IACS Institutional Repository

Indian J. Phys. 82(12), 1617-1618 (2008) 
of ^^^ 
Vol. 82 • No. 12 • December 2008 
INTENTS 
Front Title 1611 
Advisory Board and Editors 1613-1614 
From Editor's Desk 1615 
Front Contents 1617-1618 
Reviews 
The ideas of dual description of quantum chromodynamics 
H NANDAN AND H C CHANDOLA 1619-1643 
Strong Solids 
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Heliographic distribution of X-ray solar flares and their association with geomagnetic 
disturbances in relation to solar radio emissions and coronal mass ejections 
A P MISHRA, V K MISHRA, ROOPALI TRIPATHI AND B N MISHRA 1655-1664 
Condensed Matter & Materials Physics 
An evaluation of plasmon frequency associated with charge carriers for heavy 
electron systems 
PRAJWALIT SHIKHA, V K SINHA AND J D DUBEY 1665-1669 
Determination of the effective magnetic anisotropy constant of ferrite nanoparticles 
dispersed in organic matrix 
OSACIMIHAELA 1671-1679 
[ Cont'd, on next page ] 
1 6 1 8
 Contents 
Optics & Spectroscopy 
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grating 
SANJEEV KUMAR RAGHUWANSHI AND SRINIVAS TALABATTULA 1681-1687 
Plasma Physics 
Solitary waves in dusty plasmas with variable dust charge and nonthermal ions 
HAMID REZA PAKZAD AND KUROSH JAVIDAN 1689-1699 
Note 
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J D DUBEY, J P KUSHWAHA AND BHIMLAL PRASAD 1701-1705 
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Notes for Contributors 1709-1712 
Our Special Publications 1713-1714 
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English

Front Contents

Front Title

Chowdhury, Partha

Ray, P C

Ray, Saibal

Periodicity of ~155 Days in Solar Electron Fluence

Indian J. Phys. 82 (1), 123-125 (2008) 
Q^4)RTHCOMING PUBLICATIONS 
Volume 82, (2008) 
Proceedings of the National Symposium on Condensed Matter Days - 2006 (CMDAYS-06) 
organised by the Department of Physics, Tezpur University, Napaam, Tezpur-784 028, 
Assam, India held at Tezpur, India during August 29-31, 2006 Part I and Part II 
Foreword 
D MOHANTA 
Variational Monte Carlo study of magnetic correlations in bilayer t-J model 
AMAL MEDHI, SAURABH BASU AND C Y KADOLKAR 
Ion velocity dependent secondary electron induced point detects in SHI 
irradiated solids 
R BISWAL, J JOHN, D BEHERA, P MALLICK, S KUMAR, D KANJILAL AND 
N C MISHRA 
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Silicon oxinitride : a material for compact waveguide device 
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AND A CHOUDHURY 
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as electrode in supercapacitors 
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124 Forthcoming Publication 
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Complex envelope soliton in Bose-Einstein condensate with time dependent 
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Sandpile under rotational constraint 
S B SANTRA AND D DEB 
Optical and electrical properties of thermally evaporated ZnSe thin films 
S CHALIHA, M N BORAH, P C SARMAH AND A RAHMAN 
Optical and structural studies of SHI irradiated Si02 coated IV-VI 
semiconductor quantum dots 
S CHOWDHURY AND A CHOUDHURY 
Physico-chemical characteristics of groundwater of Tezpur town 
KALI PRASAD SARMA AND MONIMALA DAS 
Preparation, characterization and magnetic properties of hexagonal Y1.jrCeJVIn03 
S K SRIVASTAVA, MANORANJAN KAR AND S RAVI 
Finite-size scalling theory for anisotropic percolation models 
SANTANU SINHA AND S B SANTRA 
Forthcoming Publication 
Chaotic behaviour of population on a square lattice 
TAPATI DUTTA AND SUJATA TARAFDAR 
Microscopic inhomogeneity induced thermal fluctuation in high temperature 
superconductors 
D BEHERA, R BISWAL, U K MOHAPATRA AND N C MISHRA 
Titanium nitride nano-structure by DC magnetron sputtering plasma 
SANKAR MONI BORAH, ARUP RATAN PAL, HEREMBA BAILUNG AND 
JOYANTI CHUTIA 
Synthesis and characterizations of quality PbS quantum dots in SBR matrix 
N DUTTA, D MOHANTA AND A CHOUDHURY 
A plausible explanation of non-zero isotope shifts in superconductors 
B BANERJEE AND S K ROY 
Developement of LDPE/Ti02 composite as substrate material in microwave 
integrated circuits 
A GOGOI AND N S BHATTACHARYYA 
Study of antiferromagnetism in cuprates through Raman active phonon peaks 
K C BISHOYI AND G C ROUT 
Electrical characterization of multiwalled carbon nanotubes synthesized by 
DC-plasma enhanced chemical vapor deposition technique 
SK F AHMED, M K MITRA AND K K CHATTOPADHYAY 
Electrochemical synthesis and characterization of co-polymers of aniline and 
pyrrole 
A M PHARHAD HUSSAIN AND A KUMAR 


Forthcoming Publications

W/fl/i -A Pltys. 82 (2), 143-144 (2008) 
^ C 
Vol.82 • No. 2 • February 2008 
ONTENTS 
Proceedings (Part I) of the National Symposium entitled Condensed Matter Days-2006 
(CMDAYS-06) organised by the Department of Physics, Tezpur University, P.O.-Nappam, 
Tezpur-784 028, Assam, India held at Tezpur, India, during August 29-31,2006 
Front Title , 137 
Advisory Board and Editors 139-140 
From Editor's Desk 141 
Front Contents 143-144 
Foreword 
DMOHANTA 145 
Ion velocity dependent secondary electron induced point defects in 
SHI irradiated solids 
R BISWAL, J JOHN, D BEHERA, P MALLICK, S KUMAR, D KANJILAL 
AND N C MISHRA 147-154 
Effect of Ho and Nd substitution in La067Ca033MnO3 CMR manganites 
D K MISHRA, P K MISHRA, D R SAHU, D BEHERA AND B K ROUL 155-161 
Structural and optical properties of 150 MeVTr11 irradiated doped 
nanostructures 
U DAS, D MOHANTA, A K BORDOLOI, F SINGH, A TRIPATHI, 
D K AVASTHI AND A CHOUDHURY 163-168 
[ Cont'd on next page j 
144 Contents 
Magnetic and martensitic transformations in Ni-Fe-Ga alloy 
M RAGHUSANKAR, SIDANANDA SARMA, S RAVI AND A SRINIVASAN 169-172 
Orientation of carbazole molecule in the mixed Langmuir-Blodgett films 
S BISWAS, S A HUSSAIN AND D BHATTACHARJEE 173-177 
Steady state photo carrier grating technique for measurement of 
charge carrier diffusion length 
PURABI GOGOI, TAUFIQUE A ANSARI AND PRATIMA AGARWAL 179-184 
Complex envelope soliton in Bose-Einstein condensate with time 
dependent scattering length 
AYAN KHAN, RAJNEESH ATRE AND PRASANTA K PANIGRAHI 185-188 
Physico-chemical characteristics of ground water of Tezpur town 
KALI PRASAD SARMA AND MONIMALA DAS 189-193 
Preparation, characterization and magnetic properties of hexagonal 
Mn03 
S K SRIVASTAVA, MANORANJAN KAR AND S RAVI 195-200 
Y,.,Ce, 3 
Chaotic behaviour of population on a square lattice 
TAPATI DUTTAANDSUJATATARAFDAR 201-207 
Titanium nitride nano-structure by DC magnetron sputtering plasma 
SANKAR MONI BORAH, ARUP RATAN PAL, HEREMBA BAILUNG 
AND JOYANTI CHUTIA 209-213 
Study of antiferromagnetism in cuprates through Raman active 
phonon peaks 
K C BISHOYI AND G C ROUT 215-222 
Electrical characterization of multiwalled carbon nanotubes synthesized 
by DC-plasma enhanced chemical vapor deposition technique 
SK F AHMED, M K MITRAAND K K CHATTOPADHYAY 223-228 
Forthcoming Publications 229-230 
Notes for Contributors 231-234 
Our Special Publications 235-236 


