On the behaviour of hot fluids in the presence of magnetic field
Mihir B. Banerjee1, J. R. Gupta1, R. G. Shandil1,
Joginder S. Dhiman2 and Vinay Kanwar3
1Former Professor,
Department of Mathematics, Himachal Pradesh University, Shimla-171005.
2Department of
Mathematics, Himachal Pradesh University, Summerhill,
Shimla-171005.
3Department of
Mathematics, University Institute of Engineering and Technology, PU,
Chandigarh.
*Corresponding Author
E-mail:
In his masterly contributions to the theory
of magnetohydrodynamic thermal convection (or magnetoconvection)
Chandrasekhar (1952, 1961) laid down the guiding principles of the subject
matter of the following conjectural assertions of which the first two were
obtained by making use of the method of Pellew and Southwell (1940) while the rest were speculated through an
extremely simple solution of the governing equations that fails to satisfy any
plausible set of boundary conditions on the magnetic field (except in the case
of the (vi) for which the latter part of the remark does not apply) and pertains,
according to him, to the case of dynamically free boundaries which occurs only
in real physical situations. These assertions are;
(i)
The method of Pellew
and Southwell (1940) is not quite strong enough to
establish the validity or otherwise of the principle of exchange of stabilities
for the problem.
(ii)
There exists a parameter regime for the
problem defined by the Chandrasekhar number
(iii)
A sufficient condition for the validity of
the principle of exchange of stabilities for the problem is that the magnetic Prandtl number
(iv)
There exists a parameter regime for the
problem defined by the Chandrasekhar number
(v)
There exists a parameter regime for the
problem defined by the Chandrasekhar number
(vi)
If the principle of exchange of stabilities
is valid for the problem then the asymptotic behavior of the critical Rayleigh
number and the associated wave number for
and it
appears that the same power laws with the same constants of proportionality
hold for situations with more general boundaries such as any combination of a
dynamically free and a rigid boundary.
(vii)
If overstability is valid for the problem then the asymptotic behaviours of the critical Rayleigh number and the
associated wave number and frequency
The sufficient condition for the validity of
the principle of exchange of stabilities for the problems as given in (iii) is
known as the Thompson-Chandrasekhar criterion in the literature on magnetohydrodynamic thermal convection (Thompson (1951)
obtained it first in the context of inviscid fluids).
Subsequent contributions to the problem were
made primarily to strengthen the fundamental steps enunciated by Chandrasekhar
and widen their domain of applicability especially with regard to the
consideration of more general boundaries in the theoretical formulation.
Gibson (1966) emphasized that since no such theorem
as that established by Pellew and Southwell
for the hydrodynamic case exists in the hydromagnetic
case the possibility of overstability must be
considered. He further argued that the boundary conditions assumed by
Chandrasekhar in his investigations on (ii) and (vii) that pertain, according
to him, to the case of dynamically free boundaries are not correct since they
do not include the boundary conditions on the magnetic field and himself
investigated the problem wherein the thermally perfectly conducting boundaries
could be dynamically free or rigid that might be electrically perfectly
conducting or insulating. Gibson concluded from his analysis that while the
validity of asymptotic forms of
Sherman and Ostrach
(1966) examined the validity of the principle of exchange of stabilities for the
problem wherein the fluid is confined within an arbitrary enclosed region with
thermally perfectly conducting or insulating rigid boundary surfaces that are
electrically perfectly conducting while the uniform magnetic field is applied
in an arbitrary direction and contributed mainly through a restatement of (ii)
and the establishment of the truth of Thompson-Chandrasekhar criterion as given
in (iii) in the limit when
More recently Kumar et al. (1986) have
investigated the problem wherein the thermally perfectly conducting boundaries
are rigid and electrically perfectly conducting and making use of a slight
variant of the well known Fourier-Chandrasekhar technique concluded that overstable motions exist in the limit when
This was essentially the state of knowledge
in the field of linear magnetoconvection that was
largely dominated either by oversimplified mathematical calculations which did
not conform to reality or by mathematical conjectures that were arrived at
through the use of non-rigorous mathematical analysis and left unproven and
uninvestigated when we took up the challenges of the problem at Simla in the early eighties. The need to derive a sufficient condition for the validity of
the ‘exchange principle’ for more general boundary condition on the velocity
and magnetic field than considered by Chandrasekhar, in the parameter space of
the system alone, which would reduce to Pellew and Southwell’s result in the hydrodynamic case and correlate
it to Chandrasekhar’s conjecture concerning the two energies was indeed
pressing after Chandrasekhar’s investigations especially since Nature as well
as Laboratory were unmistakably pointing towards the existence of a sufficient
condition for the validity of ‘exchange principle’ that would be satisfied unless the conditions
prevailing were extraordinary. From the very beginning we did not see anything
wrong with the method of Pellew and Southwell which leads to equation
as a necessary condition for the existence of
the overstability and apparently blocks the way for
further progress towards any conclusive result while on the contrary we viewed
the seemingly unfavourable sign attached with the
magnetic field terms as a natural indicator of the possible fact that a
sufficient condition for the validity of the ‘exchange principle’ is going to
crucially depend upon the magnitude of
Derivation
of Governing Equations and Boundary Conditions of the Magnetohydrodynamic
Simple Bénard Instability Problem
Consider a liquid that has the property of
electrical conduction and suppose that magnetic fields are prevalent. The basis
equations that express the interaction between the liquid motions and the
magnetic fields are, of course, contained in the equations of Maxwell and
appropriately modified hydrodynamical equations, and
for our considerations here we shall present them under a fundamental
simplification which is possible namely the simplification arising out of the
neglect of displacement current in the Maxwell’s equations whose justification
is derived from the fact that we are not concerned herein with effects that are
related, in any way, to the propagation of electromagnetic waves. A closely
related further possibility of ignoring any direct reference to the electrical
charge density is offered not because it is small in itself but because its
variations affect the equation expressing the conservation of charge only by
terms of order (c being the velocity of light) which is of this order we can
ignore with justification.
Making use of above simplifications the basic
magnetohydrodynamical equations that govern the magnetohydrodynamic simple Bénard
problem can be written in Cartesian coordinates as (cf. Chandrasekhar (1961))
The equation of continuity
The equation of motion
where,
The equation of heat conduction
The equation of state
The equation of magnetic induction
where
The solenoidal
character of the magnetic field
It may be noted that in writing down the
above equations we have defined only those symbols that occur for the first
time while those symbols that have occurred earlier in the text shall retain
their same meanings and this procedure shall be generally followed in
subsequent discussions as well.
