At the beginning, let us be satisfied with only two
components of energy, which were analysed so far
and are mentioned in Fig.A-25
(gradually, one at a time in the following chapters, we shall ad to them
other components )
and let us write down the differential equation, by the integration of which
it will be possible to solve practical problems.
Fig.A-26
An infinitesimal lenght
of the integration path
for the derivation of the differential equation.
The starting point is consideration of energetic budget
and its changes during elementary transition
along the axis
from the initial position
(Fig.A-26) to an infinitesimally near position
.
Since there is
, the well known formula for
differentiation of a sum leads to
.
If the point
is located at height
and the fluid there is exposed to pressure
(Fig.A-27), then during the transition to point
pressure increases by
and height by .
If we now use the formula
of differentiating a product, there is
- of course, for
there is
and for the present model of incompressible fluid
there is also
Fig.A-27
- which means there is .
After the zero terms are droppped from the differential form of the
energy conservation equation,
,
the resultant differential equation is obtained
This equation is, of course, applicable only to
the case of fluid at rest
(it does not contain any expression for the
change of kinetic energy). Nevertheless, even condition in
fluid at rest are technologically important
and the following chapter [B]
is devoted just to this
situation.
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This is page Nr. A12 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS" Any comments and suggestions concerning this text may be mailed to the author
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