- 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.
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"
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