Peube J-L. Fundamentals of fluid mechanics and transport phenomena

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Physics of Energetic Systems in Flow 113

Figure 3.2. Momentum balance: (a) closed system; rocket + gas,

(b) open system; rocket alone

At the instant

t + dt

the rocket mass has changed from

m + dm

, whereas the

velocity in the fixed reference frame has increased from

to ,

VdVand the mass

–dm

of gas has been ejected. The mass ejected, initially moving at a velocity

has

now changed to



at time

t + dt

in the fixed reference frame. Between these

two instants, the momentum variation of the mass

can be calculated as:











dmUVdmVmUVdmVdVdmmpd

 

[3.15]

Equality [3.15] highlights:

 the change in momentum





Vmdpd

of the rocket (of variable mass











VmdVmVdVdmmpd



 the momentum

of the mass –

ejected from the rocket:





UVdmdp



We have, per unit time:







Vmd

   [3.16]

114 Fundamentals of Fluid Mechanics and Transport Phenomena

The time derivative of the momentum

of the rocket is not equal to the

quantity of acceleration

The quantity

dtpd

is associated with a matter which is constantly being

generated at the exit of the rocket exhaust: it is a

momentum flux.

In order to simplify matters, we will consider that the only external force is

constituted by gravity; taking account of [3.16], equation [3.10] for the momentum

at constant mass

can be written:



or:

gmqU



[3.17]

Equation [3.17] shows that we can apply the fundamental dynamic relation to a

system of variable mass by considering the

quantity of acceleration

for that system,

the ejection of burned gases generating a force on the rocket opposed to the

momentum flux

(relative to the rocket) generated by the propulsion system.

We can verify that the opposite of this force, acting for a time dt, corresponds to

the impulsion

dtqU

necessary to give to the mass

dtq

ejected in time

variation in velocity equal to .

This force is the

rocket thrust

The preceding balance can also be written in the non-Galilean reference frame of

the rocket (whose acceleration is

). In this reference frame, the rocket ejects

momentum

per unit time (momentum flux) which balances the inertial force

 and the gravitational force

Physics of Energetic Systems in Flow 115

A momentum flux is associated with the existence of a force. This observation is

the basis of many concepts in fluid mechanics. We will see it again when we study

the momentum flux theorem.

The preceding results can be extended to moments (of momentum, or of

quantities of acceleration) taken about a fixed point O or about the inertia center of

the system.

3.2.4.3.

Kinetic energy balance

The variation of kinetic energy

of a system comprising mass

can be

calculated in a similar manner to that used to calculate the momentum. We have:







222

UVq

By taking the scalar product of equation [3.17] with the velocity

we find the

kinetic energy equation of the rocket:

VgmqVU



This gives, for the kinetic energy variation

Vgm

 [3.18]

The term

Vgm

is the power of the gravitational force. The power

fint

of the

internal forces is here equal to

which represents the quantity of kinetic

energy per unit time created by combustion in the fixed reference frame of the

rocket where chemical energy (reaction enthalpy) is transformed into mechanical

energy.

This preceding reasoning will be used in a more general manner in Chapter 4,

where the balance equations in a moving fluid will be established. We will use

Eulerian variables in order to clearly explain the phenomena associated with the

concept of convection.

116 Fundamentals of Fluid Mechanics and Transport Phenomena

3.2.5. Thermodynamics of a system in motion

3.2.5.1. Energy equation

Consider a material system, described by means of the variables of a certain

number p of sub-systems. The internal energy of each component is the energy of its

matter in a reference frame in which it is at rest. The differential [3.5] dE of the

internal energy can be expressed as a function of the extensive quantities X

and the

intensive quantities Y

previously defined:





iiii

dXYdSTdE

Now suppose that the system components are in motion. The total energy

contained in the system must take account of all energy forms, including its kinetic

energy E

. In other words, the mechanical properties constitute just one particular

aspect of the physical phenomena to be considered. We denote as E

this “total”

energy in order to distinguish it from the internal energy, denoted E:

EEE 

The conservation principle is thus applied, not to the internal energy alone, but to

the total energy, whose variation

E' is equal to the quantities of heat Q and of

work W received from the exterior, including the work

mec

W received by the system

and coming from the external forces:

QWEEE

 '' '

The extensive and intensive scalar variables, and in particular the internal

energy, do not depend on the reference frame chosen: for example, the tension of an

elastic string and its stretching, the pressure of a gas and its volume, the number of

moles of a chemical species and its chemical potential, etc. Interpretations of

internal energy in diverse physical theories are often made using forces whose

details are of little importance, but as these are internal forces, their resultant

moments are zero; we have seen (section 3.2.2.2) that the power

fint

of the internal

forces of a system (or the corresponding work) is independent of the reference frame

chosen for its evaluation. This power of the internal forces is in fact already

described in the differential dE of the internal energy (work done by pressure forces,

etc.).

