CIUPTER I
INTRODUCTION
livery solution of the equations of hydrodynamics or of hydromagnotics may not be r e a l is a b l e in p r a c t ic e . This is due to the fact that the flow may bo unstable to
in f in it esiraal or f i n i t e amplitude d istu rb an ces. Lin ear s t a b i l i t y theory deals with the question of s t a b ility to in fin it e s im a l disturbances, while the question of
s t a b i l i t y to f i n i t e amplitude disturbances f a l l s in the domain o f nonlinear s t a b ilit y theory.
The hydrodynamic or hydromagnetic system, whose s t a b i l i t y is being stu d ied , can be described by a set of non-dimensional parameters. The s t a b i l i t y of the system
depends on the numerical v alu es of these parameters. The parameter space can be d ivided into stable and unstable zo n e s. In the stable zone, any a rh itra ry disturbance to th e system must tend asym ptotically to zero as time tends to i n f i n i t y . In the unstable zone there must be at le a st one p ar tic u la r mode of disturbance which grows with tim e.
The locus of p o in ts which separates the s t a b le zone( s) from the unstable zo n e (s ) d efines the states of marginal
s t a b i l i t y of the system. States of m arginal s t a b il it y ( or marginal states) can be of two t y p e s . In the f i r s t ty p e , t r a n s it io n from s t a b i l i t y to i n s t a b i l i t y takes
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pla~c v ia a marginal so.ite exhibiting a stationary pattern of motions. In th is case the marginal state is said to be stationary and th e p rin c ip le o f exchange of s t a b i l i t i e s is said to be v a l i d . In the second type, the tr a n s itio n from s t a b i l i t y to in s t a b ilit y takes place v ia a marginal state exh ibiting o s c illa to r y motion with a certain d e fin it e c h aracteristic frequency. In t h is c a se , re ferr e d to as o v erstability (Chandrasekhar [l]) , a
slight displacement provokes restoring forccs so strong as to overshoot the corresponding p o sitio n on the other side of eq uilibrium . However, th is c la s s if i c a t i o n o f the
marginal states is v a l i d for d is s ip a t iv e systems.
The to tal volume of s c ie n t ific lit e r a t u r e d e a lin g w ith lin e a r hydrodynamic and hydromagnetic s t a b il it y is con. id e r a b le . Hence the; discussion which follows is re s tr ic t e d to only those papers which have direct relev a n ce to th e problems investigated in th is t h e s i s .
(a) CENTRIFUGAL INSTABILITY
The s t a b il it y of c y lin d r ic a l Couettc flow of an incom pressible in v is c id f lu id was fir s t inv estig ated by R a y le ig h [2] who showed that a s t r a t i f ic a t i o n o f angular
momentum about an axis is stable i f and only i f it increases monotQnically outward. This necessary and
s u ff ic ie n t condition fo r s t a b i l i t y is known as R a y l e i g h 's
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c r it e r i o n . The viscous s t a b ility of two-dimensional stagnat ion-point flow was investigated by Gortlcr [3] and HSmmerlin [4] who showed that the flow is u n sta b le.
This result may be related to R a y le ig h 's problem because in t h is case the c ircu la tio n (product o f local v e l o c it y and the local radius of curvature) increases as the lo ca l centre of curvature is approached normal to the curved stream lines.
(b ) THERMAL B IS T A B ILIT Y
The problem of the onset of thermal in s t a b ilit y (r e fe r r e d to as Benard convection) in a h orizontal la y e r of f l u i d , heated from below in a fie ld of gravity was examined th e o re tic a lly by Rayleigh [ 5 ] . His analysis was confined to the case of two free boun d aries. He showed that the sta b ility of trie lay er i s determined by the num erical value o f a non-dimensional parameter, re fe r r e d to as Rayleigh number, which depends on the temperature g ra d ie n t, the depth of the la y er, the a c cele ra tio n due to gra v ity and on the c o e ffic ie n t of thermal expansion, the thermal d i f f u s i v i t y and the kinem atic v isco sity of the f l u i d . If the R a y leig h number is below a c e rta in v a l u e , the layer is s t a b le . In s t a b il it y sets in as soon as th e Rayleigh number exceeds th is c r i t i c a l v a l u e . He a lso proved that the m arginal state i s st a tio n a ry , P ellew and Southwell [6] extended R a y l e i g h 's a n aly sis to in c lu d e
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other types of boundary conditions and proved that even in these cases the p r in c ip le of exchange of s t a b i l i t i e s is v a l i d . They also obtained a v ar ia tio n a l p rin c ip le for th is problem and used it for approximate calcu la tio n of the c r i t i c a l Rayleigh number. Chandrasekhar [7] and Chandrasekhar and Elbert [8] extended R a y le ig h 's problem to include the effect of r ig id ro ta tio n . They obtained a v a r ia tio n a l p r in c ip le for this problem .and calculated approximate v alu es of the c r it i c a l Rayleigh number as a function of the Taylor number ( a non-dimensional parameter which gives a measure of the rotatio n r a t e ). They showed that in t h is case the marginal state may be o sc illa to ry
p a r t ic u la r ly for f l u i d s w ith low Prandtl number.
