On the dispersion of a dye with a harmonically varying concentration in the
hydromagnetic flow in a channel
V. M. Soundalgekar and A. S. Gupta
Citation: J. Appl. Phys. 48, 5344 (1977); doi: 10.1063/1.323570
View online: http://dx.doi.org/10.1063/1.323570
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Published by the American Institute of Physics.
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On the dispersion of a dye with a harmonically varying
concentration in the hydromagnetic flow in a chaonel
V. M. Soundalgekar
Department of Mathematics, Indian Institute of Technology, Powai, Bombay-400 076, India
A. S. Gupta
Department of Mathematics, Indian Institute of Technology, Kharagpur-2, India
(Received 3 November 1976; accepted for publication 14 February 1977)
An analysis of the dispersion of a dye with a harmonically varying concentration in a
magnetohydrodynamic channel flow has been carried out. An estimate is obtained for the axial distance
along which the fluctuations in concentration decay in the case of low-frequency input. It is observed that
the decay distance for the fluctuations in concentration increases with an increase in the Hartmann number
M.
PACS numbers: 47.65.+a, 52.30.+r
The dispersion of nonoscillatory distributions of a
dye in a nonconducting viscous liquid flowing in a circular pipe under laminar conditions was discussed by
Taylor. 1,2 The extension of this problem to an electrically conducting liquid flowing in a parallel-plate channel permeated by a uniform transverse magnetic field
was investigated by Gupta and Chatterjee 3 and
Soundalgekar and Gupta. 4 Carrier5 studied the longitudinal dispersion of a dye in a pipe containing fluid in
a steady flow when the distribution of concentration is
prescribed as a harmonic function of time at a fixed
cross section of the pipe. The same problem was considered by Chatwin 6 who showed that for high frequencies the concentration pattern is transported downstream at the maximum fluid velocity.
In this paper, we discuss the longitudinal dispersion
of a dye in the flow of an electrically conducting liquid
in a parallel-plate channel permeated by a uniform
transverse magnetic field. The concentration of the dye
varies harmonically with time at a certain section of the
channel and the effect of the magnetic field on the decay
distance for the fluctuations in concentration is studied.
This problem is likely to have bearing on the dispersal
of soluble materials in flow of electrically conducting
liquids. The technique of Carrier will be used in the
following analysis which is limited to low frequencies.
Consider the laminar pressure flow of an electrically
conducting liquid of conductivity CJ and permeability Ile
between two infinite parallel plates y = ± 0 in the presence of a uniform transverse magnetic field Ho along
the y axis. This flow, studied by Hartmann (see Cowling7), has a velocity along the x axis (parallel to the
plates) given by
-Z(M cothM _ M COShMY/O)
sinhM'
u" -
(1)
where Co and c l are constants.
Neglecting the effect of axial molecular diffusion, the
equation of concentration C for the dye is
2
i:lc
OC
i:l c
ar
+UV(y)ox =Di:ly
(4)
2 ,
where u" = UV( y) and U is the average velocity,
U
= (1/20)
f.:
u" dy,
(5)
with
V(Y)=(M cothM _ M
c~~:~/O)) (M cothM -1)-1.
(6)
In writing Eq. (4), it is tacitly assumed that the concentration of the dye is low enough to ignore the
buoyancy effects. Since the walls are impermeable,
i:lC=O
oY
aty=±o.
(7)
Now Eq. (4) has solutions of the form
C=
Co
+ exp[iw(t -
Xx/U)]f(Y),
(8)
where Y=y/o. Substitution of Eq. (8) into Eq. (4) gives
(9)
where
n=w0 2 /D,
Vl(Y)=V(y).
(10)
Equation (9) is subject to the boundary condition
df =0
dY
at Y=± 1
(11)
which follows from Eqs. (7) and (8).
where
M
CJ
= ll.,HoO ( pv
)1/2 •
(2)
Here P is the pressure gradient along the plates and M
is the Hartmann number. A dye is dispersed in this
liquid and its concentration at the section x = 0 is prescribed as a sinusoidal function of time such that
5344
(3)
atx=O,
c=co+clexp(iwt)
J. Appl. Phys. 48(12), December 1977
Since for a Hartmann flow Vl(Y):;' 0 everywhere, we
may say, following the analysis of Chatwin, 6 that the
eigenvalue X in Eq. (9) satisfies the inequality
(12)
Im(X)<O.
This shows that all solutions of Eq. (9) satisfying Eq.
(11) are spatially decaying. If Xu X2 , ' " are the eigen-
0021-8979/77/4812-5344$01.10
© 1978 American Institute of Physics
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5344
values corresponding to the eigenfunctions fuf2' ••• ,
then it can be shown that fuf2' ••• will form a complete
set. Thus, we may take
TABLE I. Values of <I>(M).
