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Overview
An introductory chapter covers fundamental notions from the continuum hypothesis to steady-state flow. Succeeding chapters explore conservation of mass, forces acting on a fluid in equilibrium, dynamic equations of motion, irrotational motion, integration of Euler's equation in special cases, and flows representable by harmonic functions. Additional topics include two dimensional flows, rectilinear vortices, general vortex motion, flows with a free boundary, and compressible fluids.
Product Details
| ISBN-13: | 9780486812830 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 05/13/2016 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 256 |
| File size: | 34 MB |
| Note: | This product may take a few minutes to download. |
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Read an Excerpt
Introduction to Fluid Dynamics
By Edward B. McLeod Jr.
Dover Publications, Inc.
Copyright © 2016 Edward B. McLeod, Jr.All rights reserved.
ISBN: 978-0-486-81283-0
CHAPTER 1
FUNDAMENTAL NOTIONS
1. The continuum hypothesis
All matter is made up of elementary molecules which are in a continual state of agitation. A molecule is the smallest part of a substance which possesses identifiable chemical and physical properties. The molecule is too small to be observed by any optical instrument in use at the present time. A cubic centimeter of air contains something of the order 2.7 X 1019 molecules. At a given instant of time a region containing matter contains points which are on a molecule and also points which are in empty space.
It simplifies many physical investigations to assume that certain material mediums are continua. By a continuum we mean a medium in which every sub-region however small is completely filled with matter. In this text we shall be concerned with problems for which fluids can be treated as continua. In order that our continuum hypothesis provides an accurate description of the behaviour of the medium under investigation it is necessary that very small volumes contain very large numbers of molecules and that the molecules move in paths that differ very little from the mean path. The continuum hypothesis can be applied effectively in most of the problems in classical hydrodynamics and also in many problems associated with the flow of gases. However, we must resort to the application of the theory of probability to the motion of molecules in order to attack problems which occur in the theory of rarefied gases or problems associated with the diffusion of gases and liquids. A further account of the justification for treating fluids as a continuum can be found in reference 4 at the end of the chapter.
The molecular or microscopic theory of matter is useful in the theoretical calculation of such common physical quantities such as viscosity, elastic moduli, etc. The use of the microscopic rather than the macroscopic theory of matter provides a rational rather than an empirical basis for determining these physical constants.
2. Density in a continuous medium
Let R be a region of volume, V, filled with matter having a total mass, M. We shall define the average density of R to be the ratio, M/V. We shall define the density of a point in a continuum by the following process:
Let S be any Sub-region of R which contains the point P in its interior; then the density at the point P is the limit of the average density of S as S shrinks to P.
The density is a point function, e, defined at every interior point of R. The above definition does not apply to a region filled with molecules since the above limiting process would cause the ratio M/V to approach infinity or zero, depending on whether or not the point, P, is contained within a molecule; in such a case the density must be some kind of function which involves the probability that the point P is contained within a molecule. In either case the density is a function whose volume integral taken over R is equal to the total mass contained in R. If the continuum hypothesis is accurate, it is necessary that the average density of the sub-region S, when S has small macroscopic dimensions be nearly equal to the density at P defined by the microscopic theory.
3. Stress in a continuous medium
The forces that act on a body of continuously distributed matter are either body forces or surface forces. The former is a type of force which is proportional to the mass (or possibly the volume) of the body on which it acts. Gravity is a common example of a body force. A surface force is one which acts on the boundary of the body and is proportional to the surface area.
Consider now the force that acts on a segment of a material plane contained within a body of continuously distributed matter. If the body is in equilibrium there is a force acting on one side of the plane which is equal in magnitude and opposite in direction to that on the other side; let F denote this force vector, and let n denote the unit vector normal to the plane on the side on which F acts; let A denote the area of the plane segment; and let P be a point on the plane segment. We define the stress to be the limit of the ratio, F/A, as the plane segment shrinks to the point, P. The stress is a vector, S which depends on P and n. It can be shown that the stress vector is not arbitrarily prescribable as a function of n, but is instead a linear vector function of n which can be represented by a dyadic, or tensor. We shall not have the occasion in this text to use the stress vector in its general form.
If the stress vector is in the same direction as n it is said to be a tension. If it is oppositely opposed to n, it is called a compression. If the stress vector lies within the plane whose normal is n, it is called a shear. We can resolve any stress into normal and shear components.
4. Definition of a fluid
A fluid is characterized by the following definition:
A fluid is a continuous medium which has the property that when it is in equilibrium, the stress at every point is one of compression.
Consequently we may assume that the stress at a point, P has the form, — pn, where p is a positive scalar that depends on the point P and the direction, n. We shall show in the third chapter that p is actually independent of n.
