Fluid Mechanics: Analytical Methods

Preface

Mathematical physics was brought into existence by the development of mechanics. It originated in the study of the planetary motions and of the falling of heavy bodies, which led Newton to formulate the fundamental laws of mechanics, as early as 1687. Even though the mechanics of continuous media, first as solid mechanics, and later as fluid mechanics, is a more recent development, its roots can be found in Isaac Newton’s Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), several pages of which are dedicated to the falling streams of liquid.

Applications of fluid mechanics to irrigation problems date back to Antiquity, but the subject gained a key status during the industrial revolution. Energetics was vital to the development of knowledge-demanding, specialized industrial areas such as fluid supply, heat engineering, secondary energy production or propulsion. Either as a carrier of sensible heat or as the core of energy production processes, fluid is ubiquitous in all the high-technology industries of the century: aeronautics, aerospace, automotive, industrial combustion, thermal or hydroelectric power plants, processing industries, national defense, thermal and acoustic environment, etc.

Depending on the target audience, there are various approaches to fluid mechanics. Covering this diversity is what we are striving for in this book.

Whatever the degree of difficulty of the approached subject, it is important for the reader to reflect on it while being fully aware of the laws to be written in one form or another. Various approaches to fluid mechanics are illustrated by examples in this book.

First of all, the student will have the opportunity to handle simplified tools, providing him/her with a convenient first approach of the subject. On the other hand, the practitioner will be provided with elementary dimensioning means.

Other problems may justify or require a more complex approach, involving more significant theoretical knowledge, in particular of calculus. This is once again a point on which students and practitioners who already master these subjects can converge.

A third approach, which is essential for today’s physics, especially when dealing with problems that are too complex to be accurately solved by simple calculations, resorts to numerical methods. This book illustrates these remarks.

Problem resolution relies in each chapter on reviews of fundamental notions. These reviews are not exhaustive, and the reader may find it useful to go back to textbooks for knowledge consolidation. Nevertheless, certain proofs referring to important points are resumed. As already mentioned, what matters is that the reader has a good grasp of what he/she writes.

Given that we target wide audiences, the deduction or review of general equations can be found in the appendices, to avoid the book becoming too cumbersome.

The attempt to effectively address audiences with widely varied levels of knowledge, expertise or experience in the field may seem an impossible task.

Drawing on their experience of teaching all these categories of audiences, the authors felt motivated and encouraged to engage in this daring enterprise.

This volume gathers examples of relatively simple approaches to academic problems as well as practical ones. In principle, this work is accessible to all potential readers.

The first chapter recalls the basis of dynamics by focusing on the mechanics of point power. Both the state of fluidity, as well as the main properties of fluids are defined. The problems for writing force, surface and volume when applied to a fluid volume, are discussed. Finally, a strategy for resolving problems in mechanics is approached from a general point of view.

The second chapter covers fluid in equilibrium. The study of incompressible fluid statics under simple forces of gravity or hydrostatics, is completed by that of other forces derived from a potential, such as inertia forces. Compressible fluid statics are also covered.

The third chapter is dedicated to describing flows. The Eulerian vision is favored here. The geometric elements of kinematics are defined. The geometry of flows is established, based on the data of a flow’s Eulerian speed. This chapter also provides the opportunity for a first physics principle to be outlined and developed: the principle of continuity.

The first chapter examines the structure of surface forces. This is also where one will find a definition of perfect fluids where the action of viscosity can be overlooked. The fourth chapter is dedicated to processing these flows, in which the Bernoulli theorem is central. Although this theorem is limited by the underlying hypotheses, its strength is observed in how easily one can obtain pertinent orders of magnitude in a large range of phenomena.

When the speed of a fluid varies significantly in a confined space, which is an instance of the barrier between fluids and solids, viscosity becomes a major phenomenon. This is particularly found in pipelines and all components of a hydraulic circuit. In such a situation, one is often only concerned with the loss of mechanical energy that results from fluid friction. This ‘head loss’ will be calculated in the fifth chapter.

As a general rule, propulsion studies result from a momentum exchange between fluid and a wall. Euler’s theorems apply to both perfect flows and viscous fluids and allow one to determine, with a simple knowledge of kinematic fluid passing through boundaries, the resulting moments of a system of forces when applied to a fluid. The sixth chapter will demonstrate how this powerful tool can be applied to determine different types of thrusting.

This work is aimed at students enrolled in engineering schools and technical colleges or in University Bachelors or Masters programs. It is also meant to be useful to the professionals whose activity requires knowledge or mastery of tools related to fluid mechanics.

Michel LEDOUX

Abdelkhalak EL HAMI

November 2016

1

Mechanics and Fluid

1.1. Introduction

The mechanics of fluids is a type of mechanics: it looks at the movement of matter when under the influence of forces. Matter here is in the fluid state.

This chapter is approached from the perspective of the foundations of the mechanics of point power. It will also later define what fluid is and which of this matter’s main characteristics are useful to know. These characteristics shall then be brought to life in later chapters.

