The assumption of one-dimensionality leads to acceptable resuts if the investigated flow takes place in cavities with rather simple geometry, with gradual and smooth changes of cross sections (there should be no sharp edges that cause separation of flow and generate two-dimensional effects such as backflow and vortical motions. The cavities (pipes) should not be very long, because of the tendency do develop complex velocity profiles by friction on long walls.
Practical integration of the Bernoullis' theorem is here shown on more simple cases (cf Fig.C-2) with only two terms in the equation:


The addition of the new term means that a question may arise how to integrate it. If fact, this brings no real problem as the integral is the the simple, well-kown case

... we shall, however, mostly use nit this indefinite integral but its definite counterpart, integrated within the limits which will be written in symbolic way as and , as is shown in the following example:
Fig.C-3 An illustrative example for explanation of the concept: Evaluation of efflux (otflow) velocity of gas leaving a vessel under the influence of pressure difference. To avoid problems with unsteadiness, it is assumed that pressure in the vessel is kept constant by the blower. Because of very large corss section F inside the vessel, the gas velocity at position X is negligibly small. Working with gas here means the positional changes are negligible, too. As a result, there is a one-to-one correspondence between pressure and velocity. Note that we neglect losses in this chapter - the real efflux velocity will be lower.


The unequivocal, one-to-one dependence between velocity and pressure changes may be used for experimental determination of fluid flow velocity. If we invert the problem of Fig.C-3, it is possible to find the pressure increase caused by stopping fluid to halt. In the inlet of the Pitot tube (Fig.C-4 ... note that fluid cannot pass through the pipe because its other end is blocked by a manometer to which the tube is connected)


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This is page Nr. C03 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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