Post on 23-Jan-2017
TECHNICAL PAPER
On absolute linear instability analysis of plane Poiseuille flowby a semi-analytical treatment
Nemat Dalir • S. Salman Nourazar
Received: 16 November 2013 / Accepted: 25 April 2014
� The Brazilian Society of Mechanical Sciences and Engineering 2014
Abstract The absolute linear hydrodynamic instability of
the plane Poiseuille flow is investigated by solving the Orr–
Sommerfeld equation using the semi-analytical treatment
of the Adomian decomposition method (ADM). In order to
use the ADM, a new zero-order ADM approximation is
defined. The results for the spectrum of eigenvalues are
obtained using various orders of the ADM approximations
and discussed. A comparative study of the results for the
first, second and third eigenvalues with the ones from a
previously published work is also presented. A monotonic
trend of approach of decreasing relative error with the
increase of the orders of ADM approximation is indicated.
The results for the first, second and third eigenvalues show
that they are in good agreement within 1.5 % error with the
ones obtained by a previously published work using the
Chebyshev spectral method. The results also show that
the first eigenvalue is positioned in the unstable zone of the
spectrum, while the second and third eigenvalues are
located in the stable zone.
Keywords Adomian decomposition method � Linear
instability � Plane Poiseuille flow � Orr–Sommerfeld
equation
List of symbols
a Initial condition in Eq. (16) (value of second
derivative of u with respect to z at z = -1)
b Initial condition in Eq. (16) (value of third derivative
of u with respect to z at z = -1)
c1 Lower boundary limit
c2 Higher boundary limit
D Differential operator in Eq. (3)
ei Coefficients of u1(z) in Eq. (21)
fi Coefficients of u2(z) in Eq. (21)
g Zero order approximation
h Source term in Eq. (3)
i Index for imaginary; indicator of imaginary part
L Linear invertible part of differential operator D
L-1 Inverse of operator L
m Index of u in Eq. (17)
N Nonlinear part of differential operator D
r Index for real
R Reynolds number
S Rest linear part of differential operator D
u Dependent variable in Eq. (3)
U Velocity of mean (base) flow
z Transversal coordinate in Eq. (1), independent
variable in Eq. (3)
Greek symbols
a Axial wave-umber
ar Real axial wave-umber
k Frequency
ki Imaginary part of frequency kkr Real part of frequency ku Amplitude of velocity disturbance
K Coefficient in Eq. (14)
Technical Editor: Francisco Ricardo Cunha.
N. Dalir � S. S. Nourazar (&)
Department of Mechanical Engineering, Amirkabir University
of Technology, 158754413 Tehran, Iran
e-mail: icp@aut.ac.ir
N. Dalir
e-mail: dalir@aut.ac.ir
123
J Braz. Soc. Mech. Sci. Eng.
DOI 10.1007/s40430-014-0187-2
1 Introduction
The hydrodynamic instability involves with the response of
a laminar flow to a disturbance of small amplitude. If the
flow returns to its original laminar state the flow is defined
as stable, whereas if the disturbance grows and causes the
laminar flow to change into a different state, the flow is
defined as unstable. Instabilities often result in turbulent
flow, but they may also take the flow into a different
laminar, usually more complicated state. Stability theory
deals with the mathematical analysis of the evolution of
disturbances superposed on a laminar base flow [1]. In
many cases, the disturbances are assumed to be small in
amplitude which results in a linear equation governing the
evolution of disturbances.
From the Navier–Stokes equations, the Orr–Sommerfeld
equation is obtained if a small-amplitude disturbance is
imposed into the base flow between two parallel plates.
Thus the Orr–Sommerfeld equation is the governing
equation for the fluid instability of the plane steady parallel
flows of incompressible viscous fluids. By parallel flow it is
meant that the dependent variables for the base flow are at
most function of only one independent variable, while
steady denotes that the mean flow does not change with
time. The disturbance is described by the Orr–Sommerfeld
equation and the base flow is described by the plane
Poiseuille flow by several studies. The Orr–Sommerfeld
stability equation is as follows [2]:
d4udz4� 2a2 d2u
dz2þ a4u
� iaR ðU � kÞ d2udz2� a2u
� �� d2U
dz2u
� �¼ 0
ð1Þ
The boundary conditions are
u ¼ dudz¼ 0 at z ¼ �1; 1 ð2Þ
where u is the amplitude of the velocity disturbance, z is
the transversal coordinate, a is the axial wave-number, k is
the frequency, R is the Reynolds number and U is the
velocity of mean (base) flow and the two plates are located
at z = ±1. The Orr–Sommerfeld Eq. (1) with its boundary
conditions of Eq. (2) is an eigenvalues problem with
u = u(z) as the eigenfunction. The complex eigenfunction
u is an unknown function of z. When the complex fre-
quency k = kr ? iki, kr = Re(k), ki = Im(k) is determined
as a function of the real wave-number a = ar, an absolute
or temporal instability investigation is performed. The
disturbance is applied in space by the fixed real wave-
number a and is observed as it evolves in time through the
complex frequency k calculated as the eigenvalue. Thus
the eigenvalue problem governing the flow instability is
expressed as k = f (a, R) which yields a k = kr ? iki when
a, R are specified. The temporal growth rate is given by ki.
Thus disturbances can be grouped into three classes
depending on the sign of ki, namely ki [ 0: amplified
disturbances (absolute unstable flow); ki = 0: no change in
time (neutral); ki \ 0: damped disturbances (absolute sta-
ble flow). The base flow velocity profile can be described in
various ways (i.e. plane Poiseuille flow, plane Couette
flow, Blasius flow). However, in this study, plane Poiseu-
ille flow, with the velocity profile U(z) = 1 - z2, is con-
sidered as the base flow to compare the results of the
present work with the previously published work [3].
