Hydrostatic pressure

Hydrostatics is one of the sections of hydraulics that studies the equilibrium state of a liquid and the pressure that occurs in a liquid resting on different surfaces.

Hydrostatic pressure is the basic concept in hydrostatics. Let us consider an arbitrary volume of liquid in equilibrium. Inside this volume, mark the point A and mentally divide it in half by a plane passing through point A. On this plane, select the area with area S and center at point A. We remove one half of the volume and replace the force with which it acted on the remaining volume by counterbalancing force F. Thus, the liquid in the second half will still be at rest.

Now we begin to reduce the area S so that the point A is constantly inside it. With sufficient reduction, the point A coincides with the platform S. And the pressure at the point A will be determined by the formula P (A) = lim dF / dS for dS tending to zero.

Then the pressure exerted on the pad S will be equal to the sum of the pressures exerted on all points belonging to this surface. That is, in other words: p = F / S. The hydrostatic pressure is a value equal to the quotient of the division of the force F by the area S.

The cause of hydrostatic pressure are: the weight of the liquid itself and the pressure that is applied to the surface of the liquid. Thus, the pressure caused by the weight of the liquid itself and the external pressure are the kinds of hydrostatic pressure. If the liquid is placed in the piston, and some force is applied to it, then, naturally, the pressure inside the liquid will increase. Under normal conditions, the liquid is pressurized by atmospheric pressure. If the pressure on the surface of the liquid is below atmospheric pressure, then this pressure is called gauge pressure.

The liquid is in equilibrium if all the pressure forces acting on any sufficiently small volume of liquid are balanced with each other.

Let us consider the hydrostatic pressure and its properties:

• For any point arbitrarily taken in the fluid, the hydrostatic pressure vector is directed inside its volume and perpendicular to the area allocated in the volume.

Let us prove this property: suppose that the angle at which the force is applied to a certain area is not direct. We represent the force F as P (normal), P (tangential). Suppose that the tangential component is not equal to zero, then under its influence the liquid must flow along an inclined one, but it rests at a point. Hence the conclusion suggests that the tangent is zero and the effect of pressure occurs perpendicular to the area. The property is proved.

• The hydrostatic pressure is the same in all directions.

Let us prove this property of hydrostatic pressure: in an arbitrary volume of liquid we select a tetrahedron whose two planes coincide with the coordinate planes, and the third one is chosen arbitrarily. In the base we get a right triangle. The action of the liquid on each face is denoted by: X * (P), Y * (P), Z * (P) The liquid is in equilibrium, therefore the total result of the action of all forces is 0.

E * (x) = 0

X * (P) dz -E * (P) de sin a = 0,

E * (y) = 0, E * (z) = 0

Z * (P) dx -E * (P) de cos a = 0

It is obvious that dz = de sin a, dx = de cos a

From this: X * (P) = E * (P), Z * (P) = E * (P)

Output: X * (P) = Y * (P) = Z * (P) = E * (P)

The property is proved. Since the face was chosen arbitrarily, this equality is valid for any case.

• The hydrostatic pressure varies in direct proportion to the depth. With increasing depth, the pressure at the point will increase, and with decreasing depth of immersion - increase.

Any point of the fluid in equilibrium corresponds to the following equation: j + p / g = j (o) + p (o) / g = H, where j is the coordinate of the point, j (O) is the coordinate of the liquid surface, p and P (o) is the height of the columns, g is the specific gravity of the liquid, and H is the hydrostatic head.

As a result of the transformations, we obtain: p = p (o) + g [j (0) -j] or p = p (o) + gh

Where h is the depth of immersion of a given point, and gh is none other than the weight of a column of fluid equal in height h and having a unit area in the base area. This property of hydrostatic pressure is called Pascal's Law.