669
Internat. J. Math. & Math. Sci.
VOL. 13 NO. 4 (1990) 669-676
ON A PULSATILE FLOW OF A TWO-PHASE VISCOUS FLUID
IN A TUBE OF ELLIPTIC CROSS-SECTION
A.K. GHOSH
Department of Mathematics
Jadavpur University
Calcutta 700032, India
A.R. KHAN
Department of Mathematics
Taki Government College
Parganas, West Bengal
24
India
L. DEBNATH
Department of Mathematics
University of Central Flor[da
Orlando, Florkda 32816, U.S.A.
(Received December 2, 1988 and in revised form May 2, 1989)
A study is made of an unsteady flow of an incompressible viscous fluid with
embedded small inert spherical particles contained in a tube of ellptlc cross-sectlon
The
due to a periodic pressure gradient acting along the length of the tube.
ABSTRACT.
solutions for the fluid velocity and the particle velocity are obtained for large and
It is shown that the effect of particles on the flow is significant in
small times.
the small-time solution while the large-tlme solution shows no effect of the particles
on the flow.
INTRODUCTION.
Considerable attention has been given to pulsatile flows of fluids in a tube of
various cross-sectlon due to its increasing importance in the study of blood flow in
Womersley (1955) has studied the pulsatle flow of a viscous fluid in a
arteries.
tube of circular cross-sectlon due to a given pressure gradient.
Similar problems
have been investigated for the unsteady flow of a viscoelastic liquid by Waiters and
King (1970
viscous
Khamrui (1955) has obtained solutions for a periodic flow of a
1971).
liquid
in
pressure gradient.
a
tube
of
elliptic section under the
influence
of a periodic
Later on, Ghosh and Khamrui (1978) have investigated the pulsatile
flow model of a viscoelastic fluid in a channel of elliptic cross-sectlon.
of these works,
In spite
there seems to be no study of pulsatile flow of a two-phase viscous
liquid in a tube of elliptic section due to a periodic pressure gradient.
The main
objective of this paper is to investigate such problem in order to determine the fluid
velocity as well as the particle velocity, and to examine the effects of particles on
A.K. GHOSH, A.R. KHAN AND L. DEBNATH
670
It
the flow.
shown that. the effect of particles on the flow i significant in the
[s
small-time solution while the large-tlme solution contains no effect of particles on
tlle flow.
2.
MATHEMATICAL FORUMULATION.
Based upon the Saffman (1962) two-phase ftuid model,
of
motion
an
incompressible
fluid
viscous
with
the equations of unsteady
ident[cal
embedded
small
inert
spherical particles are
u
8__v.
v(2ux2
+
z
t
]_ (u
+ --) +
(v-u)
y2
(2.1)
v)
(2.2)
x
8t
where u,v are the components of the fluid and the particle velocity in the direction
of z-axis which is taken along the length of the tube.
The last term on the right
hand side of equation (2.1) represents the force exerted by tile particles on the flow
whle
the
the
on
term
right hand side of
equation
mN
exerted by the fluid on the particles,
the
particles
and
particles.
the
respectively
the
fluid,
the
is
mass
time
a
particle,
is
a slm[lar
force-term
o
----is the ratio of the mass density of
known
commonly
relaxation
of
k
(2.2)
of
as
the
mass
concentratlon
part[cles,
the number density of
m,
N, k,
particles,
the
dust
@ and
are
of
the
Stokes
resistance coefficient, the density and the kinematic viscosity of the fluid.
The
flow is generated
from rest due to the periodic pressure gradient acting
along the length of the tube as
3p
3z
P e
imt
(2.3)
where P is a constant and
is the frequency.
The initial conditions are u(x,y,O)
3.
0 and v(x,y, O)
0
(2.4ab)
THE LAPLACE TRANSFORMED SOLUTION.