1.1.1. Mechanics: what to remember

1.1.1.1. Who is afraid of mechanics?

For some curious reason, this branch of physics appears frightening to many students, a curse that thermodynamics also shares. Somewhat recoiled from, the mechanical engineer occupies a special place in the academic world. Some people even wonder whether mechanical engineers are actually physicists who have a strong handle on mathematics, or are in fact mathematicians lost among physicists. These classifications have not been made any simpler by the addition of digital calculations.

It cannot be stressed enough that the appearance of mechanics gave birth to mathematical physics.

By pairing movement with mathematics, the Neoplanitician, Galileo, created kinematics. And then, with a stroke of genius, although perhaps slightly mythically, Isaac Newton created dynamics by incorporating the fall of an apple and the Moon’s trajectory into one vision.

Descartes must not be left out of this Pantheon of emerging physics, for he created momentum, was engaged in heated debates with Newton and Leibnitz on this subject as well as others, and discovered kinetic energy through life force. Leibnitz and Newton were also the precursors to the differential approach in mechanics.

1.1.1.2. Principles to remember

Like a game of chess, the starting rules of mechanics are the simplest. And, like a game of chess, not all paths lead to an easy victory.

a) Remember that a position vector is defined as a vector that links the starting point to another point in space. The coordinates of are evidently the point’s three coordinates:

[1.1]

By definition, the point’s speed is the derivative of the position vector in relation to the time:

[1.2]

which, when passing, accelerates the position vector’s second derivative:

[1.3]

Remember that a vector is derived with regard to a scalar by deriving its components:

[1.4]

b) In 1687, Isaac Newton’s Philosophiae Naturalis Principia Mathematica outlined three laws, which indeed can be reduced into two:

1) The principle of inertia;

2) Fundamental dynamics law;

3) The principle of action and reaction.

Let us take these three principles further:

Law no. 2. Let us begin with the fundamental dynamics principle, when applied to a constant mass (m) material point:

The acceleration that a body undergoes in an inertial frame of reference is proportional to the resulting forces that it undergoes, and is inversely proportional to its mass.

In modern notation (the notion of the vector was acquired in the 20th Century), this is written as:

[1.5]

NOTE: Vectorial notation reminds us that a given speed contains three pieces of information: a direction (instantaneous movement support), a route and an hourly speed. A speed cannot be reduced to the datum of m.s–1. A speed vector not only tells me that my car is traveling at V = 130 km.hr−1 (hourly speed), but it also tells me that I am on a highway between Paris and Rome (direction) and that I am going from Paris to Rome (route). However, I would still need the position vector to tell me where the next exit is.

Therefore, an acceleration is also a vector, and there is no reason why it is not collinear to the speed. Central acceleration in a circular movement is (or should be) known to all secondary school students.

Law no. 1. The principle of inertia was actually discovered by Galileo: In the absence of an external force, all material points continue in a uniform, straight-lined movement.

NOTE: This is what Captain Haddock realizes in the Explorers on the Moon, the illustrated Tintin adventure story by the famous Belgian author, Hergé.

This principle of inertia is in fact a consequence of the fundamental dynamics principle. If the result of forces applied to a material point is zero, then:

[1.6]

and:

[1.7]

[1.8]

It means a uniform straight-lined movement.

Law no. 3. If the first principle can be reduced to the second, the third principle of action and reaction is independent: Every body A exerting a force on a body B undergoes a force of equal intensity, but in the opposite direction, exerted by body B:

[1.9]

When solving a problem, to write that every force has an equal and opposite reaction is to write something new with regard to the fundamental dynamics principle.

These principles have been rewritten in various different forms, which lead to equations that are often much more directly applicable. A few of these equations are given in the following sections.

1.1.2. Momentum theorem

We can rewrite the fundamental dynamics principle by noting that mass is invariable:

[1.10]

A momentum vector has also been introduced:

[1.11]

And the fundamental dynamics principle is also found to be rewritten in terms of momentum:

[1.12]

In the course of mechanics, it is demonstrated that this equation applies in material points to the center of a group’s mass, whether it is continuous or discontinuous and alterable or otherwise. m is therefore replaced by the total mass of the system’s points and then represents the resultant of the forces applied to these points. This is what constitutes the center of mass theorem.

NOTE: It goes without saying that we do not intend to write a digest here on the course of fluid mechanics.

It would be impossible to attempt to reproduce a complete mechanics course. However, we must insist upon the consequences of these principles which will be directly applied when establishing fluid mechanics theorems. We will build upon the mechanics of point power, and if the reader deems it necessary, they can refer to a dedicated textbook to study system mechanics, which constitutes a more complex domain. Furthermore, in the appendix, we can find a reminder of fluid mechanics equations for a continuous fluid system. This script will be used when demonstrating Euler’s first theorem.

We observe that while mass becomes variable with speed, it is this expression that remains valid in particular mechanics. This is also the case for relativist dynamics.