Singh, K D

Mathew, Alphonsa

Indian J. Phys. 82 (4) 435-445 (2008) 
# 
Injection/suction effect on an oscillatory hydromagnetic 
flow in a rotating horizontal porous channel 
K D Singh* and Alphonsa Mathew 
Department of Mathematics (ICDEOL), Himdchal Pradesh University, Shimla 171 005, Himachal Pradesh India 
E-mdil kdsinghshimld^gmail corn 
Received 11 July 2007, accepted 15 January 200S 
Abstract : This paper studies the effects of injection/suction on an oscillatory flow of a viscous incompressible 
fluid in a porous channel The porous channel with constant injection/suction rotates about an axis perpendicular 
to the plates of the channel A magnetic field of uniform strength is also applied perpendicular to the plates The 
upper plate is allowed to oscillate in its own plane with the velocity U*(f) whereas the lower plate is at rest The 
effects of coriohs force and the magnetic field on the flow are studied 
Keywords : Oscillatory, rotating, porous channel, injection/suction, hydromagnetic 
PACS Nos. : 47.35 Tv, 47.65 -d 
1. Introduction 
Magnetofluid-dynamics (MFD) or Magnetohydrodynamics (MHD) has gained considerable 
importance because of its wide ranging applications in physics and engineering. In 
astrophysics and geophysics it is applied to study the stellar and solar structures, 
interplanetary, and inter stellar matter, solar storms and flares, radio propagation 
through the ionosphere etc. In engineering it finds its application in MHD generators, 
Ion propulsion, MHD bearings, MHD pumps, MHD boundary layer control of reentry 
vehicles eta MHD has attracted the attention of large number of scholars due to its 
diverse applications. Attia and Kotb [1] studied the MHD flow between two parallel 
porous plates. Yen and Chang [2] analyzed the effects of wall electrical conductances 
on the magnetohydrodynamic couette flow. A magnetohydrodynamic flow in a duct has 
also been studied by Chang and Lundgren [3]. In the recent years a number of 
studies have appeared in the literature on the fluid phenomena on earth involving 
rotation to a greater or lesser extent viz. Vidyanidhu and Nigam [4], Gupta [5], Jana 
'Corresponding Author © 2 0 0 8 I A C S 
436 K D Singh and Alphonsa Mathew 
and Datta [6]. Injection/suction effects have also been studied extensively for horizontal 
porous plate in rotating frame of references by Gupta [7], Mazumder [8], Mazumder 
et al [9], Soundalgekar and Pop [10] for different physical situations. The fluid flows 
in rotating channels have been studied by a number of scholars viz. Mazumder [n j 
Ganapathy [12], Singh [13]. Singh et al [14] also obtained an exact periodic solution 
of oscillatory Ekman boundary layer flow through a porous medium bounded by two 
horizontal flat plates. In this paper it is proposed to study the injection/suction effects 
on a hydromagnetic oscillatory flow in a horizontal porous channel in a rotating 
system. 
2. Mathematical analysis 
We consider an oscillatory flow of a viscous incompressible and electrically conducting 
fluid between two insulating infinite parallel porous plates at a distance d apart A 
constant injection velocity, w0> is applied at the lower stationary plate and the same 
constant suction velocity, i% is applied at the upper plate which is oscillating in its 
own plane with a velocity U*(t*) about a non-zero constant mean velocity U0. Choose 
the origin on the lower plate lying in x*-y* plane and x*-axis parallel to the direction 
of motion of the upper plate. The z*-axis taken perpendicular to the planes of the 
plates, is the axis of rotation about which the entire system is rotating with a constant 
angular velocity 11*. A transverse magnetic field of uniform strength S0 is also applied 
along the axis of rotation. The value of this uniform magnetic field is assumed to be 
unaltered by making the necessary assumptions that guarantee the neglection of the 
induced electric and magnetic fields. Hall effect, electrical and polarization effects are 
also neglected. Since the plates are infinite in extent, all the physical quantities except 
the pressure, depend only on z* and t*. Denoting the velocity components u*, v* w* 
in the x* y*, z* directions, respectively, the flow in the rotating system is governed by 
the following equations : 
iv* = 0 , 0) 
ut +wuz~ px/p + vuzz + 2ft v - oBfiu lp (2) 
vt + w uz = -py/p + vuzz - 2J? U - oBlv lp (3) 
where v is the Kinematic viscosity, t is the time, p is the density and p* is the 
modified pressure, a is the electric conductivity. The boundary conditions for the 
problem are 
< / = i / = 0 , w* = wQ% at z * = 0 , 
iniection/suction effect on an oscillatory hydromagnetic flow etc. 437 
/ / - 0, ww = w0t at z = d, 
where u/* is the frequency of oscillations and * is a very small positive constant. 
The integration of the continuity eq. (1) under boundary conditions (4) for w* 
gives, iv* = vv0. Substituting w* = wQ and eliminating the modified pressure gradient, 
under the usual boundary layer approximations, i.e. from eq. (2), we get, 
U* + <iB$U/P - -pjp. 
Also from eq. (3), we obtain 
2<7 V - -p* /p 
Substituting the above pressure gradients in eqs. (2) and (3), we get 
* * * * * • CfB/\ I * * \ 
ut 4 w0u7 = vu„ + U, + 211 v —*- U/ - U I, (5) 
"/ + w 0 i ^ = n ^ 2i7 [u - U J — ^ — . (6) 
introducing the following non-dimensional quantities 
// z/d, t - J t \ u^u*/U0, i> = i / / l / 0 l S2=-Sl*d2/r the rotation parameter, 
a ~Sd2/v the frequency parameter, s = w0d/v the injection/suction parameter and 
M B
°
d\\ ^ e H a r t m a n n number, we get, 
uuy + st/f/ = % + u;l/f 4 2£iv -- Af (ty - L/)f (7) 
a;/;, f S/;;/ = vm - 2J?(u - U) - W2/A (8) 
The corresponding tranformed boundary conditions become 
u = i/ = 0 af 7/ - 0, | 
u = L/(f) = 1 + 5 cos f, i/ = 0 a f r ; = 1 . J ^ 
Eqs. (7) and (8) can now be combined into a single equation, by introducing a 
complex function q = u + /^, as 
u/g, + sg„ = qnn + u , ^ - 2/tf(g - L0 - M2(q - 10, (10) 
and the boundary conditions (9) can also be written in complex notations as 
q = 0 at r/ = 0, 
q = (U)(t) = l + l(e* + e-*)at i/ = 1. ( 1 1 ) 
438 KD Singh and Alphonsa Mathew 
In order to solve eq. (10) subject to the boundary conditions (11), we look for a 