Applying the usual Boussinesq
approximation (1903), which in essence
amounts to neglecting terms which are of order 10-3 at most as
compared to 1 for variations in temperature of order 10 (say), we obtain, respectively, from
equations (0.1)-(0.6) the following equations namely;
and
where,
Let us consider a viscous finitely heat and
electrically conducting Boussinesq liquid of infinite
horizontal extension and finite vertical depth statically confined between two
horizontal boundaries
The initial stationary state solution of the
governing equations (0.1)-(0.6)
with
where,
Let the initial state described by equation
(A214) be slightly perturbed so that the perturbed state is given by
and
where,
Then, the linearized
perturbation equations of continuity, motion, heat conduction, magnetic
induction, solenoidal character of the magnetic field
and state are given by
and
We now analyze an arbitrary perturbation into
a complete set of normal modes and then examine the stability of each of these
modes individually. For the system of equations (0.17)-(0.26) the analysis can be made in terms of
two dimensional periodic waves of assigned wave numbers. Thus, we ascribe to
all quantities describing the perturbation a dependence on
and making use of (0.27) in
equations (0.17)-(0.26), we obtain
where,
Eliminating
We obtain, upon
dropping the double dashes and the asterisks for simplicity in writing, the
governing magnetohydrodynamical equations in their
non-dimensional forms as follows:
We now seek solutions of equations (0.38)-(0.40) which satisfy the
appropriate boundary conditions on
(i)
In the first case when the medium adjoining
the liquid is a perfect conductor, it follows that no magnetic field can cross
the boundary and we must require that
Resolving into normal modes, we have from
equation (0.41) that
and non dimensionalizing equation (0.42) with the help of the
transformations (0.37) and dropping the asterisk for convenience in writing, we
obtain
We thus have the boundary conditions;
(ii)
In the second case when the medium adjoining
the liquid is electrically nonconducting, the
equation governing the magnetic field vector
where,
Resolving into normal modes, we have from
equation (0.45) that
and nondimensionalizing
equation (0.46) with the help of the transformations (0.37) and dropping the
asterisk for convenience in writing, we obtain
where,
The general solution of equation (0.47) is
given by
and since we require
that there are no sources of perturbations at ‘infinity’, we have from equation
(0.48) that
and
where
Now, continuity of the magnetic field vector
is continuous on a plane boundary
adjoining an electrically finitely conducting medium.
Nondimensionalizing equation (0.51) with
the help of the transformations (0.37) and dropping the asterisk for
convenience in writing we obtain
is continous
on a plane boundary adjoining an electrically finitely conduction medium. Therefore,
and
since
In the particular case when the medium
adjoining the liquid is electrically nonconducting,
the boundary conditions given by equations (0.53) reduce to
Remark
1: If we take
1.
Review of the Simple
Bénard Instability Problem
Consider a double eigenvalue problem for p described
by a system of ordinary linear and homogeneous differential equations with
linear and homogeneous boundary conditions, namely (see Remark 1 and
Chandrasekhar (1961))
and
with
where z is the real independent variable such
that
Operating on both sides of equation (1.1) by
the operator
Further, since equations (1.1) and (1.2) are
valid everywhere in
Equation (1.4) is an ordinary linear and
homogeneous differential equation of order six with constant coefficients and
consequently its general solution can be written as
where, ci’s
are arbitrary constants that are to be determined by the appropriate boundary
conditions on w and wi’s constitute a
fundamental set of solutions of equation (1.4).
Applying the six linear and homogeneous
boundary conditions given by equations (1.5) on the above solution for w we
obtain six algebraic linear and homogeneous equations for the determination of
the six unknowns ci’s and since we are
interested in non-trivial solutions of equation (1.4) that satisfy the boundary
conditions specified by equations (1.5), we must have the coefficient
determinant of the above system of algebraic equations equation to zero. This leads to the characteristic equation of
the problem which can be written in the form
where f is an appropriate
complex valued function in general (since p is complex in general) of the
arguments as shown and therefore equation (1.7), in general, has a real part
and an imaginary part which can be written as
and
where fr
and fi are real and imaginary parts of
f. Now, if
and
which satisfy equations
(1.8) and (1.9).
Since two quantities are to be determined
instead of a single one for non-trivial solutions of appropriate differential
equations and boundary conditions, we call it a double eigenvalue
problem. One could have thought that the
problem should still be called an eigenvalue problem
and not a double eigenvalue problem since pr
and pi are the real and imaginary parts of a single quantity p which
is to be determined but a little consideration would yield that it cannot be
done so uniformly because in the domain of applications of such system of
equations and boundary conditions we have occasions when the problem is
regarded as an eigenvalue problem for (say) R and pi
for prescribed values of
A few remarks at this point that concern the
general mathematical nature of the problem will be helpful for the later
development of the arguments. One could
obtain from equations (1.1), (1.2) and (1.3) the following single equation
(that is similar to equation (1.4) but with
with
and it is clear, of
course, that the eigenvalues for p, as functions of
and
with
and thus showing that by
a different scaling a non-self-adjoint differential
system may be transformed into a self-adjoint
one. If now we obtain from equations
(1.1a), (1.2a) and (1.3a) a single equation and appropriate boundary conditions
for W or
A complete solution of the eigenvalue problem, which effectively is self-adjoint, consists in the determination of the countable
infinity of eigenvalues
A precise calculation of the eigenvalues of linear and homogeneous differential systems
or the development of meaningful characterizations of the eigenvalues
of such systems in the absence of a precise calculation, are mathematical
problems in their own right and thus from the point of view of a mathematician
there is obviously a problem contained in equations (1.1), (1.2) and
(1.3). Several important mathematical
methods have been constructed in the recent past and the present, for example
the Rayleigh- Ritz method, the Galerkin method, the
Chandrasekhar method, the method of weighted residuals (Finlayson (1972)) and
etc. to investigate such problems and a challenging problem in this context is
a welcome since it always has the potential to become the starting point of a
new mathematical method which could possibly be exploited in other allied
domains of enquiry. From the point of
view of a physicist, however, equations (1.1), (1.2) and (1.3) have a special
charm in that contained in these equations are motions in the form of cellular
structures that manifest in a class of hydrodynamical
systems which had defied a theoretical explanation for a long time. These equations were, in fact, written by
Lord Rayleigh (1916), a Noble Prize winning physicist, in a pioneering
contribution to the Philosophical Magazine when he was trying to understand and
theoretically explain some of the experimental observations of the French
experimentalist Henery Bénard
(1900). Bénard
was experimenting in a definitive manner to demonstrate and quantitatively
characterize the onset of first thermal motions in liquids, and for a appreciation of his findings We pose and analyze the
consequences of the following thought experiment. Consider a horizontal layer of liquid in
which an adverse temperature gradient is maintained by heating it from below. We qualify the temperature gradient thus
maintained as adverse since on account of thermal expansion the liquid at the
bottom will be lighter than the liquid at the top and this will give rise to a
top heavy arrangement which is potentially unstable. On account of this latter instability there
will be a natural tendency on the part of the liquid to redistribute it-self
and remedy the weakness in its arrangement.
