if we substitute depth of the centre of gravity for the variable depth into the formula shown in Fig.B-14. Since in most practical situations the shape of the panel is regular - circles and rectangles are perhaps the most common situations - with easily determined CG in the geometric centre of the shape, such a substitution is certainly a natural one.
On the other hand, the problem of locating the point in which the force acts is no more simple. The point is called centre of pressure or hydrostatic centre and is marked as C in Fig.B-17. Derivation of its position starts by considering the moment condition relative to the axis - the intersection of panel plane with the liquid surface plane (Fig.B-17). The force on its arm generates a moment which is equal to the integral of elementary forces acting at their particular arms :

If we insert the expression for from the previous page, the result is :
- the position is evaluated as the ratio of the inertia moment and static moment .

Definitions:




We may use the Steiner Theorem, according to which

- where is the inertia radius.

For the distance between the centre of gravity and centre of pressure insertion of the Steiner Theorem leads to:
...so that, as a result:

The horizontal position of the hydrostatic centre - in the direction - is usually no problem: in the most often encountered symmetric shapes, is located on the vertical axis of symmetry. In nonsymmetric cases the position is found from the moment condition relative to axis :
... so that

Fig.B-18
3) The case of a curved wall
- may be mathematically quite demanding. For the present purpose, it will be sufficient to use a simple method of substitution plane as shown in Fig.B-18. The curved wall (such as e.g. the convex cover in the case A) is separated by an imaginary plane from the rest of fluid. Then the hydrostatic force acting upon this plane is evaluated by the approach from Fig.B-17. The hydrostatic effect of fluid in the space between this substitution plane and the wall may be shown to be equal to weight, gravity force . Evaluating it is a question of calculating the volume of this space. The resultant force is finally obtained as the vector addition of the two components. In the case B the substitution plane is located ourside the space occupied by the liquid. As a result, the vertical component evaluated as the hydrostatic effect of fluid between the wall and the substitution plane is negative - acting upwards. It is lift , the magnitude of which is equal to the corresponding gravity force . This is nothing else but the


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This is page Nr. B07 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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