Numerical methods have previously been used by
several researchers to solve the Orr–Sommerfeld equation
to investigate the absolute linear instability of various
flows. Orszag [3] applied the Chebyshev spectral method
to solve the Orr–Sommerfeld equation. He found that
for a = 1, R = 10,000 the exact eigenvalue for the
most unstable mode of the plane Poiseuille flow is
0.23752649 ? 0.00373967i. He also found the stable
eigenvalues. Shkalikov and Tumanov [4] studied the
spectrum of the Orr–Sommerfeld equation, in particular
for Couette flow. Mamou and Khalid [5] proposed a finite
element solution of the Orr–Sommerfeld equation using
Hermite elements and validated their results for the plane
Pouseille flow. Bera and Dey [6] used the Orr–Sommer-
feld equation to examine the linear instability of boundary
layer flow subject to uniform shear using a spectral col-
location method. Makinde and Mhone [7] investigated the
temporal development of small disturbances in magneto-
hydrodynamic Jeffery–Hamel flows to understand the
stability of hydro-magnetic steady flows in convergent/
divergent channels at very small magnetic Reynolds
number. Meseguer and Mellibovsky [8] studied the effi-
ciency of a spectral Petrov–Galerkin method for the linear
and nonlinear stability analysis of the Hagen-Poiseuille
flow. They formulated the problem in solenoidal primitive
variables for the velocity field and eliminated the pressure
term from the scheme. Broadhurst and Sherwin [9] con-
sidered the numerical implementation of the parabolised
stability equations using a spectral/hp-element discretisa-
tion and presented the numerical stability of the govern-
ing equations. Prusa [10] analyzed the influence of choice
of boundary condition (no-slip and Navier’s slip boundary
conditions) on stability of Hagen-Poiseuille flow. He
concluded that Navier’s slip boundary condition has a
destabilizing effect on the flow. Giannakis et al. [11]
developed and tested spectral Galerkin schemes to solve
the coupled Orr–Sommerfeld and induction equations for
parallel incompressible MHD in free-surface and fixed-
boundary geometries. Elcoot [12] investigated the Kelvin–
Helmholtz instability for parallel flow between two
dielectric fluids in porous media. They introduced a
J Braz. Soc. Mech. Sci. Eng.
123
non-linear perturbation technique which was based on the
Fourier transform and the multiple scales method. Drag-
omirescu and Gheorghiu [13] studied the linear stability
problem of an electro-hydrodynamic convection between
two parallel walls which is an eighth-order eigenvalue
problem supplied with hinged boundary conditions for the
even derivatives up to sixth order. You and Guo [14]
investigated the effects of an electrical double layer
(EDL) together with boundary slip on the micro-channel
flow stability. They used a new electrical current density
balance mode to compute the conduction current when
the effect of EDL is considered. Saravanan and Brindha
[15] employed the energy method to investigate the sta-
bility of a steady convective flow in a heat-generating
fluid arising due to the effects of buoyancy, shear and
pressure gradient. Malik et al. [16] performed linear sta-
bility analysis of a dielectric fluid confined in a cylin-
drical annulus of infinite length under microgravity
conditions. They concluded that a radial temperature
gradient and a high alternating electric field imposed over
the gap induce an effective gravity that can lead to a
thermal convection. Asthana et al. [17] studied the Kevin–
Helmholtz instability in a porous medium using viscous
potential flow theory. They found that medium porosity
has stabilizing effect on the flow. Modica et al. [18]
attempted to bring simulations and experiments into better
agreement for the Rayleigh–Taylor instability by extend-
ing the classic purely hydrodynamic model to include
self-generation of magnetic fields and anisotropic thermal
conduction. Hagan and Priede [19] proposed a technique
for avoiding physically spurious eigen-modes that often
occur in the solution of hydrodynamic stability problems
by the Chebyshev collocation method. They applied their
method on the solution of the Orr–Sommerfeld equation
for plane Poiseuille flow. The fluid instability has also
some applications in computer code verification. For
instance, Gennaro et al. [20] performed the verification
and accuracy comparison of various commercial CFD
codes using hydrodynamic instability.
Recently, semi-analytical methods such as the Adomian
decomposition methods have been developed and applied
to many problems. Wazwaz [21] presented an efficient
algorithm of the Adomian decomposition method for
approximate solutions of higher order boundary value
problems with two-point boundary conditions. Wazwaz
[22] applied the modified Adomian decomposition method
for analytical treatment of nonlinear differential equations
that appear on boundary layers in fluid mechanics. Somali
and Gokmen [23] applied the Adomian decomposition
method to the nonlinear second-order Sturm–Liouville
problems and demonstrated the eigenvalues and the
behavior of eigenfuctions. Hayat et al. [24] investigated the
magneto-hydrodynamic boundary layer flow by employing
the modified Adomian decomposition method and the Pade
approximation. They developed the series solution of the
governing non-linear problem. Lin et al. [25] proposed a
new Adomian decomposition method using an integrating
factor and solved nonlinear models by this method to get
more reliable and efficient results. More literature survey
on the semi-analytical methods and also on the hydrody-
namic instability makes it clear that the semi-analytical
methods have not been used for investigating the hydro-
dynamic instability of various flows.
