Application of Adomian Decomposition Method(ADM)

28 Pages·2014·1.93 MB·English

Gen. Math. Notes, Vol. 20, No. 2, February 2014, pp.22-49
ISSN 2219-7184; Copyright (cid:13)c ICSRS Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Numerical Study of a Non-Linear
Peristaltic Transport: Application of
Adomian Decomposition Method(ADM)
Kh. S. Mekheimer1, K.A. Hemada2, K.R. Raslan3,
R.E. Abo-Elkhair4 and A.M.A. Moawad5
1Mathematics & Statistics Department, Faculty of Science
Taif University, Hawia 888, Taif, KSA
E-mail: kh [email protected]
2,3,4,5Mathematical Department, Faculty of Science (Men)
Al-Azhar University, Nasr City 11884, Cairo, Egypt
2E-mail:[email protected]
3E-mail:kamal [email protected]
4E-mail:[email protected]
5E-mail:ali [email protected]
(Received: 8-11-13 / Accepted: 15-12-13)
Abstract
The aim of this problem is to study the peristaltic motion of a Newtonian,
homogenous and incompressible viscous fluid in a two dimensional channel by
using Adomian Decomposition Method. The expressions for stream function,
velocity and vorticity are obtained without any approximation for the governing
equations and also for the choice of the problem parameters values, but we used
Adomian Decomposition Method as approximated one. The effects for arbitrary
values of wave number (α), amplitude ratio (φ) and Reynolds number (R )
e
on flow characteristics are studied in detail.
Keywords: ADM, Peristaltic transport, Numerical Study.
Numerical Study of a Non-Linear... 23
1 Introduction
When a progressive wave resulting from area contraction and expansion of an
extensible tube propagates along the length of the tube, a fluid contained in
the tube is mixed and transported in the direction of the wave propagation as
if it were squeezed out by the moving wall. This phenomenon, called ”peri-
stalsis”, is an inherent property of any tubular organ of the human body such
as the ureter, the gastro-intestinal tract, or the small blood vessels. In the
last decades the mechanism of this mixing and transporting peristaltic motion
has acquired general interest in the field of hydrodynamics and a number of
studies have been undertaken with respect to peristaltic flows by applying a
simple hydrodynamic model represented by sinusoidal waves. Some of them
are summarized in the review by Jaffrin & Shapiro (1971) [1]. Since these
studies have been developed in connection with the function of organs of the
human body, many of them have been concerned with the flow within a range
of small Reynolds numbers. Moreover, peristaltic pumping has been quite
recently utilized for the transport of such fluids as blood, slurries, corrosive
fluids, and it is desirable to prevent them from coming into contact with the
mechanical parts of the pump. So it is necessary that the peristaltic flow at
moderate Reynolds numbers should be analyzed.
In the range of finite Reynolds number, a theoretical analysis of the peri-
staltic flow is extremely difficult because of the nonlinearity due to the inter-
action between the moving wall and the flow field.Fung & Yih (1968)[2] first
performed a study of peristaltic flow including small nonlinear effects. They
treated the case of a two-dimensional channel with a small ratio of amplitude
to wavelength of the peristaltic wave, and their analysis was based upon per-
turbations of the ratio of wave amplitude to mean channel width. Shapiro,
Jaffrin & Weinberg (1969)[3] obtained a linear solution of peristaltic flow by
considering the limiting case in which the wavelength was infinite and the
fluid inertia could be neglected, and by applying Stokes’ approximation to
the flow. This linear solution was modified later by Jaffrin (1973)[4] in his
consideration of small nonlinear effects. He made a perturbation analysis so
as to obtain the series solutions for both the case of long wavelength at zero
Reynolds number and the case of small Reynolds number at infinite wave-
length. The applicable range of these series solutions was studied through a
comparison with experimental results (Weinberg, Eckstein & Shapiro 1971)[5].
