While in deriving differential equation it is essential to work
with specific values of extensity variables,
when the equations are integrated, we shall receive total
magnitudes of such quantities and we cannot escape the necessity to have
notations for them. In the case of extensity variables, the total value
depends upon
the amount of fluid taking part in the particular process.
In flowing fluid,
delimitation of a certain amount (mass) of fluid
is not straightforward. It is useful to define a reference amount
of a quantity passing through an inlet - and similarly also outlet
- cross section of an element
during infinitesimal time period. This definition
leads to flow or flowrate values
of extensity variables.
It may be noted that matter existing in space and time is the basic
object of our investigation in mechanics and the variables
describing them (mass, time, distance)
are neither extensive nor intensive quantities, but form a special
basic group of protensive quantities. While specific values are
defined by differentiation with respect to mass, the flowrate values are
defined by differentiation with respect to time.
Flowrate values are here denoted by adding a time derivative
operator to the (upper-case) total value symbol
cf. Fig.A-13. In this version of the textbook, the operator is a dot above
the symbol.
There is a number of disadvantages to this the operator and
the multiplicative used in earlier
editions of the present textbook is better in several respects -
apart from its being not traditional. It is the force of tradition
which forced the author to use "dot" form here. The problem with
the dot is that it does not print well,
and especially on lower quality paper used in earlier editions it was easily misread
for accidental faults. The dot also interferes with the dash above the symbol
of time-mean values. Symbol
has
also the advantage of multiplicative character: it may be e.g. it may be extracted
in front of parentheses.
In modern fluid mechanics it is common to work with nondimensional quantities.
Because of their large number used here, it became useful to apply several different
notations, depending upon their character:
Although im this textbook we shall learn a large number of relations and formulas
of fluid mechanics,
there are, among them, two relations of exceptional position, deserving a name of a "law".
In fact, the whole present textbook will be in its first half subdivided into chapters
according to terms gadually inserted into one of this laws.
The laws are called by old names derived from their originators
(who, however, introduced and used them in quite different form).
Both laws are nothing else but special cases of conservation of mass and energy.
They were, however, discovered a long time before the concepts of mass an energy
became known (and that is why they have been - and sometim,es still are -
used in forms in which the energy and mass are not recognisable at all).
We shall use them in a form of differenial equantions for an infinitesimal
length element of one-dimensional flow (the were, however, introduced
and used in a form which has nothing to do with a differential equation).
Perhaps the most instructive derivation is from energetic balance on
a pipe element.
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Vaclav TESAR : "BASIC FLUID MECHANICS"