This tendency of the liquid, however, will be inhibited by its own
viscosity and as a consequence, we expect that the temperature gradient which
is maintained will have to exceed a certain value before the first thermal motions
can manifest themselves. One of the findings of Bénard
who subjected horizontal liquid layers bounded below by metallic plate with the
upper surface left free to be in contact with the ambient air to uniform
vertical adverse temperature gradients is precisely the above but in addition
to it he discovered another fact that is as fundamental as his above finding
and cannot be easily derived by making use of a common sense analysis of the
type employed above, and more relevantly in the present context it is this
latter finding that has a dominant bearing on the main theme of the present article. It will suffice here to summarize the
principal findings of Bénard and other contemporary
experimentalists and these are, first, a certain critical adverse temperature
gradient must be exceeded before the onset of first thermal motions, and
second, the motions that ensure on surpassing the critical temperature gradient
have a stationary cellular character.
What actually happens at the onset of motions is that the layer becomes
reticulated and reveals its dissection into cells and if the experiment is
performed with sufficient care the cells become equal, hexagonal and properly
aligned.
Rayleigh described these Bénard
motions as the motions that would manifest on account of the hydrodynamical instability suffered by the initial
stationary state in which a static layer of liquid of infinite horizontal
extension and finite vertical depth is under the action of a uniform adverse
vertical temperature gradient so that the temperature decreases linearly with
height while the density increases linearly with height with an appropriate
supporting hydrostatic pressure in the force field of gravity. It can be easily checked that the above
purely conducting state of the liquid constitutes an exact solution of the
equations that express the conservation of mass, momentum and energy (with
constant thermal conductivity) and an appropriate equation of state together
with the required boundary conditions in a relevant framework. To analyze the hydrodynamic instability of
the above stationary state of the liquid Rayleigh added infinitesimally small
perturbations to it which are inevitably present in any experiment, argued that
such a disturbed state should also be a solution of the same governing
equations of which the initial stationary state is a solution which is
justified, simplified the equations governing the perturbations on the basis of
the Boussinesq approximation which is relevant to the
context, obtained the linearized equations governing
the perturbations which is also justified since one is interested in the
characterization of the onset of first thermal motions only and not beyond it,
analyzed the perturbations with the help of the usual normal mode analysis
which for the problem under consideration is in terms of two dimensional
periodic waves of assigned wave numbers and is in accordance with the Fourier
series methods which is justified in the present context, and with ent as the t-dependence of the perturbations,
where
Multiplying both sides of equation (1.1) by
Taking the complex conjugate of both sides of
equation (1.2), we have
Multiplying both sides of equation (1.13) by
Substituting for
Evaluating the definite integrals
and
It is important to note here that equations
(1.16), (1.17) and (1.18) hold good whether the boundaries are both dynamically
free or both rigid or any one of them dynamically free and the other rigid as
can be easily seen.
Making use of equations (1.16), (1.17) and
(1.18) we can write equation (1.5) as
Equating the imaginary part of both sides of
equation (1.19) and rearranging the resulting equation, we obtain
Equation (1.20) implies that
The linear theory of hydrodynamic instability
of a liquid layer heated underside as propounded by Rayleigh, together with
reality and completion brought to it by Pellew and Southwell, has served the purpose of a foundation on which
scores of experimental and theoretical investigations stand, and it would be
fairly accurate to state that this line of thought is complete in almost all
its aspects notwithstanding its limitations.
At this point we leave the considerations of
the simple Bénard instability problem and the
associated Pellew and Southwell’s
‘method of proof’ to join with it again a little later in the context of a more
general physical problem wherein the liquid has the property of electrical
conduction and magnetic fields are prevalent.
2. Magnetohydrodynamics
and magnetohydrodynamic simple Bénard
instability problem
Hans Alfvén (1942) a
Swedish physicist had made a discovery in the dynamics of electrically
conducting liquids pervaded by magnetic fields.
The motion of electrically conducting liquids across the magnetic lines
of force generates electric currents and the associated magnetic fields modify
the original magnetic fields and thus the motions contribute to the changes in
the original magnetic fields. On the
other hand, additional mechanical forces are generated which act on the current
carrying liquid elements when they traverse magnetic lines of force and thus
the magnetic fields contribute to the changes in the liquid motions. This twofold interaction between the motions
and the fields which are responsible for pattern of behavior that are often
unexpected and striking constitute essentially the subject matter of magneto
hydrodynamics. With a masterly physical
insight Alfvén chose Faraday’s method of regarding
magnetic action as represented by the magnetic lines of force; realized that
conducting liquids of large dimensions would behave as perfect conductors;
derived for such situations his famous theorem of frozen-in-fields which
implies that in a moving perfectly conducting liquid the magnetic lines of
force are frozen in the liquid; argued that the magnetic lines of force must
then behave as taut material strings possessing both inertia and tension with a
mass per unit length being equal to the density
This twofold magnetohydrodynamic
interaction was considered by Alfvén (1950) and Walen (1949) in discussing processes in the Sun, by Fermi
(1949), Richtmeyer and Teller (1949) in a theory of
the origin of cosmic rays, and by Elasser (1946) and
Bullard (1949) in an explanation of terrestrial magnetism. Bullard (1949) proposed a reasonably simple
scheme of thermal convection currents and the magnetic fields within the core
of the earth which might be able to maintain the observed field at the surface by
the self-inductive mechanism originally proposed by Larmor,
and showed that the magnetic field is of prime importance in determining the
flow.
As a first step toward investigating the hydrodynamical aspects of the process suggested by Bullard
a study was made by Thompson in 1951 in the Philosophical Magazine, of the slow
thermal convection currents set up in a plane layer of an electrically conducting
liquid placed in a homogeneous magnetic field and heated from below which in
effect amounts to a re-examination of the simple Bénard
instability problem for the case when the liquid considered is an electrical
conductor and an external magnetic field is impressed on the liquid. Thompson modified the analysis of Rayleigh
and Jeffreys to include the ponderomotive
effects of the magnetic field by adding to the hydrodynamical
and the heat flow equations Maxwell’s equations for the electromagnetic fields and
one of the fundamental questions to which he addressed himself to was whether
the ‘exchange principle’ was valid for this more general problem or
otherwise. His analysis which was unlike
that of Pelew and Southwell yielded the result that
if the uniform applied magnetic field is oppositely aligned with the direction
of the non-magnetic body force then a sufficient condition for the validity of
the ‘exchange principle’ is this that the thermal diffusivity
3.
The governing equations and Thompson’s condition for the
‘exchange principle’
The corresponding double eigenvalue
problem for p obtained as a result of
an imposition of a uniform magnetic field acting parallel to gravity on the
simple Bénard configuration of an electrically
conducting liquid are given by the following system of ordinary linear and
homogeneous differential equations with linear and homogeneous boundary
conditions. These are (cf. equations (0.38) - (0.40))
and
with
and
where
For a non-viscous liquid the above equations
and boundary conditions become
and
with
and
where
Operating on both sides of equation (3.9) by
the operator
Assuming a particular simple solution for w
in the form
which satisfies the
boundary conditions prescribed for w,
and substituting this solution for w
in equation (3.15), we obtain the characteristic equation
Assuming that overstability
is valid so that
Separating the real and imaginary parts of
equation (3.18), we obtain
and
Rewriting equations (3.19) and (3.20) in the
forms
and
and substituting for
Equation (3.23) enables us to draw at once
one conclusion and that is that the solutions describing overstability
cannot occur if
Because in that case
Thus, a sufficient condition for the
validity of the ‘exchange principle’ is that the thermal diffusivity
4.