For the limiting case of large Reynolds number with small amplitude and long
wavelength, Hanin (1968)[6] analyzed the flow by applying the boundary-layer
equation, and Ayukawa, Kawai & Kimura (1981)[7] obtained experimental re-
sults as well as an approximate solution of potential flow on the basis of source
distributions. In these two works, the peristaltic pumping was investigated
with reference to engineering applications. On the other hand, with respect
24 Kh. S. Mekheimer et al.
to numerical investigations, we have the analyses of Tong & Vawter (1972)[8]
and Brown & Hung(1977)[9]. The former uses the finite-element method that
is available for the case of large amplitude, and the latter applies the finite-
difference method by employing orthogonal curvilinear coordinates. Recently
Kh. S. Mekheimer & et al. ([10]-[13]) obtained a linear solution of peristaltic
flow by considering the limiting case in which the wavelength was infinite.
But each of them has its disadvantage in requiring enormous and complicated
calculations and/or a restriction on the geometry of the calculating region.
In this paper, the Navier-Stokes equations are solved numerically by using
Adomian Decomposition Method on the peristaltic flow induced by an infinite
train of sinusoidal waves in a two-dimensional channel.The second section will
give brief summary of the solution method which will be used. But the third
section will explain in detail the form of the problem and find a solution by
using Adomian Decomposition Method to get a stream function, velocity and
vorticity of the fluid and in the fourth section, we will discuss in detail the
results we have obtained and The effects of wave number (α), amplitude ratio
(φ) and Reynolds number (R ) on flow characteristics .
e
2 Adomian Decomposition Method(ADM)([14]-
[18])
In this section we will discuss the Adomian Decomposition Method. The Ado-
mianDecompositionMethodhasbeenreceivingmuchattentioninrecentyears
in applied mechanics in general, and in the area of series solutions in particu-
lar. The method proved to be powerful, effective, and can easily handle a wide
classoflinearornonlinear, ordinaryorpartialdifferentialequations, andlinear
or nonlinear integral equations. The decomposition method demonstrates fast
convergence of the solution and therefore provides several significant advan-
tages. The Adomian Decomposition Method was introduced and developed
by George Adomian in and is well addressed in the literature. A considerable
amount of research work has been invested recently in applying this method
to a wide class of linear and nonlinear ordinary differential equations, partial
differential equations and integral equations as well. The Adomian Decompo-
sition Method consists of decomposing the unknown function u(x,y) of any
equation into a sum of an infinite number of components defined by the de-
composition series:
∞
(cid:88)
u(x,y) = u (x,y), (1)
n
n=0
where the components u (x,y) (n ≥ 0) are to be determined in a recursive
n
manner. The decomposition method concerns itself with finding the com-
Numerical Study of a Non-Linear... 25
ponents (u ,u ,u ,···) individually. As will be seen through the text, the
0 1 2
determination of these components can be achieved in an easily way through
a relation that usually involve simple integrals. In this study, we solve for
(L (u)), hence the other linear operators, Simply become R operator in the
y
partial differential equation as follows:
L (u)+R(u)+N(u) = g(x,y), (2)
y
where N(u) represents the nonlinear term and L (∗) = ∂n(∗), therefore:
y ∂yn
L (u) = −R(u)−N(u)+g(x,y). (3)
y
Then we operate both sides the inverse operator (assuming it exists) which
is an n-fold integration operator from 0 to y, since
(cid:90) y
L−1(∗) = (∗)dy,
y
0
(cid:124) (cid:123)(cid:122) (cid:125)
n−times
so:
L−1(L (u)) = −L−1(R(u))−L−1(N(u))+L−1(g(x,y)). (4)
y y y y y
Then, we get:
L−1(L (u)) = u−Φ ,
y y y
where
(cid:88)n−1 K (x)yj
j
Φ = ,
y
j!