1.1.3. Kinetic energy theorem

Forced movement implies work. Here we will give mechanics an energetic dimension. The work of a force when applied to a material point during a time dt provides calculated work from the force and this point’s small movement :

[1.13]

is a small vector, which indicates not only the small distance traveled, but also the carrying line of this movement or direction, and the movement’s route. It is linked to speed by:

[1.14]

Remember the dynamic relation:

[1.15]

The work is written as:

[1.16]

It can be observed that

[1.17]

Finally, it becomes:

[1.18]

The work performed has helped to increase the quantity carried by the material point. This is how kinetic energy appears:

[1.19]

1.1.4. Forces deriving from a potential

In a frame of reference Oxyz, where Oz is vertical, the force of gravity applied to a mass of m = 1 kg will have the following components:

[1.20.a]

[1.20.b]

[1.20.c]

Furthermore, the operating gradient is defined by associating the vector with a function f(x, y z) by:

[1.21.a]

[1.21.b]

[1.21.c]

Therefore, can be written in the form of a gradient:

[1.22]

which, by definition, implies the following about the gradient:

[1.23.a]

[1.23.b]

[1.23.c]

By identifying:

[1.24]

Therefore, it can be said that is derived from the potential ϕG. It is worth at least being aware of this.

In general terms, it is said that a force is derived from a potential ϕ(x, y, z) when

[1.25]

This property is not universal: in particular, friction forces or electromagnetic forces are not derived from a potential.

1.1.5. Conserving the energy of a material point

The work performed by a force derived from a potential during a time period of dt is written as:

[1.26]

By developing the scalar product, this can be rewritten in the Cartesian form:

[1.27]

The exact total differential is seen to appear ϕ on the time dt, meaning the variation dϕ between the starting point at t and the arrival point at t + dt:

[1.28]

By coupling the equations together, we obtain:

[1.29]

which can be rewritten as:

[1.30]

Thus, the total energy appears:

[1.31]

Sum of the kinetic energy and the potential energy, which is conserved when the material point is moving.

NOTE: Remember that when part or all of the forces is or are not derived from a potential, the mechanical energy of the material point is not conserved. The mechanical work of the forces which is not derived from a potential is generally transformed into another form of energy. Thus, friction transforms mechanical energy into thermal energy. This enters into the domain of thermodynamics. The mechanical energy (work) is no longer conserved, but the first principle applies to the two forms of energy: work and heat.

These relations for the material point recalled here have been extended into finite volumes of matter. Curious readers may refer to more elaborate mechanical courses. The aim of this chapter lies in the need for the readers to place themselves within the framework of a basic general culture of mechanics.

All of the notions that have been recalled here will be useful when we begin interpreting Bernoulli’s theorem.

1.2. The fluid state

The term fluid state refers here to the way in which all of the states of matter used to be understood: solid, liquid, gas and plasma, a classification that has been recognized more recently.

In this group, fluid mechanics applies to the last three of these states.

Solid mechanics deals with alterable and unalterable elastic solids with a blurred boundary and a few creep or pasty rheology problems.

NOTE: It is important not to confuse this expression, which can be traced back to the oldest fluid state, with the notion of state in thermodynamics, which relates to a set of thermodynamic variables which we will discuss later.

When approached from the mechanics perspective, this fluid state prompts us to:

a) define this state in terms of its nature, its physical qualities and its movements;

b) describe the forces that can be applied to a fluid: what are they and how are they written?

1.2.1. Fluid properties

1.2.1.1. The first property of fluid is its continuity

Physically, continuity signifies that fluid density, regardless of how small it may be, contains matter. This allows a density to be defined, like the ratio of a small fluid density dm to the small volume dω that it occupies:

[1.32]

For those who like mathematics, we observe that physical continuity connects a notion of continuity for the mass occupying a given volume. This mass dm(dω) also has a derivative called density. In mathematical terms, the expression is:

[1.33]

NOTE: Herein lies a paradox. The mechanical engineer attributes this continuity property to fluid. We know that at the smallest scale of physics, matter is not continuous. Moreover, if there was no fluid discontinuity at the molecular level, we would not be able to determine its essential properties: possible compressibility, existence of pressure and temperature, thermal conduction and matter diffusivity when mixed.

A paradox is merely a poorly asked question. There are at least six or seven orders of magnitude (powers of 10) between the molecular phenomena and the mechanics of a fluids physicist. Admittedly, continuity is just a modeling tool, but it is robust. At the pipeline level, everything happens as if the fluid was continuous.

1.2.1.2. Compressibility

Density has been defined as a local property. There are many cases where this value of ρ is constant in all fluids. Therefore, it can be said that fluid is incompressible. This will be our definition of incompressibility here. Incompressible is synonymous with ρ = Cte.

There are other cases where the density varies from one fluid point to another. Therefore, it can be said that fluid is compressible. This situation is mainly concerned with gas. But the compressibility of liquids may cause certain problems: there is writing on static fluids at the deepest