PhD Thesis
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Item Intrinsic Features of Nonlinear Waves in Dusty Plasmas(University of Rajshahi, 2024) Salam, Md. Abdus; Ali, M. Zulfikar; Akber, Md. AliDusty plasmas are electrically conducting ionized gases (macroscopically quasi-neutral) that comprise positively and negatively charged dust particles in addition to ions and electrons. Dusty plasmas exist in the Earth's magnetosphere, cometary tails, planetary rings, asteroid rings, rotating stars, and many other astronomical environments. In dusty plasmas, various types of nonlinear waves, such as solitary waves, shock waves, etc., may be propagated. The presence of dust particles makes the plasma system more complex. Besides, the characteristics of nonlinear waves can be substantially affected by different forces and plasma parameters. We theoretically investigate the characteristics of dust-ion-acoustic solitary and shock waves, where the plasma species follow different particle distributions. The higher-order nonlinear and dispersive (or dissipative) effects on the waves are examined. We also investigate various kinds of effects, such as magnetic, adiabatic, parametric, etc., while it is illustrated that how they change the wave characteristics. An inhomogeneous KdV-type or modified KdV-type equation is obtained for demonstrating the solitary waves, whereas an inhomogeneous modified Burgers-type equation is obtained for the shock waves. The reductive perturbation method is extensively used for incorporating the higher-order effects into the KdV, modified KdV, and modified Burgers equations. The re-normalization technique, the Abel’s theorem, and the method of variation of parameters are used for adding higher-order nonlinear and dispersive (or dissipative) effects into the solutions. We deal with the theoretical investigation of the combined effect of the magnetic field and plasma rotation on the nonlinear features of obliquely propagated dust-ion-acoustic solitary waves in a magnetized dusty plasma. From the investigation, it is found that the overall impact of the magnetic field, oblique rotation, electron temperature, and dust concentration has a crucial role in changing the amplitude, width, and phase speed of the dust-ion-acoustic solitary waves. The results are expected to be helpful in describing the rotating flows of magnetized plasma that are believed to exist in the rotating stars, the pulsar magnetosphere, and other rotating astronomical objects. We also explore the dynamic behaviors of multi-solitons as well as multi-shocks that propagate in a magnetized dusty plasma. The simplified Hirota method and the Cole-Hopf transformation are applied to construct the multi-soliton or multi-shock solutions. We observe that the magnetic field has a decreasing effect on the multi-soliton amplitudes and widths, whereas the dust concentration has an increasing effect on the amplitudes and widths of both types of waves. The obtained results might be helpful to describe the solitary and shock waves propagated in the Earth's mesosphere, Jupiter's magnetosphere, cometary tails, etc., in which dust particles are commonly seen.Item Interaction Phenomena of Nonlinear Waves in Unmagnetized Plasmas(University of Rajshahi, Rajshahi, 2019) Alam, Mohammad Shah; Talukder, Mamunur Rashid; Ali, M. HossainThis dissertation is concerned with the study of interaction phenomena of nonlinear waves in unmagnetized plasmas. The plasma system considered is fully ionized, collisionless and homogeneous and/or inhomogeneous that contains multi-component plasma species under different situations. To investigate the physical issues of the interaction phenomena of nonlinear waves the nonlinear evolution equations are derived. The extended Poincaré-Lighthill-kue (ePLK) method is used to derive the nonlinear evolution equations. The interaction phenomena pertaining to plasma parameters on the production of ion-acoustic solitary waves, ion-acoustic shock waves and rogue waves and their consequences on phase shifts and amplitudes are investigated in different plasma situation. The interaction processes among the waves (such as ion-acoustic solitary waves, ion-acoustic shock waves) for single and multi-soliton plasmas are also studied considering the analytical solutions to the nonlinear evolution equations under some assumptions to discuss the characteristic of the waves in the plasmas that are observed in astrophysical, space and laboratory plasmas. In chapter one, some important physical terms that are relevant to the plasma phenomena are briefly discussed. Chapter two discusses the interaction phenomena of ion-acoustic multi-solition and the production of rogue waves in an unmagnetized plasmas composing non-relativistic as well as relativistic degenerate electrons and positrons, and inertial non-relativistic helium ions. The interaction phenomena are investigated by deriving two-sided Korteweg-de Vries (KdV) equations with their corresponding phase shifts employing extended Poicaré-Lighthill-Kuo (ePLK) method and to study the properties of rogue waves the nonlinear Schrödinger equation (NLSE) is obtained from the modified KdV (mKdV) equation.