j=0
and K (x) are the integration constants which are evaluated from the given
j
conditions, so:
(cid:88)n−1 K (x)yj
u(x,y) = j +L−1(g(x,y))−L−1(R(u))−L−1(N(u)). (5)
j! y y y
j=0
In the use of this method, the solution of the partial differential equation
is written as:(1), where u is a function involving the initial (or boundary)
0
conditions, the forcing function g(x,y) and integral constants. The other com-
ponents of u are determined by recursive relation. The nonlinear term N(u),
such as (u2,u3,sin[u],···), etc. can be expressed by infinite series of the so-
called Adomian polynomials A given in the form
n
∞
(cid:88)
N(u) = A (u ,u ,u ,··· ,u ), (6)
n 0 1 2 n
n=0
26 Kh. S. Mekheimer et al.
where A are specially generated Adomian polynomials for the specific nonlin-
n
earity. TheAdomianpolynomialsA fornonlineartermN(u)canbeevaluated
n
by using the following expression
1 dn (cid:88)n
A = (N( ζiu ))| , n ≥ 0. (7)
n n!dζn i ζ=0
i=0
So the solution can now be written as:
(cid:88)∞ (cid:88)n−1 K (x)yj (cid:88)∞ (cid:88)∞
u(x,y) = u = j +L−1(g(x,y))−L−1(R( u ))−L−1( A )),
n n n
j!
n=0 j=0 n=0 n=0
(8)
where the first two terms from the last equation are identified as u in the
0
assumed decomposition series and the other terms represent the components
after u . Then we get the following recursive relation:
0
(cid:88)n−1 K (x)yj
u = j +L−1(g(x,y)),
0
j! (9)
j=0
u = −L−1(R(u ))−L−1(A ), n ≥ 0.
n+1 n n
From which all components of the decomposition are identified and computed.
3 Statement of Physical Model
A Newtonian, homogenous and incompressible viscous fluid with no restriction
either on the geometry or on non-linearity is considered. In the fixed frame
(X,Y) the central axis of a two dimensional channel is taken along the X-axis
(see figure 1). The wall of the domain is defined by
2π
H(X,T) = h−(cid:15)cos[ (X −CT)], (10)
λ
where, T is time, (cid:15) is the wave amplitude, h is the mean distance of the wall
from the central axis, λ is the wave length and C is the velocity with which
the infinite train of sinusoidal wave progresses along the wall in the positive
X-direction. The flow which is unsteady in the fixed frame (X,Y) appears
steady in the moving frame (x(cid:48), y(cid:48)) which travels in the positive X-direction
with velocity “C”.
The relations between the two frames are:
x(cid:48) = X −CT, y(cid:48) = Y, u(cid:48) = U −C, v(cid:48) = V. (11)
Numerical Study of a Non-Linear... 27
Figure 1: Geometry of two-dimensional peristaltic channel
Where (U,V) and (u(cid:48),v(cid:48)) are components of velocity in the fixed and moving
frames respectively. In the fixed frame the equations governing the flow (with
pressure “P ”, viscosity “µ” and density “ρ”) are:
∂U ∂V
+ = 0, (12)
∂X ∂Y
DU ∂U ∂U ∂U ∂P ∂2U ∂2U
ρ[ ] = ρ[ +U +V ] = − +µ[ + ], (13)
DT ∂T ∂X ∂Y ∂X ∂X2 ∂Y2
and
DV ∂V ∂V ∂V ∂P ∂2V ∂2V
ρ[ ] = ρ[ +U +V ] = − +µ[ + ], (14)
DT ∂T ∂X ∂Y ∂Y ∂X2 ∂Y2
and the boundary conditions are:
∂U
V = 0, = 0, at Y = 0,
∂Y
(15)
dH 2πC 2π
U = 0, V = = − (cid:15)sin[ (X −CT)] at Y = H.