solution of the form 
Qiv, 0 = <7o iv) + \ {<h [v) e" + Q2 {ri) eH*} (12) 
substituting (12) into (10) and (11) comparing the harmonic and nonharmonic terms, 
we get 
Qi ' - sq ; - ( / 2 + Af2)q0 = - ( / 2 + /W2), (13) 
g,"- sq[ - (m2 + M2) q, = - (m2 4 M2),
 (14) 
Q^-sq^-(n 2 + W2)q2 = - ( n 2 + W2), ( 1 5 ) 
where I2 = /2/2, m2 = /' (2/2 + u) and n 2 = / (2/2 - a>). 
The corresponding transformed boundary conditions 
Qo = <7i = Qa = 0 af T/ = 0, 
qo=qA=qz=l at ^ = 1- (16) 
The solutions of eqs. (13) to (15) under the boundary conditions (16) are obtained as 
qo(i7) = 1-(fltf,4l*,)-
 e
(
"
+ r
' ">)/(e"-e '*), (17) 
Q1 („) = 1 - («!*"«"> - e ( r^•»>)/( e* _ e '<) , (18) 
q2 („) = 1 - (e('s "6"> - e^ ' 5 " 1 )/(e* - e'«), (19) 
where 
r1 = (s + >/s2 + 4(/2 + M2))/21 r2 = ( s - > / s 2 + 4(/2 + /W2)y21 
f3 = ( s + > / s 2 + 4 ( m 2 + W 2 ) y 2 , r 4 = ( s - > / s 2 + 4(m2 + /W2)y2, 
r5 = (s + > / s 2 +4(n 2 + /W2)y2> r. = ( s - ^ s 2 + 4(n2 + M2))/2. 
3. Results and discussion 
Now for the resultant velocities and the shear stresses of the steady and unsteady 
flow, we write 
"o {v) ~ i»o (v) = Qo {v) (20) 
Injection /suction effect on an oscillatory hydromagnetic flow etc 439 
and 
"i M - '"1 M = Qi (?) «* + Q2 iv) e~i{. (21) 
The solution (17) corresponds to the steady part which gives uQ as the primary and 
u0 as the secondary velocities for the steady flow are given by 
R0 = V u o+^o. eo = tan"1 ( V " o ) • (22) 
in Figure 1a, the resultant velocity R0 is shown graphically for samll and large values 
of rotations when the suction velocity and the magnetic field strenght vary. This figure 
shows that for small rotation Q (curves I, II) and increase in the injection/suction 
parameter s leads to a decrease in the resultant velocity R0 everywhere in the 
a 
1.2 
1 
0.8 
06 
04 
02 
n I 
a 
\l!///[] 
Ai//u\ { 
VII 
VIII 
'/ 
w 
V 
L-
y ' V 
l 
- L 1 
VI 
J , 
" 1 " 
s 
2 
4 
2 
2 
2 
4 
2 
2 
r 
il 
5 
5 
10 
5 
25 
2 5 
50 
25 
L 
M 
2 
2 
2 
4 
2 
2 
2 
4 
— i — 
-J 
1 
II 
in H 
IV 
V 
VI J 
VII "j 
VIII 
_J_ I 
0 1 0 2 0 3 04 0 5 0 6 0 7 0 8 0 9 1 
- 01 
0 1 0.2 0.3 0 4 0 5 0.6 0 7 0 8 0.9 1 
Figure 1(a & b). Resultant velocity R0 and phase angle 0O due to u0 and t/0. 
440 KD Singh andAlphonsa Mathew 
channel, however, the resultant velocity R0 decreases near the stationary plate but 
increases near the oscillating plate, when rotation is large (curves V, VI) An increase 
in the small values of rotation J7, an increase in R0 is observed in the whole width 
of the channel (curves I, III), whereas increase in large rotation reveals that m the 
lower half of the channel resultant velocity R0 increases but then it decreases in the 
upper half of the channel (curves V, VII) The increase in the magnetic field strength 
represented by the Hartmann number M leads to an increase in R0 everywhere in the 
channel for small rotations (curves I, IV), but for large rotations there is a slight 
increase near the stationary plate, but a significant decrease in R0 thereafter (curves 
V, VIII) For large rotations R0 rises within a very short distance from the stationary 
plate to the value unity and oscillate about it 
Figure 1b, shows that for small rotations ii the phase angle 0Q for the steady 
flow increases with the injection/suction parameter s but decreases with the increase 
of rotation parameter J? and the Hartmann number M in the whole width of the 
channel (curves I, II, III, IV) For large rotations the increase in injection/suction 
parameter s gives rise to an increase in the phase lead everywhere in the channel 
except a slight decrease in the phase lag near the oscillating plates (curves V VI) 
Also the increase in rotation il (curves V, VII) leads to a decrease in phase lead and 
phase lag in the lower half of the channel and upper half of the channel respectively 
A phase lag is also evident from this figure near the upper oscillating plate for large 
rotations The increase in the magnetic field strength represented by the Hartmann 
number M leads to a decrease in 00 for large rotations near the stationary plate but 
a slight increase in 0O thereafter (curves V, VIII) 
The amplitude and the phase difference of shear stresses at the stationary plate 
(7/ = 0) for the steady flow can be obtained as, 
Tor ^ V r o * + T o y °0r - tan 1 ( r 0 y / r 0 x ) (23) 
where 
TOX + /r0y - {dq/dn\ 0 = [ryer* - r2e" )/(er1 - er*) (24) 
Here r0x and r0y are, respectively, the shear stresses at the stationary plate due to 
the primary and secondary velocity components. The numerical values of the amplitude 
r0r of the steady shear stress and the phase difference of the shear stresses at the 
stationary plate (77 = 0) for the steady flow are presented in Table 1 This table clearly 
shows that r0r increases with the increase of J7 for all values of rotation large or 
small It is also evident from the values in this table that r0f decreases with increase 
of injection/suction parameter s Also the increase in the magnetic field strength 
represented by Hartmann number M leads to an increase of rQr for all values Q large 
or small, however, the increase is insignificant for large values of rotations The phase 
difference 0Or increases with the increase of i? for small rotations but decreases with 
infection/suction effect on an oscillatory hydromagnetic flow etc 441 
Table 1 . Values of r0r and 00t for various s J7 and M 
Q 
5 
5 
10 
5 
25 
25 
50 
25 
M 
2 
2 
2 
4 
2 
2 
2 
4 
T~0r 
2 52905 
1 9657 
3 80846 
3 48486 
6 38222 
5 74232 
9 30726 
6 48027 
#0r 
0 7663 
0 9042 
0 82642 
0 34098 
0 8409 
0 93402 
0 83454 
0 71149 
<> for large rotations. With the increase of injection/suction parameter s, 00r increases 
•or both small and large rotations. Also the increase of Hartmann number leads to a 
decrease of 0Qr for any values of rotation small or large 
The solutions (18) and (19) together give the unsteady part of the flow The 
unsteady primary and secondary velocity components u^i) and /^(A/), respectively, for 
the fluctuating flow can be obtained as 
uA (77,*)= {Realg, (?/) + Realg2 (//)}cos t- {lm g, ( / / ) - Im g2 (//)}sinff (25) 
v, (?y, f) = {Realg, (?/)-Realg2 (v)} sm t + {\m q^ (?/)+lmqf2 (?/)}cosf, (26) 