The kinetic energy and the total energy of a system depends on the reference

frame chosen for their evaluation.

Physics of Energetic Systems in Flow 117

For a system which receives from exterior sources of a purely mechanical kind,

we thus have:

()

cmecth

E+E +

[3.19]

mec

and

being the mechanical and thermal powers provided to the system from

the exterior.

Consider first a mechanical system comprised of a material “particle” which is

animated, for the sake of simplicity, by a macroscopic translational motion of

velocity V. Now suppose that this particle receives, from the exterior, the

mechanical power FV

mec

. P and the thermal power P

. The balance equation for

the (total) energy of the material system can be written:

mecth



Now consider a system of n material particles of velocities .

Letting E denote

the internal energy of the system (which is not in equilibrium), the balance equation

can be written, taking account of motion, as:

mecth



[3.20]

Subtracting the kinetic energy equation [3.12] from [3.20], we obtain the energy

equation (E, the internal energy) of the material system:

intfth



[3.21]

The rate of internal energy variation of a system is equal to the difference

between the thermal power which it has received and the power of its internal forces.

3.2.5.2.

The entropy form of the energy equation

It is useful to explain the preceding phenomena using an example. Consider the

system already discussed in section 3.2.2.3 (Figure 3.1), comprised of two masses

linked by elastic under tension



and equipped with a viscous friction system.

The tension force

derives from a potential.

118 Fundamentals of Fluid Mechanics and Transport Phenomena

The masses are subjected to external forces

(j=1,2); we assume that gravity

does not directly intervene. The viscous friction system is assumed to have no mass

and the heat released by the friction heats the two masses which are assumed to be at

the same temperature. The dynamic equation of motion for the masses in the

reference frame of the inertia center can be written:



fext

FFm



fext

FFm

in which the tension



of the elastic link and the viscous friction force

F ,

aligned with the axis Ox, are, by definition, the forces applied on each mass:





xxd



By adding the terms of the two equations after multiplication of each by its

corresponding velocity



2,1

, we obtain the kinetic energy equation. The

power of the internal forces can be immediately calculated:





fVVF

 

int

P [3.22]

Energy equation [3.21] can be written:



P



[3.23]

By specifying the differential of the internal energy

AdTdSdE



for the

system, we finally obtain the energy balance equation in its entropic form:

T P



[3.24]

We see that in this entropic

form of the energy equation the mechanical terms

have been eliminated and

all that remains is the irreversible part of the motion

which is transformed into heat

. The dissipation function



is a heat source which

is always positive. It appears in equation [3.24] as an

internal entropy source.

The preceding developments can be repeated for a system of

n particles at the

same temperature, between which there exist elastic forces and friction forces. We

thus obtain an entropic form of the energy equation, analogous to [3.24], with a

dissipation function for all the internal frictions of the system. The reader will note

that this reasoning is only valid if the entropy of the system studied is a function of

Physics of Energetic Systems in Flow 119

the other extensive quantities, i.e. if the system is in a state of thermostatic

equilibrium at every instant.

We will see these developments (sections 3.1 and 3.2) once again when we study

flowing continuous media, albeit with a formalism which is naturally more complex.

3.3. Kinematics of continuous media

3.3.1. Lagrangian and Eulerian variables

A problem of flowing matter involves the study of the influences of external

conditions on the fluid medium, specifically the many different ways forces exerted

on the medium (presence of fixed boundaries with respect to the observer, moving

boundaries such as propellers blades, turbine blades, etc., pressure differences

between two reservoirs, external forces such as gravity and electrical volume forces

due to some external device, etc.). The properties of a flow thus result from the

action of external causes which modify the mechanical and physical properties of

the matter (velocity and acceleration, internal stresses, pressure, temperature,

chemical composition, etc.) and inversely the flowing medium exerts stresses on

walls or modifies the boundary properties (stresses induced, temperature, chemical

properties, etc.).

The questions posed and the results expected from a study, an experiment or the

operation of a device or system can vary considerably. Here are some examples:

– in a water treatment station or in a chemical reactor, we are naturally

concerned with the product being treated; the practical problem is thus to design

walls, materials and diverse processes so as to obtain the desired result concerning

the product which is treated;

– in meteorological applications, the objective is to predict the weather at a given

place and time, i.e. to predict the motion, and the physical and chemical properties

of large air masses (velocity, temperature, humidity, presence of pollutants, etc.);

– for a boat, a plane or a vehicle in motion, the essential properties are the

external forces exerted on the solid boundaries in contact with the flow when the

vehicle is displaced at a given velocity. The same is true for the flow of a river or

fluids in an industrial pipe network for which suitable constructions are necessary

(dams, turning vanes, pipes, open channels, pumps, turbines, etc.).

