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In this thesis we are studying possible invariants in hydrodynamics and hydromagnetics.
The concept of flux preservation and line preservation of vector fields, especially
vorticity vector fields, have been studied from the very beginning of the study of fluid
mechanics by Helmholtz and others. In ideal magnetohydrodynamic flows the magnetic
fields satisfy the same conservation laws as that of vorticity field in ideal hydrodynamic
flows. Apart from these there are many other fields also in ideal hydrodynamic and magnetohydrodynamic flows which preserves flux across a surface or whose vector lines
are preserved.
A general study using this analogy had not been made for a long time. Moreover
there are other physical quantities which are also invariant under the flow, such as
Ertel invariant. Using the calculus of differential forms Tur and Yanovsky classified
the possible invariants in hydrodynamics. This mathematical abstraction of physical
quantities to topological objects is needed for an elegant and complete analysis of
invariants.Many authors used a four dimensional space-time manifold for analysing fluid flows.
We have also used such a space-time manifold in obtaining invariants in the usual three
dimensional flows.In chapter one we have discussed the invariants related to vorticity field using
vorticity field two form w2 in E4. Corresponding to the invariance of four form w2 ^ w2
we have got the invariance of the quantity E. w. We have shown that in an isentropic
flow this quantity is an invariant over an arbitrary volume.In chapter three we have extended this method to any divergence-free frozen-in
field. In a four dimensional space-time manifold we have defined a closed differential
two form and its potential one from corresponding to such a frozen-in field. Using this
potential one form w1
, it is possible to define the forms dw1 , w1 ^ dw1 and dw1 ^ dw1
.
Corresponding to the invariance of the four form we have got an additional invariant
in the usual hydrodynamic flows, which can not be obtained by considering three
dimensional space.In chapter four we have classified the possible integral invariants associated with
the physical quantities which can be expressed using one form or two form in a three
dimensional flow. After deriving some general results which hold for an arbitrary dimensional
manifold we have illustrated them in the context of flows in three dimensional
Euclidean space JR3. If the Lie derivative of a differential p-form w is not vanishing,then the surface integral of w over all p-surfaces need not be constant of flow. Even
then there exist some special p-surfaces over which the integral is a constant of motion,
if the Lie derivative of w satisfies certain conditions. Such surfaces can be utilised
for investigating the qualitative properties of a flow in the absence of invariance over
all p-surfaces. We have also discussed the conditions for line preservation and surface
preservation of vector fields. We see that the surface preservation need not imply the
line preservation. We have given some examples which illustrate the above results.
The study given in this thesis is a continuation of that started by Vedan et.el. As
mentioned earlier, they have used a four dimensional space-time manifold to obtain
invariants of flow from variational formulation and application of Noether's theorem.
This was from the point of view of hydrodynamic stability studies using Arnold's
method.
The use of a four dimensional manifold has great significance in the study of knots
and links. In the context of hydrodynamics, helicity is a measure of knottedness of
vortex lines. We are interested in the use of differential forms in E4 in the study of
vortex knots and links. The knowledge of surface invariants given in chapter 4 may
also be utilised for the analysis of vortex and magnetic reconnections. |
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