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Some Analytical Solutions of Laminar and Incompressible Flows of Viscid Fluids

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Some Analytical Solutions of Laminar and Incompressible Flows of Viscid Fluids DSpace Institution DSpace Repository http://dspace.org Mathematics Thesis and Dissertations 2017-10-11 Some Analytical Solutions of Laminar and Incompressible Flows of Viscid Fluids Tadesse, Zenebe http://hdl.handle.net/123456789/7899 Downloaded from DSpace Repository, DSpace Institution's institutional repository Some Analytical Solutions of Laminar and Incompressible Flows of Viscid Fluids By Tadesse Zenebe Mesfin Department of Mathematics Collage of Science Bahir Dar University September, 2017 Bahirdar, Ethiopia 1 Some Analytical Solutions of Laminar and Incompressible Flows of Viscous Fluids A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics. By Tadesse Zenebe Mesfin Adivisor: Dr. Eshetu Haile Department of Mathematics Collage of Science Bahir Dar University September, 2017 Bahirdar, Ethiopia 2 The Dissertation Entitled “Some Analytical Solutions of Laminar and Incompressible Flows of Viscous Fluids” by Tadesse Zenebe is approved for the Degree of Master of Science in Mathematics. Board of Examiners Name Signature Adviser: Dr. Eshetu Haile ----------------- Examiner 1:-------------------- --------------- Examiner 2:-------------------- ------------------ Date--------------- 3 Acknowledgements First, I would like to thank Dr. Eshetu Haile for his continuous support by giving source of reading material, unreserved advice and guidance by reading my project and giving constructive comments which helped me to arrange the report. I am grateful to Tewodros higher preparatory and secondary school teachers who gave me helpful advice and comments. Finally, I want to thank information technology professional of our school for helping computer related work. i Abstract This project work is concerned about some simple analytical solutions of laminar and incompressible viscous fluid flows. Particularly it deals about the velocity distributions of some simple fluid flow problems (Couette and Poisueille) flows. Solving non-linear Navier- Stokes equations are complex and difficult. To reduce this complexity and difficulty we have considered one-dimensional, laminar and incompressible viscous fluid flow problems. we have been used mat lab codes to solve those fluid flow problems analytically. ii Table of Contents page List of Figures iv CHAPTER ONE:INTRODUCTION AND PRELIMINARIES 1 1.1 Introduction 1 1.2 Preliminaries 3 1.2.1 Definition of Basic Terminologies. 3 1.2.2 Types of Fluid Flow 5 1.2.3 The Concept of Continuum 8 1.2.4 Governing Equations 9 1.2.5 Streamlines, path lines, stream tubes and filaments 11 CHAPTER TWO:CONSERVATION LAWS 12 2.1 Lagrangian and Eulerian Description of Motion 12 2.2 Fluid Element, Fluid Particle and Their Properties 13 2.3 Mass Balance 15 2.4 Momentum Balance 16 2.5 Energy Equation 17 2.6 Viscous Stress Tensor 20 2.7 Navier-Stokes Equations 20 2.8 Summary of Equations in Conservation Form 21 CHAPTER THREE:SOLUTIONS FOR LAMINAR FLOW OF INCOMPRESSIBLE VISCOUS FLUIDS 3.1 Steady-Laminar Flow Between Two Parallel Plates In Relative Motion (Couette Flow) 22 3.2 steady State, Laminar Flow between Stationary parallel plates(poiseuille flow) 24 3.3 Axially Moving Concentric Cylinders (Couette Flow) 26 3.4 Flows between Rotating Concentric Cylinder (Couette Flow) 28 3.5 Steady Laminar Flow Through A Cylindrical Pipe. 30 3.6 Combined Couette - Poiseuille Flow between Parallel Plates 33 CONCLUSION 35 References 36 Appendix 37 iii List of figures Contents pages Fig.1 Lagrangian description of fluid motion 12 Fig.2 Eulerian description of fluid motion 13 Fig.3 Fluid element for conservation laws 13 Fig.4 Mass balance 15 Fig.5 Viscous stress aligned with the direction of coordinate axis 16 Fig.6 Forces aligned in the x- coordinate direction 17 Fig.7 Work done by surface stress in x- direction 18 Fig.8 Energy flux due to heat conduction 19 Fig.9 Fluid flow between two parallel plates in relative motion of the above plate 23 Fig.10 Velocity profile of Couette flow between parallel plates 23 Fig.11 Fluid flow between two fixed parallel plates 24 Fig.12 The velocity profile of fluid flow between two fixed parallel plates 25 Fig.13 Axially moving concentric cylinders 26 Fig.14 Velocity profile of a fluid flow between axially moving concentric cylinders 27 Fig.15 Velocity distribution between stationary outer and rotating inner cylinders 29 Fig.16 Fluid flow through a cylindrical pipe 30 Fig.17 Velocity profile for steady-laminar flow through a cylindrical pipe 32 Fig.18 Combined couette-p0iseuille flow between parallel plates 33 Fig.19 Velocity profile of combined Couette – Poiseuille flow between parallel plates 34 iv CHAPTER ONE INTRODUCTION AND PRELIMINARIES 1.1 Introduction When a fluid is flowing through a closed channel such as a pipe or between two flat plates, either laminar flow or turbulent flow may occur depending on the velocity and viscosity of the fluid. Laminar flow tends to occur