Extension to Viscous Case and Chandrasekhar’s First
Method
Almost during the time that Thompson’s paper
appeared, the same physical configuration was also being investigated by one of
the great men of science of our times, Chandrasekhar, a Nobel Prize winner in
physics, who happened to look at the freshly arrived December 1951 issue of the
Philosophical Magazine containing Thompson’s paper while he himself was
completing his own paper for the press.
A comparison between the papers of Thompson and Chandrasekhar shows that
both of them were seeking answers on two fundamental points of enquiry, namely
(i) does the ‘exchange principle’ hold in this more
general situation also? And (ii) does the magnetic field postpone the onset of
two papers go. On the other hand,
whereas Thompson was primarily motivated from the point of view of terrestrial
applications as a consequence of which he was tempted to ignore the viscous
terms from the governing equations for the benefit of achieving mathematical
simplification while analyzing point (i),
Chandrasekhar in addition to terrestrial applications had astrophysical
applications also in mind on account of which he retained the viscous terms
throughout the analysis as it was known then that viscosity does at times play
a rather important role in astrophysical problems.
Since the answer to point (ii) crucially
depends upon the answer to point (i), Chandrasekhar
analyzed point (i) in great detail in his fundamental
paper in the Philosophical Magazine in 1952, implementing every possible
investigation that he could bear upon it as his paper shown and devised in the
process two different methods of attack to the problem, and since this part of
his work is closely connected with the subsequent investigations presented in
this article we given below a brief description of it in all its essentials.
Chandrasekhar’s first method consists in
combining the equations (3.1), (3.2) and (3.3) in terms of the single dependent
variable w and then assuming a particularly simple solution for it for the case
when both the boundaries are dynamically free.
This leads him directly to the characteristic equation of the problem
and subsequently to a sufficient condition for the validity of the ‘exchange
principle’. The various steps of this
scheme of calculations are shown below.
Operating on both sides of equation (3.1) by
the operator
Assuming a particularly simple solution for w in the form
which satisfies the
prescribed boundary conditions for w,
for the case when both the boundaries are dynamically free, and substituting
this solution for w in equation
(1.4.1), we obtain the characteristic equation
Assuming that overstability
is valid so that
Separating the real and imaginary parts of
equation (1.4.4), we obtain
and
Rewriting equations (1.4.5) and (1.4.6) in
the forms
and
and substituting for
Equation (1.4.9) enables us to draw one
conclusion at once and that is that solutions describing overstability
cannot occur if
Because in that case
and this is precisely Thompson’s
condition as mentioned earlier.
However, the solution for w as given by
equation (4.2) is not correct mathematically (and Chandrasekhar is aware of it)
since it would then imply from equations (3.3) and (4.2) that
which in turn would imply
from equations (3.3) and (4.2) that
since
and
The general solution of equation (4.14) is
given by
where A and B are constants of
integration, and equation (4.15) then implies
Equations (4.19) and (4.20) yield that A = 0 = B while equations (4.17) and (4.18)
imply that A = 0 = B cannot be
allowed. In other words, equations
(4.14) and (4.15) do not admit of a solution for
5.
Chandrasekhar’s Second Method and His Conjecture
Chandrasekhar’s second method, which was
designed to remove the deficiencies of the analysis as given under the first
method, consists in extending the proof of the ‘exchange principle’ developed
by Pellew and Southwell to
include the effects of electrical conductivity of the liquid and a uniform
applied magnetic field in the vertical direction. The various steps of this scheme of
calculation are shown below.
Multiplying both sides of equation (3.1) by
The definite integral on the left hand side
and the first definite integral on the right hand side of equation (5.1) can be
expressed, with the help of appropriate boundary conditions in the forms
exactly as in the Pellew and Southwell’s
case. For the second definite integral
on the right hand side of equation (5.1), we have on integrating by parts once
and making use of the boundary conditions given by equations (3.4)
Taking complex conjugate of both sides of
equation (3.3), we get
Substituting for
Evaluating the definite integral
Making use of equations (1.16), (1.17), (1.18),
(5.2), (5.3), (5.4) and (5.5), we can write equation (5.1) as
Equating the imaginary part of both sides of
equation (5.6) and rearranging the resulting equation, we get
It is clear from equation (5.7) that one
cannot conclude from it, as in the Pellew and Southwell’s case wherein
6.
A Sufficient Condition for the ‘Exchange Principle’
The need to derive a sufficient condition for
the validity of the ‘exchange principle’ for more general boundary conditions
on the velocity and magnetic field, than considered by Chandrasekhar, in the
parameter space of the system alone which would reduce correlate it to the
Chandrasekhar’s conjecture concerning the two energies was indeed pressing
after Chandrasekhar’s investigations especially since the Nature as well as the
Laboratory were unmistakably pointing towards the existence of a sufficient
condition for the validity of the ‘exchange principle’ that would be satisfied
unless the conditions prevailing were extraordinary.
This was essentially the state of knowledge
in the field of linear magnetoconvection that was
largely dominated either by oversimplified mathematical calculations which did
not conform to reality or by mathematical conjectures that were arrived at
through the use of non-rigorous mathematical analysis and left unproven and
uninvestigated when we took up the challenges of the problem at Shimla in the early eighties. From the very beginning we did not see
anything wrong with the method of Pellew and Southwell which leads to equation (5.7) and apparently
blocks the way for further progress towards any conclusive result while on the
contrary we viewed the seemingly unfavourable sign
attached with the magnetic field terms as a natural indicator of the possible
fact that a sufficient condition for the validity of the ‘exchange principle’
is going to crucially depend upon the magnitude of
(a) Case of perfectly conducting boundaries (Banerjee et al 1984)
In this case, since both the horizontal
boundaries are perfectly conducting, boundary conditions given by equations (3.7)
hold good and not those given by equations (3.8). Thus the double eigenvalue
problem for p is given by the
following system of equations and boundary conditions:
and
with
Multiplying both sides of equation (6.4) by
-
Equating the imaginary part of both sides of
equation (6.5) and rearranging
Multiplying equation (6.3) by
Integrating the left hand side of equation (6.7)
by parts for a suitable number of times and making use of the boundary
conditions on
Equating the real part of both sides of
equation (6.8), we have
Further, Real part of
Assuming
Combining inequality (6.11) with the
Rayleigh-Ritz inequality (Schultz (1973)), namely
we derive
and further, combining
inequality (6.11) with inequality (6.13) we obtain
Making use of equation (6.6) and inequality (6.14),
we get
and therefore we must
have
or
and which cannot hold
good if
Thus, if
(b) Case of insulating boundaries (Banerjee et al 1985a)
In this case, since both the horizontal
boundaries are insulating, boundary conditions given by equations (3.8) hold
good and not those given by equations (3.7).