In the present paper, the semi-analytical treatment of
the Adomian decomposition method (ADM) is used for
solving the Orr–Sommerfeld equation to investigate the
absolute linear instability of the plane Poiseuille flow. The
instability problem is solved by defining a new zero-order
ADM approximation. It is shown that this new definition
proves to be useful. The results for the first, second and
third eigenvalues using various orders of ADM approxi-
mations are obtained and then compared with the Ors-
zag’s [3] numerical solution using the Chebyshev spectral
method. The comparison is shown to give an excellent
agreement within the relative error of 1.5 % at the eighth
order approximation of ADM. Accurate results of the
spectrum of eigenvalues are also obtained using the ADM
which show a monotonic trend of approach towards the
previously obtained numerical results [3].
2 The basic idea of ADM
Consider the general ordinary differential equation (ODE)
with boundary conditions [24]:
DuðzÞ ¼ hðzÞ; uðc1Þ ¼ 0; uðc2Þ ¼ 0; ð3Þ
where D is the differential operator, u is the dependent
variable, z is the independent variable, h is the source term
and c1, c2 are the lower and higher boundary limits,
respectively (c1 \ c2) which are constants. By supposing
that L is the linear invertible part of the D, N is the non-
linear part of D and S is the rest of D which is linear:
Luþ Nuþ Su ¼ h: ð4Þ
Equation (4) is rewritten as
Lu ¼ h� Nu� Su: ð5Þ
Considering L-1 as the inverse of operator L, and then
applying L-1 on both sides of Eq. (5) gives
uðzÞ ¼ gðzÞ þ L�1h� L�1ðNuÞ � L�1ðSuÞ; ð6Þ
where g(z), i.e. the zero-order approximation of solution, is
due to the boundary condition for lower limit in Eq. (3), i.e.
u(c1) = 0. It should be noted that if L is the first-order
differential operator, then L-1 is as follows:
J Braz. Soc. Mech. Sci. Eng.
123
Lu ¼ ou
oz) Lð�Þ ¼ oð�Þ
oz) L�1ð�Þ ¼
Zz
c1
ð�Þdz ð7Þ
In order to solve Eq. (6), the Adomian decomposition method
(ADM) [10] states that the dependent variable u(z) and the
nonlinear terms Nu should be written as infinite series:
uðzÞ ¼X1m¼0
umðzÞ; NuðzÞ ¼X1m¼0
AmðzÞ; ð8Þ
where Am are defined as [11] AmðzÞ ¼1m!
dm
dkm NuPm
i¼0 kiuiðzÞ� �h i
k¼0. Substitution of the infinite
series (8) in Eq. (6) gives
X1m¼0
umðzÞ ¼ gðzÞ þ L�1h� L�1X1m¼0
AmðzÞ !
� L�1 SX1m¼0
umðzÞ !
ð9Þ
Due to the ADM, from Eq. (9), any term of u(z) except
u0(z), is determined by recurrence relations as follows [12]:
u0ðzÞ ¼ gðzÞ;umþ1ðzÞ ¼ L�1h� L�1ðAmðzÞÞ � L�1ðSumÞ; m� 0 ð10Þ
3 Application of the ADM on the Orr–Sommerfeld
equation
In order to use the ADM, the Orr–Sommerfeld equation,
Eq. (1), is rewritten as
d4udz4¼ 2a2 d2u
dz2� a4u
þ iaR ðU � kÞ d2udz2� a2u
� �� d2U
dz2u
� �ð11Þ
The linear differential operator L is defined as the fourth-
order derivative with respect to z as
Lð�Þ ¼ d4ð�Þdz4
ð12Þ
which gives the inverse operator L-1 as
L�1ð�Þ ¼Zz
�1
Zz
�1
Zz
�1
Zz
�1
ð�Þdzdzdzdz; ð13Þ
which is a fourfold integral operator. Application of the
L-1 from Eq. (13) on both sides of Eq. (11) gives the
following outcome:
uðzÞ ¼ gðzÞ þ L�1
� 2a2 d2udz2� a4uþ iaR ðU � kÞ d2u
dz2� a2u
� �� d2U
dz2u
� �� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
K/
ð14Þ
where g(z) represents the zero-order approximation. The
Ku in Eq. (14) can be rewritten as
Ku ¼ 2a2 þ iaRðU � kÞ� � d2u
dz2
� a4 þ ia3RðU � kÞ þ iaRd2U
dz2
� �u ð15Þ
It is obvious that the inverse operator, Eq. (13), and its
derivatives are zero at z = -1; thus g(z) and its first
derivative are zero at z = –1. Nevertheless, by the con-
sideration of (d2u/dz2)|z=-1 = a, (d3u/dz3)|z=-1 = b,
through the ADM [21] and the use of the boundary con-
ditions for lower limit in Eq. (2), i.e. u(z = -1) = (du/
dz)|z=-1 = 0, the function g(z) can be defined as
gðzÞ ¼ 1
6bz3 þ 1
2ðaþ bÞz2 þ aþ 1
2b
� �zþ 1
2aþ 1
6b
� �
ð16Þ
According to the ADM, the solution of Eq. (14) is
expressed as an infinite series for the dependent variable
u(z) as
uðzÞ ¼X1m¼0
umðzÞ ð17Þ
Substituting Ku from Eq. (15) into Eq. (14) and the use of
infinite series of Eq. (17) gives
X1m¼0
umðzÞ ¼ gðzÞ þ L�1
�2a2 þ iaRðU � kÞ� �X1
m¼0
d2um
dz2
� a4 þ ia3RðU � kÞ þ iaRd2U
dz2
� �X1m¼0
umðzÞ
0BBBB@
1CCCCA
ð18Þ
The base flow of the present problem is assumed to be the
plane Poiseuille flow. The plane Poiseuille flow has the
velocity profile U(z) = 1 - z2, where if substituted in
Eq. (18), it gives
X1m¼0
umðzÞ ¼ gðzÞ þ L�1
�2a2 þ iaRðð1� z2Þ � kÞ� �X1
m¼0
d2um
dz2
� a4 þ ia3Rðð1� z2Þ � kÞ � 2iaR� �X1
m¼0
umðzÞ
0BBBB@
1CCCCA
ð19Þ
where g(z) is the zero-order approximation in the ADM
first component, u0(z). The recursive relations of Eq. (19)
due to the ADM are expressed as
J Braz. Soc. Mech. Sci. Eng.
