, š‘ is the pressure, šœ‚ is the viscosity, and ī‚ī‚š‘¤=āˆ‡Ć—Ģƒš‘¢ is the vorticity vector.

Substituting (2.1) into (2.2), we get the following Continuity equation:šœ•š‘¢š‘Ÿ+š‘¢šœ•š‘Ÿš‘Ÿš‘Ÿ+šœ•š‘¢š‘§šœ•š‘§=0.(2.3)š‘Ÿ-component of N.S equation:šœ•ī‚€šœŒšœ•š‘Ÿ2ī€·š‘¢2š‘Ÿ+š‘¢2š‘§ī€øī‚+š‘āˆ’šœŒš‘¢š‘§Ī©(š‘Ÿ,š‘§)=āˆ’šœ‚šœ•Ī©(š‘Ÿ,š‘§)šœ•š‘§.(2.4)š‘§-component of N.S equation:šœ•ī‚€šœŒšœ•š‘§2ī€·š‘¢2š‘Ÿ+š‘¢2š‘§ī€øī‚+š‘āˆ’šœŒš‘¢š‘§1Ī©(š‘Ÿ,š‘§)=šœ‚š‘Ÿšœ•(š‘ŸĪ©(š‘Ÿ,š‘§))šœ•š‘Ÿ,(2.5) where Ī©(š‘Ÿ,š‘§)=šœ•š‘¢š‘Ÿ/šœ•š‘§āˆ’šœ•š‘¢š‘§/šœ•š‘Ÿ is the vorticity function.

Now we define a function which is known as generalized pressure,šœŒĢ‚š‘=2ī€·š‘¢2š‘Ÿ+š‘¢2š‘§ī€ø+š‘.(2.6) Using (2.6), (2.4) and (2.5) take the following form:šœ•šœ•š‘ŸĢ‚š‘āˆ’šœŒš‘¢š‘§Ī©(š‘Ÿ,š‘§)=āˆ’šœ‚šœ•Ī©(š‘Ÿ,š‘§),šœ•šœ•š‘§šœ•š‘§Ģ‚š‘āˆ’šœŒš‘¢š‘§1Ī©(š‘Ÿ,š‘§)=šœ‚š‘Ÿšœ•(š‘ŸĪ©(š‘Ÿ,š‘§)).šœ•š‘Ÿ(2.7) We now introduce the stream functions, š‘¢š‘Ÿ(š‘Ÿ,š‘§,š‘”)=(1/š‘Ÿ)(šœ•šœ“/šœ•š‘§),š‘¢š‘§(š‘Ÿ,š‘§,š‘”)=(āˆ’1/š‘Ÿ)Ɨ(šœ•šœ“/šœ•š‘Ÿ),and obtain the following results: 1Ī©(š‘Ÿ,š‘§)=āˆ’š‘Ÿšø2šœ“,(2.8) where šø2=šœ•2/šœ•š‘Ÿ2āˆ’(1/š‘Ÿ)(šœ•/šœ•š‘Ÿ)+šœ•2/šœ•š‘§2,šœ•1šœ•š‘ŸĢ‚š‘āˆ’šœŒš‘Ÿ2šœ•šœ“šøšœ•š‘Ÿ2šœ‚šœ“=āˆ’š‘Ÿšœ•šøšœ•š‘§2šœ•šœ“,1šœ•š‘§Ģ‚š‘āˆ’šœŒš‘Ÿ2šœ•šœ“šøšœ•š‘§2šœ‚šœ“=āˆ’š‘Ÿšœ•šøšœ•š‘Ÿ2šœ“.(2.9) Eliminating Ģ‚š‘, from (2.9), we obtainšœ•ī€·āˆ’šœŒšœ“,šø2šœ“/š‘Ÿ2ī€ø=šœ‚šœ•(š‘Ÿ,š‘§)š‘Ÿšø4šœ“.(2.10) We consider viscous incompressible fluid, squeezed between two large planar and parallel plates, separated by a distance 2š‘‘. The plates are moving towards each other with velocity š‘ˆ. The surfaces of both plates are covered by special material with slip length (slip coefficient)š›½. For small values of š‘ˆ the gape distance 2š‘‘ between the plates varies slowly with the time š‘”, so the flow can be taken as quasisteady [29ā€“32]. See Figures 1, 2, 3, and 4.