dT λ λ
The stream function Ψ and the vorticity Ω in the fixed frame are introduced
as:
∂Ψ ∂Ψ ∂V ∂U
U −C = , V = − , Ω = − , (16)
∂Y ∂X ∂X ∂Y
and in the moving frame (x(cid:48),y(cid:48)) the stream function ψ(cid:48) and vorticity ω(cid:48) are
defined by
∂ψ(cid:48) ∂ψ(cid:48) ∂v(cid:48) ∂u(cid:48)
u(cid:48) = , v(cid:48) = − , ω(cid:48) = − . (17)
∂y(cid:48) ∂x(cid:48) ∂x(cid:48) ∂y(cid:48)
28 Kh. S. Mekheimer et al.
Using equations (11) in [(12) to (15)], we obtain the governing equations in
the moving frame as:
∂u(cid:48) ∂v(cid:48)
+ = 0, (18)
∂x(cid:48) ∂y(cid:48)
∂u(cid:48) ∂u(cid:48) ∂p(cid:48) ∂2u(cid:48) ∂2u(cid:48)
ρ[u(cid:48) +v(cid:48) ] = − +µ[ + ], (19)
∂x(cid:48) ∂y(cid:48) ∂x(cid:48) ∂x(cid:48)2 ∂y(cid:48)2
and
∂v(cid:48) ∂v(cid:48) ∂p(cid:48) ∂2v(cid:48) ∂2v(cid:48)
ρ[u(cid:48) +v(cid:48) ] = − +µ[ + ]. (20)
∂x(cid:48) ∂y(cid:48) ∂y(cid:48) ∂x(cid:48)2 ∂y(cid:48)2
Also, the configuration of the peristaltic wall can be represented by:
2π
η(cid:48)(x(cid:48)) = h−(cid:15)cos[ x(cid:48)], (21)
λ
and the no-slip condition or the symmetry condition on the planes y(cid:48) = 0 and
y(cid:48) = η(cid:48)(x(cid:48)) can be expressed as follows:
∂u(cid:48)
v(cid:48) = 0, = 0, at y(cid:48) = 0,
∂y(cid:48)
(22)
2πC 2π
u(cid:48) = −C, v(cid:48) = − (cid:15)sin[ x(cid:48)], at y(cid:48) = η(cid:48)(x(cid:48)).
λ λ
Moreover, since the both planes y(cid:48) = 0 and y(cid:48) = η(cid:48)(x(cid:48)) constitute the stream-
(cid:82)
lines in the wave frame, the flow rate (q(cid:48) = u(cid:48)dy(cid:48)) in the wave frame is
constant at all cross-sections of the channel. Thus the following equation can
be obtained as:
ψ(cid:48) = 0, at y(cid:48) = 0, ψ(cid:48) = q(cid:48), at y(cid:48) = η(cid:48)(x(cid:48)). (23)
The dimensionless variables defined by:
x(cid:48) y(cid:48) u(cid:48)
x = , y = , u = ,
λ h C
λ ψ(cid:48) h
v = v(cid:48), ψ = , ω = ω(cid:48), (24)
Ch Ch C
q(cid:48) h2 η(cid:48)
q = , p = p(cid:48), η = ,
Ch µcλ h
will be used; and the amplitude ratio “φ”, the wave number “α”, and the
Reynolds number “R ”, are defined by
e
h ρCh (cid:15)
α = , R = , φ = . (25)
e
λ µ h
Numerical Study of a Non-Linear... 29
Equation of motion and boundary conditions in the dimensionless form be-
come:
∂u ∂v
+ = 0, (26)
∂x ∂y
∂u ∂u ∂p ∂2u ∂2u
R α[u +v ] = − +[α2 + ], (27)
e ∂x ∂y ∂x ∂x2 ∂y2
and
∂v ∂v ∂p ∂2v ∂2v
R α3[u +v ] = − +α2[α2 + ]. (28)
e ∂x ∂y ∂y ∂x2 ∂y2
Also, the configuration of the peristaltic wall can be represented by:
η(x) = 1−φcos[2πx], (29)
and the no-slip condition or the symmetry condition on the planes y = 0 and
y = η(x) can be expressed as follows:
∂u
v = 0, = 0, ψ = 0 at y = 0,
∂y (30)
u = −1, v = −2πφsin[2πx], ψ = q at y = η(x),
and the stream function ψ and vorticity ω are defined by:
∂ψ ∂ψ ∂v ∂u
u = , v = − , ω = α2 − . (31)
∂y ∂x ∂x ∂y
By eliminating the pressure from equations (27) and (28), we get:
∂2u ∂2v ∂2u ∂2v
R α[u( −α2 )+v( −α2 )]
e ∂x∂y ∂x2 ∂y2 ∂x∂y
(32)
∂2 ∂2 ∂u ∂v
= [(α2 + )( −α2 )].
