This chapter presents a possibility to solve fluid flows by another method, different from the Bernoullis' Theorem approach used so far. It is a relatively new method: while differential analysis began with Euler in 18th century, the control volume approach (though used from time to time in an ad hoc manner was not developed on a rigorous basis until the 1940s). The new method is particuraly suitable for those problems in which the task is to evaluate fluid force action on bodies past which or through which fluid flows. Essential part of this action is often given by momentum rate of change (according to Newton's " golden rule of mechanics"). The force balance equation, which is the central tool in the new method, is often desrcibed also as "momentum change theorem" . In the present text, it is derived from Bernoullis' Theorem. It should be said, however, than in general the force balance is not equivalent to the energy conservation condition: the difference between them is due to energetic changes (such as heat transfer, which is not studied in the present text) that do not generate force action.
One of the advantages of the method is its applicability to multi-dimensional flows (and, in fact, it is one of the basic tools there). Here, however, we shall need it only in one-dimensional form. The one-dimensional flow may be either (really more or less one-dimensional) duct flow, or it may be an imagined flow tube (streamtube), in a flow which is multidimensional.
Fig.H-1 Derivation of the force balance
equation of an infinitesimally small flow tube.
We follow changes taking place according to Fig.H-1 on an infinitesimal tube length . The cross section of the tube is of finite size at the starting position. It changes to + on the length . To make the initial considerations easier to follow, horizontal = 0 flow is assumed in Fig. H-1 so that we are not distracted by the gravitational force , which behaves quite simply and may be added to the final results as shown in Fig. H-2. For energy conservation, the following expression is then valid:

- the infinitesimal work of outer forces icludes also the effect of frictional and other drag forces. In the elementary tube shown in Fig.H-1, only a part of the drag between fluid and the body is acting. It is evaluated as the power of the action divided by the fluid flow velocity .


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This is page Nr. H01 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

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