Fig.A-25 The two components
of fluid energy taken
into account
in the simplest form
the differential equation.
Diferential equation:
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 to his address tesar@fsid.cvut.cz

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