Department of Applied Mathematics

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Laminar Flow of Incompressible Viscous Newtonian Fluid
(University of Rajshahi, 2009) Haque, Md. Abdul; Rahman, M. Zillur; Alam, M. Shamsul
The 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………………………………
A Study on Turbulence and MHD Turbulence
(University of Rajshahi, 2009) Aziz, Md. Abdul; Sarker, M. Shamsul Alam; Azad, M. Abul Kalam
Turbulent 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.
Study on Fiber Motion in Turbulent Flow
(University of Rajshahi, 2010) Ahmed, Shams Forruque; Sarker, M. Shamsul Alam
Turbulence means agitation, commotion and disturbance. This definition is, however too general and does not suffice to characterize turbulent fluid motion in the modern sense. Osborn Reynolds in the study of turbulent flows, named this type of motion "sinuous motion". The use of the word "turbulent" is to characterize a certain type of flow, namely the counterpart of streamline motion. In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Turbulence occurs nearly everywhere in nature. It is characterized by the efficient dispersion and mixing of vorticity, heat, and contaminants. In flows over solid bodies such as airplane wings or turbine blades, or in confined flows through ducts and pipelines, turbulence is responsible for increased drag and heat transfer. Turbulence is therefore a subject of great engineering interest. On the other hand, as an example of collective interaction of many coupled degrees of freedom, it is also a subject at the forefront of classical physics. Origin of turbulence is a central role in determining the state of fluid motion played by the Reynolds number. In general, a given flow undergoes a succession of instabilities with increasing Reynolds number and, at some point, turbulence appears more or less abruptly. It has long been thought that the origin of turbulence can be understood by sequentially examining the instabilities. In 1937, Taylor and Von Karman [29] gave the definition, "Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces 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. 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 coordinates. But turbulent motion is irregular in the sense that it is possible to describe it by the laws of probability. It appears possible to indicate distinct average values of various quantities, such as velocity, pressure, temperature etc. If turbulent motion were entirely irregular, it would be inaccessible to any mathematical treatment. Therefore, it is not sufficient to say that turbulence is an irregular motion. According to J.O. Hinze [11], the turbulent flow is "Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation with space and time coordinates, so that statistically only distinct average values can be discerned." The addition "with space and time coordinates" is necessary; it 1s 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. Again, turbulent motion is not irregular in space alone, because a steady flow with an irregular flow pattern might then come under the definition of turbulence. According to the definition of Taylor and Von Karman [29] there are two distinct types of turbulence, wall turbulence and free turbulence. Wall Turbulence: Turbulence generated by a viscous effect due to presence of a solid wall is designated by wall turbulence. Free Turbulence: Turbulence in the absence of wall generated by the flow of layers of fluids at different velocities is called free turbulence. Turbulent flow occurs in our daily life. If we observe the smoke rising out of a chimney of a factory or a cigarette, we find that upto a certain length from the chimney or the cigarette, the smoke has a regular shape and after that its shape becomes irregular and if we see still farther then the smoke becomes completely irregular. Again, if a drop of ink is dropped in a glass of water, we find a similar phenomenon, i.e, a regular ink thread falling for a short distance after which it spreads and a vortex type motion can be observed. Ultimately the thread splits into several vortices and motion becomes irregular. The flows with such irregular motions are usually called turbulent flows. Turbulent flow also occurs in large arteries at branch points, m diseased and narrowed (stenotic) arteries and across stenotic heart valves………………………………………..
Massive Particle Tunneling from Black Hole Spacetime
(University of Rajshahi, 2013) Hossain, Md. Ilias; Rahman, M. Atiqur
We 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.
A Study on Turbulent and Magneto-hydrodynamic turbulent Flow in Incompressible Fluid
(University of Rajshahi, 2014) Mumtahinah, Mst.; Azad, Md. Abul Kalam
The first chapter is a general introductory chapter and gives the general idea of turbulence, distribution functions and their principal concepts. Some results and theories which are needed in the subsequent chapters have been included in this chapter. Types and examples of turbulence, different stages of Reynolds number, Reynolds equation, averaging rules, Coriolis effect etc have been briefly discussed. Distribution functions, Joint distribution functions, equation of motion of dust particles, spectral representation of turbulence and Fourier Transformation of the Navier-Stockes equation have also been discussed. Lastly, a brief review of the past researchers related to this thesis have also been studied in this chapter. Throughout the work we have considered the flow of fluids to be isotropic and homogeneous. The notions generally adopted are those used by Taylor, Vonkarman, Hinze, Reynolds, Deissler, Sarker, Kisore, Batchelor, Coriolis and Lundgren. The Second chapter consist of two parts. In part A, we have studied the decay of temperature fluctuations in dusty fluid homogeneous turbulence prior to the final period considering correlations between fluctuating quantities at two- and three- point. In this part we have tried to solve the correlation equations by converting it to spectral form by taking their Fourier transform. Lastly, by integrating the energy spectrum over all wave numbers, the energy decay law of temperature fluctuations in homogeneous turbulence before the final period in presence of dust particle is obtained. In part B, we have studied the decay of temperature fluctuations in dusty fluid homogeneous turbulence before the final period in presence of Coriolis force and have considered correlations between fluctuating quantities at two- and three- points by neglecting the fourth order correlation in comparison to the second and third order correlations. The correlation equations for two- and three- point in a rotating system in presence of dust particles are obtained and these equations are converted to spectral form by taking their Fourier transforms. Finally by integrating the energy spectrum over all wave numbers, the energy decay law of temperature fluctuations in homogeneous dusty fluid turbulence before the final period in presence of Coriolis force is obtained. The Third chapter consists of two parts. In part A, we have studied the joint distribution functions for simultaneous velocity, temperature, concentration fields in turbulent flow undergoing a first order reaction in presence of Coriolis force. The various properties of the constructed joint distribution functions have been discussed. In this chapter we have tried to derive the transport equations for one and two point joint distribution functions of velocity, temperature, concentration in convective turbulent flow due to first order reaction in presence of coriolis force. In part B, we have an attempt to derive the transport equation for the joint distribution function of certain variables in convective turbulent flow undergoing a first order reaction in a rotating system in presence of dust particles. Equations for the evolution of one- point and two- point joint distribution function for velocity, temperature and concentration in convective turbulent flow field undergoing first- order reaction in a rotating system in presence of dust particles have been derived. Finally we have made a result with comparison of the equation for one- point distribution function in the case of zero coriolis force in the absence of the dust particles and negligible diffusivity. In Chapter four, we have studied the statistical theory of certain variables for three- point distribution functions in MHD turbulent flow in a rotating system in presence of dust particles. In this chapter we have made an attempt to derive the transport equations for evolution of distribution functions for simultaneous velocity, magnetic, temperature and concentration fields in MHD turbulent flow due to Coriolis force in presence of dust particles and various properties of the distribution function have been discussed. In Chapter five, we have made an attempt to discuss the summary about the whole thesis.
Solitary Wave Solutions of NLEEs in Plasma Physics and Engineering
(University of Rajshahi, 2015) Khan, Md. Ashrafuzzaman; Akbar, Md. Ali
Although 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.
Interaction Phenomena of Nonlinear Waves in Unmagnetized Plasmas
(University of Rajshahi, Rajshahi, 2019) Alam, Mohammad Shah; Talukder, Mamunur Rashid; Ali, M. Hossain
This 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.-------
Intrinsic Features of Nonlinear Waves in Dusty Plasmas
(University of Rajshahi, 2024) Salam, Md. Abdus; Ali, M. Zulfikar; Akber, Md. Ali
Dusty 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.