123
u0ðzÞ ¼ gðzÞ;
umþ1ðzÞ ¼ L�1 2a2 þ iaRðð1� z2Þ � kÞ� � d2um
dz2
�
� a4 þ ia3Rðð1� z2Þ � kÞ � 2iaR� �
umðzÞ�;
for m� 0
ð20Þ
The first three components of u(z), using Eq. (20), are
obtained as follows:
u0ðzÞ ¼1
6bz3 þ 1
2ðaþ bÞz2 þ aþ 1
2b
� �zþ 1
2aþ 1
6b
� �;
u1ðzÞ ¼e1z9 þ e2z8 þ e3z7 þ e4z6 þ e5z5 þ e6z4 þ e7z3
þ e8z2 þ e9zþ e10;
u2ðzÞ ¼f1 þ f2zþ f3z2 þ f4z3 þ f5z4 þ f6z5 þ f7z6 þ f8z7
þ f9z8 þ f10z9
þ f11z10 þ f12z11 þ f13z12 þ f14z13 þ f15z14 þ f16z15:
ð21Þ
Here the coefficients ei and fi are functions of the real axial
wave-number a, the Reynolds number R, the complex
frequency k and the initial conditions a and b. Since the
coefficients fi include lengthy terms, it was not possible to
indicate them in here. However, the coefficients ei are
obtained as follows.
e1 ¼�1
18144ia3Rb;
e2 ¼�1
3360ia3Rðaþ bÞ;
e3 ¼�1
2520ia3Rbþ 1
5040a4b� 1
5040ika3Rb� 1
840ia3Ra
þ 1
1260iaRb;
e4 ¼1
1080ia3Rbþ 1
720a4aþ 1
720a4b� 1
720ika3Ra
� 1
720ika3Rb;
e5 ¼�1
60iaRb� 1
60iaRaþ 1
240ia3Rbþ 1
120ikaRb
þ 1
120a4aþ 1
240a4b� 1
240ika3Rbþ 1
120ia3Ra
� 1
120ika3Ra� 1
60a2b;
e6 ¼�1
144ika3Rbþ 1
144ia3Rb� 1
18iaRb� 1
48ika3Ra
þ 1
144a4bþ 1
48ia3Ra� 1
12iaRaþ 1
24ikaRa
� 2
24a2aþ 1
24ikaRb� 2
24a2bþ 1
48a4a;
e7 ¼�1
36ika3Raþ 7
1080ia3Rb� 2
12a2bþ 1
40ia3Ra
� 1
6iaRc1 þ
1
36a4a� 1
144ika3Rb� 1
12iaRb
þ 1
12ikaRbþ 1
6ikaRa� 1
3a2aþ 1
144a4b;
e8 ¼1
48a4a� 1
240ika3Rbþ 1
60ia3Ra� 1
48ika3Ra
� 1
2a2aþ 1
240a4b� 1
6iaRa� 1
15iaRbþ 1
280ia3Rb
þ 1
4ikaRaþ 1
12ikaRb� 2
12a2b;
e9 ¼1
120a4aþ 11
10080ia3Rb� 1
720ika3Rbþ 1
720a4b
þ 1
24ikaRb� 1
120ika3Ra� 1
12a2b� 1
36iaRb
þ 1
168ia3Raþ 1
6ikaRa� 1
12iaRa� 1
3a2a;
e10 ¼�1
5040ika3Rb� 1
720ika3Ra� 1
60iaRa� 1
12a2a
� 1
60a2bþ 1
1120ia3Raþ 1
24ikaRaþ 1
120ikaRb
� 1
210iaRbþ 13
90720ia3Rbþ 1
5040a4bþ 1
720a2a:
ð22Þ
4 Results and discussion
The absolute fluid instability of the plane Poiseuille
flow is taken into account by considering real constant
given values of the axial wave-number (a) and the
Reynolds number (R). To achieve this purpose, we use
exactly the same data as Orszag [3] used, i.e. a = 1,
R = 10,000, to compare the ADM results for the
complex eigenvalue k with the ones obtained numeri-
cally by Orszag [3]. The numerical result of the first
three eigenvalues using Chebysev’s spectral method [3]
is compared with the relevant eigenvalues obtained in
the present work using the semi-analytical Adomian
decomposition method. In order to acquire the eigen-
values, the truncated solution u(z) = u0 ? u1 ? u2
? ��� ? um is considered for the first eighth-order
ADM approximations. Then the boundary conditions at
the higher boundary limit (z = 1), i.e. u(z = 1) = (du/
dz)|z=1=0, are applied on the u. This course of action
gives a system of two homogeneous linear algebraic
equations as functions of k, a and b. Then since the
system is homogeneous, the determinant is equated to
zero, which will give the eigen-condition equation that
can be solved for k. It should be mentioned that these
procedures have been incorporated into a MATHEM-
ATICA package code.
J Braz. Soc. Mech. Sci. Eng.
