6. The basic hydrostatic equation

Let us consider the basic case of equilibrium of a fluid when the only body force acting on it is the force of gravity and develop an equation which would enable us to determine the hydrostatic pres­sure at any point of a given fluid volume. In this case, of course, the free surface of a liquid is a horizontal plane.

Referring to Fig. 6, acting on the free surface of the liquid in the vessel is a pressure p₀. Let us determine the hydrostatic pressure p at an arbitrary point M at a depth h from the surface.

Taking an elementary area dS with point M as its centre, erect a vertical cylindrical fluid element of height h and consider the equilibrium conditions for this element. The pressure of the liquid on the base of the cylinder is external with respect to the latter and is normal to the base, i. e., it is directed upward.

Summing the forces acting vertically on the cylinder, we have

pdS —p_odS —γhdS = 0,

where the last term represents the weight of the liquid in the cylin­der. The pressure forces acting on the sides of the cylinder do not enter the equation as they are normal to the side surface. Elimina­ting dS and transposing,

p = p_o + hγ. (2.2)

This is the hydrostatic equation with which it is possible to cal­culate the pressure at any point of a still liquid. The hydrostatic pressure, it will be observed, is com­posed of the external pressure p₀ acting on the boundary surface of the liquid and the pressure exerted by the weight of the overlying layers of the liquid.

The value of p₀ is the same for any point of a liquid volume. Therefore, taking into account the second prop­erty of hydrostatic pressure, it may be said that a liquid transmits pressure equally in all directions (Pascal's law).

It will be also observed from Eq. (2.2) that pressure in a liquid in­creases with depth according to a linear law and is the same for all points at a given depth.

*These equations have the form:

Asurface layer where the pressure is the same at all points iscalled a surface of equal pressure, or equipotential surface. In the case considered the equipotential surfaces are horizontal planes, the free surface being one of them.

Let us take at an arbitrary elevation a horizontal datum level from which a vertical coordinate z is to be measured. Denoting the coordinate of point M as z and the coordinate of the free surface of the liquid as z₀, and substituting z₀— z for h in Eq. (2.2), we obtain

But M is an arbitrary point, hence, for the stationary fluid ele­ment

The coordinate z is called the elevation. The term , which is also a linear quantity, is called the pressure head, and the sum is called the piezometric head.

The piezometric head is thus constant for the whole volume of a stationary fluid.

These results can be obtained in more definite form by integrating the differential equilibrium equations for a fluid (see Appendix).