5. Liquids and gases
Before giving a general definition of a liquid and a gas let us consider how water and air behave when confined within a container having rigid boundaries. When water is poured into a tank, a portion of the medium is bounded by the rigid walls of the container but there is also a horizontal surface which is not in contact with the rigid boundary; such surface is called a free surface. On the other hand if air is put into a closed tank it will expand so that the tank is completely filled with air, regardless of the size or shape of the tank.
We adapt these basic properties for the following definitions:
(1) A gas is a fluid which when confined in a bounded region having rigid boundaries,expands to fill its container.
(2) A fluid is said to be a liquid if it is possible to Confine it in such a way that it exhibits a free boundary surface while it is in equilibrium.
When the stress which acts on a fluid is increased, a decrease in volume can be observed. All fluids are to some degree Compressible. In liquids the compressibility is so slight that we may usually regard them as Incompressible. At very great ocean depths the density is considerably more than it is near the surface and therefore the compressibility in this case is not negligible. Also the compressibility of liquids is a significant feature when we are investigating the propagation of sound waves through liquids.
Gases, however, show very large changes of density when stress is applied, so that compressibility generally cannot be neglected. We shall demonstrate in a later chapter that a gas may be regarded as incompressible when the magnitude of the velocity of flow is small compared to the velocity of sound.
In contrast to fluids we shall call a continuous medium capable of exhibiting a tensile, or shearing stress while it is in equilibrium a solid. Solids experience various different types of deformations when stress is applied. A solid which does not deform at all under stress is called a rigid body. An absolutely rigid body does not exist in nature, but in some instances a rigid body can be used as a convenient mathematical model.
When a solid is stressed it may deform in such a way that it will assume its original form after the stress is removed. Such a deformation is called an elastic deformation. It may happen that after a body is stressed beyond a certain limit equilibrium can still be maintained but when the load is removed the body assumes a shape different from its original shape. This type of deformation is called Strain-hardening. Strain-hardening is a type of plastic deformation. The maximum stress for which the deformation is elastic is called the yield-point. There is also the possibility that when the yield-point is exceeded, the medium does not remain in equilibrium but a flow occurs. When the body is unloaded so that the stress is again below the yield-point, the body will again harden and assume a state of elastic equilibrium. This type of deformation is said to be perfectly-plastic. Types of deformation in solids are described in detail in reference 2.
Fluids in contrast, have a zero yield-point. A fluid in equilibrium will begin to flow if a deviating stress of arbitrarily small magnitude is applied.
There are materials that exhibit both solid and fluid properties. The type of deformation depends on the rate at which stress is applied. The general study of deformation is called rheology.
6. The three states of matter
For a continuum, the three states of matter are defined by the manner in which the medium reacts to stress. For media containing molecules the solid, liquid or gaseous state is determined by the manner in which the molecules are moving. A medium can be made to change from one state to another by appropriate changes of temperature.
In the solid state the molecules occupy fixed positions, and are held in equilibrium by the balance between Newtonian attractive forces and electrostatic repulsive forces. The atoms within the molecules are in a state of vibration.
In the liquid state there is a tendency for molecules to adhere to one another, but they are also free to slide over one another. The molecules are known to migrate in clusters.
In a gas the molecules move freely in space but are continually colliding with one another. When the gas is in equilibrium the mean velocity of the molecules is zero.
7. Temperature
Temperature is a quantitative measurement of "warm" and "cold" sensations which the human nervous system experiences. The centigrade scale is conveniently chosen so that the freezing point of water is 0° C and the boiling point of water is 100° C. Gases are known to expand or contract with a raising or lowering of the temperature. It can be deduced that a so-called ideal gas will contract to zero-volume, or infinite density if the temperature is approximately — 270° C. This point may be taken to be the zero point of an absolute temperature scale.
The microscopic theory of molecules yields a theoretical meaning of temperature, while for the macroscopic or continuum theory temperature is merely an empirical parameter. The temperature of a fluid is proportional to the average kinetic energy of its moving molecules.
8. Viscosity
The definition of a fluid which we have stated implies that there is a possibility that a shearing stress can occur when the fluid is in motion. This dissipative phenomenon or frictional resistance between the particles is called viscosity. It has been demonstrated experimentally that this resistance is an increasing function of the relative velocity of the particles. The motion of a sufficiently small body of flowing fluid can be resolved into a component that resembles a rigid body motion, one called the rate of strain, and one whose smallness is of a higher order of magnitude than the dimensions of the body. More precisely we may say that the viscosity is a parameter which increases when the rate-of-strain is increased.