123
Here the procedure explained above is indicated for the
first-order ADM approximation of u. It is assumed that u is
approximated by u = u0 ? u1. Then the two boundary
conditions u(z = 1) = (du/dz)|z=1=0 are applied on uwhich gives:
52
63bþ 152000
63ikbþ 52000
9ikaþ 34
45a� 32000
7ia
� 1312000
567ib ¼ 0;
32000
3ikaþ 34
45bþ 52000
9ikb� 2
5a� 232000
21ia
� 400000
63ib ¼ 0;
8>>>>>>>>>><>>>>>>>>>>:
ð23Þ
which can be re-ordered as:
34
45� 32000
7iþ 52000
9ik
� �a
þ 52
63� 1312000
567iþ 152000
63ik
� �b ¼ 0;
� 2
5� 232000
21iþ 32000
3ik
� �a
þ 34
45� 400000
63iþ 52000
9ik
� �b ¼ 0;
8>>>>>>>>>>>><>>>>>>>>>>>>:
ð24Þ
Eq. (24) is a system of two homogeneous linear algebraic
equations. Since it is homogeneous, the system has a
solution if the determinant equates zero, i.e.,
34
45� 32000
7iþ 52000
9ik
� �34
45� 400000
63iþ 52000
9ik
� �
� 52
63� 1312000
567iþ 152000
63ik
� �
� � 2
5� 232000
21iþ 32000
3ik
� �¼ 0; ð25Þ
which is the equation of eigen-condition. Here the eigen-
condition is a second-order linear algebraic equation in k.
Solving Eq. (25) gives the two roots as follows:
k1 ¼ 0:39662415þ 0:00005191i;
k2 ¼ 1:14124260þ 0:00016852i;ð26Þ
where only the first root, 0.39662415 ? 0.00005191i, is
acceptable as an eigen-value. If the second-order ADM
approximation for u is taken into account, i.e. the approx-
imation of u is considered to be u = u0 ? u1 ? u2, then
its equation of eigen-condition gives four roots as follows:
k1 ¼ 1:14754394� 0:00010754i;
k2 ¼ 0:32602906þ 0:00012854i;
k3 ¼ 0:72624279þ 0:00002869i;
k4 ¼ 0:86867262þ 0:00065560i;
ð27Þ
in which only the second root, 0.32602906 ? 0.00012854i,
may be accepted as an eigenvalue. When u is approxi-
mated by u = u0 ? u1 ? u2 ? u3 which are the third-
order ADM approximation for u, six roots are found from
the relevant eigen-condition equation as
k1 ¼ 0:98075999� 0:00049907i;
k2 ¼ 1:12207738� 0:00025474i;
k3 ¼ 0:78447423� 0:05150667i;
k4 ¼ 0:27443272þ 0:00008555i;
k5 ¼ 0:78408212þ 0:05234203i;
k6 ¼ 0:64296165þ 0:00020925i;
ð28Þ
Among six roots of (28), only the third and fourth
roots, i.e. 0.27443272 ? 0.00008555i and 0.78447423
- 0.05150667i, are recognized as two eigenvalues. How-
ever, the roots for the case of fourth-order ADM approxi-
mation for u, i.e. u = u0 ? u1 ? u2 ? u3 ? u4,
using the relevant eigen-condition equation, are obtained
as
k1 ¼ 0:75129208� 0:06996643i;
k2 ¼ 0:82495217� 0:04098886i;
k3 ¼ 1:09394231� 0:00042828i;
k4 ¼ 0:23911480þ 0:00006589i;
k5 ¼ 0:75108018þ 0:07112940i;
k6 ¼ 0:58336542þ 0:00033610i;
k7 ¼ 0:82191491þ 0:04293999i;
k8 ¼ 1:04208368þ 0:00041633i;
ð29Þ
where roots 0.23911480 ? 0.00006589i, 0.82495217
- 0.04098886i, 0.75129208 - 0.06996643i are acceptable
as eigenvalues. By considering the fifth-order ADM
approximation of u, i.e. u being approximated by
u = u0 ? u1 ? u2 ? u3 ? u4 ? u5, the relevant eigen-
condition equation roots are obtained as:
k1 ¼ 0:66335869� 0:00094393i;
k2 ¼ 0:80890671� 0:06726736i;
k3 ¼ 0:86278237� 0:04241511i;
k4 ¼ 1:07234816� 0:01887326i;
k5 ¼ 0:23842576þ 0:00005713i;
k6 ¼ 0:53873696þ 0:00040909i;
k7 ¼ 1:07197041þ 0:01863580i;
k8 ¼ 1:16531234þ 0:00081859i
k9 ¼ 0:86302774þ 0:04337901i;
k10 ¼ 0:80901005þ 0:06776103i;
ð30Þ
where only the roots of 0.23842576 ? 0.00005713i,
0.86278237 - 0.04241511i, 0.80890671 - 0.06726736i
can be recognized as eigen-values. For the sixth-order
J Braz. Soc. Mech. Sci. Eng.