-------Item Massive Particle Tunneling from Black Hole Spacetime(University of Rajshahi, 2013) Hossain, Md. Ilias; Rahman, M. AtiqurWe investigate the Hawking radiation from different kind of black holes by massive particle tunneling process near the event horizon of the black hole in de Sitter and anti-de Sitter spaces. We calculate the imaginary part of the action from the relativistic Hamilton-Jacobi equation avoid by exploring the equation of motion of the radiation particle in Pain leave coordinate system in order to explore the Hawking non-thermal and purely thermal radiations. The thesis is organized as follows: In chapter 1 we give a brief discussion about our work of studying of massive particle tunneling from black hole space-time. In chapter 2 we review the relativistic Hamilton-Jacobi equation to perform our prime work. In chapter 3 to 10 we investigate the Hawking non-thermal and purely thermal radiations using massive particles tunneling process by employing Hamilton-Jacobi method for Schwarzschild-de Sitter (SdS), Schwarzschild-anti-de Sitter (SAdS), Reissner-Nordström-de Sitter (RNdS), Reissner-Nordström-anti-de Sitter (RNAdS), Kerr-de Sitter (KdS), Kerr-anti-de Sitter (KAdS), Kerr-Newman-de Sitter (KNdS) and Kerr-Newman-anti-de Sitter (KNAdS) black holes. We express the position of all kind of black holes in an infinite series in terms of black hole parameters so that the space-time metric becomes dynamical and derive the new line elements. Taking into account the energy conservation, the angular momentum conservation and the unfixed background spacetime. When self-gravitation interaction is considered, the derived emission/radiation spectrums are not purely thermal and the tunneling rates are related to the change of the Bekenstein-Hawking entropy, which satisfy an underlying unitary theory. Our new process provides an interesting correction to the Hawking pure thermal radiation of the black hole and in the limiting case, the results are accordant with that obtained by Parikh and Wilczek’s method of the black hole.Item A Study on Turbulence and MHD Turbulence(University of Rajshahi, 2009) Aziz, Md. Abdul; Sarker, M. Shamsul Alam; Azad, M. Abul KalamTurbulent motions are very common in nature. The theory of turbulent motion has received considerable attention in recent developments of high-speed jet aircraft, plasma physics and chemical engineering. The formation of a turbulent boundary layer is one of the most frequently encountered phenomena in high-speed aerodynamics. Turbulence occurs nearly everywhere; in the oceans, in the atmosphere, in rivers even in stars and galaxies. It occurs when an airplane hits an air pocket. Much like there are currents in the ocean, there are currents in the air. Winds disturbed by thunderstorms or mountains are just one of the many causes of turbulence. In turbulent flow, the motion of the fluid is steady so far as the temporal mean values of velocities and the pressures are concerned where as actually both velocities and the pressures are irregularly fluctuating. The velocity and pressure distributions in turbulent flows as well as the energy losses are determined mainly by turbulent fluctuations. The essential characteristic of turbulent flows is that the turbulent fluctuations are random in nature. It is common experience that the flow observed in nature such as rivers and winds usually differ from stream flow or laminar flow of a viscous fluid. The mean motion of such flows does not satisfy the Navier-Stokes equations for a viscous fluid. Such flows, which occur at high Reynolds numbers, are often termed turbulent flows. Atmospheric scientists define "turbulence" as "a state of fluid flow in which the instantaneous velocities exhibit irregular and apparently random fluctuations." Those "irregular fluctuations" of the flow create the bumps. With sufficient disturbances the result is known as turbulence. The instability of laminar flow at a high Reynolds numbers, are causes disruption of the laminar pattern of fluid motion. In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. Turbulence is one of those few things that many don't understand. It's not a hard concept at all. At least, the technical people understand the meaning of turbulence. The use of the word "Turbulence" to characterize a certain type of flow, namely, the counterpart of streamline motion 1s comparatively recent. Reynolds,0. [112] made the first systematic experimental investigation of turbulent flow. The turbulent motion of fluid was described by Reynolds [112], one of the pioneers in the study of turbulent flows as "sinuous motion" because fluid particles in turbulent flow appeared to follow sinusoidal or irregular paths. The word "Turbulence" means: agitation, commotion, disturbance etc. Turbulence is rather a familiar notion; yet it is not easy to define in such a way as to cover the detailed characteristic comprehended in it and to make the definition agree with the modern view of it held by professionals in this field of applied science. Taylor and Vonkarman [146) suggested that, "Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surface or even when neighboring streams of the same fluid flow past or over one another". According to this definition, the flow has to satisfy the condition of irregularity. But this irregularity is a very important feature. Because of irregularity, it is impossible to describe the motion in all details as a function of time and space co-ordinates. But fortunately turbulent motion is irregular in the sense that it is possible to describe it by laws of probability. It appears possible to indicate distinct average values of various quantities, such as velocity, pressure, temperature, etc and this is very important. It is not sufficient just to say that turbulence is an irregular motion yet we do not have clear-cut definition of turbulence. In 1975, Hinze [51] gave the definition, "Turbulent fluid motion is an irregular condition of flow in which various quantities show a random variation with time and space co-ordinates, so that statistically distinct average values can be discerned". Turbulence is a form of movement which is characterized by an irregular or agitated motion. Both liquids and gases can exhibit turbulence, and a number of factors can contribute to the formation of turbulence. The addition "with time and space co-ordinates" is necessary; it is not sufficient to define turbulent motion as irregular in time alone. For instance, the case in which a given quantity of a fluid is moved bodily