# Analytic Approximate Solutions for Unsteady Two-Dimensional and Axisymmetric Squeezing Flows between Parallel Plates

Authors: Mohammad Mehdi Rashidi; Hamed Shahmohamadi; Saeed Dinarvand
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

The flow of a viscous incompressible fluid between two parallel plates due to the normal motion of the plates is investigated. The unsteady Navier-Stokes equations are reduced to a nonlinear fourth-order differential equation by using similarity solutions. Homotopy analysis method (HAM) is used to solve this nonlinear equation analytically. The convergence of the obtained series solution is carefully analyzed. The validity of our solutions is verified by the numerical results obtained by fourth-order Runge-Kutta.


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## Body

## 1. Introduction

The
problem  of  unsteady squeezing of a viscous incompressible fluid between two
parallel plates in motion normal to their own surfaces independent of each
other and arbitrary with respect to time is a fundamental type of unsteady flow
which is met frequently in many hydrodynamical machines and apparatuses. Some practical examples of squeezing flow include polymer
processing, compression, and injection molding.  In addition, the lubrication
system can also be modeled by squeezing flows. Stefan [1] published a classical
paper on squeezing flow by using lubrication approximation. In 1886, Reynolds
[2] obtained a solution for elliptic plates, and Archibald [3] studied this
problem for rectangular plates. The theoretical and experimental studies of
squeezing flows have been conducted by many researchers [4, 4–14]. Earlier
studies of squeezing flow are based on Reynolds equation. The inadequacy of
Reynolds equation in the analysis of porous thrust bearings and squeeze films
involving high velocity has been demonstrated by Jackson [13], Ishizawa [14].  The general study
of the problem with full Navier-Stokes equations involves extensive numerical
study requiring more computer time and larger memory. However, many of the
important features of this problem can be grasped by prescribing the relative
velocity of the plates suitably. If the relative normal velocity is
proportional to (1−αt)1/2, where t is the time and α a constant of dimension [T−1] which characterizes unsteadiness, then the
unsteady Navier–Stokes equations
admit similarity solution.In 1992, Liao [15] employed the basic ideas of
the homotopy in topology to propose a general analytic method for nonlinear problems,
namely, homotopy analysis method (HAM) [16–21]. Based on
homotopy of topology, the validity of the HAM is independent of whether or not
there exist small parameters in the considered equation. Therefore, the HAM can
overcome the foregoing restrictions and limitations of perturbation methods [22].
Furthermore, the HAM always provides us with a family of solution expressions
in the auxiliary parameter ℏ, the convergence region, and the rate of each
solution might be determined conveniently by the auxiliary parameter ℏ. The HAM also avoids discretization and
provides an efficient numerical solution with high accuracy, minimal
calculation, and avoidance of physically unrealistic assumptions. Besides, the
HAM is rather general and contains the homotopy perturbation method (HPM) [21],
the Adomian decomposition method (ADM) [23], and δ-expansion method. In fact,
HPM and ADM are always special cases of HAM when ℏ=−1. The convergence of HAM solution
series is dependent upon three factors, that is, the initial guess, the
auxiliary linear operator, and the auxiliary parameter ℏ. However, as a special case of homotopy
analysis method when ℏ=−1, the convergence of HPM solution series is only
dependent upon two factors: the auxiliary linear operator and the initial guess. So, given the initial guess and the
auxiliary linear operator, HPM cannot provide other ways to ensure that the
solution is convergent. HAM provides us
with a family of solution expression in the auxiliary parameter ℏ and the solution given by ADM is only
one of them.In
recent years, the HAM has been successfully employed to solve many types of
nonlinear problems such as the nonlinear equations arising in heat transfer [24],
the nonlinear model of diffusion and reaction in porous
catalysts [25], the chaotic dynamical systems [26], the nonhomogeneous
Blasius problem [27], the generalized three-dimensional
MHD flow over a porous stretching sheet [28], the wire
coating analysis using MHD Oldroyd
8-constant fluid [29], the axisymmetric flow and heat transfer of a second-grade fluid past a
stretching sheet [30], the MHD flow of a second-grade
fluid in a porous channel [31], the generalized Couette
flow [32], the Glauert-jet problem [33], the Burger and regularized long wave equations [34], the laminar
viscous flow in a semiporous channel in the presence of a uniform magnetic
field [35], and other problems. All of these successful applications verified the validity,
effectiveness, and flexibility of the HAM.In
this paper, we use homotopy analysis method to investigate the problem of
unsteady squeezing of a viscous incompressible fluid between two parallel
plates. The paper is organized as follows. In Section2, the mathematical
formulation is presented. In Section 3, we extend the application of the HAM to
construct the approximate solutions for the governing equation. The convergence
of the obtained series solution is carefully analyzed in Section 4. Section 5
contains the results and discussion. The conclusions are summarized in Section
6.