at lower velocities and large fluid viscosity, below a threshold at which it becomes turbulent. Turbulent flow is a less orderly flow regime that is characterized by eddies or small packets of fluid particles which result in lateral mixing [5]. In non-scientific terms, laminar flow is smooth while turbulent flow is rough. In fluid dynamics the dimensionless Reynolds number is an important parameter in the equations that describe whether fully developed flow conditions lead to laminar or turbulent flow. In fully developed flow the velocity profile and momentum does not change in the direction of fluid flow. The phenomenon of laminar, transitional and turbulent flow regime was first investigated in an experiment by injecting some dye streaks into the flow in a glass pipe by the British engineer Osbourne Reynolds (1842–1912) did over a century ago. From the experiment he observes that the dye streak forms a straight and smooth line at low velocities when the flow is laminar, has bursts of fluctuations in the transitional regime and zigzags rapidly and randomly when the flow becomes fully turbulent. These shows in laminar flow the viscous force are much more important than the inertial forces and are usually counterbalanced by pressure or gravitational effects. If the Reynolds number is very small, much less than 1, then the fluid will exhibit Stokes or creeping flow, where the viscous forces of the fluid dominate the inertial forces. The specific calculation of the Reynolds number and the values where laminar flow occurs will depend on the geometry of the flow system and flow pattern. The common example is flow through a pipe. For such systems, laminar flow occurs when the Reynolds number is below a critical value of approximately 2,040, though the transition range is typically between 1,800 and 2,100 [2]. A common application of laminar flow is in the smooth flow of a viscous liquid through a tube or pipe. In that case, the velocity of flow varies from zero at the walls to a maximum along the 1 cross-sectional center of the vessel. The flow profile of laminar flow in a tube can be calculated by dividing the flow into thin cylindrical elements and applying the viscous force to them [4]. If the fluid is assumed to be of constant density, vanishes from the continuity equation and the second viscosity coefficient disappears, from Newton's law, the Navier- Stokes equation are not greatly simplified through, if the first viscosity µ allowed to vary with temperature and pressure (and hence with position). However, if we assume that µ is constant, many terms vanish and leaving us with a much simpler Navier-Stokes equation for constant viscosity. This is an excellent point of departure in the theory of incompressible viscous flow. In a compressible fluid, accelerations are important, and continuity is not trivial. There for density is an important variable in case of compressible fluid. A mathematical theory of fluid motion had been worked out by Leonhard Euler, Joseph Louis Lagrange, Alfred George Greenhill, C.V. Coates, Gröbli and others during the eighteenth and early nineteenth centuries. Fluid flows are governed by partial differential equations which represent conservation laws for the mass, momentum, energy and the interrelation ship between the flow variables and their evolution in time and space. The basic differential equations describing the flow of Newtonian fluids are commonly called the Navier–Stokes equations, named in honor of the French mathematician L. M. H. Navier (1785–1836) and the English Mechanician Sir G. G. Stokes (1819–1903), who were responsible for their formulation. These equations of motion provide a complete mathematical description of the flow of incompressible Newtonian fluids. In Parallel flows only one velocity component is different from zero, two-dimensional, incompressible fluid have this characteristic. Examples for which analytical solutions exist are parallel flow through a straight channel Couette flow and Hagen-poiseuille flow i.e. flow in a cylindrical pipe. 2 1.2 Preliminaries 1.2.1 Definition of Basic Terminologies. Fluid:-is any material that unable to prevent the deformation caused by a shear stress (or a it is any substance that deforms continuously when subjected to a shear stress, no matter how small). This continuing deformation in shear is the characteristics of all fluids. In a fluid at rest all shear stresses must be absent. In the Eulerian description, a fluid is an indivisible continuous material that moves through space under the action of external and internal forces. Fluid Mechanics:-is concerned with understanding, predicting, and controlling the behavior of a fluid. The field of fluid mechanics has historically been divided into two branches, fluid statics (fluid at rest) and fluid dynamics (fluid in motion). Fluid statics or hydrostatics, is concerned with the behavior of a fluid at rest or nearly so. Fluid Dynamics:-is "the branch of applied science that is concerned with the movement of liquids and gases," according to the American Heritage Dictionary.
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