The resultant velocity or amplitude and the phase difference ot the unsteady flow are 
given by 
ft1 = ylu* + vf9 0, = tan 1 (i/^uO (27) 
For the unsteady part, the resultant velocity or the amplitude, RA and the phase angle, 
tf are presented graphically in Figure 2a,b for the two cases of rotation, ft small 
(5 10) and i? large (25,50) Figure 2a shows that with increase of injection/suction 
parameter s the resultant velocity ft, decreases (curves I, II) for small rotations but 
increases (curves VI, VII) for large rotations except in the vicinity of the stationary 
plate For small values of i? the amplitude R, increases (curves I, III) with increase 
of rotation, however, for large values of rotation it, the amplitude R\ increases (curves 
VI, VIII) in the lower half of the channel and decreases in the upper half The 
resultant velocity R^ increases with increase of Hartmann number M for small rotations 
(curves I, IV) but decreases with M for large rotations throughout the width of the 
channel (curves VI, IX), Keeping all the parameters s, 4? and M fixed, an increase in 
the frequency of oscillations u leads to a decrease of the resultant velocity ft, (curves 
'. V) for small rotations but to a slight increase of ft, (curves VI, X) for large values 
°f rotations. 
Figure 2b exhibits that the phase angle 0, increases (curves I, II) with the 
442 K D Singh and Alphonsa Mathew 
Figure 2(a & b). Resultant velocity R} and phase angle 6, due to u, and vv 
increase of injection/suction parameter s in the whole width of the channel for small 
rotations; however, for large rotations although it increases in the lower part of the 
channel, it decreases in the upper part (curves VI, VII). In both the cases of small or 
large rotations 0^ decreases with the increase of Q (curves I, III and VI, VIII). For 
small rotations 0, decreases with the increase of Hartmann number M (curves I, IV) 
but for large rotations it oscillates (curves VI, IX). The increase of frequency of 
oscillations LJ leads to an increase in the lower half of the channel and a decrease 
in the upper half of the channel for small rotations (curves I, V) but for large rotations 
the phase angle 0A oscillates (curves VI, X). It is also noticed that with the increase 
m/ection/suction effect on an oscillatory hydromagnetic flow etc. 443 
of frequency of oscillations u the phase lead turns into phase lag near the upper 
oscillating plate where it becomes approximately zero. 
For the unsteady part of the flow, the amplitude and the phase difference of 
shear stresses at the stationary plate (77 = 0) can be obtained as 
ru + /vly = (aui/a^o +' (dv i/ar>Uo (28) 
which gives 
Tu = y[rlc + r?y, 0U = tan'1 (r,y/ru) (29) 
The amplitude r1r of the unsteady shear stress is shown graphically in Figure 3. The 
figure clearly shows that the amplitude r1r increases for both small and large values 
14 I 1 1 I 1 1 1 1 
12 
10 
8 
6 
4 
2 
10 15 20 25 30 35 
Figure 3. The amplitude r u of unsteady shear stress for t = ?r/4. 
of rotation parameter Q (curves I, III and V, VII). For small values of ft the figure also 
shows that r1r decreases with the increase of injection/suction parameter s and then 
increases for large values of frequency u\ however, ru decreases with the increase of 
injection/suction parameter $ for large rotations and for any value of frequency of 
oscillations u (curves I, II and V, VI). The amplitude r1r increases with the Hartmann 
vn s 2 
4 
2 
2 
2 
4 
2 
2 
il 
5 
5 
10 
5 
25 
25 
5 0 
25 
M 
2 
2 
2 
4 
2 
2 
2 
4 
1 
II 
III 
IV 
V 
VI 
VII 
VIII 
444 K D Singh and Alphonsa Mathew 
number M for any value of the frequency and small rotations (curves I, IV), but f0r 
large rotations TU decreases except for very small frequency (curves V, VIII). The 
numerical values of the phase difference 6U are listed in Table 2. 
Table 2. Values of 0U for various s, tt, Mand uj. 
s U M u 
5 
5 
10 
5 
25 
25 
50 
25 
2 
2 
2 
4 
2 
2 
2 
4 
0 
0 76221 
0.90106 
0.82669 
0 34096 
0.84088 
0.9340 
0.83454 
0.71149 
5 
1 0511 
1.2438 
0.9707 
0.4345 
0.8960 
0.9947 
0.8613 
0 7631 
10 
1.3258 
1.55176 
1.12026 
0.5248 
0.95152 
1.05522 
0.88819 
0.81493 
15 
1.52791 
1 54498 
1.28195 
0.60515 
1.00771 
1.11601 
0.91511 
0.86727 
20 
-1.4857 
-1 16751 
1.45032 
0.6731 
1.6532 
1 17776 
0.94217 
0.9204 
25 
-1.39355 
-1.03643 
-1.56719 
0.73041 
1.12523 
1.24129 
0.96946 
0.97459 
30 
-1.32107 
-0,93663 
-1.49379 
0.78008 
1 18871 
1.30769 
0.99705 
1.03003 
35 
-1.25953 
-0.85806 
-1.44407 
0.8247 
1.25773 
1.3784 
1.0250 
1.0866 
It is interesting to note that there remains a phase lead with the increasing 
frequency of oscillations for large rotations; however, the phase lead changes to phase 
lag with increasing frequency of oscillations for small values of rotation Q. In the case 
of small rotations, 0U increases for all values of frequency of oscillations with the 
increase of injection/suction parameter s. It is also evident from this table that with the 
increase of Hartmann number M, 01r decreases initially for small frequency of 
oscillations, but increases later on for large values of frequency of oscillations. 
Acknowledgment 
The authors are highly thankful to the referee for his valuable suggestions which led 
to a definite improvement of the paper. 
References 
[1 ] HA Attia and N A Kotb Acta Mech. 117 215(1996) 
[2] J T Yen and C C Chang Z Angew. Math. Phys. (ZAMP) 15 400 (1964) 
[3] C C Chang and T S Lundgren Z. Angew, Math. Phys. (ZAMP) 12 100 (1961) 
[4] V Vidyanidhu and S D Nigam J. Math Phys. Sci. 1 85 (1967) 
[5] A S Gupta Acta Mech. 13 155 (1972) 
[6] R N Jana and N Datta Acta Mech. 26 301 (1977) 
[7] A S Gupta Phys. Fluids 15 930 (1972a) 
[8] B S Mazumder PhD Thesis (IT Kharagpur (India) 
[9] B S Mazumder, A S Gupta and N Datta Int. J. Heat Mass Transfer 19 523 (1976) 
injection/suction effect on an oscillatory hydromagnetic flow etc. 
[10] V M Soundalgekar and I Pop J. Appl. Math. Mech. (ZAMM) 53 718 (1973) 
pi] B S Mazumder J. Appl. Mech. 581104 (1991) 
[12] R Ganapathy J. Appl. Mech. 61 208 (1994) 
[13] K D Singh J. Appl. Math Mech (ZAMM) 80 429 (2000) 
[14] K D Singh, M G Gorla and Hans Raj Indian J. Pure Appl. Math. 36151 (2005) 


