Ph.D.http://hdl.handle.net/20.500.12228/482024-03-29T09:50:39Z2024-03-29T09:50:39ZOn MHD Flows of Viscous Incompressible FluidsRahman, Fouziahttp://hdl.handle.net/20.500.12228/2962018-09-18T09:20:37Z1986-08-01T00:00:00ZOn MHD Flows of Viscous Incompressible Fluids
Rahman, Fouzia
The presented thesis entitled “ON MHD FLOWS OF VISCOUS INCOMPRESSIBLE FLUIDS” is being presented for the award of the degree of Doctor of Philosophy in Mathematics. It is the outcome of my researches conducted in the Department of Mathematics, Banaras Hindu University during the years 1983-86 under the esteemed guidance of Dr. Newal Kishore, Reader in the Department of Mathematics, Banaras Hindu University, Varanasi, India.
The whole thesis consists of six chapters. The first chapter is introductory, giving the general description and fundamental equations of magnetohydrodynamics, free conviction flow, flow through porous media, rotating fluid flow, oscillatory flow and flow with Hall currents. Lastly a brief review of the past researchers related to the thesis have been given. Throughout the work we are considering the flows of electrically conducting viscous and incompressible fluids. The magnetic Reynolds number is assumed small for all the problems except the problems discussed in chapter two.
The second chapter has been divided into parts. Part A of this chapter deals with the flow between two
infinite, non-conducting, parallel porous flat: plates, when the lower plate is injecting fluid and the upper one is absorbing it. The flow is subjected to a uniform transverse magnetic field and the magnetic Reynolds number of the flow is sufficiently large so as to include the effect of induced magnetic field. The expressions for the velocity and induced magnetic fields have been obtained by using Laplace transform technique. The effect of the magnetic parameter on the velocity and induced rnagnetic field has been studied. It is found that the velocity decreases with increase in M in the lower region between the plates and increases with increase in M in the upper region. The induced magnetic field decreases with increase in M. In part B of this chapter, the effect of uniform transverse magnetic field on unsteady MHD free convictive flow past an impulsively started infinite vertical non-conducting plate has been discussed. Here also, the magnetic Reynolds number is assumed to be sufficiently large to take account of the induced magnetic field. There is constant heat flux at the plate. Expressions for the velocity and induced magnetic have been obtained by Laplace transform technique. The effect of the different parameters on the flow have been discussed with the help of tables.
In part A of the third chapter, the effect of a uniform transverse magnetic field on the steady free convective flow through a porous medium, occupying a semi-infinite region of space and bounded by a steadily moving vertical porous plate has been studied. The flow is subjected to constant suction. Approximate solutions to the equations relevant to the problem have been obtained. The influence of the different parameters on the velocity and temperature fields have been discussed with the help of graphs and tables.
The problem considered in part 3 of this chapter is an extension of the problem considered in part A. Here we have taken into account the effect of rotation on the flow. Due to rotation the flow become three dimensional. Approximate solutions to equations relevant to the problem have been obtained. Effects of the various parameters on the primary velocity, secondary velocity, the components of skin friction and the temperature have been discussed.
The fourth chapter is concerned with the unsteady free convective flow past on impulsively started infinite vertical porous plate in presence of a uniform transverse magnetic field. The free stream is assumed to oscillate in time about a constant mean. The flow is subjected to content suction velocity and there is constant heat flux at the plate. Approximate solutions for the mean flow and transient flow have been obtained and the results have been discussed with the help of tables and graphs.
In the fifth chapter we have studied the effects of flail currents on the unsteady MHD free convective flow past an impulsively started infinite vertical porous plate in presence of a uniform transverse magnetic field. The p1ate temperature is assumed to oscillate in time about a constant mean and the flow is subjected to constant suction at the plate. Approximate solutions for the mean flow and transient flow have been obtained. The inf1uence of the various parameters on the mean and transient flows has been discussed with the help of tables and graphs.
In the last chapter, an attempt has been made to study the effects of rotation and Hall currents on the unsteady MHD free convective flow through porous medium occupying a semi-infinite region of space and counted by an infinite vertical porous plate in presence of a transversely applied uniform magnetic field. The plate is assumed to oscillate in time about a constant mean and there is constant heat flux at the plate. Approximate solutions for the mean flow and transient flow have been obtained and the results have been discussed with the help of graphs and tables.
This thesis is submitted to the Department of Mathematics, Faculty of Science, Banaras Hindu University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, August 1986.; Cataloged from PDF Version of Thesis.; Includes bibliographical references (pages 167-170).
