- may be of importance only if it has nonzero
Fig.F-2
rate of change in time, acceleration:
This result is analogical to what we have learned in chapter [E] about unsteady flows - where there was the time-varying relative velocity . Note that the infinitesimal path , which in
analogical
Fig.F-3
expressions in chapter [E]
coincided with an infintesimal segment of the integration axis , has here a more general meaning: it is the projection of into the direction of the acting acceleration.
The simplest situations that require taking this term into account arise in calculations of pressure inside moving vessels - such as e.g. vehicle fuel tanks. In these cases fluid remains at rest relative to vessel walls, = 0, which means that there is = 0 and also = 0, because . In the Bernoullis's Theorem equation remain only the terms:

In an upwards motion, as in the example given in Fig.F-3, the carry-away acceleration is of opposite direction to the gravity acceleration . The corresponding terms, therefore, have opposite signs, which means they add when moved to the same side of the equation. As a result, the pressure in liquid increases when compared with similar situation in a stationary vessel.
Fig.F-4
In general, if the directions of the two accelerations and are at an angle to each other, the problem may belong into the class of multidimensional problems that are not treated in the present text. Sometimes, however, the solution is rather simple and does not require any special multidimensional solution methods. The shape of the liquid surface may be calculated from the condition of

- cf. Fig.F-4. This is derived from the fact that on the surface there is everywhere the same pressure . In the example shown in Fig.F-4 the carry-away acceleration acts in the horizontal direction. Application of the equation of the surface leads to the conslusion than slope of the surface is equal everywhere, so that the surface is plane. Pressure at a given location is found by one-dimensional integration along any integration path, which starts at on the surface. Particularly recommendable are the vertical or the horizontal integration paths. Both lead, of course, to the same result. In the horizontal integration case, there is zero variation of the height and the position energy term disappears. Moreover, there is some simplification since the projection coincides with the the intehration path . On the other hand, in the vertical integration case (perpendicular to the acting acceleration), the projection is zero and it is the carry-away acceleration term that disappears.


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This is page Nr. F02 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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