123
ADM approximation for u, i.e. with u = u0 ? u1 ? u2
? u3 ? u4 ? u5 ? u6, the 12 roots of eigen-condition
equation are obtained as follows:
k1 ¼ 0:79452091� 0:00062926i;
k2 ¼ 0:86289067� 0:06696342i;
k3 ¼ 1:09246133� 0:01887326i;
k4 ¼ 1:03041658� 0:00116829i;
k5 ¼ 0:91712153� 0:03713498i;
k6 ¼ 0:23797422þ 0:00004978i;
k7 ¼ 0:49279325þ 0:00006843i;
k8 ¼ 0:71741196þ 0:00125416i;
k9 ¼ 1:09186193þ 0:01775982i;
k10 ¼ 0:91329473þ 0:03895159i;
k11 ¼ 1:17501560þ 0:00049561i;
k12 ¼ 0:86925685þ 0:06863910i;
ð31Þ
where only 0.23797422 ? 0.00004978i, 0.91712153
- 0.03713498i, 0.86289067 - 0.06696342i are taken to
be eigenvalues. When u is approximated by u = u0 ?
u1 ? u2 ? u3 ? u4 ? u5 ? u6 ? u7, which are the
seventh-order ADM approximation of u, eigen-condition
roots are found as follows:
k1 ¼ 1:19380782� 0:02699236i;
k2 ¼ 0:84683305� 0:00104921i;
k3 ¼ 0:94078642� 0:03675124i;
k4 ¼ 0:34452091� 0:08662926i;
k5 ¼ 0:90145390� 0:06613121i;
k6 ¼ 1:26125794� 0:00009357i;
k7 ¼ 0:23768787þ 0:00004365i;
k8 ¼ 0:35286961þ 0:08879383i;
k9 ¼ 0:68543296þ 0:00071908i;
k10 ¼ 0:47725644þ 0:00008572i;
k11 ¼ 1:19036170þ 0:02715631i;
k12 ¼ 0:94674964þ 0:03725437i;
k13 ¼ 0:90756419þ 0:06210643i;
k14 ¼ 0:99685653þ 0:00085934i;
ð32Þ
in which the three roots of 0.23768787 ? 0.00004365i,
0.94078642 - 0.03675124i, 0.90145390 - 0.06613121i
can be accepted as eigenvalues. For the eighth-order ADM
approximation of u, i.e. u = u0 ? u1 ? u2 ? u3 ? u4 ?
u5 ? u6 ? u7 ? u8, the roots are obtained as
k1 ¼ 0:38257378� 0:08257890i;
k2 ¼ 0:45537619� 0:00047173i;
k3 ¼ 1:29326743� 0:00210695i;
k4 ¼ 0:95662987� 0:03691370i;
k5 ¼ 0:92684819� 0:06440772i;
k6 ¼ 1:08251583� 0:00006212i;
k7 ¼ 1:22659235� 0:03455131i;
k8 ¼ 0:86365101� 0:00311448i;
k9 ¼ 0:23759293þ 0:00004172i;
k10 ¼ 1:22481017þ 0:03644291i;
k11 ¼ 0:95873618þ 0:03211574i;
k12 ¼ 0:78157662þ 0:00007218i;
k13 ¼ 0:39946965þ 0:08625781i;
k14 ¼ 0:91948472þ 0:06336409i;
k15 ¼ 0:62787846þ 0:00083195i;
k16 ¼ 1:15449129þ 0:00068925i;
ð33Þ
Among the 16 roots of (33), only the three roots
0.23759293 ? 0.00004172i, 0.95662987 - 0.03691370i
and 0.92684819 - 0.06440772i may be considered as
eigenvalues.
It should be pointed out that to find the acceptable
eigenvalues among the roots of the eigen-condition equa-
tions, the following procedure is applied: at first step, all the
72 roots (2 ? 4 ? 6 ? 8 ? 10 ? 12 ? 14 ? 16 = 72)
obtained for various orders of ADM approximations from
1st to 8th order are plotted in a ki - kr diagram (a diagram
with ki as vertical axis and kr as horizontal axis). Since the
ADM is basically a series solution, thus with the addition of
each term of approximation (i.e. increase of the order of
ADM approximation) the new solution must give a trend of
approach toward an exact value of the solution. This sig-
nificant point is used to obtain the eigenvalues among the
roots in the ki - kr diagram such that the roots which would
finally bring an eigenvalue must show a monotonic trend of
approach. Thus, at the second step, the ki - kr diagram is
very carefully checked for any pattern of roots showing a
monotonic trend of approach with the increase of the order
of ADM approximations, and three patterns are found.
Those roots with patterns are considered to be the eigen-
values among all the roots of eigen-conditions.
The first, second and third eigenvalues which are
obtained semi-analytically for the first- to eighth-order
ADM approximations among the roots of relevant eigen-
condition equations are indicated in Table 1. It may be
observed that the first eigenvalue monotonically approaches
from 0.39662415 ? 0.00005191i to 0.23759293 ?
0.00004172i with the increase of the order of ADM
J Braz. Soc. Mech. Sci. Eng.
123
approximation from first to eighth. Thus the first eigenvalue
obtained by the eighth-order ADM approximation, i.e.
0.23759293 ? 0.00004172i, can be considered as the first
eigenvalue achieved using the semi-analytical treatment.
This first eigenvalue is an unstable one because its imaginary
part has positive sign (ki [ 0). This eigenvalue could be
compared with the first eigenvalue given by Orszag [3], i.e.
0.23752649 ? 0.00373967i. The comparison shows a rela-
tive error of only 0.54 %. For the second eigenvalue, a
monotonic trend of approach from 0.78447423
- 0.05150667i to 0.95662987 - 0.03691370i for the third-
to eighth-order ADM approximation is observed. Hence the
value of 0.95662987 - 0.03691370i could be taken as the
second eigenvalue obtained by ADM. It can be seen that
this is a stable eigenvalue (ki \ 0). The second eigenvalue
obtained by Orszag [3] is 0.96463092 - 0.03516728i which
when compared to the ADM result, i.e. 0.95662987
- 0.03691370i, presents an error of 0.98 %. Table 1 also
shows that with the increase of order of ADM approxima-
tions from fourth to eighth the third eigenvalue approaches
from 0.75129208 - 0.06996643i to 0.92684819
- 0.06440772i in a monotonic trend. Therefore,
0.92684819 - 0.06440772i is the third eigenvalue obtained
using the Adomian decomposition method which in com-
parison to the third eigenvalue obtained by Orszag [3], i.e.