835268.fig.005

The boundary conditions are as follows:

ā€ƒā€ƒ (i)Atš‘§=š‘‘,š‘¢š‘Ÿ=š›½šœ•š‘¢š‘Ÿšœ•š‘§,š‘¢š‘§=āˆ’š‘ˆ.(ii)Atš‘§=0,š‘¢š‘§=0,šœ•š‘¢š‘Ÿšœ•š‘§=0.(2.11) Now we use the transformation šœ“(š‘Ÿ,š‘§)=š‘Ÿ2š‘“(š‘§).(2.12)

By virtue of šø2=šœ•2/šœ•š‘Ÿ2āˆ’(1/š‘Ÿ)(šœ•/šœ•š‘Ÿ)+šœ•2/šœ•š‘§2 and (2.12), the compatibility equation (2.10) and the boundary conditions equation (2.11) becomeš¹š‘–š‘£šœŒ(š‘§)+2šœ‚š¹(š‘§)š¹ā€²ī…žī…ž(š‘§)=0,š¹(0)=0,š¹ī…žī…žš‘ˆ(0)=0,š¹(š‘‘)=2,š¹ī…ž(š‘‘)=š›½š¹ī…žī…ž(š‘‘).(2.13) Now by introducing dimensionless parametersš¹āˆ—=š¹š‘ˆ/2,š‘§āˆ—=š‘§š‘‘š›½,š›¾=š‘‘,š‘…=šœŒš‘‘š‘ˆšœ‚,(2.14) and dropping ā€œ*ā€ for simplicity, the boundary value problem (2.13) become,š‘‘4š¹š‘‘š‘§4š‘‘+š‘…š¹3š¹š‘‘š‘§3=0,(2.15) with boundary conditionsš¹(0)=0,š¹ī…žī…ž(0)=0,š¹(1)=1,š¹ī…ž(1)=š›¾š¹ī…žī…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 š‘…=1,š›¾=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 equationšæ(š¹(š‘§))+š‘€(š¹(š‘§))+š‘(š¹(š‘§))=š‘”(š‘§),(3.1) where šæis the operator of the highest order derivative with respect toš‘§, šæ=š‘‘4/š‘‘š‘§4, š‘…is the reminder of the linear term, and the nonlinear term is represented byš‘(š¹(š‘§)). Operating šæāˆ’1 on both sides of (3.1) we get the following: š¹(š‘§)=š›¼0+š›¼1š‘§+š›¼2š‘§22!+š›¼3š‘§33!+šæāˆ’1(š‘”(š‘§))āˆ’šæāˆ’1š‘€(š¹(š‘§))āˆ’šæāˆ’1š‘(š¹(š‘§)),(3.2) whereš›¼š‘–:š‘–=0,1,2,3, the constants can be determined by using initial or boundary conditions.

The unknown function š¹(š‘§)can be expressed by an infinite series of the formš¹(š‘§)=āˆžī“š‘›=0š¹š‘›(š‘§),(3.3) whereš¹0(š‘§)=š›¼0+š›¼1š‘§+š›¼2š‘§2/2!+š›¼3š‘§3/3!+šæāˆ’1(š‘”(š‘§)) and š¹š‘›+1=āˆ’šæāˆ’1š‘€(š¹š‘›(š‘§))āˆ’šæāˆ’1(š“š‘›), š‘›=0,1,2,ā€¦. The nonlinear term š‘(š¹(š‘§)) is decomposed by an infinite series of polynomial given byš‘(š¹(š‘§))=āˆžī“š‘›=0š“š‘›,(3.4) where š“š‘› are the so-called Adomian polynomials that can be determined by the formulaš“š‘›=1š‘‘š‘›!š‘›š‘‘šœ†š‘›īƒ¬š‘īƒ©š‘›ī“š‘–=0šœ†š‘–š¹š‘–īƒŖīƒ­šœ†=0.(3.5)

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

The solutionāˆ‘š¹(š‘§)=āˆžš‘›=0š¹š‘›(š‘§) is approximated by the truncated series of orderš¾, that is,ī‚š¹(š‘§)=š¾ī“š‘›=0š¹š‘›(š‘§).(3.6) In our caseš‘”(š‘§)=0, š‘€(š¹(š‘§))=0.