Injection/Suction Effect on an Oscillatory Hydromagnetic Flow in a Rotating Horizontal Porous Channel

Chatterjee, Prasanta

Kundu, Sanjib Kumar

Indum J. Phys. 82 (4) 447-457 (2008) 
^m 
Large amplitude solitary waves in four component 
dusty plasma 
Prasanta Chatterjee* and Sanjib Kumar Kundu 
Department of Mathematics, Siksha Bhavana, Visva Bharati, Santiniketan 731 235, West Bengal, India 
E-mail prasantachatterjeel<#rediffmail com 
Received 25 April 2007 accepted 16 December 2007 
Abstract : Nonlinear dust acoustic waves are studied in a four component dusty plasma The existence of 
soliton solution is determined by pseudo-potential approach It is shown that in small amplitude approximation our 
result reproduces the result obtained by Sayed and Mamun [Phys Plas 14 014501 (2007)] 
Keywords : Solitary waves, dusty plasma, pseudopotential, four components 
PACS Nos. : 52 35 Fb, 52 35 Qz 
1. Introduction 
Dust and plasmas are omnipresent in the universe. It plays significant role in space 
plasma, astrophysical plasma, laboratory plasma and environment. The presence of 
dusty plasmas in cometary tails, asteroid zones, planetary ring, intersteller medium, 
lower part of earth's ionosphere and magnetosphere [1-8] makes this subject increasingly 
important. Dusty plasmas also play a vital role in low temperature physics, radio 
frequency plasma discharge [9], coating and etching of thin films [10], plasma crystal 
[11] etc. 
Nonlinear wave phenomena like soliton, shocks and vortices in dusty plasmas 
have also been studied by several investigators for the last two decades or so [12-
22]. Bliokh and Yarashenko [14] first theoretically observed such waves while dealing 
with waves in Saturn's ring. Later the discovery of dust-acoustic wave (DAW) [15,16], 
dust ion-acoustic wave (DIAW) [17,18] and dust lattice (DL) waves [19,20], gave a 
new impetus to the study of waves in dusty plasmas. Due to the dust grain dynamics 
few new eigen modes like Dust-Berstain-Greene-Kruskal (DBGK) mode, Shukla-Verma 
^Corresponding Author © 2008 IACS 
448 Prasanta Chatterjee and Sanjib Kumar Kundu 
mode [21], Dust-drift mode [22] etc. are also introduced. 
Dust-acoustic solitary waves in the one dimensional and unmagnetized plasma 
have been investigated by several authors. However, most of them considered the 
three-component dusty plasma system consisting of ions and electrons and negatively 
charge dust particles [23-25]. But both negative as well as positive dust particles also 
present in different areas of space [26-28]. Fortov et al [29] explained the mechanism 
by which a dust grain can be positively charged. Chow et al [28] also explained the 
situations under which smaller dust particles become positively charged and larger 
particles become negatively charged. It was also investigated that both positively and 
negatively charged dust present simultaneously in different space plasmas [28-30j 
Recently Sayed and Mamun [31] investigated solitary waves in four component 
plasmas where they considered both positively and negatively charged dust particles 
To obtain the solitary wave solution they used Reductive Perturbative Technique (RPT) 
But few years ago, Malfliet and Wieers [32] reviewed the studies of solitary waves in 
plasma and found that the RPT is based on the assumption of smallness of amplitude 
and so this technique can explain only small amplitude solitary waves. But there are 
situations where the excitation mechanism gives rise to large amplitude waves, and to 
study such situation one should employ a non-perturbative technique. Sagdeevs [33. 
pseudo-potential method is one such method to obtain solitary wave solution. This 
method has been successfully applied in various cases [34,35]. 
In this paper, we consider a four component unmagnetized dusty plasma system 
consisting of Boltzmann distributed electrons and ions and also positively (smaller size) 
and negatively (larger size) charged dust grains. The existence of solitary waves is 
studied by Sagdeev's pseudo-potential technique. It is shown that in small amplitude 
approximation our result reproduces that of Sayed and Mamun [31]. 
The organization of this paper is as follows. In Section 2 basic equations are 
written for four component dusty plasma and Sagdeev's pseudo-potential is derived 
Conditions for the existence of soliton solution are also discussed in Section 3. Small 
amplitude approximation solutions are given in Section 4. Section 5 is kept for result 
and discussions and Section 6 is kept for conclusion. 
2. Basic equations 
We consider a four-component dusty plasma consisting of Boltzmann distributed ions 
and electrons and also negatively and positively charged dust grains. The basic 
equations are (see Ref. [31]) 
at dxK ' v 
Large amplitude solitary waves in four component dusty plasma 449 
| M A <„,„,). o.
 (3, 
+ u. — - = -oili — /4v 
L -^ = ^ - (1 - / i f f Me) n2 + M e ^ ' ~ /^e"1", (5) 
where ^ and n2 are the negative and positive number density normalized by the 
equilibrium values n10 and n20 respectively, i/j and u2 are negative and positive dust 
7k T k T 
fluid speed normalized by 1 B ' . V, the electric potential is normalized by - £ - i . x 
and t are normalized by \D = (Z,kBTl/4nZ?e2n,0) and u;p1 = (m1/47rZ1e n10) . 
n , ^ t 0 = ^ , Mf i = - ^ - , / ; , = Jk_t <, = ! . , z, and Z2 are the number of 
Z, m2 Z^nw Z,nw Te 
electrons or protons residing on a negative and positive dust particle respectively, m^ 
and m2 are masses of the negative and positive dust particle respectively. T, and Te 
are ion and electron temperatures respectively, kB is the Boltzmann constant and e is 
the charge of the electrons. In order to search for solitary waves which solves eqs. 
(1) to (5), we introduce a linear substitution £ = x - Mt admitting only solution which 
depend in space and time in the form of the wavy variable x - Mt. By 
d d
 M d ** d 
substitution— = — and TTt" d£ e q s ' ( 1 H 5 ) reduce to 
. . du. du. dil> 
AMdu7 du7 ndty 
450 Prasanta Chatterjee and Sanjib Kumar Kundu 
- = n, - (1 - /x, + PL,) n2 + n,eT* - ^e"* , 
The boundary conditions are :ip, uu u 2 - > 0 , n,, n 2 - + 1 , n, 
From (6) we get, 
M 
n. = . 
1
 Af-u, 
Similarly from (8) we get, 
M 
2
 M-u2 
From (7) we get, 
(10; j 
/*, a n d n 9 - / J e a s 
1
 2 
and from (9) 
(Hi 
(12) 
(13) 
a(ii> = Mu2 , 
Now using (11)—(14) in (10) we get 
cfy; dV{4>) 
d? d\j> 
where 
VW) = M2 1 + 24>) 
M2 
1/2 
1 - 1 - 2a/ty)' 
1/2 
M' 
+ i k ( i _ e ^ ) + M l ( i - e - * ) . 
(14) 
(15) 
Multiplying both side of eq. (15) by 2 — and integrating w.r.t. £ with the boundary 
d£ 
conditions)^ _• oo, V - » 0 and - ^ O w e get, 
amplitude solitary waves in four component dusty plasma 451 
VW + -
(dfi>) 
[da = 0. (17) 
3. pseudo-potential and soliton solution 
Eq. (15) can be considered as a motion of a particle (whose pseudo-position is i/t at 
pseudo-time 0 with pseudo-velocity (di>/d0 in a pseudo-potential well 1/(0). That is 
why Sagdeev's potential is called pseudo-potential. Here the pseudo-particle starts at 
cfy> 
a position tp = 0 with a small velocity -37 and it will be reflected back at some y = y>m 
and then come back to t/> = 0. 
Whether the solitary wave solution of eq. (15) exists or not, can be determined 
from the nature of the pseudo-potential l/(y). To discuss the possibility of solution we 
may consider the cases in classical mechanics when a conservative force field is 
given. It is known that between to single roots (0 and ipm here) of 1/(0) if l/(y>) < 0 
(hen 0 is periodic and that leads to a periodic nonlinear waves. But when a single root 
on the one side of the interval is considered, and a double root on the other end, a 
solitary wave is generated (see Ref. [6]). Due to initial conditions incorporated in l/(y>), 
the double root is in y = 0. It takes an infinite large time to get away from it and the 
</ reaches a maximum or minimum in ym and then again taking infinite long time to 
returned to 0. Hence, conditions for the existence of soliton solution are 
(i) vtyt) = 0 at <0 = 0 and 0 = ipm 
(ii) 
d \ / 
v=o 
0 but — 
aw 
* 0 . 
V-t/^ 
There is also one requirement that the double root at tp = 0 corresponds to a local 
maximum at y = 0. 
Hence another condition is 
d2V 
(iii) dip2 < 0 . (/'=0 
Also, it can be noted that physically complex V will not be allowed because this 
