Object of interest in the present chapter are phenomena
associated with fluid inertia. In the equation of energy conservation these phenomena give
rise to another additional term - however, in contrast to the new terms in previous
chapters that made it possible to obtain better approximation to reality, the present
new term brings capability to treat flows occuring only in some situations: situation where
flow velocity varies with time. There are many flows which are steady, and for them the
new term is not needed.
Unsteadiness may give rise to processes which it is possible to describe only by two- or three-dimensional
approach we do not follow in this chapter. We shall be, therefore, able to treat only
rather simple cases and we shall accept rather crude simplifications. This is also
a consequence of the fact that until relatively recently unsteady flows were
extremely difficult to investigate experimentally - this requires fast velocity and
pressure sensors, fast signal processing and fast data logging. It was difficult to
verify the results and even large calculation errors went unnoticed. As an example of the
problems encountered in this area, let us note that we shall here treat unsteady flows as
piston flows, of equal velocity across the whole pipe cross section. In fact, oscillations at higher
frequencies generates velocity profiles (in laminar case described by the Bessel functions) with several
local maxima and local minima.
Until this point, we were in this textbook interested in study of changes
of velocity
along the pipe axis
. In the present chapter, the interest is widened to
include also the dependence on another variable, time. We shall assume general functional dependence
. The infinitesimal velocity change
in the conservation equation
is now the total differential
|
 |
- the first of the above right-hand terms represents the spatial, path change
...this is the change we have studied so far.
- the second term represents the time change; it is possible to re-write is as


... where
is fluid acceleration.
When inserted into the kinetic energy change expression
, the second term becomes

. Of course, velocity equals

Here we shall denote as
path in the direction of acceleration.
At this point, it is equal to our single spatial co-ordinate
. Since there is
=
,
the new variable seems to be superfluous. At some later point, however, (rudimentally)
multidimensional problems arise and it will be necessary ti discriminate between the two
paths.
Inserting
into 
makes possible to cancel
in the numerator and denominator. What we obtain is an
an expression
with the only differential
- this, of course, is just an expression for a spatial change.
Whatever an origin there was for the new term, it now does not represent anything else but
a change along our single and only axis
.
The new term gets in line with other terms we have considered so far:
,
,
and
.
Another way of stating this result is that introduction of acceleration
(this step is, in fact, due to Newton)
at least formally eliminates the time variable and converts the partial differential equation into
just an ordinary one. This separability, as a matter of fact, is only formal
and we shall not be spared the necessity to
perform the integration along the time axis (at least as long as there is not the
acceleration directly the computed variable). Nevertheless, there is the
advantage of the two integration being performed independently.
The unsteady one-dimensional flow studied in the present chapter
is, as a result, described by the differential equation

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Vaclav TESAR : "BASIC FLUID MECHANICS"
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