4. 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, imaging 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, р_ц and p_z be the respective hydrostatic pressures acting on the faces normal to the x,y and z axes, and let p_n be the hydroctatic pressure on inclined face of area dS. All the pressure are, of course, normal to respective areas.

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

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

F_x = dxdydzρX.

The equilibrium equation for the tetrahedron takes the form

dydzp_x — p_ndS cos (p_n, x) + dxdydzρX = 0.

Dividing through by dydz [which is the projection of the area of the inclined face dS on plane yOz and is therefore equal to dS cos(p_n,x)], 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

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

and , or

(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.

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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. In 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.