Now appling ADM on (2.15) and (2.16), for š‘›=0,1,ā€¦,5.

We obtain š¹0š‘§(š‘§)=š“36š¹+šµš‘§1ī‚µ1(š‘§)=āˆ’120š“šµš‘§5+š“2š‘§7ī‚¶š¹50402(š‘§)=š“šµ2š‘§7+š“16802šµš‘§9+š“226803š‘§111108800ā‹®.(3.7)

Considering the Adomian 5th-order solution,ī‚š¹(š‘§)=š¹0(š‘§)+š¹1(š‘§)+ā‹…ā‹…ā‹…+š¹5ī€·š‘§(š‘§)+š‘‚20ī€ø.(3.8)

The boundary conditions at š‘§=1 are used to get the following values of š“and šµ.š“=1.74911,šµ=0.71896.(3.9)

By substituting these values our solution isī‚š¹(š‘§)=0.718965š‘§+0.291518š‘§3āˆ’0.0104795š‘§5āˆ’0.0000688452š‘§7+0.0000701133š‘§9āˆ’3.49247Ɨ10āˆ’6š‘§11āˆ’4.73949Ɨ10āˆ’7š‘§13+5.95015Ɨ10āˆ’8š‘§15āˆ’8.90909Ɨ10āˆ’10š‘§17āˆ’1.65991Ɨ10āˆ’9š‘§19ī€·š‘§+š‘‚20ī€ø.(3.10)

3.2. New Iterative Method

The basic idea of new iterative method [15, 16]. Consider the following nonlinear general differential equation:šæ(š¹(š‘§))+š‘€(š¹(š‘§))+š‘(š¹(š‘§))=š‘”(š‘§),(3.11) where šæis the operator of the highest order derivative with respect toš‘§, šæ=š‘‘4/š‘‘š‘§4, š‘€ is the reminder of the linear term, and the nonlinear term is represented byš‘(š¹(š‘§)). Operating šæāˆ’1 on both sides of (3.11) we get,š¹(š‘§)=š›¼0+š›¼1š‘§+š›¼2š‘§22!+š›¼3š‘§33!+šæāˆ’1(š‘”(š‘§))āˆ’šæāˆ’1š‘€(š¹(š‘§))āˆ’šæāˆ’1š‘(š¹(š‘§)),(3.12) where š›¼š‘–:š‘–=0,1,2,3, are constants to be determined by using initial or boundary conditions.

The unknown function š¹(š‘§)can be expressed by an infinite series of the formš¹(š‘§)=āˆžī“š‘›=0š¹š‘›(š‘§),(3.13) where š¹0(š‘§)=š›¼0+š›¼1š‘§+š›¼2š‘§2/2!+š›¼3š‘§3/3!+šæāˆ’1š‘”(š‘§) and š¹š‘›+1=āˆ’šæāˆ’1š‘€(š¹š‘›(š‘§))āˆ’šæāˆ’1(šŗš‘›), š‘›=0,1,2,ā€¦ The nonlinear term š‘(š¹(š‘§)) is decomposed by an infinite series of polynomials given byš‘(š¹(š‘§))=āˆžī“š‘›=0šŗš‘›,(3.14) where šŗš‘›āˆ‘=š‘(š‘›š‘˜=0š¹š‘˜āˆ‘)āˆ’š‘(š‘›āˆ’1š‘˜=0š¹š‘˜) and šŗ0=š‘(š¹0(š‘§)).

The solutionāˆ‘š¹(š‘§)=āˆžš‘›=0š¹š‘›(š‘§), is approximated by the truncated series of order š¾, that is,ī‚š¹(š‘§)=š¾ī“š‘›=0š¹š‘›(š‘§).(3.15)

In our case š‘”(š‘§)=0 and š‘€(š¹(š‘§))=0.

Now using NIM on (2.15) and (2.16), for š‘›=0,1,ā€¦,5.