Thus, the double eigenvalue problem for p is given by the following system of
equations and boundary conditions:
and
with
and
Multiplying both sides of equation (6.19) by
Evaluating the definite integral on the left
had side and the first definite integral on the right hand side of equation (6.24)
as before, we get
and
For the second definite integral on the right
hand side of equation (6.24), we have integrating by parts once and making use
of the boundary conditions given by equations (6.22)
Taking complex conjugate of both sides of
equation (6.21), we get
and substituting for
Evaluating the definite integral
Making use of equations (6.25), (6.26), (6.27),
(6.29) and (6.30), we can write equation (6.24) as
Equating the imaginary part of both sides of
equation (6.31), rearranging the resulting equation and cancelling
Multiplying both sides of equation (6.21) by
Integrating by parts once the definite
integral on the right hand side of equation (6.33) and making use of the
boundary conditions given by equations (6.22), we get
Making use of equations (6.3) and (6.34),
equation (6.33) can be written as
Equating the real part of both sides of
equation (6.35), we obtain
Further, - Real part of
Assuming
and hence
Making use of the boundary conditions on w
given by equations (6.22), we have by the Rayleigh-Ritz inequality
Combining inequality (6.32) with inequalities
(6.39) and (6.40), we obtain
and therefore, we must
have
and which cannot hold
good if
Thus, if
Another important conclusion that can be
derived from the foregoing analysis is that the same sufficient condition is
valid also for the case when both the horizontal boundaries are perfectly
conducting, since in this case boundary conditions given by equations (3.7)
yield that
and thus the proof given
in case (b) is uniformly applicable to perfectly conducting as well as
insulating boundaries and improves upon the contents of inequality (6.18) as
shown by inequality (6.43).
However, surprisingly, Banerjee
et al did not see the connection between their own work and Chandrasekhar’s
conjecture concerning the two energies and thus the second question raised in
section 1.5 remained partially answered only.
This gap in the literature on magnetoconvection
has been recently completed by Banerjee et al (1985b)
who presented a simple mathematical proof to establish that Chandrasekhar’s
conjecture is valid in the regime
7.
Resolution of Chandrasekhar’s Conjecture Concerning the
Two Energies (Banerjee et al 1985b)
Consider the case when both the horizontal
boundaries are insulating so that the double eigenvalue
problem for p is given by the system of equations (6.19), (6.20) and (6.21) and
boundary conditions (6.22) and (6.23).
Multiplying both sides of equation (6.21) by
It follows from inequality (7.1) that
Making use of boundary conditions on w given
by equations (6.22), we have by Rayleigh-Ritz inequality
Combining inequalities (7.2) and (7.3), we obtain
and hence
Therefore, if
We thus have the result that the total
kinetic energy associated with a neutral or unstable perturbation is greater
than or equal to its total magnetic energy in the regime
In the case when both the horizontal
boundaries are perfectly conducting so that the boundary conditions given by
equations (3.7) hold good and not those given by equations (3.8), we have
and it then follows from
the above analysis that inequality (7.6) holds good in the regime
The foregoing result when considered in the
context of the result given in section (1.6) shows the link between
Chandrasekhar’s conjecture and the validity of the ‘exchange principle’ which
was missing in the literature, and settles the conjecture in the regime
Banerjee et al. (1989) and Dhiman (1995) further extended the results of Banerjee et al. (1985b) respectively for hydromagnetic thermohaline
convection and rotatory magneto thermohaline
convection problems of Veronis’ and Stern’s types.
8.
Solutions for the Case When ‘Exchange Principle’ is Valid
In his pioneering work on the initiation of magnetoconvection wherein the boundaries are dynamically
free and the ‘exchange principle’ is valid Chandrasekhar (1961) derived
analytically, consequences of importance regarding the inhibiting effect of the
magnetic field and the asymptotic dependence of the critical Rayleigh number
When both the boundaries are dynamically free
and the ‘exchange principle’ is valid the neutral state will be characterized
by
and
with
(both the boundaries dynamically free)
or
(both the boundaries rigid)
(if the regions outside the liquid are perfectly conducting )
and
(if the regions outside the liquid are insulating)
Eliminating
while eliminating
We must seek solution of equations (8.9)
which satisfy the boundary conditions
in deriving which we
have made use of equation (8.8) and the boundary conditions given by equations (8.4).
If follows from equation (8.9) and (8.10)
that
and then by
differentiating equation (8.9) with respect to z for an even number of times we can likewise successively conclude
that all even derivatives of w must
vanish
where A is a constant.
Substituting the above solution for w in equation (8.9), we obtain the
characteristic equation
which clearly shows the
inhibiting effect of the magnetic field on the onset of instability. As a function of
and with
and thus the asymptotic behaviours of
of which the first
result constitutes Chandrasekhar’s celebrated
The relevant governing equations and boundary
conditions for the situation under consideration are given by equations (8.1)-(8.10)
with
In that case, eliminating
Multiplying equation (8.16) by
Evaluating the various definite integrals by
making use of the remaining equation (8.2), namely
and boundary conditions (8.4)
– (8.7), we get
Multiplying equation (8.18) by
Integrating the left hand side of equation (8.20)
by parts once and making use of the boundary conditions on
(Cauchy-Schwartz
inequality)
It follows from inequality (8.21) that
or
Combining inequalities (8.21) and (8.22), we get
Making use of the boundary conditions on w given by equations (8.4), we have by
the Rayleigh-Ritz inequality
so that inequality (8.23)
reduces to
Combining equation (8.19) with inequality (8.25),
we obtain
and therefore, we must
have
or
and this establishes the
result of Banerjee et al. However, the result
Motivated by the above problem, Dhiman and Kumar (2012) reinvestigated the problem of onset
of thermal convection in an electrically conducting fluid layer heated from
below in the presence of magnetic field and tried to validate the Chandrasekhar’s
conjecture regarding
Following Finlayson (1972),
and applying the Galerkin’s method to find the
critical value of Rayleigh number by taking a single term in the expansion for
Therefore, taking
where
The various suitable trial functions chosen
for the different combinations of boundary conditions are as below:
Case I:
( both dynamically free boundaries)
Case II:
( both rigid boundaries)
Case III:
(Lower rigid and upper free
boundary)
Now, considering each of the case of boundary
conditions above and find the value of the critical Rayleigh number as follows;
Case
I: When both boundaries are dynamically
free
Evaluating the value of
The minimum of
Now, for
which implies that for a2 = 4.92
which is very close to the
critical value of Rayleigh number
obtained by Chandrasekhar (1961) for Bénard
problem (657.511) in absence of magnetic field.