∂x2 ∂y2 ∂y ∂x
Using equations (31) in [(30) and (32)], we get the governing equations and
boundary conditions as:
∂ψ ∂3ψ ∂3ψ ∂ψ ∂3ψ ∂3ψ
R α[( )( +α2 )−( )( +α2 )]
e ∂y ∂x∂y2 ∂x3 ∂x ∂y3 ∂x2∂y
(33)
∂2 ∂2 ∂2ψ ∂2ψ
= [(α2 + )( +α2 )].
∂x2 ∂y2 ∂y2 ∂x2
30 Kh. S. Mekheimer et al.
For simplify, we get:
∂4ψ ∂ψ ∂3ψ ∂3ψ ∂ψ ∂3ψ ∂3ψ
=R α[( )( +α2 )−( )( +α2 )]
∂y4 e ∂y ∂x∂y2 ∂x3 ∂x ∂y3 ∂x2∂y
(34)
∂4ψ ∂4ψ
−[α4 +2α2 ],
∂x4 ∂x2∂y2
and the no-slip condition or the symmetry condition on the planes y = 0 and
y = η(x) can be expressed as follows:
∂2ψ ∂ψ
ψ = 0, = 0, = 0 at y = 0,
∂y2 ∂x
(35)
∂ψ ∂ψ
ψ = q, = −1, = −2πφsin[2πx] at y = η(x).
∂y ∂x
4 ADM Solution
For the solution of equation (34) with the boundary conditions (35) , we use
Adomian Decomposition Method, we write equation (34) in operator form
∂ψ ∂3ψ ∂3ψ ∂ψ ∂3ψ ∂3ψ
ψ =L−1(R α[( )( +α2 )−( )( +α2 )])
y e ∂y ∂x∂y2 ∂x3 ∂x ∂y3 ∂x2∂y
(36)
∂4ψ ∂4ψ
−L−1([α4 +2α2 ]).
y ∂x4 ∂x2∂y2
Since
(cid:90) y
L−1(∗) = (∗)dy,
y
0
(cid:124) (cid:123)(cid:122) (cid:125)
4-times
So, Now we can decompose ψ as:
∞
(cid:88)
ψ = ψ . (37)
n
n=0
Substituting ψ into equation (36), and using the boundary equations (35),
then we get the following recursive relation:
(cid:18) (cid:19) (cid:18) (cid:19)
η(x)+3q η(x)+q
ψ = y −y3 ,
0 2η(x) 2η3(x)
∂ψ ∂3ψ ∂3ψ
ψ = Φ +L−1(R α[( n−1)( n−1 +α2 n−1)
(n≥1) y,n y e ∂y ∂x∂y2 ∂x3 (38)
∂ψ ∂3ψ ∂3ψ ∂4ψ ∂4ψ
−( n−1)( n−1 +α2 n−1)])−L−1([α4 n−1 +2α2 n−1]),
∂x ∂y3 ∂x2∂y y ∂x4 ∂x2∂y2
since Φ = c (x)+yc (x)+y2c (x)+y3c (x),
y,n 0,n 1,n 2,n 3,n
Numerical Study of a Non-Linear... 31
where c . c , c , c are the integration constants which are evaluated
0,n 1,n 2,n 3,n
from the given conditions.