0.93631654 - 0.06320150i, gives a relative error of
1.43 %.
Table 2 indicates the relative error for the first eigen-
value obtained by the ADM for various orders of ADM
approximations compared to the first eigenvalue given by
Orszag [3]. The relative errors are calculated based on the
magnitudes of the complex eigenvalues. The results for the
first eigenvalue obtained by the ADM show that with the
increase of the orders of ADM approximations the relative
error reduces with a fast trend. This observation makes it
clear that a fast monotonic trend of approach towards the
numerical results is achieved by the ADM. The least of
relative error for the first eigenvalue is 0.54 %. Table 3
presents the second eigenvalue obtained using various
orders of the ADM approximations and the relative error in
comparison to the second eigenvalue obtained by Orszag
[3], i.e. 0.96463092 - 0.03516728i. It may be observed
that, with the increase of the order of ADM approximations
from third to eighth, the relative error decreases somewhat
monotonically from 18.63 to 0.98 %. Table 4 shows the
third eigenvalue for various orders of ADM approxima-
tions and the relative error in comparison to the third
eigenvalue of Orszag [3]. With the increase of the order of
ADM approximations from forth to eighth, the relative
error diminishes from 24.61 to 1.43 %. Hence the mono-
tonic trend of approach of ADM results toward the
numerical result is also achievable in this case.
Figure 1 illustrates the spectrum of eigenvalues obtained
by various orders of approximations of the ADM. The
Table 1 First, second and third
eigenvalues for various orders
of ADM approximations
Order of appro. First eigenvalue Second eigenvalue Third eigenvalue
1st order 0.39662415 ? 0.00005191i – –
2nd order 0.32602906 ? 0.00012854i – –
3rd order 0.27443272 ? 0.00008555i 0.78447423 - 0.05150667i –
4th order 0.23911480 ? 0.00006589i 0.82495217 - 0.04098886i 0.75129208 - 0.06996643i
5th order 0.23842576 ? 0.00005713i 0.86278237 - 0.04241511i 0.80890671 - 0.06726736i
6th order 0.23797422 ? 0.00004978i 0.91712153 - 0.03713498i 0.86289067 - 0.06696342i
7th order 0.23768787 ? 0.00004365i 0.94078642 - 0.03675124i 0.90145390 - 0.06613121i
8th order 0.23759293 ? 0.00004172i 0.95662987 - 0.03691370i 0.92684819 - 0.06440772i
Table 2 First eigenvalue and the relative error in comparison to the
first eigenvalue given by Orszag [3], i.e. 0.23752649 ? 0.00373967i,
for various orders of ADM approximations
Order of ADM
approximation
First eigenvalue Relative
error (%)
1st order 0.39662415 ? 0.00005191i 66.89
2nd order 0.32602906 ? 0.00012854i 37.08
3rd order 0.27443272 ? 0.00008555i 15.36
4th order 0.23911480 ? 0.00006589i 1.58
5th order 0.23842576 ? 0.00005713i 1.12
6th order 0.23797422 ? 0.00004978i 0.83
7th order 0.23768787 ? 0.00004365i 0.67
8th order 0.23759293 ? 0.00004172i 0.54
Table 3 Second eigenvalue and the relative error in comparison to
the second eigenvalue given by Orszag [3], i.e. 0.96463092
- 0.03516728i, for various orders of ADM approximations
Order of ADM
approximation
Second eigenvalue Relative
error (%)
3rd order 0.78447423 - 0.05150667i 18.63
4th order 0.82495217 - 0.04098886i 14.42
5th order 0.86278237 - 0.04241511i 10.49
6th order 0.91712153 - 0.03713498i 4.81
7th order 0.94078642 - 0.03675124i 2.64
8th order 0.95662987 - 0.03691370i 0.98
J Braz. Soc. Mech. Sci. Eng.
123
spectrum of eigenvalues is in fact the eigenvalues on the
(kr, ki) plane. It can be seen that the first, second and third
eigenvalues of the ADM semi-analytical treatment
approach the eigenvalues given by Orszag [3]. It is also
observed that as the orders of ADM approximations
increase a monotonic approach towards the Orszag’s [3]
numerically obtained eigenvalues is achieved. It is also
seen that, while the first eigenvalue is in the unstable zone,
ki [ 0, the second and third eigenvalues are in the stable
zone, ki \ 0. Figure 2 shows the relative errors in terms of
various orders of ADM approximation for the first, second
and third eigenvalues obtained by the ADM. It is apparent
that the relative error reduces monotonically with the
increase of the order of the ADM approximations. It may
also be noticed that, at any order of ADM approximations
from fourth to eighth, the relative error for the first
eigenvalue is lower than the relative error for the second
eigenvalue. The relative error of the second eigenvalue is
also lower than the relative error of the third eigenvalue.
Thus the relative error results in the conclusion that the
ADM gives more accurate results for the first eigenvalue
than the second and third eigenvalues. Figure 3 demon-
strates the neutral curve for the Orr–Sommerfeld equation
with the plane Poiseuille base flow using the ADM. In
order to obtain the neutral curve, the eigenvalues k are
computed for many values of the axial wave-number a and
the Reynolds number R. As it can be seen, the values of aand R are considered in the ranges of 0 B a B 1.4 and
500 B R B 100000, respectively. In fact, the neutral curve
outlines the boundary between stable and unstable solu-
tions. The neutral curve is sometimes called the thumb
curve or the hoop curve. All points inside the hoop have
ki [ 0 which shows an unstable flow. All points outside the
hoop correspond to a value of k with a negative imaginary
part (ki \ 0), i.e. which indicate that the flow is unstable.