Table 1: The values of
critical Rayleigh numbers
|
Chandrasekhar Results |
Results by Galerkin Method. |
|||||
|
Q |
ac |
ac2 |
Rc |
ac |
ac2 |
Rc |
|
0 |
2.233 |
4.9863 |
657.511 |
2.2270 |
4.9594 |
664.5249 |
|
5 |
2.432 |
5.9146 |
796.573 |
2.432 |
5.9408 |
804.5119 |
|
10 |
2.590 |
6.7081 |
923.070 |
2.5959 |
6.7387 |
931.8232 |
|
20 |
2.826 |
7.9863 |
11554.19 |
2.8330 |
8.0257 |
1164.3691 |
|
50 |
3.270 |
10.6929 |
1762.04 |
3.2775 |
10.7420 |
1775.7072 |
|
100 |
3.702 |
13.7048 |
2653.71 |
3.7098 |
13.7624 |
2671.9889 |
|
200 |
4.210 |
17.7241 |
4258.49 |
4.2196 |
17.8049 |
4284.2424 |
|
500 |
4.998 |
24.9800 |
8578.28 |
5.0094 |
25.0936 |
8622.1088 |
|
1000 |
5.684 |
32.3079 |
15207.0 |
5.6968 |
32.4533 |
15273.621 |
|
5000 |
7.585 |
57.5322 |
63135.9 |
7.6275 |
58.1792 |
63337.962 |
|
6000 |
7.839 |
61.4499 |
74632.1 |
7.8793 |
62.0840 |
74863.275 |
|
10000 |
8.588 |
73.7537 |
119832 |
8.6252 |
74.3941 |
120171.78 |
Case II:
When both boundaries are rigid.
Evaluating the value of
The minimum of
Now, for
which implies that for
which is very close to the
critical value of Rayleigh number
obtained by Chandrasekhar for Bénard problem
(1715.1) in absence of magnetic field.
Table 2: The values of
critical Rayleigh numbers
|
Chandrasekhar Results |
Results by Galerkin Method. |
|||||
|
Q |
ac |
ac2 |
Rc |
ac |
ac2 |
Rc |
|
0 |
3.13 |
9.7969 |
1715.1 |
3.1165 |
9.7127 |
1749 |
|
10 |
3.25 |
10.5625 |
1953.7 |
3.2599 |
10.6270 |
1996 |
|
50 |
3.68 |
13.5424 |
2811.4 |
3.6689 |
13.4615 |
2906.8351 |
|
100 |
4.00 |
16 |
3767.6 |
4.0068 |
16.0547 |
3950.0581 |
|
200 |
4.45 |
19.8025 |
5499.9 |
4.4553 |
19.8501 |
5886.1131 |
|
500 |
5.16 |
26.6256 |
10122 |
5.2098 |
27.1424 |
11199.0 |
|
1000 |
5.80 |
33.64 |
17116 |
5.8954 |
34.7560 |
19425.0889 |
|
2000 |
6.55 |
42.9025 |
30139 |
6.6782 |
44.5984 |
34993.4982 |
|
4000 |
7.40 |
54.76 |
54712 |
7.5617 |
57.1787 |
64752.959 |
|
6000 |
7.94 |
63.0436 |
78405 |
8.1279 |
66.0634 |
100934.2345 |
|
8000 |
8.34 |
69.5556 |
101622 |
8.5531 |
73.1572 |
126786.3153 |
|
10000 |
8.66 |
74.9956 |
124523 |
8.8970 |
79.1578 |
150292.1626 |
Case III:
When
one boundary is rigid and the other is free
Evaluating the value of
The minimum of
Now, for
which implies that
for a2 =7.13
which is very close to the
critical value of Rayleigh number
obtained by Chandrasekhar for Bénard problem
(1112.7) in absence of magnetic field.
It is to note that the Case IV (Lower
boundary free and upper boundary rigid) of the boundary conditions has not
treated separately, since the results of Case III of the boundary conditions
are same as that of Case IV. Further,
this combination of boundary conditions is also difficult to realize
physically.
Table 3: The values of
critical Rayleigh numbers
|
Chandrasekhar Results |
Results by Galerkin Method |
|||||
|
Q |
ac |
ac2 |
Rc |
ac |
ac2 |
Rc |
|
0 |
2.68 |
7.1824 |
1112.7 |
2.6698 |
7.1276 |
1138.7003 |
|
2.5 |
2.75 |
7.5625 |
1179.4 |
2.7356 |
7.4835 |
1209.341 |
|
12.5 |
2.97 |
8.8209 |
1428.3 |
2.9518 |
8.7132 |
1475.8114 |
|
25 |
3.17 |
10.0489 |
1712.7 |
3.1567 |
9.9647 |
1784.4809 |
|
50 |
3.45 |
11.9025 |
2231.3 |
3.4556 |
11.9401 |
2354.6292 |
|
125 |
4.00 |
16.000 |
3600.2 |
4.0012 |
16.00987 |
3886.9737 |
|
250 |
4.50 |
20.2500 |
5627.5 |
4.5226 |
20.4535 |
6193.4502 |
|
500 |
5.10 |
26.0100 |
9318.5 |
5.1317 |
26.3345 |
11350.548 |
|
1000 |
5.75 |
36.0625 |
16143 |
5.8274 |
33.9585 |
18381.22 |
|
2000 |
6.50 |
42.2500 |
28893 |
6.6124 |
43.7231 |
33368.889 |
|
5000 |
7.65 |
58.5225 |
64861 |
7.7981 |
60.8107 |
73901.542 |
|
10000 |
8.65 |
74.8225 |
122155 |
8.8194 |
77.7820 |
144167.93 |
From the results shown in Tables 1-3 for
various cases of boundary conditions, it is apparent that all three cases
exhibit the same general features. With a2 determined as a solution
of cubic equations (8.35), (8.37) and (8.39), equations (8.32), (8.34) and (8.36)
respectively give the required critical Rayleigh number
Chandrasekhar while analyzing the magnetoconvection problem by following usual variational techniques predicted that the lowest
characteristic value of
The celebrated
Following the variational method adopted by
Chandrasekhar for thermal convection problem and proceeding analogously, one
can easily prove the stationary property of the functional
where ,
Further, it is remarkable to note that the above result is uniformly
valid for all cases of boundary conditions. This establishes the variational principle for the magnetoconvection
of a fluid layer heated from below.
We shall now prove
In view of inequality
(8.27), it suffices to prove that
To prove above inequality, let us consider a trial function
which obviously satisfies the boundary conditions
where the origin has been shifted to the
midway for convenience in computation.
Now, using the above defined value of
whose general solution is obtained as
where,
and
and
where,
and
Now, evaluating integrals
Substituting the
above values of integrals
Inequality (8.51) can
be written in a convenient form as;
where
Now, using the values
of
Taking
for each case of
boundary conditions.
Therefore, inequality (8.51), in view of above value of
which for sufficiently
large values of
So, combining
inequalities (8.27) and (8.55), we have
This establishes the
9.