1986-08-01T00:00:00ZA Study on Distributive NearlatticesRahman, Md. Bazlarhttp://hdl.handle.net/20.500.12228/2892018-09-18T09:21:19Z1994-08-01T00:00:00ZA Study on Distributive Nearlattices
Rahman, Md. Bazlar
This thesis studies the nature of distributive nearlattices. By a nearlattice S we will always mean a (lower) semilattice which has the property that any two elements possessing a common upper bound, have a supremum, Cornish and Hickman in their paper [14], referred this property as the upper bound property, and a semilattice of this nature as a semilattice with the upper bound property. Cornish and Noor in [15] preferred to call these semilattices as nearlattices as the behaviour of such a semilattice is closer to that of a lattice than an ordiary semilattice. In this thesis we give several results on nearlattices which certainly extend and generalize many results in lattice theory. In chapter 1 we discuss ideals, congruences and other results which are basic to this thesis. We include some characterizations of distributive and modular nearlattices. We generalize the separation properties given by M. H. Stone for distributive lattices. We also show that the set of prime ideals of a nearlattice S is unordered if and only if S is semiboolean.
Chapter 2 discusses the skeletal congruences of a distributive nearlattice. Skeletal congruences on distributive lattices have been studied extensively by Cornish in [11]. Here we extend several results of Cornish for nearlattices. We also introduce the notion of disjutive nearlattices. A distributive nearlattice S with 0 is called disjunctive if for 0 ≤ a < b there is an element x ϵ S such that x ˄ a = 0 and 0 < x ≤ b. Then we give several characterizations of disjunctive nearlattices and semiboolean algebras using skeletal congruences. Finally we show that a distributive naerlattice is semiboolean if and only if θ -----> ker θ is lattice isomorphism of Sc(S) onto KSc(S)
whose inverse is the map J´ ---> θ (J).
In chapter 3, we discuss on normal and n-normal nearlattices. Normal lattices have been studied by several authors including Cornish [8] and Monteiro [34]; while n-normal lattices have been studied by Cornish [9] and Davey [16]. In proving some of the results we have used Principle of Localization, which is an extension of lecture note of Dr. Noor on localization. This technique is very interesting and quite different from those of the previous authors.
Chapter 4 studies the multiplier extension (meet translation) of a distributive nearlattjce. Previously multipliers on semilattices and lattices have been studied by several authors e.g, Szasz and Szendrje [54,55,56] Kolibiar [29],Cornjsh [101 and Niemenen [37] on a latice. In a more recent paper, Noor and Cornjsh in [39] studied them on nearlatticee. Here we extend some of their work. We also give a categorjcai result, where we see that the multiplier extension has a functorial character which is entirely different from that of the Lattice Theory, c.f. Cornish [10, theorem 2.41. In section 2 of this chapter we discuss multipliers on sectionally pseudocomplemented distributive nearlattices which are sectionally in B՛n, -1 ≤ n ≤ Ꞷ and generalize a number of results of [10]. We show that S is sectionally in Bn if and only if M(S), the lattice of multipliers is in Bn. Finally we show that for 1 ≤ n < Ꞷ, above conditions are also equivalent to the condition that S is sectionally pseudocomplemented and for any n+1 minimal prime ideals
P1,P2,. ........... ,Pn+1,
P1 V P2 V ........... V Pn+1 = S.
This thesis is submitted to the Department of Mathematics, Rajshahi University in partial fulfillment of the requirements for the degree of Doctor of Philosophy, August 1994.; Cataloged from PDF Version of Thesis.; Includes bibliographical references (pages 117-124).
1994-08-01T00:00:00ZStudy of Principal n-ideals of a LatticeAzad, Md. Abul Kalamhttp://hdl.handle.net/20.500.12228/2822018-09-18T09:22:07Z2006-07-01T00:00:00ZStudy of Principal n-ideals of a Lattice
Azad, Md. Abul Kalam
This thesis studies extensively the Principal n-ideals of a lattice. The idea of n-ideals in a lattice was first introduced by Cornish and Noor in studying the kernels around a particular element n, of a skeletal congruence on a distributive lattice. Then Latif and Ayub Ali in their thesis studied thoroughly on the n-ideals and established many valuable results. For a fixed element n of a lattice L, a convex sublattice of L containing n is called an n-ideal. If L has a "0", then replacing n by 0, an n-ideal becomes an ideal and if L has a "1" then it becomes a filter by replacing n by I. Thus, the idea of n-ideals is a kind of generalization of both ideals and filters of lattices. The n-ideal generated by a finite number of elements of a lattice is called a finitely generated n-ideal, while the n-ideal generated by a single element is known as a principal n-ideal. Latif in his thesis has given a neat description on finitely generated n-ideals of a lattice and has provided a number of important results on them. For a lattice L, the lattice of all n-ideals of L and the lattice of all finitely generated n-ideals of L are denoted by In (L) and Fn (L) respectively, while Pn (L) represents the set of principal n-ideals of L. In this thesis, we devote ourselves in studying several properties on Pn (L) and Fn (L) which will certainly enrich many branches of lattice theory. Our results in this thesis generalize many results on normal, relatively normal, m-normal and relatively m-normal lattices. We also introduce the concept of n-annulets and α -n-ideal in studying Pn (L).