implies complex dust density which is not allowed. Hence there exists a nonzero ym> 
the maximum (or minimum) value of ip where V(ipm) = 0 (i.e. Vtym) crosses the y axis 
from below). Then ipm is the amplitude of the solitary wave. If ipm is positive then the 
solitary wave is called compressive solitary wave and if V;m is negative then the solitary 
wave is called refractive solitary wave. Obviously, V{4>) is negative in the interval 
(0, ym). 
452 Prasanta Chatterjee and Sanjib Kumar Kundu 
4. Small amplitude approximation 
To obtain KdV type solution we obtain the small amplitude approximation of V(</>)
 We 
expand V(tp) about v> = 0. Using the boundary condition V -> 0 a n d — - > Q
 a s 
dip 
V>2
 A V' 3 
V> — 0, we get 
V ( 0 ) - A y + * 2 ~ , (18) 
when 
4> = ^ j 7 + (1 - /*, + A*.) - j j j j r - - ffW2 - Mi/2. (20) 
Hence the KdV type soliton solution is given by 
0 = t/>0 sec r r - (21) 
o 
when 
. 34 
V
'
0 =
 ~ " ^ ' (22) 
is the amplitude of the solitary wave and 
2 b = V 1 ^ (23) 
is the width of the solitary wave. Sayed and Mamun [31] studied the same model for 
small amplitude solitary wave using RPT and they obtained the KdV equation as 
when 
= 1 [/., _ + \ a2 /?2 _ g _ V«(^^2 _ „ )| 
1/3 
e = 3L_ 
where V0, the phase speed of the DA wave is given by 
(25) 
(26) 
amplitude solitary waves in four component dusty plasma 453 
VS 2 _ 
1 + (l + /i.-A«,)"fl 
<T/te + /X, 
(27) 
To get the steady state solution they use a transformation £ = // - L/0T and the usual 
boundary conditions, where U0 is the velocity of the frame of the transformed 
coordinate. By usual technique one can obtain 
de B «w-£ A01 
whence 
V,(0i) = - ^ 0 i 2 + 3^V'i 
(28) 
(29) 
Now to compare the small amplitude approximation of V(*p) of eq. (18) with the values 
of Vi(4>) of eq. (29) obtained by RPT (Ref. [31]), we first replace M by V0 + U0 when 
UQ IS small. Then Keeping only first-order terms (in U0) it can easily be verified that 
\Z(c) in eq. (18) reduces to the M,(</',) given in eq. (29). Hence the l/t(</>,) obtined by 
RPT in Ref. ([31]) is nothing but a small amplitude approximation of V(</') of eq. (18). 
5. Results and discussion 
Figure 1 shows the plot of V(V) vs. ib for v = 1.57, 1.75 and 1.925. Other parameters 
are and a = .01, ft = 50, //, = .5, fie = .2, a = .5. It is seen that V(fi) crosses the 
c axis for negative values of </' for 1.57 < v < 1.925. Hence rarefractive solitary 
waves exists for 1.57 < u < 1.925. For r = 1.75, !/(</;) crosses the </> axis at 
c = -1.35. Hence |0min| = </»0 = 1.35 is the amplitude of the rarefractive solitary waves 
for the above set of parameters with v = 1.75. It is also seen from this figure that 
the amplitude of the solitary waves increases with the increase of velocity. 
- 0 5 
Figure 1. Plot of V\tp) vs. $ for t; = 1.57, 1.75 and 1.925 Other parameters are and a = .01, (3 = 50, / i , * .5, 
/^. = .2,(TS .5. (read ip in place of y, £ in place of z and tf in place of b tn all Figures). 
454 Prasanta Chatterjee and Sanjib Kumar Kundu 
Figure 2 shows the plot of V/(V) vs. il> for v = 1.85, 2.85 and 3.41. J - 15Q 
Other parameters are the same as those in Figure 1. It is seen that V(IJJ) crosses the 
^ axis for positive values of ^ for 1.85 < v < 3.41. Hence compressive solitary waves 
exists for 1.85 < v < 3.41. For v = 2.85, l/(V) crosses the t/> axis at ^ = 2.45. Hence 
m^ax = V'o = 2.45 is the amplitude of the compressive solitary waves. It is also seen 
from this figure that the amplitude of the solitary waves increases with the increase 
of velocity. 
The shape of the solitary wave is obtained from the formula 
/
•</ 1 
<t> J-2VM 
and Figure 3 depits the soliton solution 0(0 plotted against £ for t> = 2.85. 
The other parameters are same as those in Figure 2. 
Figure 2. Plot of V(V>) vs V for v =1 85, 2 85 and 3 41 fi = 150 Other parameters are same as those in Figure 1 
-7.5 -5 -2.5 2 5 5 7.5 
Figure 3. Plot of r/> vs. £ for v = 2.85. Other parameters are same as those in Figure 2. 
large amplitude solitary waves in four component dusty plasma 455 
To see the effect of /? on the amplitude of the solitary wave Figures 4(a) and 
4(b) are drawn. For Figure 4(a) v is taken as 2.5. Other parameters are same as 
those in Figure 3. Here it is seen that for compressive solitary waves, the amplitude 
of the solitary waves increase with the increase of 0. 
For Figure 4(b) t; - 1.75. Other parameters are same as those in Figure 1. 
Here it is seen that the amplitude of the rarefractive solitary waves decreases with the 
increase of 0. Hence 0 has a significant effect on the shape of solitary waves. It can 
also be shown that the amplitude of the solitary waves also depends upon other 
parameters. 
yO 
4 r 
100 150 200 250 300 350 400 
Figure 4a. ipQ is plotted vs. ft for v = 2.5. Other parameters are same as those in Figure 3. 
y0 
.ft 
20 40 60 80 100 120 
Figure 4b. ip0 is plotted vs. ft for v « 1.75. Other parameters are same as those in Figure 1. 
6. Conclusion 
Existence of both the rarefractive and compressive solitary waves in four component 
dusty plasma are investigated using Sagdeev's pseudo-potential approach. It is also 
seen that in small amplitude approximation our result completely agrees with the RPT 
456 Prasanta Chatterjee and Sanjib Kumar Kundu 
result obtained by Syaed and Mamun [31]. The shape of the solitary wave is done 
using the integration (30). It is shown that 0 has a significant effect on the amplitude 
of the solitary wave. This technique can be extended to the study of non-thermal 
distribution of electrons in four component plasmas. Work in this direction is m 
progress. 
Acknowledgment 
This work is supported by UGC under SAP(DRS) programme. Authors are grateful to 
the referee for his valuable comments which helped to improve this paper. 
References 
[1 ] M Horanyi and D A Mendis J Astrophysics 294 357 (1985) 
[2] P K Shukla, D A Mendis and T Desai Advances in Dusty Plasmas (Singapore World Scientific) (1997) 
[3] C K Goertz Rev Geophys 27 271 (1989) 
[4] P K Shukla and A A Mamun Introduction to Dusty Plasma Physics (Bristol Institute of Physics) (2002) 
[5] A Bouchule Dusty Plasmas (New York Wiley) (1999) 
[6] F Verheest Waves in Dusty Space Plasmas (2000) 
[7] D A Mendis and M Rosenberg Annual Rev Astronomy and Astrophys 32 419 (1994) 
[8] F Verheest Space Sci Rev 77 267 (1996) 
[9] J H Chu, J B Du and I Lin J Physics D27 296 (1994) 
[10] G S Selwyn Jpn J Appl Phys Part 1 32 3068 (1993) 
[11] H Thomas, G E Morfill and V Dammel Phys Rev Lett 73 652 (1994) 
[12] D P Sheehan, M Canlo and W Heidbnnk Rev Scient Instrum 61 3871 (1990) 
[13] R N Carhle, S Geha, J F O'Hanlon and J C Stewart Appl Phys Lett 59 1167 (1991) 
[14] P V Bhokh and V V Yarashenko Sov Astron (Eng tran ) 29 330 (1985) 
[15] N N Rao, P K Shukla and M Y Yu Planet Space Sci 38 543 (1990) 
[16] A Barkan, R L Merlino and N D' Angelo Phys Plasmas 2 3563 (1995) 
[17] P K Shukla and V P Silin Phys Scnpta 45 508 (1992) 
[18] R L Merlino. A Barkan, C Thompson and N D'Angelo Phys Plasmas 5 1607 (1998) 
[19] F Melandso Phys Plasmas 3 3890 (1996) 
[20] B Farokhi, P K Shukla, N L Tsindsadze and D D Tskhakaya Phys Lett A264 318 (1999) 
[21] P K Shukla and R K Verma Phys Fluids B5 236 (1993) 
[22] P K Shukla, M Y Yu and R Bharuthram J Geophys Res 96 21343 (1992) 
[23] S Mahmood and H Saleem Phys Plasmas 10 47 (2003) 
[24] S Maitra and R Roychoudhury Phys Plasmas 10 1 (2003) 
[25] P Chatterjee and R K Jana Z Naturforschung 60a 275 (2005) 
[26] D A Mendis and M Horanyi in Cometary Plasma Processes Monograph 61 (Washington * American 
Geophysical Union) DC p17 (1991) 
[27] M Horany, G E Morfill and E Grun Nature (London) 363 14 (1993) 
m amplitude solitary waves in four component dusty plasma 457 
[28] V W Chow, D A Mendis and M Rosenberg J. Geophys Res. 98 19065 (1993) 
[29] V E Fortov, A P Nafedov and 0 S Vaulina ef a/, J. Exp. Theor Phys. 871087 (1998) 
r30] A A Mamun and P K Shukla Geophys. Res. Lett. 29 1870 (2002) 
[31 ] F Sayed and A A Mamun Phys. Plasmas 14 014501 (2007) 
[32] W Malfliet and E Wieers J. Plasma Phys. 56 441 (1996) 
[33] R Z Sagdeev In Rev. PI. Phys. (ed) M A Leontovich (New York : Consultant Bureau) Vol 4 23 (1996) 
[34] P Chatterjee and R Roychoudhury Can. J. Phys. 75 337 (1997) 
[35] p Chatterjee and R Roychoudhury Phys. Plasmas 6 406 (1999) 