The problem of thermp.l in s t a b ilit y in spheres and in spherical s h e ll s , heated w ith in , was examined by
Chandrasekhar [ 9 , 1 0 ] . Bisshopp [ l l ] analysed the problem fo r spheres, heated w ithin and subject to r i g i d ro tatio n about a diameter as the a x i s . His a n a l y s is , however, was con fined to axisymmctric d istu rb an ces. The a n aly sis for non-axisymmetric disturbances was ca rr ie d out by
Roberts [12] who showed th?„t except for th e smallest Taylor numbers, i n s t a b il i t y f i r s t sets in as a non- axisymmetric d istu rb an ce.
Horton and Rogers [1 3] and iapwood [14] considered the problem of thermal i n s t a b i l i t y in a h o r izo n ta l la y e r
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o f i l.uid in a porous medium, duo to a uniform adverse temperature gradient-. The problem of thermal in s t a b il it y in flu id - fille d porous spheres heated -within was examined by Pradhan and Patra [ 1 5 ] .
H u rle, Jakeman and P ik e [l6] made the analysis of P ello w and Southwell [6] more r e a l is t ic by co nsidering,
fo r the case of a f l u i d layer bounded by two r ig id su rfa ce s, that the media bounding the flu id have a f i n i t e thermal
co n du c tiv ity , so that the bounding surfaces are no longer at constant temperaturs as had been assumed by P ellow and Southwell [ 6 ] , In fact the assumption of constant temper
atures at the bo undaries is v a lid when t h e ir co n d u c tiv itie s are very la r g e .
(c) THERMQCONVECTIVE WAVES
Transverse thermal waves in an isothermal viscous f l u id are known to b e strongLy damped even in the absence of gra v ity forces (Carslaw and Jaeger [17 ]) . It was f i r s t shown by Luikov and Berkovsky [18] that in a viscous
l i q u i d heated from below , such that Benard convection does not set in , tran sverse plane waves may propagate almost undamped under certa in c o n d itio n s . These waves a re known as thermoconvective waves. Subsequently, undamped propagation of thermoconvective waves was shown to be p o s s ib le in the presence of a magnetic f i e l d by
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Takashima [19] only und^r astrophysical co nditions and in certain v isc o ela stic f l u i d s by Gupta and Gupta [ 2 0 ] .
Keck \nd Schneider [21J studied the cffec t of com pressibi
l i t y on the propagation of thermoconvcctivo -waves. Moreover, they considered a h o rizo ntally bounded la y e r and showed
that the horizontal w alls do not prevent the existence of weakly damped thermoconvcctivo waves provided that the R ayleigh number is l a r g e .
(d) HYDROHA.GNETK STABILITY OF NON-DISSIPATIVE FLOWS
In a th e o re tic a l study o f the in v is c id s t a b ilit y of p a r a lle l shear flow of a d e n sity - stra tifie d f l u i d ,
M iles [22] and Howard [23] derived a s u ffic ie n t co n d itio n for s t a b ilit y of the flow in a gravity f i e l d . They showed that i f a certain non-dimensional param eter, known as the Richardson number based on the density-gradient and the
shear in the p a r a l l e l flow everywhere exceeds 1 / 4 , then the flow is stable to in fin it e s im a l d istu rb a n c es.
