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Item On MHD Flows of Viscous Incompressible Fluids(Banaras Hindu University, Varanasi, India, 1986-08) Rahman, Fouzia; Kishore, Dr. NewalThe 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.Item A Study on Distributive Nearlattices(Rajshahi University, Rajshahi, Bangladesh, 1994-08) Rahman, Md. Bazlar; Noor, Prof. Dr. A.S.A.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.