## 2. Mathematical formulation

Let
the position of the two plates be atz=±ℓ(1−αt)1/2, where ℓ is the position at time t=0 as shown in Figure 1. We assume that the
length 1 (in the two-dimensional case) or the diameter D (in the axisymmetric
case) is much
larger than the gap width 2z at any time that the end effects can be
neglected. When α is positive the two plates are squeezed until
they touch at t=1/α. When α is negative the two plates are separated. Let u, v, and w be the velocity components in the x, y,
and z directions, respectively. For two-dimensional flow, Wang
introduced the following transforms [36]: (2.1)u=αx[2(1−αt)]f′(η),w=−αℓ[2(1−αt)1/2]f(η), where (2.2)η=z[ℓ(1−αt)1/2]. Substituting (2.1) into the unsteady two-dimensional Navier-Stokes equations yields a
nonlinear ordinary differential equation in form (2.3)f′′′′+S{−ηf′′′−3f′′−f′f′′+ff′′′}=0, where S=αℓ2/2ν (squeeze number) is the nondimensional
parameter. The flow is
characterized by this parameter. It should be mentioned that ν is the kinematic
viscosity. The boundary conditions are such that on the plates the
lateral velocities are zero and the normal velocity is equal to the velocity of
the plate, that is, (2.4)f(0)=0,f′′(0)=0,f(1)=1,f′(1)=0. Similarly, Wang’s
transforms [36] for axisymmetric flow are (2.5)u=αx[4(1−αt)]f′(η),v=αy[4(1−αt)]f′(η),w=−αℓ[2(1−αt)1/2]f(η). Using transforms (2.5), unsteady
axisymmetric Navier–Stokes equations
reduce to (2.6)f′′′′+S{−ηf′′′−3f′′+ff′′′}=0, subject to the boundary
conditions (2.4).Figure 1
Schematic diagram of the problem.Consequently, we should solve the
nonlinear ordinary differential equation(2.7)f′′′′+S{−ηf′′′−3f′′−βf′f′′+ff′′′}=0, where (2.8)β={0,axisymmetric,1,two-dimensional, and subject to boundary
conditions
(2.4).


## 3. HAM solution

To investigate the explicit and totally analytic solutions of (2.7) by using HAM,
we choose (3.1)f0(η)=12(3η−η3), as initial approximation of f(η), which satisfies the boundary conditions (2.4).
Besides, we select the auxiliary linear operator ℒ(f) as (3.2)ℒ(f)=f′′′′. It is easy to check that this
operator satisfies the following equation: (3.3)ℒ(c1η3+c2η2+c3η+c4)=0, where ci,1≤i≤4, are
arbitrary constants. Based on
(2.7), we are led to define the following nonlinear operator: (3.4)𝒩[φ(η;p)]=∂4φ(η;p)∂η4+S(−η∂3φ(η;p)∂η3−3∂2φ(η;p)∂η2−β∂φ(η;p)∂η∂2φ(η;p)∂η2+φ(η;p)∂3φ(η;p)∂η3). Using these operators, we can construct the so-called zeroth-order
deformation equation as (3.5)(1−p)ℒ[φ(η;p)−f0(η)]=pℏ𝒩[φ(η;p)], where p∈[0,1] is an
embedding parameter and ℏ is an auxiliary nonzero parameter. It should be emphasized that one has
great freedom to choose the initial guess, the auxiliary linear operator and
the auxiliary parameter ℏ. However, (3.5), the original equation of HAM, is the origin of
the mathematical term “homotopy" (parameter ℏ is the head letter of “homotopy"). In addition, if ℏ=−1, (3.5) will always change to original equation of HPM. The boundary conditions
for (3.5) are (3.6)φ(0;p)=0,∂2φ(0;p)∂η2=0,φ(1;p)=1,∂φ(1;p)∂η=0. Obviously, when p=0 and p=1, the above zeroth-order deformation equation
has the following solutions: (3.7)φ(η;0)=f0(η),φ(η;1)=f(η). As p increases from 0 to 1, φ(η;p) varies from f0(η) to f(η). Now expanding φ(η;p) by its Taylor  series in terms of p, one would
obtain (3.8)φ(η;p)=f0(η)+∑m=1+∞fm(η)pm, where (3.9)fm(η)=1m!∂mφ(η;p)∂pm|p=0. As pointed out by Liao [19], the convergence of
series (3.8) strongly depends
upon the auxiliary parameter ℏ. Assume that ℏ is selected such that series (3.8) is
convergent at p=1, then due to (3.7), the final series solution becomes (3.10)f(η)=f0(η)+∑m=1+∞fm(η). For the mth-order deformation equation, we differentiate (3.5) m times with
respect to p, divide by m!, and then set p=0. The resulting deformation equation at the mth-order is (3.11)ℒ[fm(η)−χmfm−1(η)]=ℏRm(η), with the following
boundary conditions (3.12)fm(0)=0,fm′′(0)=0,fm(1)=0,fm′(1)=0, where (3.13)Rm(η)=∂4fm−1(η)∂η4+S(−η∂3fm−1(η)∂η3−3∂2fm−1(η)∂η2+∑n=0m−1(−β∂fn(η)∂η∂2fm−1−n(η)∂η2+fn(η)∂3fm−1−n(η)∂η3)),χm={0,m≤1,1,m>1. We use the symbolic
software MATHEMATICA to solve the system of linear equations (3.11) with the boundary
conditions (3.12), and successively obtain (3.14)f1(η)=ℏS560{(37+15β)η−(73+33β)η3+(35+21β)η5−(−1+3β)η7},f2(η)=ℏS15523200{(1025640+415800β+1025640ℏ+415800ℏβ+153060ℏS+126789ℏSβ+25875ℏSβ2)η−(2023560+914760β+2023560ℏ+914760ℏβ+349010ℏS+325392ℏSβ+73998ℏSβ2)η3+(970200+582120β+970200ℏ+582120ℏβ+227304ℏS+281358ℏSβ+79002ℏSβ2)η5−(−27720+83160β−27720ℏ+83160ℏβ+20196ℏS+92268ℏSβ+40392ℏSβ2)η7+(10780ℏS+8085ℏSβ+10395ℏSβ2)η9−(378ℏS−1428ℏSβ+882ℏSβ2)η11},⋮ Therefore,  like (3.10), the analytical
solution of the problem can be expressed as an infinite series of the form (see
[37]) (3.15)f(η)=f0(η)+limM→∞[∑m=1M(∑n=12m+2am,nη2n−1)],f(η)=limM→∞[∑m=0M(∑n=12m+2am,nη2n−1)].


