Internat. J. Math. & Math. Sci.
(1990) 139-144
139
VOL. 13 NO.
EIGENVALUES OF THE TIME
DEPENDENT FLUID FLOW PROBLEM I.
EL-SAYED M. ZAYED and EL-SAYED F. ELSHEHAWEY
Department of Mathematics and Computer Sciences
United Arab Emirates University
Ai-Ain, P.O. BOX !5551
United Arab Emlrates
(Received April 19, 1988 and in revised form November 19, 1988)
ABSTRACT.
The direct and inverse boundary value problems for the linear unsteady
viscous fluid flow through a closed conduit of a circular annular cross-section R with
arbitrary tlme-dependent pressure gradient under the third boundary conditions have
been Investigated.
KEY WORDS AND PHRASES. Inverse boundary value problems, unsteady viscous flow, closed
conduits, annular cross-section.
1980 AMS SUBJECT CLASSIFICATION CODE. 76D05, 31A25.
I.
INTRODUCTION.
The problems of unsteady fluid flow through a closed conduit of annular cross-
(see, for example Bansal [I], Muller [2],
and
Wadhwa [6]).
In this paper we shall
[5],
[4],
[3],
Szymanskl
Sanyal
OJalvo
combine the Ideas of OJalvo [3] and Zayed [7] to solve the third boundary value
section R have been solved by many authors
problem (2.1)
are
considered.
requires
the
(2.3) mentioned in Section 2, where the direct and inverse problems
For inverse problems, we shall follow Zayed’s work [7], which
determination
of
the geometrical
properties
of
the
circular
annular
cross-sectlon R from the complete knowledge of Its spectrum, Spec (R), that is
Spec (fl)
{0
<
I 2<3
<
k
(I.I)
’’’+
using the asymptotic expansion of the spectral function
0(t)
tr(e-tV2)
(R)E e-t
(1.2)
k=1
for small positive t, where
is the Laplace operator in R
2.
Note that the spectrum
(I.I) is discrete and consists of all %’s such that there exist non-zero solutions
of
;
each % is written In the spectrum of
as a number of times equal
140
to
E.M. ZAYED AND E.F. ELSHEHAWEY
its
multiplicity,
fnct[on
that
{:
is dim
Note also that
}.
the
spectral
(1.2) for the vibrating membranes has been studied by many authors using the
[7], Gottlleb [8,9], Kac [I0], Sleeman
heat equation approach (see, for example Zayed
[II,12],
and Zayed
and Stewartson [13]).
In this paper, the authors have arrived at the fact that the same analysis of the
spectral function (1.2) for the vibrating annular membrane case, holds for the case of
unsteady viscous flow through conduits, even though the governing partial differential
for unsteady viscous flow through conduits of annular cross-sectlon
2.
.
This fact enables us to determine the spectral function (1.2)
equation is different.
FORMULATION OF THE MATHEMATICAL PROBLEM.
In this section, we discuss the following Inltial-boundary value problem of a
circular annular cross-sectlon
with radii a and b; b
>
a, for unsteady viscous fluid
flow when the pressure gradient F(t) is an arbitrary function of time t:
u
-F(t) + v
t
u
{- It -f (r--r-)}
in fl, t
> O,
(2.1)
together with the initial condition
u(r,0)
e.
0, a
<
r
(2.2)
b,
and the artlflcial third boundary conditions
(_
r
where u
+
the
is
u
lU
(-
r=a
kinematic
+
O,
"f2u) r=b
coefficient
(2.3)
of
viscosity,
it
is
and
Y1
and
Y2
are
positive
constants.
[3],
Following the method of Oja]vo
problem (2.1)
u(r,t)
(r) k(t)
I
kffil
where
(r)
+ V(r) F(t),
(2.4)
2
(-2 r- drd)
,
the solution of
are the etgenfunctions of the problem
d
in which
easily seen that,
(2.3) can be written in the form
/
[d__
(r)
dr k
+
<
(r)
k (r)
’1 (r)]r=a
=[ d
rr k (r)
+
r
Y2
<
(2.5)
b
(r)]r=b
O,
(2 6)
1,2,3,... are the corresponding eigenvalues, while Vir) satisfies the
k
problem
d
2
d
(r2 + -) V(r)
[d--V(r)
dr
+
,
IV(r)]r=a
a
d
[-
<
r
<
V(r) +
(2.7)
b,
Y2V(r)]rfb
O.