We obtain the following: š¹0š‘§(š‘§)=š“36š¹+šµš‘§1ī‚µ1(š‘§)=āˆ’120š“šµš‘§5+š“2š‘§7ī‚¶š¹50402(š‘§)=āˆ’š“šµ2š‘§7+š“16802šµš‘§9+š“226803š‘§11āˆ’š“11088002šµ2š‘§11āˆ’š“19008003šµš‘§13āˆ’š“384384004š‘§153962649600ā‹®.(3.16) Considering the NIM 5th-order solution,ī‚š¹(š‘§)=š¹0(š‘§)+š¹1(š‘§)+š¹3(š‘§)+š¹4(š‘§)+š¹5ī€·š‘§(š‘§)+š‘‚20ī€ø,(3.17) we have the following:ī‚š¹(š‘§)=šµš‘§+š“š‘§36āˆ’1120š“šµš‘§5+ī‚µāˆ’š“2+5040š“šµ2ī‚¶š‘§16807+ī‚µš“2šµāˆ’22680š“šµ3ī‚¶š‘§241929+ī‚µš“3āˆ’1108800241š“2šµ2+39916800š“šµ4ī‚¶š‘§38016011+ī‚µāˆ’71š“3šµ+239500800131š“2šµ3āˆ’207567360š“šµ5ī‚¶š‘§658944013+ī‚µāˆ’1051š“4+2179457280001357š“3šµ2āˆ’251475840004759š“2šµ4ī‚¶š‘§8717829120015+ī‚µ2243š“4šµāˆ’1140023808000359š“3šµ3+50523782400179š“2šµ5ī‚¶š‘§13028843520017+ī‚µ21919š“5āˆ’84475764172800071999š“4šµ2+167094918144000104977š“3šµ4āˆ’30035827261440031š“2šµ6ī‚¶š‘§64981357056019ī€·š‘§+š‘‚20ī€ø.(3.18) Using the boundary conditions at š‘§=1, we get the following value of š“ and šµš“=1.74911,šµ=0.71897.(3.19) The approximate solution is as follows:ī‚š¹(š‘§)=0.718965š‘§+0.291518š‘§3āˆ’0.0104795š‘§5āˆ’0.0000688448š‘§7+0.0000701132š‘§9āˆ’3.49247Ɨ10āˆ’6š‘§11āˆ’4.73948Ɨ10āˆ’7š‘§13+5.95015Ɨ10āˆ’8š‘§15āˆ’8.34647Ɨ10āˆ’11š‘§17āˆ’1.18034Ɨ10āˆ’9š‘§19ī€·š‘§+š‘‚20ī€ø.(3.20)

3.3. HPM

To illustrate the basic idea of homotopy perturbation method [17ā€“21], we consider the following nonlinear differential equation:š“(š¹)=š‘“(š‘Ÿ).(3.21)š“ is a general differential operator, the operator š“ can usually be divided into two parts šæand š‘, where šæ is linear, and š‘ is nonlinear:š“=šæ+š‘.(3.22) sošæ(š¹)+š‘(š¹)āˆ’š‘“(š‘Ÿ)=0,š‘ŸāˆˆĪ©.(3.23)š‘“(š‘Ÿ)is a known analytic function.

With the boundary condition šµ(š¹,šœ•š¹/šœ•š‘›)=0,š‘ŸāˆˆĪ“.

šµ is a boundary operator, and Ī“is the boundary of the domain Ī©.

Now we construct the following homotopy:š»ī€ŗšæī€·š¹(š‘£,š‘)=(1āˆ’š‘)(š‘£)āˆ’šæ0[]ī€øī€»+š‘š“š‘£āˆ’š‘“(š‘Ÿ),(3.24) where š‘āˆˆ[0,1]is an embedding parameter and š¹0 is the first approximation that satisfied the boundary condition. To get an approximate solution, we expand š¹(š‘Ÿ,š‘)in Taylorā€™s series aboutš‘ in the following manner:š¹(š‘Ÿ)=š‘£0(š‘Ÿ)+āˆžī“š‘š=1š‘£š‘š(š‘Ÿ)š‘š‘š.(3.25) Plugging (3.25) into (3.24) and then equating the coefficient of like powers ofš‘, we get the following problems which are directly integrable.