## 4. Convergence of HAM solution

The series
solution contains the auxiliary parameterℏ. The validity of the method is based on such an
assumption that series (3.8) converges at p=1. It is the auxiliary parameter ℏ which ensures that this assumption can be
satisfied. In general, by means of the so-called ℏ-curve, it is straightforward to choose a proper value
of ℏ which ensures that the series solution is
convergent. For the different values of the squeeze number S, the ℏ-curves obtained by the 15th-order approximation for
the axisymmetric (β=0) and two-dimensional (β=1) cases are shown in Figures 2 and 3,
respectively. From these figures, the valid regions of ℏ correspond to the line segments nearly
parallel to the horizontal axis. Figures 2 and 3 elucidate that the size of the
valid region strongly depends on S. In fact, the interval for admissible values
of ℏ shrinks toward zero by increasing the squeeze
number. As mentioned above, the homotopy analysis method is rather general and always
contains the homotopy perturbation method (HPM) [21] and the Adomian
decomposition method (ADM) [23] when ℏ=−1. From Figures 2 and 3, ℏ=−1 is not valid for the large values of S.Figure 2
Theℏ-curves of f′(0) given by the 15th-order approximation for the
axisymmetric case for the different values of the squeeze number S.Figure 3
Theℏ-curves of f′(0) given by the 15th-order approximation for the
two-dimensional case for the different values of the squeeze number S.