We apply the Laplace transform of u(x,y,t) and v(x,y,t) wth respect to t defined
by (Mylnt-U and Debnath
u(x,y,s)
1987])
f,0 e-St u(x,y,t)dt
to solve the differential system (2.1)
(3.1)
(2.3).
The transformed equation for
u(x,y,s) is given by
2 u + )2
x2
y2
S
(i+_k+s)u
l+s
P
pv (s-i c)
(3.2)
PULSATILE FLOW OF A TWO-PHASE VISCOUS FLUID
2
3y
into
p s(s-i)(l+k+s)
2+ 2
}x
+
P(
U
Substituting u
q
2
2F
(3.2),
671
we obtain
(3.3)
0
where
q
s(l+k+sz)
2
(3.4)
v(l+sT)
We next introduce the elliptic coordinate (K,n)defined by x+iy
c cosh(+i)
2
2 I/2
(a
in (3.3) to transform equation (3.3) in the form
b
where c
2- + 2
2
-3q2
3
442
where
where a
the
(),
82
}n
2
2 (cosh
cos 2) U
2
(3.5)
0
2 2
--q c
Separating
equation for
2
by
variables
-
using U
and a Mathleu equation for
22cosh
(a
+ (a
+
22cos
2)
2n)
is a constant.
() (),
() as.
we
find
a
modfled
Mathleu
(3,6)
0
(3.7)
0
Since U is symmetrical with respect to the axis of the ellipse and is periodic
with
period 7,
(1947)).
is
a
periodic
then
is
the
function
modified
Ce2n( 42)
Mathieu
of
function
order
2n
Ce2n(,-2).
(see
Mclachlan
these
are
represented by expansions
Ce2n(r, _2
(-1 )n
2
(-1) n
Ce2n(-4
where A
(2n)
.
ro
(-1 )r .(2n) cos 2rr,
a2r
(3,7)
2n)cosh
(3.8)
(-1) r
are functons of 4 2
2r
The appropriate general solution for u then becomes
_P
1+sz
p s(s--i)(l+k+s)
If
o designates
u
0
P
(l+sz)
+
"
r-o
C2n Ce 2n(,-2)
Ce2n(r,-2).
(3.9)
the boundary of the ellipse, the boundary condition reads
when
(3.10)
0
That is,
O s(s-i)(l+k+sz)
ro
C2nCe2n (0 42 Cn
(n
2
4 ).
(3.11)
A.K. GHOSH, A.R. KHAN AND L. DEBNATH
672
Multiplying (3.11) by
Ce2n(n ,-62
and integrating from 0 to 2 with respect to
n
and
then using orthogonal relations, we obtain
C
2n
whe re
2(-l)nA(2n)
o
P(l+s)
Ce2n(o,- 62 I2n
ps(s-i0)(l+k+s)
(3.12)
2w
12n of Ce2n(B, -62
(3.13)
d.
Hence
--u
with
4.
I+sT
+
s(s-tto)(l+k+sT) rffio
P
0
(3.14)
2 2
2=q
8ffi
C2nCe2n ( 62) Ce2n(rt 62
c
4
SOLUTIONS FOR SMALL AND LARGE TIMES.
The form of q, that is,
clearly suggests that the Laplace inversion of (3.14)
is almost a formidable task.
So in order to give a fairly good description of the
flow, the inversion will be considered in two limiting cases of small and large values
of 8 (-
62).
Case I:
For small values of 8, we have
ce
o
(n,-8)
+8
(1
cos 2n + 8(
ce2(n,-8)
(4.1)
cos 2n)
-2
(4.2)
-)
cos 4n
and similar asymptotic expansions for the modified Mathleu functions.