Then the solution of the stream function written as:
∞
(cid:88)
ψ = ψ = ψ +ψ +ψ +···
n 0 1 2
n=0
(cid:18) (cid:19) (cid:18) (cid:19)
η(x)+3q η(x)+q 1
= y −y3 − yα(y2 −η2(x))2[R (120q2y4
2η(x) 2η3(x) 20160η9(x) e
×α2η(cid:48)3(x)+15qy4α2η(x)η(cid:48)(x)(14η(cid:48)2(x)−9qη(cid:48)(cid:48)(x))+6α2η8(x)η(3) +3α2
×η7(x)(2η(cid:48)(x)η(cid:48)(cid:48)(x)−5qη(3)(x))+η5(x)(η(cid:48)(x)(−792q −α2(54qη(cid:48)2(x)+
(783q2 +32y2)η(cid:48)(cid:48)(x)))−58qy2α2η(3)(x))+y2η3(x)(2η(cid:48)(x)(30q(6−5α2η(cid:48)2(x))
+(243q2 −35y2)α2η(cid:48)(cid:48)(x))+25qy2α2η(3)(x))+3y2η2(x)(4(−34q2 +5y2)α2
×η(cid:48)3(x)+2qη(cid:48)(x)(36q −35y2α2η(cid:48)(cid:48)(x))+5q2y2α2η(3)(x))+2η4(x)(6η(cid:48)(x)
×(−90q2 +12y2 +α2((48q2 −2y2)η(cid:48)2(x)+25qy2η(cid:48)(cid:48)(x)))+y2(−39q2 +5y2)
×α2η(3)(x))+η6(x)(54η(cid:48)(x)(−4+α2(−2η(cid:48)2(x)+qη(cid:48)(cid:48)(x)))+(270q2 +8y2)
×α2η(3)(x)))+12α2(−360qy2α2η(cid:48)4(x)−120y2α2η(x)η(cid:48)2(x)(η(cid:48)2(x)−3qη(cid:48)(cid:48)(x))
−12η2(x)(18qα2η(cid:48)4(x)+3qy2α2η(cid:48)(cid:48)2(x)+12η(cid:48)2(x)(7q −y2α2η(cid:48)(cid:48)(x))+4qy2α2
×η(cid:48)(x)η(3)(x))+4α2η6(x)η(4)(x)−3α2η5(x)(12η(cid:48)(cid:48)2(x)+16η(cid:48)(x)η(3)(x) +5q
×η(4)(x))−3η3(x)(80α2η(cid:48)4(x)−84qη(cid:48)(cid:48)(x)+12η(cid:48)2(x)(14+qα2η(cid:48)(cid:48)(x))+8y2
×α2η(cid:48)(x)η(3)(x)+y2α2(6η(cid:48)(cid:48)2(x)−qη(4)(x)))+2η4(x)(12(7+12α2η(cid:48)2(x))η(cid:48)(cid:48)(x)
+27qα2η(cid:48)(cid:48)2(x)+α2(36qη(cid:48)(x)η(3)(x)+y2η(4)(x))))]+··· .
(39)
Now, we can evaluate the velocity and vorticity from equation (31). From
the two-dimensional steady Navier-Stokes equations, introducing the stream
function ψ and vorticity ω gives the pressure terms in dimensionless form as
∂p ∂ψ∂2ψ ∂ψ ∂2ψ ∂ω
= R α[( )−( )]− ,
∂x e ∂x ∂y2 ∂y ∂x∂y ∂y
(40)
∂p ∂ψ∂2ψ ∂ψ ∂2ψ ∂ω
= R α3[( )−( )]+α2 .
∂y e ∂y ∂x2 ∂x ∂x∂y ∂x
From these equation and substituting of ψ and ω we can get the value of the
pressure gradients.
5 Numerical Results and Discussion
In a two dimensional symmetric channel with both walls moving, the stream
function, the velocity and pressure fields are evaluated for values of the four