## 5. Results and discussion

Our
main concern is the various values off(η) and f′(η). These quantities describe the flow behavior.
For several values of S, the function f(η) obtained by the different orders of approximation
for the axisymmetric and two-dimensional cases are compared with the numerical
results in Tables
1 and 2, respectively. It is worth mentioning that the numerical results have been obtained using the fourth-order
Runge-Kutta
in C++ program. We can see a very good agreement between the
purely
analytic results of the HAM and numerical results. The variation of f′(η) with the change in the positive
values of S for the two-dimensional case is plotted in Figure 4. Figure 5 shows
the influence of negative S on f′(η) for the axisymmetric case. Note that for
the large negative values of S, the results of similarity analysis are not
reliable. f′′(1) gives skin friction, and f′′′(1) represents pressure gradient. f′′(1) and f′′′(1) as functions of S are illustrated in Figures 6
and 7, respectively.Table 1
The analytic results off(η) at different orders of approximation
compared with the numerical results obtained by the
fourth-order Runge-Kutta for the axisymmetric case.
S,ℏη2nd-order4th-order6th-order7th-order8th-orderNumerical−1.5,−1.30.20.3194740.3195260.3195260.3195260.3195260.3195260.40.6036520.6038250.6038300.6038300.6038300.6038300.60.8225740.8228630.8228750.8228760.8228760.8228760.80.9565800.9567890.9568000.9568010.9568010.956801−0.5,−10.20.3025450.3025820.3025820.3025820.3025820.3025820.40.5780280.5780820.5780820.5780820.5780820.5780820.60.8007370.8007800.8007800.8007800.8007800.8007800.80.9476860.9477020.9477020.9477020.9477020.9477020.5,−10.20.2903530.2903220.2903220.2903220.2903220.2903220.40.5592990.5592530.5592520.5592520.5592520.5592520.60.7843410.7843040.7843030.7843030.7843030.7843030.80.9407170.9407040.9407030.9407030.9407030.9407031.5,−0.80.20.2810320.2810100.2810100.2810100.2810100.2810100.40.5448510.5447800.5447790.5447790.5447790.5447790.60.7714930.7713740.7713710.7713710.7713710.7713710.80.9351270.9350380.9350360.9350360.9350360.9350362.5,−0.70.20.2737670.2736830.2736820.2736820.2736820.2736820.40.5335160.5332550.5332470.5332460.5332460.5332460.60.7612990.7608680.7608480.7608470.7608470.7608470.80.9306170.9302990.9302810.9302800.9302800.930280Table 2
The analytic results off(η) at different orders of approximation
compared with the numerical results obtained by the
fourth-order Runge-Kutta for the two-dimensional case.
S,ℏη2nd-order4th-order6th-order7th-order8th-orderNumerical−1.5,−1.30.20.3328830.3335910.3336170.3336180.3336180.3336180.40.6231900.6243150.6243580.6243580.6243580.6243580.60.8382190.8392840.8393240.8393250.8393250.8393250.80.9624410.9629610.9629830.9629840.9629840.962984−0.5,−10.20.3054360.3055430.3055450.3055450.3055450.3055450.40.5823140.5824680.5824700.5824700.5824700.5824700.60.8042710.8043900.8043920.8043920.8043920.8043920.80.9490650.9491070.9491080.9491080.9491080.9491080.5,−10.20.2883470.2882610.2882600.2882600.2882600.2882600.40.5562680.5561450.5561430.5561430.5561430.5561430.60.7817680.7816700.7816710.7816710.7816710.7816710.80.9396740.9396410.9396400.9396400.9396400.9396401.5,−0.80.20.2765260.2764330.2764320.2764320.2764320.2764320.40.5379290.5377540.5377520.5377520.5377520.5377520.60.7654630.7652520.7652490.7652490.7652490.7652490.80.9326070.9324740.9324710.9324710.9324710.9324712.5,−0.70.20.2680410.2677970.2677910.2677910.2677910.2677910.40.5245580.5240570.5240450.5240450.5240450.5240450.60.7532850.7526270.7526050.7526050.7526050.7526050.80.9271660.9267240.9267040.9267030.9267030.926703Figure 4
The influence of positiveS on f′(η) for the two-dimensional case, when ℏ=−0.3.Figure 5
The influence of negative S onf′(η) for the axisymmetric case, when ℏ=−2.Figure 6
The
skin friction (f′′(1)) for the axisymmetric and two-dimensional
cases, when ℏ=−0.4. From ℏ-curves, the series solution for ℏ=−0.4 converges in the whole region 0≤S≤10.Figure 7
The pressure gradient (f′′′(1)) for
the axisymmetric and two-dimensional
cases, when ℏ=−0.4. From ℏ-curves, the series solution for ℏ=−0.4 converges in the whole region 0≤S≤10.


## 6. Conclusions

In this paper, the unsteady axisymmetric and two-dimensional squeezing
flows between two parallel plates are studied analytically using the HAM. The convergence of the results is explicitly shown. Graphical results and tables are presented to
investigate the influence of the squeeze number
S on the velocity, skin friction, and pressure
gradient. The solution obtained by means of
the HAM is an infinite power series for appropriate initial approximation,
which can be, in turn, expressed in a closed form. Unlike perturbation methods, the HAM does not depend on
any small physical parameters. Thus, it is valid for both weakly and strongly
nonlinear problems. Besides, different from all other analytic methods, the HAM
provides us with a simple way to adjust and control the convergence region of
the series solution by means of auxiliary parameterℏ. Thus the auxiliary parameter ℏ plays an important role within the frame of
the HAM which can be determined by the so-called ℏ-curves. Consequently, the present success of the
homotopy analysis method for the highly nonlinear problem of squeezing
flows verifies that the method is a useful tool for
nonlinear problems in science and engineering.



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*Source: 935095-2009-02-25.xml*