Howard [23] also derived a sem icircle theorem which
demonstrates that the complex wave v e l o c i t y of any u n stab le mode must l i e in a c e r ta in sem icircle in the complex p l a n e . Howard end Gupta [24] derived Richardson number type
c r it e r i a and c ir c l e theorems for a number of cases of non- dissipativ e hydrodynamic and hydromagnetic h e lic a l fl o w s . However, in most of their a n a l y s is , only axisymme- t r ic disturbances were c o n sid ered . Barston [25] gave a
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gen'.r'T.l method of constructing c irc lc theorems which
l o c a l i z e the complex eigenfroquencies a r is in g in the l in e a r s t a b i l i t y an aly sis of conservative steady flo w s. As
a p p lic a t io n s , he considered h elic al flow of an i n v i s c i d , incompressible flu id and rotating flow of an in v is c id , incom pressible, perfectly conducting f l u i d permeated by an axial magnetic f i e l d . In his a n aly sis axisymmetric as w ell as non-axisymmetric disturbances were considered.
U sin g B a r s t ^ n ’ s technique Ganguly and Gupta [26] obtained a number of re su lts for sta b ility of non- dissipative
hydro magnetic h e lic a l flows to a l l normal modes of d is t u r b a n c es. B a r s to n 's technique, however, y ie ld s a circ le theorem with the r a d iu s of the c i r c l e , in general,
dependent on the wav cn umber s .
The effect of a radia.1 magnetic f i e l d on the s t a b ilit y of viscous flow between two ro ta tin g e le c t r i
c a l l y non-conducting cy lin ders was studied by Antimirov and Kolyshkin [ 2 7 ] . They calcu lated the v alu es of c r i t i c a l Taylor number as a fu nction of wavenumber for d iffe re n t v a lu e s of the Hartmann number.
The s t a b i l i t y of combined r a d i a l and rotational flo w ( s p i r a l flow) of an in v is c id flu id was studied by
K azlehurst [ 2 8 ] . R e s t r ic t in g h is an aly sis to axisymmetric disturbances he found that the flow” is unstable or stab le according as it is d irected towards or away from the a x is
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of symmetry. Bahl [29] examined the s t a b il it y of v isc o u s flow between two rotatin g porous concentric c y lin d e r s . While studying the effects of suction on c r itic a l Taylor number, he fo und that injectio n at the outer cylinder
improves sta b ility but suction has an opposite e f fe c t . Chang and Sortory [30] examined the s t a b ilit y of th e flow of a viscous electrically- conducting f l u id between two concentric ro tatin g porous cylinders in the presence of an a x ia l magnetic fie ld both when the marginal state is
st at ionary or o sc i l l a t ory.
Gilman [?l] analysed the s t a b ilit y of a horizo ntal layer of a p erfec tly conducting in v isc id gas in the
p resence of a horizo ntal magnetic f i e l d . He showed that when the strength of the magnetic f i e l d and the f l u id density are functions of t h e v e r t ic a l coordinate, in s t a b i
l i t y sets in by a mechanism known as magnetic bouyancy i f the basic magnetic f i e l d strength decreases with height fa s t e r than the d en sity . Ho also examined the effect o f r i g i d rotation on the s t a b ilit y of the system. Aches^n and Gibbons L 3 2 j extended Gilm an's analysis to the case of a uniform ly rot ating gas in a cy lin drica l geometry. However, in both these in v e s tig a tio n s i n f i n i t e thermal conductivity of the flu id was assumed. In an exhaustive study A c hes011 [3 considered the s t a b i l i t y of a ro tatin g spherical body o f com pressible f l u i d permeated by a t o r o id a l magnetic f i e l d
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tak in g into account a l l d iff u s iv e phenomena (v isc o u s, thermal and ^h:nic) . Ko carried out lo c a l s t a b ility analys for 1 ow frequency disturbances. Gibbons [34] examined the v a l i d i t y of lo cal s t a b ilit y analysis fo r a p articular problem involving in s t a b il i t y due to magnetic buoyancy in a rotating plane layur of compressible f l u i d . However, he retained the low frequency approximation in his a n a l y s is .