Zeroth-order problem:š¹0(š‘–š‘£)š¹(š‘§)=0,0(0)=0,š¹ā€²ī…ž0(0)=0,š¹0(1)=1,š¹ī…ž0(1)=š¹0ī…žī…ž(1).(3.26)

First-order problem:š¹1š‘–š‘£(š‘§)=āˆ’š¹0(š‘§)š¹0ī…žī…žī…žš¹(š‘§),1(0)=0,š¹1ī…žī…ž(0)=0,š¹1(1)=0,š¹ī…ž1(1)=š¹1ī…žī…ž(1).(3.27)

Second-order problem:š¹2š‘–š‘£(š‘§)=āˆ’š¹1(š‘§)š¹0ī…žī…žī…ž(š‘§)āˆ’š¹0(š‘§)š¹1ī…žī…žī…žš¹(š‘§),2(0)=0,š¹2ī…žī…ž(0)=0,š¹2(1)=0,š¹ī…ž2(1)=š¹2ī…žī…ž(1)ā‹®.(3.28) We consider the following 5th-order solution,ī‚š¹(š‘§)=š¹0(š‘§)+š¹1(š‘§)+š¹3(š‘§)+š¹4(š‘§)+š¹5ī€·š‘§(š‘§)+š‘‚20ī€ø,ī‚š¹(š‘§)=2379640217780939164049š‘§+3309814671645081600000723651501050808628889š‘§3āˆ’248236100373381120000043087465806546383š‘§5āˆ’41115710206771200001652256496336207š‘§7+23984164287283200000121738793951š‘§9āˆ’173581664256000031017787š‘§11āˆ’8879270400000430067207š‘§13+90923728896000074925133š‘§15+122397327360000030859š‘§17āˆ’1542948126720016191589š‘§19ī€·š‘§18688959184896000+š‘‚20ī€ø.(3.29)

3.4. OHAM

According to [22ā€“24], we consider the following differential equation:š“(š¹(š‘§))=š‘“(š‘Ÿ),š‘ŸāˆˆĪ©,(3.30) with the boundary conditionsšµī‚µš¹,šœ•š¹(š‘§)ī‚¶šœ•š‘›=0,š‘ŸāˆˆĪ“,(3.31) where š“is a differential operator, šµis a boundary operator, and š‘“(š‘Ÿ)is a known function of š‘Ÿ: š‘ŸāˆˆĪ©. The operator š“ can be written as š“=šæ+š‘, wherešæ is linear and š‘is a nonlinear operator. In OHAM we first construct a homotopy equation,[](1āˆ’š‘)šæ(š¹(š‘Ÿ,š‘))āˆ’ā„Ž(š‘)šæ(š¹(š‘Ÿ,š‘))+š‘(š¹(š‘Ÿ,š‘))āˆ’š‘“(š‘Ÿ)=0,(3.32) where š‘āˆˆ[0,1]is an embedding parameter, ā„Ž(š‘)is a nonzero auxiliary function for š‘ā‰ 0 andā„Ž(0)=0šæ(š¹(š‘§))=0,forš‘=0š“(š¹(š‘§))=š‘“(š‘Ÿ),forš‘=1.(3.33) The solutionš¹(š‘Ÿ,0)=š‘£0(š‘Ÿ) of šæ(š¹(š‘§))=0 traces the solution curve š‘£(š‘Ÿ) continuously as š‘ approaches to 1, where š‘£0 is the solution of the zeroth-order problem that will come in the next few lines. We next choose the auxiliary function ā„Ž(š‘) in the following form:ā„Ž(š‘)=š‘šī“š‘–=1š‘š‘–š¶š‘–,(3.34) where š¶1,š¶2,ā€¦are the convergence controlling constants which are to be determined. To get an approximate solution, we expand š¹(š‘Ÿ,š‘)in Taylorā€™s series aboutš‘ in the following manner:š¹ī€·š‘Ÿ,š‘,š¶š‘–ī€ø=š‘£0(š‘Ÿ)+āˆžī“š‘š=1ī€·š‘Ÿ,š¶1,š¶1,ā€¦,š¶š‘šī€øš‘š‘š.(3.35) Now after substituting the auxiliary function ā„Ž(š‘) and š¹(š‘Ÿ,š‘,š¶š‘–)in homotopy equation we compare the coefficient of like powers ofš‘, to obtain the following linear equations.

Zeroth-order problem:šæš‘£0ī‚µš‘£=0,šµ0,šœ•š‘£0ī‚¶šœ•š‘›=0.(3.36)

First-order problem:šæš‘£1āˆ’ī€·1+š¶1š‘£0ī€øšæ+š¶1š‘“(š‘Ÿ)=š¶1š‘0š‘£0ī‚µš‘£,šµ1,šœ•š‘£1ī‚¶šœ•š‘›=0.(3.37)

Second-order problem:šæš‘£2āˆ’ī€·1+š¶1ī€øšæš‘£1āˆ’š¶2šæš‘£0āˆ’š¶2š‘“(š‘Ÿ)=š¶2š‘0š‘£0+š¶1š‘1š‘£1ī‚µš‘£,šµ2,šœ•š‘£2ī‚¶šœ•š‘›=0,(3.38) and so on.

