Fluid statics [hydrostatics]

Fluid statics (or hydrostatics) is a branch of fluid mechanics that studies fluids at rest, i.e., any continuous body for which Pascal’s law is valid with constant average velocity in time and vectorially homogeneous in space.

In particular, hydrostatics is that part of hydraulics that considers water in equilibrium, that is, not presenting reciprocal displacements of the points that constitute the mass.

A mass of water in equilibrium behaves as an isotropic body. Around a generic point, the pressure per unit area on a plane element is normally exerted on the element, and has the same value whatever the position of it. In other words: there are no tangential components of the pressures, i.e. the situation of perfect fluid occurs.

Water in equilibrium, as well as in most cases of movement, is considered as incompressible (the cubic compressibility modulus of water is 2.07・108 kg/m2 at a temperature of 10° C.). The magnitude of the pressure varies with continuity from point to point in the mass; and since the equations of equilibrium are summarized, in vector form, in the expression that the force acting on the unit mass of the liquid is equal to the gradient of the pressure function divided by the density, it is deduced:

  • that in an incompressible liquid the equilibrium state is possible only if the field of mass forces admits a potential;
  • that surfaces of equal pressure in the liquid at equilibrium are surfaces of equal potential in the field of mass forces;
  • that the mass forces acting on a generic volume of liquid in equilibrium constitute a mechanical system equivalent to the system of pressures exerted on the surface enclosing the said volume.

When, as is very frequent the case, the mass forces are reduced to be only the weight of the liquid, that is the field is that of gravity (field that admits the potential (-gz), being (g) the gravitational acceleration and (z) the altitude of the generic point on a horizontal reference plane), then the surfaces of equal pressure are horizontal planes, and such is the free surface, that is the boundary surface between the liquid mass and the atmospheric air, and is constant, for the whole liquid mass, the sum (z + p/omega) which is called piezometric height, in which (p) is the pressure per unit area (or unit pressure or simply pressure) and (omega) is the specific gravity of the liquid. The constancy of the piezometric height for all the heavy mass in equilibrium can also be stated by saying that the pressure difference between two points is equal to the specific gravity of the liquid multiplied by the difference in height between the two points.

If we consider a flat surface area, however lying in space, pressed by a heavy liquid in equilibrium, the pressures exerted on the individual elements constitute a system of forces parallel to each other, normal to the plane of the surface and directed in the same direction, whose resultant is equal to the area of the surface area multiplied by the pressure at the depth of the center of gravity of it, and passes through a point on the plane of the surface, the center of pressure, which is the center of gravity of the area, each element of which is affected by a coefficient proportional to the pressure on that element.

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