Solutions for the Case when Overstability
is Valid. Settlement of the Controversy
As pointed out in the concluding part of Section
1.4, the construction of the corrects solution for w and the answer regarding the validity or otherwise of the
‘exchange principle’ in magnetoconvection wherein
both the boundaries are dynamically free, has been a subject matter of intense
research activity and controversy in the recent past. In this section we take-up the problem again
and show the way the matter has been settled by Banerjee
et al (1989) in favour of the classic calculations of
Chandrasekhar. The various steps in
stepson Banerjee et al’s
scheme of calculations are presented below.
The governing equations and boundary
conditions for the magnetoconvection problem wherein
the dynamically free boundaries are thermally insulating and electrically
perfectly conducting and a uniform magnetic field acting parallel to gravity is
impressed upon the system are given by (cf.
Section 3)
with
wherein the various symbols used in the above
equations have the same meanings as given in section 1.3 with the difference
that the origin of z is translated to
be mid way between the two horizontal boundaries for the sake of convenience,
and boundary conditions that are relevant to thermally insulating boundaries
(Normand et al 1977) replace the corresponding ones that that are relevant to
thermally perfectly conducting boundaries.
Combining the above equations and boundary
conditions in an appropriate manner we derive the following systems of
equations and the associated boundary conditions in terms of w alone,
where
and
We may first observe that it follows from the
eveness of the operator L that occurs in equations (9.5), (9.7) and (9.9), and the identity
of the boundary conditions that have to be satisfied at
With
which upon integration yields
where
The requirement that the above solution for w
satisfies the boundary conditions as specified by equation (9.4) leads to a
unique determination of
and
Making use of equations (9.19) – (9.22), we
obtain the proper solution for w as
With w given by equation (9.23), equation (9.5)
becomes
where
and
Multiplying equation (9.24) by
where
Equations (9.29) provide a set of linear and
homogeneous equations for the constants
The nth
approximation to the eigenvalues of R and pi is obtained by setting the nth order determinant consisting of the first n rows and columns in the left hand side
of equation (9.30) to zero, and this corresponds to the retention of first n terms only in the Fourier expansion of
from which it follows
uniquely that the lowest eigenvalue of R and the associated value of pi are given by
Since
With w given by equation (9.23),
We complete the solution of the problem by
demonstrating that w,
To prove this we consider equation (9.5)
which can be written in an alternative form as
where
and
and for w,
Multiplying equation (9.34) by
Equating the imaginary part of both sides of
equation (9.38), we obtain
Since,
which in turn implies that
equation (9.1) is also satisfied.
The
characteristic equation of the problem that we have derived herein is identical
with the characteristic equation (1.4.3) which was first derived by
Chandrasekhar through an extremely simple solution for w (cf
equation (1.4.2)) of equation (9.5) that does not satisfy all the boundary
conditions as specified by equation (9.6) and leading to a solution for hz that fails to satisfy any plausible set of
boundary conditions required of a magnetic field. However, the present analysis shows that Chandrasekhar’s
characteristic equation (1.4.3) and the subsequent conclusions based on it was
derived by him are valid for the problem with dynamically free and thermally
insulating boundaries and therefore this problem stands completely solved. However, the problem of finding a necessary
and sufficient condition for the validity of the ‘exchange principle’ for the
relatively simpler case of magnetohydrodynamic simple
Bénard instability problem with dynamically free and
thermally perfectly conducting boundaries continues to remain unresolved and
its solution is awaited with great interest.
Gupta et al (1984, 85, 86, 88) have shown
that the scheme of calculations presented in Section 6 are of much wider
generality than the simple context in which they are given and derived
important conclusions on the characterization of the neutral states for these
more general instability problems.
Kumar et al. (1986)
have investigated the problem wherein the thermally and electrically perfectly
conducting boundaries are rigid. However, a careful analysis of Kumar et. al.’s work shows that their solution of the mathematical
double eigen value problem
is not correct since it does not satisfy, in general, the magnetic induction
equation which constitutes one of the governing equations of the problem. In
the above analysis Banerjee et.
al. (1989) have obtained a exact solution of this problem wherein the
electrically perfectly conducting and dynamically free boundaries are thermally
insulating and proved for this allied problem (on which
However, since this
was not the problem to which Chandrasekhar addressed himself, therefore Banerjee and Bhowmick
(1992) examined the magnetohydrodynamic thermal
stability problem with dynamically free and thermally and electrically
perfectly conducting boundaries and attempted to construct a exact solution of
the problem by appropriately modifying their earlier analysis (1989) of the
problem with dynamically free and thermally insulating and electrically
perfectly conducting boundaries. However, a close and critical examination of
their analysis reveals a flaw in their analysis and invalidates their claim to
have obtained a correct exact solution of the problem.
Dhiman and Sharma (2013) presented
a correct solution of the problem by appropriately rectifying the flaw in the
earlier analysis of Banerjee and Bhowmick
(1992). They showed that in the analysis of the incorrectness of the solution
may be attributed to the observation of Banerjee and Bhowmick that the relevant solutions for
so as to satisfy
equation (9.1) and the thermally perfectly conducting boundary condition,
namely
Following Banerjee
and Bhowmick (1992), the proper solutions for
is even and since it is required to vanish at
With
where
and
Using the boundary conditions on
where
Further, with
Therefore, it follows from equation (9.2)
that
where
Now, since
Equations (9.45), (9.46) and (9.51) lead to
the following unique values of
where
Substituting for
where
We now complete the
solution of the problem by demonstrating that
where
and
The expressions for
Note that
Multiply equation (9.60) by E* (the complex
conjugate of E) throughout and integrating the resulting equation over the
range of z by parts an appropriate numbers of times, using (9.60), equating the
imaginary part of the resulting equation
to zero and canceling
Equation (9.61)
clearly implies that
This proves our
contention, namely
10.
Some illustrative examples
Example 1
Consider a double eigenvalue
problem for p described by the following system of differential equations and
boundary conditions (thermohaline instability problem
of the Veronis (1965) type)
and
either
or
or
where
We prove the following results:
(a) For
the case of dynamically free boundaries, if
Proof. If
and equations (10.1) – (10.3)
become
Equations (10.5) – (10.7) for the case of
dynamically free boundaries imply that
where
Therefore a proper solution for w belonging to the lowest mode is given
by
where A is a constant.
Operating on equation (10.5) by
Substituting for w from equation (10.9) in equation (10.10), we get
Equation (10.11) can be written in an
alternative form as
Equating the real and imaginary parts of
equation (10.12), we get
and
Substituting the value of
Using the expression for R given by equation (10.15)
in equation (10.14), we get
Equation (10.16) clearly implies that
This completes the proof of the result.