(2 8)
EIGENVALUES OF TIME-DEPENDENT FLUID FLOW PROBLEM
141
If, now, V(r) from (2.7) is expressible In the form
z
v()
k--1
(t)
then
should be determined from the inltlal value problem
d
+
VXk) k (t)
k(O)
hkF(0)
(-d-
where
hk,
(2.9)
k(r),
h
k
k
1,2,3,... are
hk
some
dF(t)
(2 10)
d-
Let
constants.
when
satisfy the following conditions: (1)
0
(ill)
TI, (2
< 2 <<
3.
< TI <<
0
I,
Y2 >>
now
us
YlY2 >>
the
discuss
(ll)
"fl >>
problem
I,
I.
THE SOLUTION OF THE PROBLEM IN THE CASE
I’ 2 >>
I.
In thls case, it Is straightforward to show that
)
Y(r
0
(3.1)
[2 yI-1 y,(a2k
)o + Yo(a )]
and
-1
V(r)
=
{(r
-1
-1
[2(a1
-1
Y1
a
where
z_ are
2
a
2
2ay
2
2 b) + (a
-1
Y2 +ina
g]
b
+
(__I_
In
_r)
X
2
b )]
(3.2)
roots of the equation
0.
(3.3)
E.M. ZAYED AND E.F. ELSHEHAWEY
142
k(t)
The function
has the form
k(t)
h
HF() e k
+
e{F(O)
d}e
-t
(3.4)
o
b
/
where
hk
On
inserting
dr
(3.51
b
f
(r)
r
dr
(3.2) and (3.4) into (2.4) we arrive at the solution of our
(3.1),
YI’ Y2 >>
problem when
V(r)k(r)
r
a
I.
[7],
Using the same analysis of Zayed
the
asymptotic
through
of
annular
function 8(t) for
spectral
the
of
expansion
conduits
we deduce after lengthy calculations that
cross-sectlon fl in
the
case
unsteady
where
viscous
yi,Y2 >>
flow
can
be
written In the form:
4 wt
b
a
-I
a and r
4.
YI
-I
b2 }-F-+.._
2
(R)’ we obtain
Y2
b (See
-I )]
Y2
l’z’(3.6)
0(t) as t /0.
This is the same form as (4.7) in Zayed
Note that if
+ (b +
Y1
8( wt
-1
a
2[(a +
a2)
(b2
(t)
[7] for the vibrating annular membrane.
the results of Dirlchlet boundary conditions on r
and Winelnger [6] and Sleeman and Zayed [12]).
WadJwa
THE SOLUTION OF THE PROBLEM WHEN y| >> 1,0 <
Y2 << I.
In this case, we can show that k(r) has the same form as (3.1), while V(r) takes
the form
-I
V(r)
=-;’v
[Y2(2aYl
a
+ a2
b
(-
[2
where
rr 2
2a
2)
Y1
’1
+
In r)
a
x
2b]
(4.1)
a
+ In -6)
-]
are roots of the equation
;k J’ (b ) + Y2Jo (b )]
o
O.
[ ,IYo’(a
’ ’’
+
Yo (a
(4.2)
143
EIGENVALUES OF TIME-DEPENDENT FLUID FLOW PROBLEM
On inserting
(4.1) and (3.4) into (2.4) we arrive at the solution of our
(3.1),
Y1 >>
problem in the case where
< Y2 <<
I, 0
1.
In this case, one can show after some reduction that the asymptotic expansion
of 8(t) for unsteady viscous flow through conduits can be written as:
2
(b
4t
8(t)
a
2)
2[b- (a +
+
-I
YI
)]
Y2b
+
8(vt) 2
7
--at
+-
32
2
+ 64
2
1
f2 b----}
a
(rl:)
2
128
+ O(t)
as t
O,
(4.3)
(4.6) in Zayed [7] for the vibrating annular membrane.
we obtain the results of Dirlchlet boundary condition on r
0,
a,
I
Y2
and Neumann boundary condition on r
b. (See Sleeman and Zayed [12]).
REMARK 4.1. The solution of our problem in the case where 0 <
I << I, 2 >> can
which has the same form as
If
and
be deduced from the previous case with the interchanges
I
--/
2
and a
b.
Finally, we close this paper with the remark that the expansions of 8(t)
(area, length of the boundary, number of holes, curvatures
...) for unsteady viscous flow through conduits, which are very
determine the geometry of R
of
the
boundary,
similar to those obtained in Zayed
ACKNOWLEDGEMENT.
comments.
[7] for the vibrating annular membrane case.
The authors are grateful
to the
referee for his suggestions and
They also express their sincere thanks to the Editor Dr. Lokenath Debnath
for hls cooperation.
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2.
3.
4.
5.
6.
7.
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E.M. ZAYED AND E.F. ELSHEHAWEY
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