Large Amplitude Solitary Waves in Four Component Dusty Plasma

Borthakur, G G

Baishya, B

Indian J. Phys. 82 (4) 459-465 (2008) 
M» 
Mixing dynamics and microscopic correlation of 
associated liquids 
G G Borthakur* and B Baishya 
Department of Physics, Dibrugarh University, Dibrugarh 786 004, Assam India 
E-mail borthakur gg@gmail com 
Received 12 September 2007, accepted 15 January 2008 
Abstract : In the present work we outline the role of time-dependent probability theory in explaining the 
mixing dynamics of two liquids, as an extension of work carried out by D Bertolini et al (1998) [1] Exact form of 
microscopic correlation function and its relation to molecular structure and interaction are proposed 
Keywords : Dielectric relaxation, Debye behaviour associating liquids, dipole correlation function 
PACSNos. : 61.30.Czt76 60.Es 
1. Introduction 
The process of dielectric relaxation, particularly in solution, is a very complex and 
interesting topic of scientific discussion, in the sense that an in-depth understanding 
of the subject is not complete. A novel approach in this regard was the work carried 
out by D Bertolini et al (1998) [1], in which, they have developed a dielectric relaxation 
model as a set of first order differential equations in some very generally conceivable 
parameters. Dielectric relaxation model has been developed for two associated liquids, 
by writing differential equations for dipole correlation function, and herein lies the 
beauty of the approach. 
In this paper we have further interpreted the parameters in the aforesaid 
relaxation model, as probabilities, and also predict the forms of dipole correlation 
functions. At first, a basic introductory understanding of the relaxation model is 
presented. Then, we have found the interpretation for the microscopic correlation 
function. In the last section of this paper, we interprete the parameters in the aforesaid 
relaxation model, as probabilities. Then in the same section, we write the explicit 
'Corresponding Author © 2008 IACS 
460 G G Borthakur and B Baishya 
functional forms of microscopic correlation functions. The section concludes two 
axioms proposed by us regarding the properties of the dipole correlations. 
2. Dielectric relaxation model 
Assumptions : 
Let us first assume that that the dynamics of mixing of two liquids can be described 
by introducing some very general dynamical parameters, which take into account the 
short-range interactions occurring in the solution. These parameters, together with the 
relaxation rates of these molecules, then, will affect the relaxation process. 
In lieu of explicitly adding diffusion terms in the equations, and then directly 
developing the dielectric relaxation model, the effect is included by developing the 
relaxation model from mixing dynamics of the liquids. Owing to diffusional processes 
the existing molecular environments in a binary solution will be continuously formed 
and destroyed. In this case the mixing dynamics can be described by a set of 
differential equations, from which we tend to develop the relaxation model afterwards 
The dynamics of mixing : 
Inside a binary solution at constant temperature, three different types of microscopic 
dynamically environment coexist, in each of which different types of short range 
interactions will be dominant : These three systems are individual molecules of the 
two liquids separately, and the molecules of both the liquids. Owing to diffusional 
processes, these environments will be continuously formed and destroyed. In this case 
the mixing dynamics can be described by the following set of differential equations (1), 
'i K 0 = Y ' 1 K 0 h (*i. *) + *%• P12 (*.. t) 
P12 (*i, 0 = b,2r, (x„ t) r2 (x1f t) - anp„ (x„ t) 0) 
These equation shows that mixing of two liquids is a nonlinear process, and this 
peculiarity will lead to some difficulties, while discussing relaxation phenomena. 
Here r, and r2 are the fractions of molecules that at the time t belong to 
environments of first and second types respectively. While p12 is the fraction of 
molecules which are found in the mixed environments, x, is the molar fraction of liquid 
1; a12 is the escaping rate of a tagged molecule from the environment (Breaking rate) 
while b12 is its entering rate (forming rate). It is important to underline the basic 
difference between these parameters. The breaking rate a12 is related to short range 
interactions while the forming rate fr12 will depend also on the diffusional processes 
and could be in principle, a fraction of molar concentration since the number of 
dynamics and microscopic correlation of associated liquids 461 
molecules is conserved, and we must have 
fl(x„„+fiL^iU, 
, ( , , ) + ^ i ) = 1_,,. <2) 
These two equations describe mixing dynamics using only two parameters, which 
depend on the molecular interaction (hydrogen bond, dipolar interactions etc.) occurring 
m the solution. 
For solution, of these equations, let, 
r1(x1l0) = x1 and r2 (x1f0) = 1 - x1. 
Eq. (1) can be expanded around the equilibrium solution; neglecting non-linear terms 
and using (2) we get, 
*Pl2 (*1, 0 = &12 [ri (* l ) bf2 (*1, 0 + '2(*l)<^1 (*1, 0] ~ M P l 2 K 0 (3) 
From eq. (3) we can show that the fluctuatuations br^ (xlr t) ,6r2 (x1f t) and <Sp12 (x1f f) 
decay exponentially with a rate b i 2 \ r % ' w h e r e 
A= l + 2 a ^ _ 4 X 1 ( 1 - X 1 ) 
This decay rate is just the inverse of mixing time. 
The dielectric relaxation model : 
For a liquid mixture in its equilibrium condition, ru r2 and p12 fluctuate around their 
equilibrium values according to eq. (3). Accepting the presence of three processes, 
the dielectric relaxation model can be developed by writing three equations (similar to 
e9« (1)) for dipole correlation function (DCF). In setting up these equations, it is 
important to notice that the DCF is only affected only when a molecule goes into the 
mixed state. If we denote with ^ ( x ^ f ) , y>2(xi»0' ^ ( ^ - O * t h e normalized dipole 
correlation functions in the three different environments, we obtain 
462 GG 
vM*i,0 = - *i + % M * i ) <M*i.O + ^ r ^ 2 ( * i .O 
b,0 r-
*>12 K 0 = " (ai2 + *12 ) ^ 12 (*L 0 + - ^ [ ^ l ^ l K 0 + rJX,yp2 (X1f f)] 
¥>2 ( *1. 0 = ~ *1 + ^ T f2(^l) ^2 (*!• 0 + ^ T 1^2 (X1- 0 
(4) 
Here frlf fc2, /c12 are the relaxation rates in the respective environments, their 
values will be assumed by supposing that the rotational diffusion essentially depends 
on the interactions among the nearest molecules. 
The arguments reported here refer to the mixing of two liquids, which show no 
characteristic association in their pure state. We will consider the association of two 
associated liquids in the following sections. 
Two associated liquids : 
When a mixture is made up of two liquids which in their pure state are structured 
(e.g. by hydrogen bond) the eqs. (1) must be generalized in order to take into account 
also the structural dynamics in the two environments 1 and 2 respectively where 
interactions among molecules of same type are dominant. So let p1f r, and ft, r2 be 
the structured and non structured fractions inside environment 1 and 2 respectively 
The breaking and forming rates inside these environments are denoted by als b} and 
a2, fa? respectively. 
The equations describing the dielectric relaxation are 
^ (*„*) = - a, + -^-[72(x,) + p2(x,)} + k, ^ 1 ( x 1 l f )+ *>i\i ( * , . ' )+ ^ ^ ( M ) 
d o r-\ i ( x i . 0 = - &i + -JL[''2(*i) + P2(*i)] + 7i X^{x^tt) + a^<P^{xvt) + -&<Pl2{Xl,t) 
V?12 K 0 = ~ (ai2 + ^12)^12 (*i , 0 + %" [/2(*i) + ft(*i)] 
^[^(^ .O + ^ if^i.Ol + ^ i^O + A ^ i ^ l ^ i ^ i . 0 + ^ 2(^1.0] <5> 
V&2(X1.0 = - a 2+^ [ ' l ( * l ) + Pl(*l)] + *2 M ^ ' O + ^ K O + ^ ^ K ' ) 
*2 K 0 = - *>2 + " ^ ['l(*l) + Pl(*l)] + 72 a, Xz (x„ f) + a2^2 (x„ t) + -f <p12 (x1t f) 
dynamics and microscopic correlation of associated liquids 463 
In eq. (5) the parameters, fc,, 7,, /c12, 72 and k2 are the orientation rates of the 
molecules having dielectric correlation function equal to ^i» Xu ^12. X2 a n d v>2 
respectively. But it is important to remark that when p, » r, and p-, » r2, eq. (5) 
coincides with eq. (1), so also the corresponding dielectirc behaviour. 
3. An interpretation of dielectric correlation function (DCF) 
As has been pointed out in the beginning, the dielectric relaxation model has been 
developed for two associated liquids, by writing differential equations for dipole 
correlation functions. So, it will Be noteworthy to find some physical interpretation for 
the DCFs at this point. 
If [2] 
M
° = £ f / (6) 
is the contribution of the permanent moments to the total moment M. An autocorrelation 
function for M0 is defined as 
K(0)M°(Q) 
Also the microscopic correlation functions 
-<,.<«,
 M,,0> l i i m t h \ (8, 
\ y=i / 
Where, Af^f)-V^My(f) denotes the sum of the permanent moments in a sphere that 
is small with respect to the dielectric so that its average value is fixed. The relation 
of mis MCF to the time dependent generalization g(t) of Kirkwood correlation function 
9 is 
9(t) = 07(0 = 4r (riO),Mj(t)) = 4- Uitoy E M')) (9) 
The relation between (p^(t) a n d complex permittivity is 
i(v)-eQ e-£„ 
i(u) + 2 e + 2 I- /^(^(0)] (10) 
464 G G BorthakurandBBaishya 
For a liquid the function v?sph(0 differs from a microscopic correlation function, since 
it incorporates the long-range dipole-dipole interaction in the polar liquid The dipoie 
correlation functions implied in eq (4) are obviously the microscopic correlation 
functions The Laplace transform of 7(f) is related to the complex permittivity as 
e(u)-ej(2e+en) l ' l " < " ' 
4. Proposals 
Probabilistic interpretation 
It appears to us that r, and r2 will be the probabilities that at the time t a particular 
molecule belong to environments of first and second types respectively While p r is 
the probability that a molecule of type 1 or 2 will be found in the mixed environments 
The probabilities r1f r2 and p12 are time dependent x1 is the probability that a molecule 
chosen at random from the sample is of type 1, irrespective of the environment from 
which it has been chosen Obviously x, does not depend on time a12, dT is the 
probability that a particular bond wall be broken in time 97, while fc12<9T is the 
probability that a particular bond will be formed in time OT With such probabilistic 
interpretation, which depends on time, we argue that the dynamics can be interpreted 
as a stochastic process 
Forms of microscopic correlation functions (MCF) 
In view of eq (8), we define the functions v?1t \ u </?12, \2 and </?2 as 
MO = 
/'*(0) X>*(0] 
xi(0 = 
' A \ (12a) 
N 
VA,(0) £ > / B ( 0 
___ M 
N
 \ (12b) UA(0) $>*((» 
/WO) £ A W ) 
<M>) = ~ — / » \ (12c) 
(/WO) £/W°)) 
Mixing dynamics and microscopic correlation of associated liquids 465 
\2(0 
vMO 
s^(O) E^(O) 0 2 d ) 
/^ s(O) £>*(0)\ (12e) 
Properties of dipole correlation in liquid 
Although, our final aim is to obtain the solutions of coupled differential equation of first 
order eq (5), for which some amount of simplifications will be necessary But due to 
empirical nature of the model, it is not possible for us, at this point, to make 
reasonably good assumptions, So, let us first form following axioms regarding the 
properties of the dipole correlations before actually solving the equations 
Proposal 1 Functional form of microscopic correlation function is independent of 
molecular structure of a liquid 
Proposal 2 Functional form of microscopic correlation function only depends upon 
only the type of interaction prevailing among different types of molecules 
The logic behind these proposals are at once clear if we note that the 1st, the 2nd, 
the 4th and the 5th equation of eq (5) are similar in form while the third one is 
different Therefore the solutions of the 1st, the 2nd, the 4th and the 5th equation of 
eq (5) should be similar in form, but the third of these equations should have a 
solution which is different in form 
5 Conclusion 
The mixing dynamics, and hence the process of dielectric relaxation in a diffusion 
dominated environment is established as a stochastic process Regarding the form of 
MCF, we noted that it is independent of molecular structure of a liquid, but only 
depends upon only the type of interaction prevailing among different types of molecules 
References 
M] D Bertohnila, M Cassettan. C Ferrari and E Tomban J Chem Phys 108 6414 (1998) 
[2] C F J Bottcher and P Bordewijk Theory of Electric Polarization Vol. II (Elsevier CFJ Bottcher) (1978) 