Results (a) implies that for the case of
dynamically free boundaries a necessary condition for the validity of overstability is that
(b) if
Proof. Multiplying both sides of equation (10.1) by
Substituting for
Integrating equation (10.18) by parts a
suitable number of times and making use of boundary conditions (10.4) and the
equality
we may rewrite equation
(10.18) in the form
Equating the imaginary part of both sides of
equation (10.20) and cancelling
Multiplying equation (10.3) by its complex
conjugate, integrating the resulting equation over the vertical range of z by parts a suitable number of times
and using equation (10.19), we get
Since
Making use of boundary conditions on w and
and
Combining inequalities (10.23) – (10.25), we
get
Using inequality (10.26) in equation (10.21),
we get
Inequality (10.27) clearly implies that
This completes the proof of the result.
Result (b) in particular implies that a
necessary condition for the validity of overstability
is that
(c) For the case of dynamically free
boundaries, if the ‘exchange principle’ is valid, then a proper solution for w belonging to the lowest mode implies
that the critical Rayleigh number
Proof. If the ‘exchange principle’ is valid, then
at the marginal state
and equations (10.1) – (10.3)
become
Equations (10.28) – (10.30) for the case of
dynamically free boundaries imply that
where
Therefore a proper solution for w belonging to the lowest mode is given
by
where A is a constant.
Operating on both sides of equation (10.28)
by the operator
Substituting for w from equation (10.32) in
equation (10.33), we get
It follows from equation (10.34) that
and
Hence,
This completes the proof of the result.
(d) For
the case of dynamically free boundaries, if overstability
is valid, then a proper solution for w belonging to the lowest mode implies
that the critical Rayleigh number
and
Proof. If overstability
is valid, then at the marginal state
so that equations (10.15)
and (10.16) hold good.
and
Hence,
and
This completes the proof of the result.
Example 2
The hydrodynamic instability that manifests
under appropriate conditions in a static horizontal initially homogeneous
viscous and Boussinesq liquid layer of infinite
horizontal extension and finite vertical depth which is kept under the action
of a uniform vertical adverse temperature gradient in the force field of
gravity is known as simple Bénard in stability while
the problem of a theoretical investigation of the essential aspects of this
instability is known as simple Bénard instability
problem. One of the fundamental assumption that constitutes the theoretical framework in
which this problem is analyzed is that the liquid under consideration is
initially homogeneous which, in general, is not true in any real physical
situation although it may happen that the extent of this non-homogeneity is
small. But then the Boussinesq
approximation which is made use of in the theory with respect to small extent
of non-homogeneity introduced on account of thermal effects should have been
applied with respect to the small extent of initial non-homogeneity as well and
thus there is a need of an extended theory for the simple Bénard
instability problem wherein the liquid is initially non-homogeneous.
The hydrodynamic instability that manifests
under appropriate conditions in a static horizontal initially continuously
vertically non-homogeneous viscous and incompressible liquid layer of infinite
horizontal extension and finite vertical depth in the force field of gravity is
known as Rayleigh-Taylor instability while the problem of a theoretical
investigation of the essential aspects of this instability is known as
Rayleigh-Taylor instability problem. One
of the fundamental assumption that constitutes the theoretical framework in
which this problem is analyzed is that the upper and the lower horizontal
boundaries of the liquid under consideration are at the same temperatures
which, in general, is not true in any real physical situation although it may
happen that the extent of this temperature difference is small. But then the continuous vertical
non-homogeneity that is introduced on account of this thermal effect should
also be taken into account in the theory along with the initial continuous
vertical non-homogeneity and thus there is a need of an extended theory for the
Rayleigh-Taylor instability problem wherein the liquid is initially
continuously vertically non-homogeneous.
The hydrodynamic instability that manifests
under appropriate conditions in a static horizontal initially homogeneous
viscous and Boussinesq liquid layer of infinite
horizontal extension and finite vertical depth which is kept under the
simultaneous action of a uniform vertical temperature gradient and a
gravitationally opposite uniform vertical concentration gradient in the force
field of gravity is known as thermohaline instability
while the problem of a theoretical investigation of the essential aspects of
this instability is known as thermohaline instability
problem. However, there is one
particular case of this problem which is important from the point of view of
its applicability to real physical situations (such as the oceans) and that is
the smallness of the coefficient of mass diffusivity of the liquid under
consideration compared to its coefficient of heat diffusivity which makes the
resulting Lewis number small compared to the thermal Prandtl
number. Thus, there is a need for
working out the consequences of a thermohaline
instability problem that is simplified on the basis of the above mentioned
hypothesis.
These ideas were put forward for the first
time in 1971 and 1972 by Banerjee who gave a unified
mathematical treatment of the three well known problems of hydrodynamic
instability namely the simple Bénard instability
problem, the Rayleigh-Taylor instability problem and the thermohaline
instability problem by showing that if these problems are modified on the basis
of the ideas as given above they give rise to the same problem which non is
known in the literature as the generalized Bénard
instability problem. Some of the
consequences derived by Banerjee from this unified
problem later found relevance in the context of more general problems both from
physical as well as mathematical points of view (Banerjee
et al 1978, 1981; Gupta et al 1985) and in the following we present a brief
mathematical investigation of the generalized Bénard
instability problem.
Consider a double eigenvalue
problem for p described by the following system of differential equations and
boundary conditions (generalized Bénard instability
problem):
and
either
or
or
where
We prove the following results:
(a) If
Proof.
Multiplying equation (10.38) by
Equating the real parts of both sides of
equation (10.41), we get
If
Equation (10.43) clearly implies that
This completes the proof of the result.
Result (a) implies that for the generalized Bénard instability
problem the ‘exchange principle’ is not valid. Further, this result is uniformly valid for
all combinations of boundaries which may be rigid or dynamically free.
(b) If
Proof. Since
Equation (10.44) can be written as
Equation (10.44) clearly implies that
This completes the proof of the result.
Result (b) implies that the complex growth
rate of an arbitrary oscillatory perturbation whether stable, neutral or
unstable for the generalized Bénard instability
problem must lie inside the circle given by
Further, this result is uniformly valid for
all combinations of boundaries which may be rigid or dynamically free.
(c) For
the case of dynamically free boundaries, a proper solution for w belonging to
the lowest mode implies that the critical Rayleigh number
and
Proof. Follows
by proceeding exactly as in result (d) of Example 1. We note that the critical frequency as given
in result (c) above lies inside the circle given by equation (10.46). Further, the result contained herein also
follow from the corresponding results in result (d) of Example 1 by putting
ACKNOWLEDGEMENT:
This research paper
is dedicated to Rev. Fr. Goreux and contains the
excerpts of Key Note Address ‘Rev. Fr. Goreux
Memorial Lecture’, by Prof. Mihir B. Banerjee on UGC
sponsored, Two-day National Seminar held at Sidharth
Govt. College Nadaun, Dist. Hamirpur
(HP) on March 12-13, 2015.
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Received on
18.04.2015 Modified on 30.04.2015
Accepted on
28.05.2015 ©A&V Publications All right reserved
Research J. Science and Tech. 7(2): April-June. 2015; Page 55-103
DOI: 10.5958/2349-2988.2015.00009.1