in an irregular way; the motion of each part of the fluid is then irregular with respect to time to a stationary observer, but not to an observer moving with the fluid. Nor is turbulent motion, a motion that is irregular in space alone, became a steady flow with an irregular flow pattern might then come under the definition of turbulence.Item Laminar Flow of Incompressible Viscous Newtonian Fluid(University of Rajshahi, 2009) Haque, Md. Abdul; Rahman, M. Zillur; Alam, M. ShamsulThe thesis entitled "Laminar Flow of Incompressible Viscous Newtonian Fluid" is being presented for the award of the degree of Doctor of Philosophy in Applied Mathematics. It is the outcome of my researches conducted in the Department of Applied Mathematics, University of Rajshahi, Bangladesh. The whole thesis consists of nine chapters. The first chapter is a general introductory chapter, giving the general information about Laminar Flow of Incompressible Viscous Newtonian Fluid. In chapter II, we have described about the basic concepts of incompressible viscous Newtonian fluid. Some fundamental equations are presented in this chapter. In Chapter III a laminar flow of incompressible viscous fluid has been considered. Here two numerical methods for solving boundary layer equation have been discussed; (i) Keller Box scheme, (ii) shooting method. Runge-Kutta method is used to solve the initial value problem. The shooting method is supported by a suitable example. The Chapter IV is divided into two parts. In Part: A an attempt has been made to investigate the velocity profile of unsteady laminar flow of incompressible viscous fluid. The method of separation of variable is used to determine the solutions of the governing differential equations. Time varying pressure gradient is considered for poiseuille flow. The velocity profiles for the various types of flow are shown by the figures. In Part: B a fully developed conducting flow of incompressible viscous Newtonian fluid between two parallel plates under the action of a parallel Lorentz force is considered. Analytic solutions for this type of flow are developed. The velocity profiles are presented in figures. In Chapter V, an attempt has been made to study the flow of a viscous incompressible fluid between two parallel porous plates. In case I, we have considered the flow of conducting fluid between two fixed porous plates in presence of a transverse magnetic field. Small suction and injection are imposed on the plates. The velocity of the fluid has been obtained under the three different cases, when pressure gradient is (i) varying linearly with time (ii) decreasing exponentially with time and (iii) varying periodically with time. In case I I of this chapter an attempt have been made to study the flow of a conducting viscous incompressible fluid between two porous plates in absence of pressure gradient force. One plate is at rest and the other plate is oscillating with a constant frequency. A small suction is imposed on the oscillating plate. A transverse magnetic field is also placed on the fluid. The velocity distribution has been investigated numerically with the help of finite difference method. In Chapter VI the laminar flow of Newtonian conducting fluid produced by a moving plate in presence of transverse magnetic field is investigated. The basic equation governing the motion of such flow is expressed in non-dimensional form. Analytic solution of the governing equation is obtained by Laplace transformation. Numerical solution of the dimensionless equation is also obtained with the help of Crank-Nicholson implicit scheme. Velocity profiles of the corresponding problem are shown in the graphs. The Chapter VII is also divided into two parts. In part: A, the temperature distributions of various types of parallel flow of incompressible viscous fluid have been considered. Temperature distribution near a heated plate is also discussed. Coefficients of heat transfer for various types flow has been investigated. In part: B of this chapter, we have considered unsteady MHD flows of an incompressible viscous fluid past an infinite vertical plate. The uniform flow is subject to a transverse applied magnetic field………………………………Item Solitary Wave Solutions of NLEEs in Plasma Physics and Engineering(University of Rajshahi, 2015) Khan, Md. Ashrafuzzaman; Akbar, Md. AliAlthough the modified simple equation (MSE) method effectively provides exact solitary wave solutions to nonlinear evolution equations (NLEEs) in the field of applied mathematics, mathematical physics, plasma physics and engineering, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this dissertation, we have exposed a process as to how to implement the MSE method to solve the NLEEs for balance number two. In order to verify the process, some NLEEs have been solved by means of this scheme, and we found some fresh traveling wave solutions. When the parameters receive special values, solitary wave solutions are derived from the exact traveling wave solutions and we have analyzed the solitary wave properties by the graphs of the solutions. These solitary wave solutions include soliton, kink shape soliton, singular kink shape soliton, bell shape soliton, singular bell shape soliton, anti-bell shape soliton, singular anti-bell shape soliton, etc. The attraction of the MSE method is that it is consistent, peaceful, authentic, and we found some fresh new traveling wave solutions other than the existing methods, such as, the basic (G /G) -expansion method. We emphasize the implementation of the MSE method, how to examine the solutions to NLEEs for balance number two and also compare the solutions obtained by the MSE method and the well-known existing (G /G) -expansion method. This shows the validity, usefulness, and necessity of the MSE method and our graphical representations describe the obtained traveling wave solutions.