Fig.H-2 Derivation of the gravity force term in the force equation
for an infinitesimally small flow tube from Fig.H-1.
In Fig.H-2, the same approach as in Fig.H-1 is applied to the gravity term - again, the force is evaluated for an infinitesimal tube length . The expression for position energy is multiplied by the ratio / to obtain the infinitesimal gravitational force. Since the gravitational force is oriented and since its orientation in general is at an angle with respect to the local orientation of axis , the new term contains the cosine of the orientation angle. In the final form of the force equation, this term is added to the other terms already derived in Fig.H-1.
In the resultant equation there are thus four terms. They represent the infinitesimal forces, which are of two sorts: The gravitational force is of volume character, since it acts in the c.g. of the investigated volume. The other two left-hand-side forces are differences between effects of planar character - they act in the inlet and outlet planes. The infinitesimal drag force is also of planar character, but it may act anywhere in the outside of the elementary tube.
The force equation is used not in the differential form as shown above, but in an integrated form, valid over a closed finite volume called control volume . Because the forces of the planar character
Fig.H-3 Boundaries of a control volume
- with fluid inlet and fluid outlet.
require specification of the plane in which they act, it is necessary to specify unambiguously the boundaries of the volume over which the equation is integrated. It is customary to draw the closed boundaries of the control volume as thick dot-and-dash lines, Fig.H-3. The planar forces act only on specified segments of this boundary. It may be useful, with some advantage, to chose the boundaries so that their part passes outside the fluid. Control volume may be also chosen so that it moves together with fluid - unless, however, the motion is not a simple constant-velocity translation, the latter case may require consideration of additional forces of the volume character (centripetal or Coriolis forces).
Integration of the first (momentum) term in the force equation in one-dimensional flow leads to:

- obviously, the result contains only the momentum flowrate at the inlet and the outlet, because nowhere else the fluid velocity (= specific momentum) possesses a component normal to the volume boundary.
Similarly, a nonzero magnitude of the pressure
Fig.H-4 Pressure and the corresponding pressure force
acting in the input cross section of the elementary tube.
force

is found only at the inlet and outlet. The situation at the inlet is shown in Fig.H-4.
In the basic case of a flow in straight line, the resultant integrated form of the equation is:

If, however, the investigated flow changes its direction, so that the outlet cross section is inclined relative to the inlet cross section, it is necessary to write the equation in the vector form as:

... where is a vector having the direction normal to the area .
In the one-dimensional cases that are of interest here, it is possible to use the the mass conservation condition = so that there is = ( - )
Because the relation between the inlet and outlet velocities is specified in a unique manner by the area ratio , there is
= ( - 1 )
= () ( - 1 ) /


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This is page Nr. H02 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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