( e) HYDRO MAGNETE STABILITY OP DISSIPATIVE PLOWS
Taylor [35] extended the analysis of Rayleigh [2] to include the effect of v is c o s it y and showed that
v is c o s i t y has a s t a b il iz in g influ en ce on the flow . He showed th at, in this case, the s t a b ilit y of the flow is determined by the numerical value of a non-dimensional parp.neter, referred to as Taylor number, which gives a measure of the extent to which R a y l e i g h 's c r ite rio n is v i o l a t e d . He ca lc u la ted , under certain sim p lify ing
assum ptions, the numerical value of the c r i t i c a l Taylor number at which in s t a b il it y fir s t sets i n . The effect of a uniform a x ia l magnetic f i e l d on the s t a b i l i t y of
c y lin d r ic a l Couettc flow of a v isco us electrically- conducting fl u id was studied by Chandrasekhar [3 6]. He found that the magnetic f i e l d tends to s t a b i l i z e the flo w . Chandrasekhar [3 6] r e s tr ic t e d his analysis to th e case of a weakly conducting flu id (lo w magnetic P ran dtl number app ro xim atio n ). Kurzweg [3 7] extended h is a n aly sis
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by considering flu id s of a rb itra ry e le c t r ic a l co n d u c tiv ity .
Theoretical as well as experimental work has been done on the sta b ility of hydrodynamic as well as hydro- magnetic flows in which the basic state is o s c illa to r y . The st.ability of viscous c y lin d r ic a l Couette flow , when the ro tatio n rate of the inner cylinder is modulated sin u s o i
d a ll y , was investigated experimentally by Donnelly [ 3 8 ] . E e showed that the onset of in s t a b il i t y can be in h ib ite d by modulation. The th eoretical analysis fo r th is problem was carried out by R ile y and Laurence [ 3 9 ] . The effect of an o sc illa to ry magnetic fie ld on the s t a b ilit y of p a r a l l e l flow was examined by Drazin [40 ]. He carried out d e ta ile d s t a b i l i t y analysis for an in v is c id , p e r fe c t ly conducting f l u i d . In s t a b i l i t y of a lin e a r pinch in an o s c illa tin g magnetic fie ld was studied by Tayler [ 4 l ] .
In the light of the above l i t e r a t u r e survey the fo lio wing p ro b1em s hav e b e on inv e st ig at ed .
In chapter II the l i n e a r s t a b ilit y of steady two- dim ensional boundary la y er flow of an incom pressible
v is c o u s flu id over a f la t deformable sheet is inv estig a ted when the sheet is strctched in it s own plane with an
outward v e lo c ity pro p ortio nal to t h e d ista n c e from a p oint on i t , the analysis b ein g confined to i n f i n it e s i m a l , non
p rop agatin g , Gbrtler type d istu rb a n c es. The flow is sh^wn
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to he stab le. The s ig n ific a n c e of th is r e s u lt in polyner processing app lica tio n s is pointed out.
In chapter I I I the manner of the onset of thermal in s t a b il it y in flu id - fille d porous spheres and spherical sh e lls heatcd fron w ithin is in v e s tig a te d . The fir s t part of t h i s chapter examines the effect of r i g i d rotation of the system on the onset of i n s t a b i l i t y . A general d is t u r bance is analysed into nodes in terms of spherical
harmonics and the c r ite r io n for the onset of thermal
in s t a b ilit y is determined for the v ario u s modes. From the numerical re su lts conclusions are drawn about the mode for which i n s t a b il i t y f ir s t sets i n . The c r i t i c a l Darcy-
Rayleigh. number is obtained as a fu n c tio n of the Darcy- Taylor number. It is also found that fo r the most
unstable mode, the marginal state is characterized by a very slow westward ro tation of the entire convection
p attern around the globe. The second part of this chapter examines the effect of a f i n i t e thermal co nductivity of th e core on th e onset of thermal i n s t a b i l i t y in a, fluid- f i l l e d porous sp h erical s h e l l . It is sh^wn that as the conductivity of the core decreases, the c r i t i c a l Darcy- R a y leig h number decreases and the most unstable rn^de
tends to sh ift towards a lower wavenumber associated w ith th e polar angle in a system of spherical co ord in ates. The problem of the onset of thermal i n s t a b il i t y in f l u i d - f i l l e d
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porous spheres and sphcricnl shells is of relevance in connection with problems of g e o p h y s i c a l nnd astro ph y sics!
in t e re s t, e . g . , convection in. the e a r t h 's mantle.
