Topological Invariants in Hydrodynamics and Hydromagnetics

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.

A Study Of The Marketing Channels Of Natural Rubber With Special Reference To Co-Operative Marketing In Kerala

The study is designed to gather, record, analyse and critically evaluate data on natural rubber production, consumption and marketing in Kerala. The scope of the study also covers the processing of natural rubber by rubber growers, especially small growers. Taking into consideration Kerala’s supremacy in natural rubber cultivation, a detailed study of marketing channels of natural rubber in Kerala, by giving special reference to co-operative rubber marketing has been given importance. Attempt has also been made to collect secondary data of the last 15 years

Structure of marketing channels of selected manufacturing industries in Kerala

The study evolved from the basic premise that the existing distribution structure is not adequate or adaptive to meet the needs of the expanding manufacturing sector and the emerging mass market. The hypothised causes of the problem are the following: marketing channels are not used for strategy differentiation by manufacturers: there are too many intermediaries in the channels; the distributive institutions are not adaptive; and there is very little control over the flow of products through the channels. These assumptions about the causes of the problem have been translated into specific hypotheses and tested with data. Empirical analysis, while supporting some of these hypotheses, challenges certain widely held notions. The ensuing summary presents the important findings, in the sequence in which they are discussed in the study.

Centre-State Financial Flows and Inter—State Disparities in India

The problem of regional disparities in economic development is for India an inheritance from the colonial past. At the beginning of the First Five Year Plan (1950-51), three years after the advent of independence, the per capita State income showed considerable inter—state variations.

Marketing Channels in Ornamental Fish Trade in West Bengal

A lucrative export market and high domestic demand has made ornamental fish industry in West Bengal a potential source for income generation. The study aimed to identify: (i) the commercially important size groups of main ornamental fish varieties available in the state; (ii) the existing supply chain; (iii) major constraints for development of the industry; (iv) and to anlayse price spread of commercially important varieties; and (v) to evaluate the profitability of operation at different stakeholder levels in the marketing chain. Export market of ornamental fishes in the state followed a single supply channel while three different distribution channels existed in the domestic market. High electricity charges was the major problem faced by breeders (producers/rearers) whereas lack of technical knowledge regarding transportation was the major constraint for wholesalers. Lack of knowledge on proper health management inhibited the growth of retail industry. The fresh water catfish, angel, molly, arowana, gold fish, tetras, and gouramis showed comparatively higher breeders’ share in consumers’ rupee. Wholesalers were earning comparatively higher annual profit than the other stakeholders due to moderate initial investment and also due to the comparatively lower risk involved.