On the neutral curve itself, we have ki = 0. Modes corre-
sponding to values of k lying on the curve are known as
neutral modes. As can be seen from Fig. 3, there is a
critical Reynolds number below which there are no
unstable modes. Using the ADM, this is found to be
Rcr = 5786. The corresponding wave number at the
beginning of the neutral curve is also computed as
acr = 1.02. If the critical Reynolds number obtained by
ADM, i.e. 5786, is compared to the one given by Orszag
[3], i.e. 5772.22, a relative error of 1.11 % is found.
It should be noted that the calculation time for the
eighth-order ADM approximation of the solution of the
instability problem in the present paper is observed to be
6.93 s. This calculation time is measured when the proce-
dure for finding the eigenvalues is performed on a 16-RAM
supercomputer with a 32-GB available memory. Compar-
ison of the calculation time of present method with the one
of Orszag’s method using 50 Chebyshev polynomials, i.e.
8.24 s, shows a small decrease in calculation time by the
Ψ
Ψ
Ψ
0 0.2 0.4 0.6 0.8 1 1.2-0
-0.05
0
0.05
0.1Orszag' s solution1st order ADM appro.2nd order ADM appro.3rd order ADM appro.4th order ADM appro.5th order ADM appro.6th order ADM appro.7th order ADM appro.8th order ADM appro.
Ψ
first eigenvalue
second eigenvalue
third eigenvalue
λi
λr
Fig. 1 Spectrum of eigenvalues for the Orr–Sommerfeld equation
with the plane Poiseuille base flow using various orders of ADM
approximation and Orszag’s solution [3]
order of approximation
Rel
ativ
e er
ror
(%)
1 2 3 4 5 6 7 80
20
40
60
80
first eigenvaluesecond eigenvaluethird eigenvalue
Fig. 2 Relative errors for the first, second and third eigenvalues in
comparison to the first, second and third eigenvalues given by Orszag
[3] for various orders of ADM approximations
Table 4 Third eigenvalue and the relative error in comparison to the
third eigenvalue given by Orszag [3], i.e. 0.93631654 - 0.06320150i,
for various orders of ADM approximations
Order of ADM
approximation
Third eigenvalue Relative
error (%)
4th order 0.75129208 - 0.06996643i 24.61
5th order 0.80890671 - 0.06726736i 13.68
6th order 0.86289067 - 0.06696342i 7.95
7th order 0.90145390 - 0.06613121i 3.87
8th order 0.92684819 - 0.06440772i 1.43
J Braz. Soc. Mech. Sci. Eng.
123
use of ADM. Concerning the radius of convergence of
Eq. (21), it should be mentioned that if the convergence
theorem is applied on Eq. (21), a finite value for the con-
vergence radius is obtained. However, the ADM approxi-
mations used here are of high-order accuracy such that
errors decrease more rapidly than 1/N as N ? ?, where
N is the order of ADM approximations.
It is observed that the ADM gives analytical rapidly
convergent approximations for the solution. Approximation
of solution to the Orr–Sommerfeld equation using the ADM
also requires less arithmetic operations in comparison to the
finite difference, the finite element and the spectral meth-
ods. It is worth mentioning that the ADM used in the present
paper can be easily extended to instability problems for
various base flows such as the plane Couette flow and the
Hagen-Poiseuille flow. For this purpose, the velocity profile
of the base flow within the Orr–Sommerfeld equation and
also the boundary conditions have to be altered. This means
that the definition of the zero-order approximation of ADM
would have to be modified. The ADM can also be used to
perform a spatial instability analysis in which the Reynolds
number R and the frequency k are real given values and the
problem is solved to find the complex axial wave number
a = ar ? iai. Also another advantage of the ADM is that it
is capable of solving the temporal stability problem for
different values of the axial wave-number a and the Rey-
nolds number R.
5 Conclusions
The Orr–Sommerfeld hydrodynamic instability equation
is solved for the plane Poiseuille flow using the Adomian
decomposition method (ADM). Thus an absolute instability
investigation is performed using a semi-analytical treat-
ment. In the applied procedure, a new zero-order ADM
approximation is proposed which, by using higher limit
boundary conditions, leads to a system of two homogeneous
linear algebraic equations giving the eigenvalues. The
spectrum of eigenvalues is obtained using various orders of
ADM approximations. The results for the first, second and
third eigenvalues are also compared to the results of a
numerical method previously used by Orszag [3]. The fol-
lowing conclusions may be made:
(1) With the increase of the order of ADM approxima-
tions, a monotonic trend of approach of the ADM
results for the first, second and third eigenvalues
towards Orszag’s [3] results is observed.
(2) The relative errors for the first, second and third
eigenvalues obtained using the ADM, in comparison
to the first, second and third eigenvalues given by
Orszag [3], decrease monotonically as the orders of
ADM approximations increase.
(3) At the eighth-order ADM approximation, the ADM
results show a quite good agreement within the
relative error of 1.5 % with the results of Orszag [3].
(4) The first eigenvalue obtained for the plane Poiseuille
flow by the ADM, i.e. 0.23759293 ?0.00004172i is
an unstable one, while the second and third eigen-
values due to the ADM, i.e. 0.95662987
- 0.03691370i, 0.92684819 - 0.06440772i, belong
to the stable zone.
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