Basic of Fluid Mechanics

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Basic Concepts in Fluid Mechanics

This chapter will define a fluid and introduce important concepts, like thecontinuum hypothesis and local thermodynamic equilibrium, which enable amathematical treatment of fluid flow.

1.1 The Concept of a Fluid

In order to describe what a fluid is, two points of view are introduced: the

macroscopic and the microscopic.The macroscopic point of view consists of observing matter from the sen-sorial point of view: matter consists of what we touch and what we see. It isbasically the engineering perspective.

In contrast, the microscopic point of view consists of describing matterthrough its molecular structure.

1.1.1 The Macroscopic Point of View

Experience tells us that, whereas in the solid state matter is more or lessrigid, fluids are that state of matter characterized by its endless motion anddeformation.

However, although the above observation is very common, a more rigorousdefinition is necessary. In order to formulate such a definition, let us introducethe concept of normal and shear stress.

Normal and Shear Stress

Given a surface of a body on which a force is acting, there are two types ofstresses acting on that surface: normal stress and shear (or tangential) stress.

Definition 1.1 (Normal stress). Normal stress is the force per unit areaexerted perpendicularly to the surface over which it acts.


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6 1 Basic Concepts in Fluid Mechanics

F/A

A

Fig. 1.1. Normal and shear stresses due to the force F acting on the surface A.

Definition 1.2 (Shear stress). Shear (or tangential) stress is the forceper unit area exerted tangentially to the surface over which it acts.

Remark 1.1. The SI unit of stress is the pascal Pa = N/m2. Since the pascal isa very small unit, in engineering applications the megapascal, 1 MPa = 106 Pa,is more frequently used.

Example 1.1 (Normal and shear stress). Typical examples of normal and shearstress are, respectively, pressure and friction.

Example 1.2 (Calculation of stress components). In a horizontal plane, alignedwith the x axis, there is a stress of fs = (1, 3) MPa. Calculate the normaland tangential stress.Solution. The normal stress is the component of the stress perpendicular tothe plane. In this case, the unit normal vector to the plane is n = (0, 1), sothe normal stress is

= fs n = 3 (1.1)The tangential unit vector is t = (1, 0), so the tangential projection of the

stress can be calculated as = fs t = 1 (1.2)

y

f=s

=3

=1

x

1

3{ }

Fig. 1.2. Example 1.2. Shear and normal stresses.


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1.1 The Concept of a Fluid 7

Definition of a Fluid

With the concept of shear stress at hand, we can formally define a fluid. Next,two equivalent definitions of a fluid are presented.

Definition 1.3 (Fluid). A fluid is a substance that continually deforms un-der the action of shear stress.

Definition 1.4 (Fluid). A fluid is a substance that at rest cannot withstandshear stresses.

solid

fluid

t = 0 t = t 1

t = t 2

Fig. 1.3. Behavior of a small rectangular piece of solid and fluid under the actionof shear stress.

Differences between Solids, Liquids and Gases

In order to further clarify what a fluid is, it is helpful to compare a fluid toa solid. In Fig. 1.3 one can observe that (below the elastic limit of deforma-tion) a solid subject to a shear stress deforms until it reaches the equilibrium

deformation, maintaining its shape thereafter. Once the force is removed, thesolid recovers its original shape.

Oppositely, when subjected to a shear stress, a fluid deforms continuouslyuntil the force is relieved. A fluid does not recover its original shape when theshear stress is removed. Examples of common fluids are water, oil and air.

However, the division between solids and fluids is not always clear. Thereexist substances that behave like solids when the stress acts during a shortperiod of time, but turn into fluids when the stress prolongs in time. Twoexamples of such substances are asphalt and the earths crust. There are

other solids that behave like fluids when the stress acting upon them reachesa threshold, like toothpaste, play-dough and melted cheese.

Definition 1.5 (Generalized fluid). In general, a substance that for anycondition obeys the definition of a fluid is called a (generalized) fluid.