In chapter IV the propagation of transverse therr.nl waves in a layer of binary mixture in the presence of uniform t h e m a l and concentration gradient is stud ied. It is shown that when the thermal d i f f u s i v i t y K^ is equal to the m s s d i f f u s iv it y Kg, undamped waves do not e x ist, but when K,p > Kg, these waves nay exist provided the layer i s heated from below (such that thernohaline in s t a b il it y
does not occur) and th e concentration of the salute in the mixture decreases v e r t ic a l l y upwards. The analysis also r e v e a ls the in te restin g resu lt that when K^ < Kg,
undamped waves can propagate even i f the layer is heated from above provided the concentration increases v e r t i c a l l y upwards. It is further shown that as in a homogeneous
f l u i d , weakly damped loir frequency waves can propagate ond the regions in the parameter space whore such propa
gation is p o ssib le are d elineated . High frequency waves are shown to b e always strongly damped. This problem may have some relevance to oceanography e .g . th e s t r a t i f i c a tio n of salt in oceans might a ffect the propagation of therm oconvective waves in the presence of v a r ia t io n s in so lar h ea tin g .
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In the fir s t part of chaptcr V the lin e a r s t a b ilit y of the f l^w o f an i n e x p r e s s i b l e , in v is c id , p erfec tly conducting f l u id between two nnn-por^us coaxial c y lin d e rs having an a x ia l and an azimuthal component of v e lo c it y and permeated by a magnetic f i e l d having r a d i a l , azimuthal and a x ia l components is examined. It is shown that in such a flow kno-vjn as h e lic a l flow , the presence of a r a d ia l component of the magnetic fie l d imposes certain r e s t r i c tio n s on the basic state . It is further shown that for any basic state consistent w ith th e governing equations, the flow is stable even in the presence of non-axisymmetric d istu rb an ces. In the second part of th is .chapter, the l in e a r s t a b ilit y of the r a d ia l flow of an incom pressible,
i n v i s c i d , p e rfec tly conducting flu id between two c o a x ia l porous cylinders permeated by a r a d ia l magnetic f i e l d is examined. It is found that the r a tio of the local Alfven v e lo c it y and the flow v e lo c it y in the b a sic state must be
constant over the en tire flow f i e l d . It turns out that i f the Alfven v e lo c ity exceeds the flovr v e lo c it y , then the flow is stab le. The problem considered in the fir s t part of th is chapter has a bearing on l.JHD power generation and such a configuration occurs in magnetohydrodyn?„nically d riv en v o r t ic e s . R a d ia l magnetic f i e l d a ls o finds
a p p lic a tio n in the study of magnetic f l u x generation in the interio r of p lan ets and stars.
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In chn.ptcr V I the lin e a r s t a b ilit y of non- dissipative hydro magnetic flow of an inconpressible den sity - stra tifie d f l u id between two coaxial cylinders to non-axisy''.metric disturbances is stud ied . A su ffic ie n t condition for s t a b i
l i t y is derived fo r r i g i d l y rotating f l u id permeated by an azim uthal magnetic fie l d against a l l d istu rb a n c es. In the case of h e lic a l flow permeated by an a x i a l magnetic f i e l d , a bound on the growth ra te independent of wavenumbers is
obtained. ,
In chapter V I I the hydro magnetic s t a b il it y of a ro tatin g compressible flu id is stu d ied . In the fir s t part
of th is chapter the lin e a r s t a b i l i t y of a plane layer of a p erfe c tly conducting gas permeated by a v e r tic a lly
s t r a t i f ie d h o rizo ntal magnetic f i e l d is examined when the la y er rotates about a h orizontal a x is perpendicular to the magnetic f i e l d . Using I7K3J approxim ation, it is shown that under adiabatic conditions the layer would be unstable to perturbations whose wav el en gth-al ong the magnetic f i e l d is much larg e r than a tj'pical wavelength along a h o rizo n ta l d ire c tio n provided the d e s t a b iliz in g in flu en ce due to magnetic buoyancy overcomes the s t a b i l i z i n g influ en ce of a p o s it iv e entropy s t r a t i f i c a t i o n . In the second part of t h is chapter the s t a b i l i t y of a c y l in d r ic a l rotating mass of a p e r fe c tly conducting in v isc id gas in the presence of a to ro id al magnetic f i e l d is s t u d ie d . It i s shown that
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under i iabatic conditions, unstable low irequency waves propagate along the basic ro tatio n outside a c r it ic a l r a d iu s depending on the basic acoustic speed. Inside the c r it i c a l ra d iu s , however, unstable waves propagate against the basic ro t a tio n . The hydromagnetic s t a b ility of a rotatin g compre
ssible f l u id is of importance in several geophysical and astrophysical problems concerning planetary and stellar atmospheres.