The study of a physical problem begins with a definition of its variables. The

description of moving matter involves the localization of a material particle which is

identifiable in a given reference frame by means of coordinates. The physical

properties of this particle are associated with it. In particle mechanics, we give

120 Fundamentals of Fluid Mechanics and Transport Phenomena

numbers to the different particles studied and to the physical quantities which

characterize them. We thus define the trajectory of a particle as an ensemble of the

successive positions which it occupies over the course of time.

When studying the physics and mechanics of continuous media, this means of

description is no longer suitably adapted, since the set of the material particles of a

continuous medium is not denumerable. We thus need to identify them with respect

to a continuous ensemble which is simply the ensemble of positions x

which this

material occupies at a given reference time. The variables

g associated with the

matter are thus functions of this position

occupied at an initial instant and of time:

),(

txgg

This is called

material representation by means of Lagrangian variables.

Problems for moving continuous media where the matter must be clearly

individualized are generally treated using Lagrangian variables; this description is

often used in the mechanics and physics of solids.

The disadvantage of Lagrangian variables is that such a description does not

have any particular interest for the observer in many situations: the fisherman at the

water’s edge sees the passage of water particles which he will never again see, and

the same is true for the plumber examining a radiator, etc. As the fluid matter passes

by continuously, the identification of fluid particles is not particularly useful. This is

not the case however when we are interested in identifying the local physico-

chemical properties of the matter. Examples include the progress of a chemical

reaction in a closed reactor, the presence of clouds and the knowledge of humid air

masses in weather applications or the presence and movement of pollutants

downstream of a pollution source, etc.

Frequently, in fluid mechanics we consider a preferred observer reference frame,

which is most often associated with solid boundaries between which or in the

vicinity of which the fluid matter passes (pipe networks, solid obstacles, the wings

of an aircraft, etc.). We thus represent the matter

spatially by means of a field where

quantities associated with a material particle are functions of the coordinates

which the particles occupy at the observation time

t. This kind of representation

involves

Eulerian variables.

In physics, the study of particles contained in and subject to a force field

generally involves the combination of these two representations, the particle being

described by its coordinates as a function of time (Lagrangian variables) whereas the

external field (gravity, etc.) is defined using its spatial properties.

Physics of Energetic Systems in Flow 121

Fluids in motion are generally described using Eulerian variables, the observer

reference frame being associated with the solid boundaries. A considerable difficulty

thus appears, as the formulation of the laws of thermodynamics and mechanics

requires the

balance of extensive quantities associated with matter. In the following

sections we will write these balances by “following matter” over material domains

represented using Eulerian variables.

3.3.2. Trajectories, streamlines, streaklines

In Eulerian variables the kinematics of a flow is characterized by the velocity

field, which is a function of time:



123 or

iij

uux,t i,j ,, VVM,t

GGG

In a velocity field, we can define the following curve families:

 the streamlines at time t are lines whose tangents are the velocity vectors at

this instant.

These give a

vision of the flow at instant t and their differential equations are:

),(),(),(

txu

iii

[3.25]

a trajectory is the locus of a fluid particle; trajectories are characterized by

their spatial coordinates which are a function of time (Lagrangian description); their

differential equations (here

t is a variable) are:

txu

iii

),(),(),(

[3.26]

 a streakline at time t is the locus of particles which have passed by a given

point earlier than

t, such as the cloud of smoke issuing from a chimney or the

colored line obtained in a moving fluid by injection of a colored liquid or a trickle of

smoke. The process of flow visualization involves the generation of visible emission

lines (Figure 3.3) by injection of markers (dyes, smoke, small particles).

122 Fundamentals of Fluid Mechanics and Transport Phenomena

trajectory

fluctuating wind

streakline

Figure 3.3.

Trajectory of a smoke particle and

streakline from a chimney when wind is blowing

When a velocity field does not explicitly depend on the variable t,









jiji

xutxu

and we have a steady flow. The streamlines, which no longer

depend on time, are fixed in space: the appearance of the flow does not change. The

preceding definitions and equations show that

streamlines, trajectories and

streaklines are all the same in steady flows

EXERCISE

– Find the equations for the streamlines, the trajectories and the

streaklines of the following 2D velocity field:

u = constant, tav

cos

t. Interpret

the results.

3.3.3. Material (or Lagrangian) derivative

3.3.3.1. Material (or Lagrangian) derivative of a quantity g

Let



txg

, be the field of a continuous scalar quantity associated with a

medium of fluid particles moving with a velocity field





txu

defined in a given

reference frame. The

material derivative (also called the total or Lagrangian or

substantial derivative) is defined as the time derivative of the quantity g associated

with the material particle

of coordinates



and velocity





txu

. It can be

obtained by calculating the time derivative of

g in which the coordinates are

functions of time and describe the trajectory:

ggradV

.or 



We will denote the Lagrangian derivative

, which is by its very definition the

derivative of a

function of time only for a fluid particle. It is also often denoted

in fluid mechanics.