If the series is convergent atš‘=1 for suitable auxiliary constants š¶1,š¶2,ā€¦, thenī€·š‘£(š‘Ÿ)=š¹š‘Ÿ,š¶š‘–ī€ø=š‘£0(š‘Ÿ)+āˆžī“š‘š=1š‘£š‘šī€·š‘Ÿ,š¶1,š¶1,ā€¦,š¶š‘šī€ø.(3.39)

The result of the mth-order approximations are given byĢ†š‘£(š‘Ÿ)=š‘£0(š‘Ÿ)+š‘šī“š‘–=1š‘£š‘–ī€·š‘Ÿ,š¶1,š¶1,ā€¦,š¶š‘–ī€ø.(3.40)

Residual of the solution isš‘…ī€·š‘Ÿ,š¶1,š¶2,ā€¦,š¶š‘šī€øĢ†=š“š‘£āˆ’š‘“(š‘Ÿ).(3.41) Ifš‘…=0, Ģ†š‘£ will be the exact solution, but it does not happen specially in nonlinear problems. To find the optimal values ofš¶š‘–, many methods can be applied. We follow the method of least squares. According to the method of least squares, we first construct the functionalš½ī€·š‘Ÿ,š¶1,š¶2,ā€¦,š¶š‘šī€ø=ī€œš‘š‘Žš‘…2š‘‘š‘Ÿ,(3.42) and then minimizing it, we havešœ•š½šœ•š¶1=šœ•š½šœ•š¶2=ā‹Æ=šœ•š½šœ•š¶š‘š=0,(3.43) where š‘Žand š‘are in the domain of the problem. With these constants known, the approximate solution (of orderš‘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:š¹0(š‘–š‘£)š¹(š‘§)=0,0(0)=0,š¹ā€²ī…ž0(0)=0,š¹0(1)=1,š¹ī…ž0(1)=š¹0ī…žī…ž(1).(3.44)

First-order problem:š¹1š‘–š‘£(š‘§)=š¶1š¹0(š‘§)š¹0ī…žī…žī…ž(š‘§)+š¹0š‘–š‘£(š‘§)+š¶1š¹0š‘–š‘£š¹(š‘§),1(0)=0,š¹1ī…žī…ž(0)=0,š¹1(1)=0,š¹ī…ž1(1)=š¹1ī…žī…ž(1).(3.45)

Second-order problem:š¹2š‘–š‘£(š‘§)=š¶1š¹1(š‘§)š¹0ī…žī…žī…ž(š‘§)+š¶1š¹0(š‘§)š¹1ī…žī…žī…ž(š‘§)+š¹1š‘–š‘£(š‘§)+š¶1š¹1š‘–š‘£š¹(š‘§),2(0)=0,š¹2ī…žī…ž(0)=0,š¹2(1)=0,š¹ī…ž2(1)=š¹2ī…žī…ž(1)ā‹®.(3.46) Considering the OHAM 5th-order solution and using the method of least squares, we obtain š¶1=āˆ’0.93281. Hence the solution is ī‚š¹(š‘§)=0.718965š‘§+0.291517š‘§3āˆ’0.0104795š‘§5āˆ’0.0000688407š‘§7+0.0000700828š‘§9āˆ’3.4536Ɨ10āˆ’6š‘§11āˆ’4.87073Ɨ10āˆ’7š‘§13+5.80616Ɨ10āˆ’8š‘§15+3.31269Ɨ10āˆ’9š‘§17āˆ’5.61766Ɨ10āˆ’10š‘§19ī€·š‘§+š‘‚20ī€ø.(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 š¹ 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:š¹(š‘§)=0.718965š‘§āˆ’2.5833Ɨ10āˆ’6š‘§2+0.291529š‘§3āˆ’0.000025575š‘§4āˆ’0.0104737š‘§5+0.0000970056š‘§6āˆ’0.000294814š‘§7+0.000231011š‘§8āˆ’0.0000378459š‘§9+0.0000122062š‘§10.(3.48)