In this connection it should be mentioned that if L has a 0, then putting n = 0 we find that Fn (L) is the set of all principal ideals of L which is isomorphic to L. Thus, for every result on Fn (L) in this thesis, we can obtain a result for the lattice L with 0 by substituting n = 0. Hence the result in each chapter of the thesis regarding Fn (L) are generalizations of the corresponding results in lattice theory.
In chapter 2, we discuss some fundamental properties of n-ideals, which are basic to this thesis. Here we give an explicit description of Fn (L) and Pn (L) which are essential for the development of the thesis. Though Fn (L) is always a lattice, Pn (L) is not even a semilattice. But when n is a neutral element, Pn (L) becomes a meet semilattice. Moreover, we show that Pn (L) is a lattice if and only if n is a central element, and then in fact, Pn (L) = Fn (L). We also show that, for a neutral element n, the lattice L is complemented if and only if Pn (L) is so. In this chapter we also discuss on prime n-ideals. We give several properties and characterizations of prime n-ideals. We include a proof of the generalization of Stone's separation theorem. We also include a new proof of the result that for a distributive lattice L, Fn (L) is generalized Boolean if and only if prime n-ideals are unorderd.
Chapter 3 discusses on minimal prime n-ideals of a lattice. We give some characterizations on minimal prime n-ideals which are essential for the further development of this chapter. Here we provide a number of results which are generalizations of the results on normal lattices.
We prove that for a distributive lattice L, Fn (L) is normal if and only if each prime n-ideal of L contains a unique minimal prime n-ideal. We also show that if n is central in L, then Pn (L) is a normal lattice if and only if any two minimal prime n-ideal are comaximal which is also equivalent to < x > n ∩ <y> n = {n} implies <x> n* v <y> n*=L.
In chapter 4 we introduce the notion of relative n-annihilators <a, b >n. We characterize distributive and modular lattices in terms of relative n-annihilators. Then we generalize several results of Mandelker on annihiltors. We use these to characterize those Fn (L) which are relatively normal lattices. Among many results we have shown that for a central element n, Pn (L) is a relatively normal lattice, if and only if any two incomparable prime n-ideal are comaximal . What is more, this is also equivalent to the condition <<a >n,< b >n> v <<b >n,< a >n> = L for all a,b ϵL.
Pseudocomplemented distributive lattices satisfying Lee's identities form equational subclasses denoted by Bm , - 1 ≤ m ˂ w Cornish have studied distributive lattices analogues to Bm-lattices and relatively Bm-lattices. He referred then as m-normal lattices.Moreover, Beazer and Deavy have each independently obtained several characterizations of (sectionally) Bm -lattices and relatively Bm -lattices.
In chapter 5 we generalize their results by studying finitely generated n-ideals which form a m-normal and a relatively m-normal lattice .We show that for a central element n ϵ L, Pn(L) is m-normal if and only if for any m+1 distinct minimal prime n-ideals P0 ............., Pn of L, P0 v ................v Pm = L. In this chapter we also show that for a central element n ϵ L, Pn (L) is relatively m-normal if and only if any m+1 pairwise incomparable prime n-ideals are comaximal.
Chapter 6 introduces the concept of n-annulets and α -n-ideals of a lattice. Here we include several result on the set of n-annulets An(L) when n is a central element of L. We proved An(L) is relatively complemented if and only if Pn(L) is sectionally quasi-complemented.
This thesis is submitted to the Department of Mathematics, Khulna University of Engineering & Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, July 2006.; Cataloged from PDF Version of Thesis.; Includes bibliographical references (pages 127-131).
2006-07-01T00:00:00ZStudy of Certain Topics in Fuzzy Supra Topological SpaceMolla, Md. Yahiahttp://hdl.handle.net/20.500.12228/2792018-09-18T09:22:26Z2013-07-01T00:00:00ZStudy of Certain Topics in Fuzzy Supra Topological Space
Molla, Md. Yahia
American Mathematician Lotfi A. Zadeh in 1965 first introduced the concept of fuzzy set. He interpreted a fuzzy set on a set as a mapping from the set into the unit interval I= [0, 1], which is a generalization of the characteristic function of the set. Many mathematicians throughout the world used this set to fuzzify different areas of mathematics. Fuzzy supra topology is one of the outcomes of such fuzzification of the usual topology. In this thesis, we have studied and have introduced several results on fuzzy supra topological spaces. At first we have discussed the standard definitions and properties of fuzzy supra R0 and R1 topological spaces, which are found in the literatures. Then we have introduced some new definitions and properties for these spaces. We have also studied the Fuzzy supra T0, T1 , T2 and Fuzzy supra regular topological spaces and obtained the following properties, such as, Good extension, Initial, Reciprocal, Productivity, Hereditary and Homeomorphism, etc. Moreover we have discussed compactness of Fuzzy Supra Topological Spaces and have proposed some new definitions, theorems and proofs.
This thesis is submitted to the Department of Mathematics, Khulna University of Engineering & Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, July 2013.; Cataloged from PDF Version of Thesis.; Includes bibliographical references (pages 128-132).
2013-07-01T00:00:00Z