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



Going to another page: click
This is page Nr. E01 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

WWW server administrators: Jiri Kvarda, Zdenek Maruna ...... Contact: webmaster@vc.cvut.cz
Last change : 21.03.1997