<I>(M)
M
22.620
35.852
50.347
65.118
79.984
5
(13)
where the constants Ap can be determined such that
Eq. (3) is satisfied. We now order Ap in ascending magnitude of their negative imaginary parts such that
0< - Im(A o ) < - Im(A 1 ) < .. '. Hence, for very large
values of x,
C'" Co + Ao exp[iw(t - Aox/U)]fo(Y).
10
15
20
25
Im(A o). Denoting this distance by d and using Eqs. (2),
(10), and (24), we find
w
(14)
Thus for large x, the concentration pattern is transported at a velocity U /Re(A o) and decays in an axial distance
of order - U/Im(A o).
For n« 1, we follow the method due to Carrier. 5
Assume
-U
PD
(25)
I (A)=-2-<I>(M),
d",
w m o w pv
where
<I>(M)- M cothM -1
M2F(M)
(26)
(15)
(16)
Substituting Eqs. (15) and (16) into Eq. (9) and equating
different powers of n, we get
rPfoo -0
d¥2 -
(17)
,
~~ =i[l- AOOV1(Y)]foo,
~~2 =i{[l- AOO V 1(Y)]f01 -
(18)
A01
(19)
V 1(Y)foJ.
The normalized solution of Eq. (17) satisfying dfoo/ dY
=0 at Y=± 1 is
(20)
foo= 1.
Using Eqs. (6) and (20) in Eq. (18), integrating twice,
and utilizing the condition dfot! dY = 0 at Y:=: ± 1, we have
(21 )
and
f, (Y):=:i(
coshMY
~
).
01
M coshM - sinhM 2(M cothM - 1)
u
x
P
= -2pv
(52 _ y2).
(27)
Repeating Chatwin's analysis with the above profile,
we eventually find that the eigenvalue AO (defined earlier) is given by
(28)
(22)
Again substituting Eqs. (20)-(22) into Eq. (19),
integrating, and using df02/ dY = 0 at Y = ± I, we obtain
after a lengthy calculation
iA01
The numerical values of <I>(M) are entered in Table I.
This shows that the decay distance for the fluctuations in
concentration increases with an increase in the Hartmann number M. From a physical point of view, this
result is consistent with the fact that the Hartmann
velocity profile becomes flatter with an increase in M
as compared with the nonmagnetic case. Direct comparison of our results with those in the nonmagnetic
case (M =0) studied by Chatwin 6 is not possible since the
analysis in Ref. 6 is confined to diffusion in a circular
pipe, whereas our analysis is concerned with diffusion
in a parallel-plate channel. However, Chatwin's analysis can be easily extended to the case of a parallel-plate
channel if we replace the velocity profile (1) by the
parabolic profile in the nonmagnetic case as follows:
Thus, the fluctuations in concentration decay in an
axial distance d of the order of - U /w Im(A o) where the
average velocity U for the profile [Eq. (27)] is given by
U=P5 2 /3pv. Thus
=F(M),
(29)
where
1
(1
F(M):=: (M cothM _ 1)2 '3
2
1
+ M2 - 2 sinh2M -
3 cothM)
2M
•
(23)
(24)
As shown previously, the fluctuations in concentration decay in an axial distance of the order of - U /
J. Appl. Phys., Vol. 48, No. 12, December 1977
.
1:~
Thus from Eqs. (15), (21), and (23)
5345
This agrees with Eq. (25) in the limit M - 0 since it can
be easily shown that
M cothM - 1 35
M2F(M)
:=: 2"'
It may be noted that for a given phase difference two
lines of equal phase in concentration will be very close
to each other for high frequencies of the input and the
smoothing effect of lateral diffusion will be pronounced.
Communications
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5345
This smoothing effect will not, however, be uniform.
Phase differences will persist longest near the central
section of the channel where the liquid velocity is
maximum. Thus, when n» 1, it may be expected that
the transport velocity of the concentration pattern will
be near the maximum fluid velocity. Since the maximum
velocity in a Hartmann flow decreases with increase in
M, it follows that the velocity of transport of the concentration pattern also decreases with increasing M.
5346
J. Appl. Phys., Vol. 48, No. 12, December 1977
IG.I. Taylor, Proc. R. Soc. (London) A 219, 186 '(1953).
2G.I. Taylor, Proc. R. Soc. (London) A 225, 473 (1954).
3A. S. Gupta and A. S. Chatterjee, Proc. Cambridge Philos.
Soc. 64, 1209 (1968).
4V. M. Soundalgekar and A. S. Gupta, Rev. Roum. Phys. 15,
811 (1970).
5G.F. Carrier, Q. Appl. Math. 14, 108 (1956).
6p.C. Chatwin, J. Fluid Mech. 58, 657 (1973).
7T. G. Cowling, Magnetohydrodynam ics (Interscience, New
York, 1957).
Communications
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5346