It is of interest to remark that in liquids the viscosity decreases as the temperature increases, but the reverse situation is true for gases. These facts are readily explained by the microscopic theory. In liquids the energy dissipation is caused by the tendency of the molecules to stick to one another, while in gases energy is lost in the collision of the molecules.
To treat the theory of viscous fluids in a general way it is desirable to use tensor analysis as a principle tool. In this introductory text we shall consider only fluid flows for which the viscosity is zero. These are called perfect fluids.
9. Perfect fluids
We shall adopt the following definition:
A "perfect" fluid is one in which the stress at each point is one of compression only, whether the fluid is in motion or in equilibrium.
A perfect fluid is an idealization which assumes shearing stress to be absent at all times. The terms ideal fluid, inviscid fluid and non-viscous fluid are also in common usage.
The assumption of the absence of viscosity in many cases leads to accurate predictions. For example in the study of the flow of air about an airplane wing, viscosity must be considered in the immediate neighborhood of the wing surface, but can be neglected with small error beyond a distance that is very small compared to the dimensions of the wing. The region where viscosity must be considered is called the boundary layer. Outside the boundary layer we may apply the perfect fluid hypothesis and in most cases neglect the dimensions of the boundary layer. A detailed account of the validity of the theory of perfect fluids is given in the first chapter of reference 1.
10. The velocity field
In the motion of any continuous medium we must have a method of identifying the particles at each instant of time. By a Particle we mean a point whose position vector is given by a vector relation of the form
r = r(r0, t);
r = r(r0, t0).
A particle is so to speak a point which moves with the medium and is identifiable by a mathematical equation. The above equations determine at a time, t, the position of the particle which occupied the position r0at time t0. A space curve each of whose points is a particle moving with the medium will be called a Material Curve. We may define similarly Material Surfaces and Material regions. A region which consists of the same particles will be called a body.
The motion of a continuous medium is mathematically a one-to-one correspondence between a Euclidean space determined at a time t and another Euclidean space at time t. We shall assume that r is a twice continuously differentiable function of t, so that each particle may have a well defined velocity and acceleration. This process of identifying each particle by its position vector is called the Lagrange representation. A vector equation of this form, usually determined from auxiliary physical consideration provides us with a method of determining the position of every particle at every instant of time.
The Lagrangian representation actually yields more information about the particles than is usually necessary, and except for a few special cases involves an excessive amount of numerical calculation.
It is usually sufficient to find the velocity of the particle which occupies the position, r at time t. This representation, called the Euler representation expresses the velocity q as a vector function of r and t. This representation is used almost exclusively in fluid dynamics.
It will be assumed that the reader has had some experience with elementary operations on vector quantities. A brief outline of some of the prerequisite material is provided in the appendix following the last chapter of the text. This appendix consists of the mathematical material which is desirable for a course of study in classical mechanics. To understand the dynamical behaviour of a continuum it is necessary to have a working knowledge of the dynamics of a system of particles.
If a vector, F is a function of a point in three dimensional Euclidean space, it is called a vector field. The velocity vector q of a continuous medium determined by the Euler representation is a vector field which we shall call the velocity field, where the velocity vector, q, is determined by an equation of the form,
q = q(r,t).
For a given instant of time the Euler representation gives a method of determining the velocity of the particle at each position r, but in itself does not provide a permanent record of the behaviour of any particular particle.
11. Streamlines and paths of particles
A space curve that has the property that at some instant of time it is tangent everywhere to a velocity vector associated with some continuum is called a stream-line. (Analogous so-called vector lines may be defined for general vector fields.) If the velocity field is known we can determine the vector equation of any particular streamline (passing through a prescribed point r at a prescribed time t0) by solving the vector differential equation
dr/dθ x q(r,t0) = 0, (1.1)
where θ, is an auxiliary parameter chosen in such a way that r = r0 when θ = θ0, and is a fixed t0 parameter.
(Continues...)
Excerpted from Introduction to Fluid Dynamics by Edward B. McLeod Jr.. Copyright © 2016 Edward B. McLeod, Jr.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Contents
I. FUNDAMENTAL NOTIONS,II. CONSERVATION OF MASS,
III. FORCES ACTING ON A FLUID IN EQUILIBRIUM,
IV. DYNAMIC EQUATIONS OF MOTION,
V. IRROTATIONAL MOTION,
VI. INTEGRATION OF EULER'S EQUATION IN SPECIAL CASES,
VII. FLOWS REPRESENTABLE BY HARMONIC FUNCTIONS,
VIII. TWO DIMENSIONAL FLOWS,
IX. RECTILINEAR VORTICES,
X. GENERAL VORTEX MOTION,
XI. FLOWS WITH A FREE BOUNDARY,
XII. COMPRESSIBLE FLUIDS,
APPENDIX,