4. 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 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_QdS —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=p₀+hγ. (2.2)

This is the hydrostatic equation with which it is possible to calculate the pressure at any point of a still liquid.

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* These equations have the form:

;

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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.

A surface layer where the pressure is the same at all points is called 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

z + p/γ= const. (2.3)

The coordinate z is called the elevation. The term p/γ , which is also a linear quantity, is called the pressure head, and the sum (z + p/γ) 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).

4. Pressure head. Vacuum. Pressure measurement

The pressure head p/γ represents the height of a column of a given liquid referred to a given absolute or gauge pressure p. The pressure head referred to gauge pressure can be measured with a so-called piezometer, which is the simplest device for measuring pressure. A piezometer is a vertical glass tube the upper end of which is open to the atmosphere while the lower end is connected to a small hole in the wall of a liquid container (Fig. 7).

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Applying Eq. (2.2) to the liquid in the piezometer, we obtain

p_ab = p_atm + γh

where p_ab= absolute pressure of the liquid at the level of the attached end of the piezometer; p_atm = atmospheric pressure. From this, the height of the liquid in I ho piezometer is

where p_g = gauge pressure at the same level.

Obviously, if the free surface of a still liquid is subjected to atmospheric pressure, the pressure head at any point of the liquid is equal to the depth of that point.

Pressure in fluids is often expressed numerically in terms of head according to Eq. (2.4). One atmosphere, for example, corresponds to:

If the absolute pressure in a fluid is less than atmospheric, we have a partial or complete vacuum. The magnitude of a vacuum is measured as the difference between the atmospheric and absolute pressures:

Take, for example, a tube with a tightly fitting piston, lower one end of it into a container with a liquid and then draw the piston up (Fig. 8). The liquid will follow the piston and rise together with it to some height h above the free surface, which is subject to atmospheric pressure. For the points immediately under the piston the depth with respect to the free surface is negative. Hence, from Eq. (2.2), the absolute pressure in the liquid below the piston is

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As the piston rises the absolute pressure in the liquid decreases. The lower limit for the absolute pressure is zero; the maximum value of vacuum is numerically equal to atmospheric pressure. Hence, the maximum elevation of the liquid in our example, i. e., the maximum height of suction, can be found from Eq. (2.5), assuming p = 0 (or, more precisely, p = p_t).

Thus, neglecting the vapour pressure p_t, we have

At normal atmospheric pressure (1.033 kgf/cm²) the height h_max is 10.33 m for water, 13.8 m for gasoline (y = 750 kgf/m³) and 0.76 m for mercury.

The simplest device for measuring vacuum, or negative pressure, is a glass U-tube arranged either with one end open, like the right-hand tube in Fig. 9, or inverted with the free end submerged in a liquid, like the left-hand tube.

In laboratories, besides piezometers, various manometers and mechanical gauges are used to measure the pressure of liquids and gases.

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