In chapter V I I I the s t a b il it y o f c y lin d r ic a l Couette flow of an e le c t r ic a l ly conducting incompressible viscous flu id in the presence of a modulated a x ia l magnetic fie ld is examined. With the usual sim plify ing assumptions o f low magnetic Prandtl number and narrow gap approxim ation, it i s shown that for low frequencies and for low v alu es o f th e Chandre ekhar number, incr- ise in the am plitude of modula
tio n tends to s t a b iliz e the system.
REFERENCES
[1] S. Chandrasekhar, Hydrodynamic and hydromagnetic s t a b i l i t y , Clarendon P r e s s , O xfo rd, 1 9 6 1 .
[2] Lord R a y le ig h , P ro c. Roy. Soc. (London) A 93.
1 4 8 , ( 1 9 1 6 ) .
[3] H . G'ortler, 50 Jahre G re nzschich tf orschung, F r ie d r . View eg and Sohn, Braunschweig, p . 304 ( 1 9 5 5 ) •
[4] G. Hammerlin, 50 Jahre G-r enz sc hie h tf orschung, F r i e d r . View eg and Sohn, Braunschweig, p . 315 ( 1 9 5 5 ) .
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[5] Lord R ayleigh , P h i l . Mag. (6) 2 2 , 529 ( 1 9 1 6 ) . [6] A Pellew and R . V . Southwell, P r o c . Roy. Soc.
(London) A 1 1 6 , 312 ( 1 9 4 0 ) .
[7] S. Chandrasekhar, Proc. Roy. Soc. (London) A .212.
306 (1953) .
[8] S. Chandrasekhar and D. D . E lbert, P ro c . Roy. Soc, (London) A 2 3 1 . 198 (1 9 5 5 ) .
[ 9 J S. Chandrasekhar, P h il Mag. (7) 42> 1317 ( 1 9 5 2 ) . [10] S. Chandrasekhar, P h i l . Mag. (7 ) J4> -253 (1953) J
P h i l . Mag (7 ) 44, 1129 (1 9 5 3 )*
[11] F . E. Bisshopp, P h i l . M a g .(8) 1 3 4 2 (1958) . [12] P . H . Roberts, P h i l . Trans. Roy. Soc. (London)
A 2 6 1 , 93 (1 9 6 8 ) .
[13] C . VJ. Horton and F . T. Rogers, J r . , J . Appl. P hys.
1 6 , 367 (1 94 5) .
[14] E. R . Lapwood, P ro c. Camb. p h i l . Soc. 4 4, 508 ( 1 9 4 8 ) . [15] G. K. Pradhan and B . P a tra , Indian J . pure appl. Math.
1 4, 661 (198 3) •
[16] D. T . J . K u rle , E . Jakeman and E . R . P ik e , Proc.
R oy. Soc. (London) A 2 9 6 , 469 ( 1 9 6 7 ) .
[17] H . S. Carslaw and J . C . Jaeg er, Conduction of heat in s o lid s , Oxford U n iv e r s ity P r e s s , O x fo rd, 1 9 4 8 . [18 ] A. V . Luikov and B . H. Berkovsky, I n t . J . Heat Mass
T r an sfer, 1 2 , 741 (1 9 7 0 ) .
[19] M. Takashima, J . P hy s. Soc. Jp n . 2 % , 1 7 1 2 ( 1 9 7 2 ) .
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[ 20 3 P . S. Gupta and A. o. Gupta, Japanese J . Appl. P hy s.
1 2 , 1831 (1973) .
[21] H. Keck and Schneider, In t. J . Heat Mass Transfer 2 2 , 1501 (1979) .
[22] J . 7. M iles, J . F lu id Mcch. 1 0 , 496 ( 1 9 6 1 ) . [23] L . N . Howard, J . 'F l u i d Mech. IQ , 509 (1961) . [24] L . N . Howard and A. 3. Gupta, J . F lu id Mech. 1 4 .
463 (1962) .
[25] E. M. Barston, In t . J . Engng. S c i. 1 8 , 477 ( 1 9 8 0 ) . [26] K. Ganguly and A. S. Gupta, J . Math. A n a l. Appl.
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[ 27 J M. Y a . Antimirov and A. A. Kolyshkin,
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Copyright
IIT Kharagpur
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