Mixing Dynamics and Microscopic Correlation of Associated Liquids

From Editor's Desk

Choudhury, Dilip Kumar

Bhattacharjee, Buddhadeb

Indian J. Phys. 82 (6), 663-664 (2008) 
IMJfi! 
OREWORD (M 
Proceedings (Part II) of the 5th Biennial Conference of the Physics Academy 
of the North East (PANE) held at Department of Physics, Gauhati University, 
Guwahati-781 014, Assam, India during March 1-2, 2007. 
The conference was inaugurated by Prof. A. Choudhury, Vice Chancellor, Gauhati 
University followed by a key note address on "Use of Low Energy Accelerator" by 
Prof. Amit Roy, Director of the Inter University Accelerator Centre, New Delhi. The 
inaugural ceremony was presided over by Prof. A. Rahman, Head, Department of 
Physics, Gauhati University. The conference received a large number of contributions 
on various topical areas of physics. Altogether, 113 research contributions had been 
accepted for presentation of which 76 papers were finally presented in the form of 
poster and oral presentation. Total 175 delegates from the North East India as well 
from other parts of the country participated in it. It was a wonderful experience for 
the teachers and students community of the university to be with such a distinguished 
gathering of physicists and enthusiastic young research scholars. 
Keeping the tradition of earlier conferences of the series, Professors J. Hatibarua, 
B.K. Barua and T.D. Goswami, the three distinguished and senior physicists of North 
East India were felicitated in the conference for their life long contribution to research 
and teaching in physics. 
The history of the PANE spans little less than a decade. It was inaugurated formally 
by renowned Physicist Prof. G C Deka on April 6,1998 in the premises of IIT, Guwahati 
with the aim of promoting basic research and physics education in North East India, 
besides Providing a platform of interaction among different groups of physicists 
working in different corners of North East India. The genesis of the academy can be 
traced to a seminar celebrating the hundred years of the discovery of electron held 
at IIT, Guwahati on November 11,1997 when a small group of enthusiastic physicists 
of North East India led by Prof. K M Pathak and Y V G Murti gave a call to start such 
an organization. 
©2008IACS 
664 Foreword 
With these objectives, PANE organized the first conference of physics research 
at NT, Guwahati during 1998, to be followed by the similar ones in Cotton College 
(2000), Dibrugarh University (2002) and G.C. College, Silchar (2004). The keynote 
speakers of these earlier conferences include persons like Prof. J.V. Narliker, Prof 
Asoke Sen (FRS), Prof. S. Duttagupta and Prof. K. D. Krori. 
On physics education sector, the academy arranged a workshop on new models 
of Post Graduate Programmes of physics at Tezpur University during 1999 followed 
by another on a theme "Physics Education in Undergraduate and Postgraduate levels 
now requires some Need Based Orientation" in the year 2005 jointly held at Gauhati 
and Dibrugarh University. 
The Academy has also been awarding talented young research scholars for their 
best presentations in the conferences since 2004. They include Ms. R. Gogoi, Mr. 
S. Sengupta, Mr. P. Nath of Gauhati University and Mr. J. Choudhury of Assam 
Universiy, Silchar. 
We are thankful to the members of the advisory and organizing committees for 
their valuable suggestions for planning various aspects of the conference. We are 
also thankful to all the students and non-teaching members of this department for 
their significant contributions in organizing the conference. We acknowledge with 
gratitude the financial assistance of Gauhati University; DST, New Delhi; CSIR, New 
Delhi; BRNS, Mumbai; ASTEC, Guwahati; UCIL, Shillong; Bipin Pal Das Trust and 
Amiya Bala Bhattacharjee Trust. We express our heartfelt thanks to all the concerned 
persons at the respective funding agencies. We express our deep gratitude to the 
Vice Chancellor of Gauhati University for providing all possible help and constant 
guidance in holding the conference here at Gauhati University. 
Dilip Kumar Choudhury Buddhadeb Bhattacharjee 
Executive President, PANE Convener, LOC, 
Vth PANE Conference, 
Department of Physics, 
Gauhati University, 
Guwahati-781014, 
Assam, India 
and 
Guest Editor 


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Mixing Dynamics and Microscopic Correlation of Associated Liquids

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