Also from Mclachalan (1947), we get
A
(0)
A
o
(2)
o
-8+0( 83
A(2n)
0(Sn)
o
so that
C
o
C2
-
P
P
(I + sT)
(I
s(s-l) (l+k+sT)
P(I + sT)
p s(s-lm)(l+k+sT)
8
cosh 2
(4.4)
o
(3.14), we obtain
Substituting these is
P
(4.3)
8 cosh 2o)
2
c’
8n (s-i)
[cosh 2
o- cosh
where n is the viscosity of the fluid.
o
2
cos
cosh
cos2n + cosh 22
o
2n]
(4.5)
PULSATILE FLOW OF A TWO-PHASE VISCOUS FLUID
673
The inversion of (4.5) gives
P c
u
2
e
8 n
it
[cosh 2
-cosh
2- cos 20 +
cosh 2 cos
cosh 2
20]
(4.6)
cos t
(4.7)
Cons eq uent ly
2
c
Re{u}
[cosh 2
8 n
cos 2n +
2
cosh
cOShcosh2cos 20]
2-
B (or small s), that
This result describes the fluid velocity for small
large time t.
0
is for
Thus the large time solution does not show any effect of the particles
on the fluid flow.
The particle velocity in this case can be obtained by integrating (2.2) and has
the form
P c
8 0
v
2
e
it
-e
-t/
o-
[cosh 2=
+ l0z
cosh
cosh 2 cos
cosh 2
cos 2n +
2
20j
(4.8)
and its real part is
P c
Re {v
cos2n]
(cost+z22 slnt-t/Z)[csh 2o-Csh 2-cos20 + cosh2
cosh 2
2
800
(4.9)
Expressing in Cartesian form of (4.7) and (4.9), we obtain
200 2-b2
P
Re(v)
These
results
particles
2
a2b 2
P----
Re{u}
2
__a2__
2)
2qo
(a2+b
indicate
that
with
a
phase
cost (I
2
x__
a
2
b
cosmt+mz sint -e
+
in
lead
the
tan
-I
(4.10)
z
-t/z
(1
m2z2
=,
limit t
mz if m
the
0, and
x
2
fluid
2
bZ2
moves
for m
0, the
(4.11)
faster than the
fluid
and
the
particle move in unision in the ultimate steady state condition.
Case II:
A0]=o"
When
B
(or t is small), it follows from Mclachlan (1947) that
is very small. Consequently, the asymptotic formula
is large
and for n
IAo(2n)l
I,
gives
2
Ceo(’-B) (’Isinh-------- KoCSh[2
Ceo(0) Ceo(12)
K
o
x(0)
(2)z/2o
where
Since 2
cosh
tanh
ltan(/4
i/2)]
(4.13)
qc, we get
cosh[2 cosh
1
exp{qc cosh
2
tanh
-l{tan(
tanh
-I
-
i
-))]
It
tan( --)
(4.12)
(tanh
) I/ 2exp{qc cosh}.
A.K. GHOSH, A.R. KHAN AND L. DEBNATH
674
Hence for large values of 6 i.e. for large
(1+sT)
s(s-lm)(l+k+sz)
P
p
P
C
cosh
l+sT
p s(s-i)(l+k+s)
)}.
(4.14)
exp {-qc(cosh
o-cosh
cosh
The inversion of (4.14) yields
l+k
P
Re{u}
p[(1+k)
+
P
p[ l+k 2
-r
[mzk
(l+k+B2T2)slnt
t
mke
cos(t-)+(l+k+2z2)sln(t-6)
l+k,
(l+k+2z 2) e-tsln (z___ P ----)
[: I/-:
/v
I 2+o2
Tkp-
,O
e
l+k
,
+m2 z2
{e
+2 z2
’-
l+k
2
{mzk cost +
(-
-pt
l+k
sin
(z_ [--I I/2 )d}
I/2
)d
C
cosh
(4.15)
0
where
r
I/2
za
2v(
+m2 2)
6
z a
a
2
2v(
l+2z 2)
)
1/2,
z
c(cosh
/o
,f
d
cosh ),
f
In the case of fluid flow without particles (k=0), the velocity field (4.15) assumes
the form
Re{u}
x exp
+
P
p
slnt
+ __P
{sln[tpo
o
[-/--’v c(cosh
p cosh
P cosh
oo
oe -t
2+m2
/2v
cosh
cosh )]}
cosh
sin [c(cosh
This is identlcal with Khamrui’s result
-
c(cosh
1/2 o
o-COSh
o-
cosh )] x
+
)
(4.16)
]dp
(1955), when P is replaced by -P and
t
.