3.6. DTM

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

If š¹(š‘§) is a given function, its differential transform is defined as follows:š¹1(š‘Ÿ)=š‘‘š‘Ÿ!š‘Ÿš¹(š‘§)š‘‘š‘§š‘Ÿ||||š‘§=0.(3.49) The inverse transform of š¹(š‘Ÿ) is defined byš¹(š‘§)=āˆžī“š‘Ÿ=0š‘§š‘Ÿš¹(š‘Ÿ).(3.50) In actual application, the function š¹(š‘§) is expressed by a finite seriesš¹(š‘§)=š‘ī“š‘Ÿ=0š‘§š‘Ÿš¹(š‘Ÿ).(3.51) Equation (3.51) implies that āˆ‘š¹(š‘§)=āˆžš‘Ÿ=š‘+1š‘§š‘Ÿš¹(š‘Ÿ). is negligibly small.

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

3.6.1. Analysis of the Method

Consider a fourth-order boundary value problemš¹(4)(š‘§)=šŗ(š‘§,š¹),0<š‘§<š»,(3.52) with the boundary conditions:š¹(0)=š›¼0,š¹(š»)=š›¼1,š¹(1)(š»)=š›¼2,š¹(2)(0)=š›¼3,(3.53) where š›¼š‘–:š‘–=0,1,2,3 are given values.

The differential transform of (3.52) is as follows:š¹(š‘Ÿ+4)=šŗ(š‘Ÿ)āˆ4š‘–=1(š‘Ÿ+š‘–),(3.54) where šŗ(š‘Ÿ) is the differential transform ofšŗ(š‘§,š¹).

The transformed boundary conditions (3.53) are given byš¹(0)=š›¼0,š‘ī“š‘Ÿ=0š»š‘Ÿš¹(š‘Ÿ)=š›¼1,š‘ī“š‘Ÿ=0š‘Ÿš»š‘Ÿš¹(š‘Ÿ)=š›¼2,š›¼š¹(2)=32.(3.55) Using (3.54) and (3.55) values of š¹(š‘–)āˆ¶š‘–=4,5,ā€¦are obtained which give the following series solution up to š‘‚(š‘§š‘+1),š¹(š‘§)=š‘ī“š‘Ÿ=0š‘§š‘Ÿī€·š‘§š¹(š‘Ÿ)+š‘‚š‘+1ī€ø,(3.56) applying DTM on (2.15) and (2.16), the transformed boundary conditions and differential are as follows: š¹(0)=0,š¹(1)=š‘Ž,š¹(2)=0,š¹(3)=š‘,š¹(š‘Ÿ+4)=š‘Ÿ!īƒÆ(š‘Ÿ+4)āˆ’š‘…š‘Ÿī“š‘˜=0(š‘˜+1)(š‘˜+2)(š‘˜+3)š¹(š‘˜+3)īƒ°.š¹(š‘Ÿāˆ’š‘˜)(3.57) Using (3.57), we obtain the following values of š¹(š‘–)āˆ¶š‘–=1,2,3ā€¦,19.

For these values, the unknowns š‘Žand š‘ are determined by the following system:19ī“š‘Ÿ=0ī‚š¹(š‘Ÿ)=1,19ī“š‘Ÿ=0š‘Ÿī‚š¹(š‘Ÿ)=19ī“š‘Ÿ=0ī‚š‘Ÿ(š‘Ÿāˆ’1)š¹(š‘Ÿ).(3.58) We find, š‘Ž=1.53266andš‘=āˆ’0.570353.

Now using the inverse differential transform, the following approximate solution of š‘‚(š‘§20)is obtained:ī‚š¹(š‘§)=1.53266š‘§āˆ’0.570353š‘§3+0.043708š‘§5āˆ’0.00710856š‘§7+0.00130068š‘§9āˆ’0.000251907š‘§11+0.0000492109š‘§13āˆ’9.6216Ɨ10āˆ’6š‘§15+1.86893Ɨ10āˆ’6š‘17āˆ’3.60367Ɨ10āˆ’7š‘§19ī€·š‘§+š‘‚20ī€ø.(3.59)

4. Conclusion

In this paper we have used Adomianā€™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 Taylorā€™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 Taylorā€™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.