Chapter II hydrostatics.

5. Hydrostatic pressure

It was mentioned before that the only stress possible in a fluid at rest is that of compression, or hydrostatic pressure. The following two properties of hydrostatic pressure in fluids are important.

1. The hydrostatic pressure at the boundary of any fluid volume is always directed inward and normal to that boundary.

This property arises from the fact that in a fluid at rest no tensile or shear stresses are possible. Hydrostatic pressure can act only nor­mal to a boundary, as otherwise it would necessarily have a tensile

or shearing component.

By "boundary" is meant the real or imaginary surfaces of any ele­mentary fluid body taken within a given volume.

2. The hydrostatic pressure at any point in a fluid is the same in all directions, i. e., pressure in a fluid does not depend on the inclination of the surface on which it is acting at a given point.

To prove this statement, imag­ine inside a fluid at rest an ele­mentary body having the shape of a right-angled tetrahedron with three edges dx, dy and dz paral­lel to the respective axes of a coordinate system (Fig. 5).

Let there be acting near this elementary volume a unit body force whose components are X, Y and Z, let p_x, p_u and p_g be the respective hydrostatic pressures acting on the faces normal to the x, y and z axes, and let p_n be the hydrostatic pressure on the inclined face of area dS. All the pressures are, of course, normal to the respective areas.

Now let us develop the equilibrium equation for the elementary volume parallel to the x axis. The sum of the projections of the pres­sure forces on axis ox is

The mass of a tetrahedron is equal to its volume times its density, i. e., у dxdydzq; hence the body force acting on the tetrahedron parallel to the x axis is

,

The equilibrium equation for the tetrahedron takes the form

Dividing through by [which is the projection of the area of the inclined face dS on plane yOz and is therefore equal to], we have

.

When the tetrahedron is made to shrink infinitely, the last member of the equation, which contains the term dx, will tend to zero, the pressures p_x and p_n remaining finite.

Hence, in the limit

p_x — p_n = 0,

or

p_x = p_n

Developing similar equations for equilibrium parallel to the у and x axes, we obtain

p_y = p_n and p_z = p_n

or

p_x = p_y = p_z = p_n . (2.1)

As the dimensions dx, dy and dz of the tetrahedron were chosen arbitrarily, the inclination of face dS is also arbitrary, and, conse­quently, when the tetrahedron shrinks to a point the pressure at that point is the same in all directions.

This can also be proved very easily by means of the formulas of strength of materials for compression stresses acting along two and three mutually perpendicular directions.* For this we only have to assume the shearing stress in these formulas equal to zero, and we obtain

These two properties of hydrostatic pressure, which were proved for a fluid at rest, also hold good for ideal fluids in motion. Fn a real moving fluid, however, shearing stresses are developed which were not taken into account in our reasoning. That is why, strictly speak­ing, hydrostatic pressure does not display these properties in a real fluid.