We further note that the result (4.15) represents the small-time solution for the twophase fluid velocity.
Moreover, the presence of the parameter k and
T in
(4.15)
indicates that the fluid velocity is significantly affected by the particles when t is
very small.
This small-tlme solution also exhibits the boundary layer character of
the flow similar to that of the fluid motion without particles.
The thickness of the
boundary layer decreases with increasing values of the particle concentration.
PULSATILE FLOW OF A TWO-PHASE VISCOUS FLUID
The particle velocity, when
P
Re{v}
p(a[
+k)
[(a
{(ak
+m2 2
2
B
675
is large, is given by
sln(at
+
+
cos(at-e
-t/
+
2 (a2
l+k
+
(l+k+(a2z2)[(a e -t/sin 2 (at2
(acos(at
(e
-t/ -e
)}
+
o [e-r{(ak[(a stn((at-)+cs((at-6)+e-t/((astn-cs6)]2 + +
cosh
p
2
+
-t/z((azcos+sin 5)]
(1+k+(a2z2) [,sin((at-)-(azcos((at-8)+e
2 2
!
f’
l+k
z
-
,0
+(ak_
f
l+k
(azkp-
z(V-
m(l+k+(a2z 2) ,(e-t/Z_e-Pt)
l/z) (l2+(a 2)
-t/T e -t
(v- 1/)(u
{__z_
sin
}dl
,/v
l+k.
e
,0
l+k.
sin
l+k)
{z [%.I(V-----’) 1/2
(4.17)
’
Finally, the results corresponding to the circular cylinder are obtained by
cosh
replacing c(cosh
t
r) and
cosh ) by (a
o by
a
r
In particular, when k
where a is the radius of the cylinder.
in
(4.15) and (4.17)
0, we find from (4.16)
that
Re{u)
P sin
P
(at
/
(a
P
() 1/2 [- sin
sin
where r
2
x
2+ y2.
{(a
{(at
r)
(a
r)
(4.18)
d].
This is a well-known result for large
(a
and t
.
REFERENCES
I.
WOMERSLEY, J.R., Method for the calculation of velocity, Rate of flow and viscous
drag in arteries when the pressure gradient is known, J. Physiol. 127 (1955),
553.
2.
WALTERS,
N.D.
and
KING,
M.J., Unsteady flow of
Rheol. Acta. 99 (1970), 345-355.
an
elastlco-vlscous
liquid,
676
3.
A.K. GHOSH, A.R. KHAN AND L. DEBNATH
WALTERS,
N.l). and KING, M.J., The unsteady flow of an elast[co-vlscous Itquld ia
a straight pipe of circular cross section, J__k_. P__hy__s_._D: A_lp_[.. Phys. 4 (1971),
204-211
4.
KHAMRUI,
5.
GHOSH, A.K. and KHAMRUI, S.K., PulsattIe flow of an elasttco-Viscous liquid in a
channel of elllptc cross section, Rheol. Acta. 17 (1978), 227-230.
6.
SAFFMAN, P.G., On the Stability of Laminar flow of a dusty gas, J. Fluid Mech.
S.R., On the flow of a viscous liquid through a tube of elliptic
section under the Influence of a periodic pressure grafltent, Bull. Cal.
Math. Sot. 49 (1955), 57-60.
13 (1962), 120-128.
7.
McLACHLAN, N.W., Theor and Application of Mathleu Functions Oxford University
Press, (1947).