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1.1.2 The Microscopic Point of View

The origin of the substances behavior is based on their microscopic struc-ture. Matter is formed by moving molecules, subject to various types of inter-

molecular forces. These forces maintain the bonding of molecules, and it isthe strength of these forces that distinguishes the solid and fluid (liquid orgas) states of matter.

In a solid the inter-molecular forces are strong, allowing the molecules tostay at an approximately fixed position in space. For liquids, these cohesiveforces are intermediate, weak enough to allow relative movements betweenmolecules, but strong enough to keep the relative distance constant. Liquids,when within an open container in a gravitational field, take the shape of thecontainer and form a surface of separation with the air, called free surface.Finally, in gases the inter-molecular forces are so weak as to allow a variableinter-molecular distance. Gases tend to expand and occupy all the availablevolume.

1.2 The Fluid as a Continuum

A fluid flow is characterized by the specification of fluid variables (sometimesalso called fluid properties) such as density , pressure p, temperature T,velocity vector v, chemical concentration of the component A A, and so on.But according to the molecular structure of nature, if matter is made of voidsand fast moving particles, how can we define each one of the above fluidvariables?

For example, let us examine the fluid density. For that purpose, let us takea volume of matter V, which will have a mass m. In principle, the densitycan be calculated as the ratio between the mass m and its volume V,

=m

V(1.3)

However, depending on the size of the volume V, we will find different valuesof the density.

If V is very small, lets say microscopic, due to random molecular mo-tion, we may at one time find one molecule, at others three, etc. Therefore,the value of the density will vary from one measurement to the next. Thistype of uncertainty is called microscopic uncertainty and is caused by thediscontinuous and fluctuating nature of matter.

On the other hand, if the sampling volume is very large, such as a room,statistically speaking the number of molecules inside is going to be constant.

However, due to variations of density inside the volume, the average densitymight differ from the actual density at the center of the room. This type ofuncertainty is called macroscopic uncertaintyand is caused by spatial variationof the fluid variables.


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1.2 The Fluid as a Continuum 9

VV*

Fig. 1.4. Behavior of the measured density as a function of the sample volume.

As a consequence, in order to calculate a reasonable value of the densitywe need a specific size of the sampling volume V, not too large nor tootiny. This volume also needs to contain enough molecules to be able to attainstatistically meaningful averages. Thus, the density at a point in space isdefined as

= limVV

m

V(1.4)

It has been estimated that for a stable measurement, this volume must containaround 106 molecules. Therefore, the size of V must be 3

V/ 100

where is the distance between molecules [14]. For instance, for air at ambienttemperature, a volume V of the order of 109 mm3 contains about 3 107molecules [25], a number sufficiently large to attain a correct value of density.

Furthermore, in order to be able to employ differential calculus, it willbe assumed that the above definition of density yields a continuous and con-tinuously differentiable function. The substances that are treated with thishypothesis are called continuum media and are studied in the branch of phys-ics called continuum mechanics.

In conclusion, the continuum hypothesis allows us to model discontinu-

ous matter as continuous. Certainly, every hypothesis has a range of validity.The continuum hypothesis is valid as long as a characteristic length of theflow L is much larger than that of V, i.e. 3

V


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1.3 Local Thermodynamic Equilibrium

Fluid dynamics is tightly linked to thermodynamics, which studies the equi-librium states of substances. This equilibrium implies that the properties of

matter are constant in space and time, which is rarely the case in movingfluids.However, as in the continuum hypothesis, under certain conditions it can

be assumed that each piece of fluid is in thermodynamic equilibrium. Sincemolecular collisions are the mechanism responsible for equilibrium, it canbe assumed that there exists local thermodynamic equilibrium if within acharacteristic distance for property variations L = c/|c| (where c is anythermodynamic variable) there are enough collisions [8]. This condition can beexpressed as

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Basic of Fluid Mechanics