Let △ S be any area on the ABCD plane, use △ P to represent the total force acting on the area △ S, and the limit of the ratio △ P/△ S is called fluidStatic pressure intensity。Therefore, the force on the surface perpendicular to the unit area of the fluid is called the static pressure strength of the fluid.Traditionally, the hydrostatic pressure is called static pressure, or pressure, and the static pressure on an area is called total force.[1]
Chinese name
Hydrostatic pressure
Foreign name
Hydrostatic pressure
Definition
△P/△S
Substantive
Hydrostatic pressure per unit area
Features
The static pressure in all directions at any point is equal
In the static solution as shown in Figure 1, take any point K, and take a small area △ A around it, then the adjacent fluid will have a force on it, set as △ F.When the small area tends to zero, the stress at point K is:
Where, p is the stress in the static fluid, called the static pressure, and the unit is
Or Pa, sometimes called static pressure.[2]
Fig. 1 Surface force acting on static fluid
Characteristics of hydrostatic pressure
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Hydrostatic pressure has two important characteristics.
The first characteristic: the direction of hydrostatic pressure is along the internal normal of the action surface, or vertically pointing to the action surface.
Because the shear stress in a static liquid is equal to zero, and because the fluid cannot bear tension, it can only bear pressure. Therefore, the only surface force acting on the fluid is the static pressure p pointing to the internal normal direction of the action surface.
The second characteristic: the hydrostatic pressure at any point in the static fluid is independent of the direction of the action surface, that is, the hydrostatic pressure at any point in all directions is equal.[2]
Hydrostatic balance equation
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Differential equation of fluid equilibrium
Take the micro element hexahedron with side length of dx, dy, dz in the static fluid, as shown in Figure 2. The length of each side of the micro parallelepiped, dx, dy, dz, is parallel to the corresponding coordinate axis.To analyze the force acting on the tiny hexahedron.
Fig. 2 Microelement hexahedron in static fluid
Acting onParallelepipedThere are surface force and mass force.Because there is no shear stress in the static fluid, there is only pressure acting on the surface of the micro parallelepiped.Let the fluid pressure at point A be p.According to the characteristics of hydrostatic pressure, the fluid pressure at a point is equal in all directions, so the hydrostatic pressure at each point on any one of the three vertical planes containing point A, such as ABCD plane parallel to the xy plane, is also equal to p.
Since pressure is a continuous function of coordinates, that is, p=f (x, y, z), the function f is expanded according to Taylor series, and the first two terms of the series are taken, then the pressure expression of points on abcd on the other side of the hexahedron parallel to the xy coordinate plane can be obtained
For the pressure on other surfaces of the hexahedron, the corresponding expression can also be written using the above method.
Acting on tinyParallelepipedThe mass force on is G, which may be in any direction in general. Its projection on the x-axis is dxdydz ρ X, where ρ is the density of the fluid, dxdydz is the volume of the tiny parallelepiped, and X is the projection of the unit mass force on the x-axis.
Similarly, the projections of the mass force on the x axis and on the y axis can be written out.Since the micro parallelepiped is at rest, the sum of projections of forces acting on it on any coordinate axis is equal to zero.For X axis:
In the above formula, density is ρ, divide the items in the above formula by ρ dxdydz, that is, the force per unit mass is
。Using the same method for y-axis and z-axis, we can get:
The above formula is the differential equation of fluid balance, also known as Euler equation.[1]
Fluid balance equation in gravity field
When the fluid is in the gravity field, the area mass is only gravity.The three components of the mass force on the unit mass fluid should be X, Y, Z=- g (the z-axis is positive upwards). Multiply the Euler equilibrium equation by dx, dy, dz and add them together to get:
Since p=f (x, y, z), then:
The three components of the mass force acting on the unit mass fluid are taken into the above equation to obtain:
dp=-ρgdz
yesIncompressible fluid, ρ is a constant. If the above equation (dp=- ρ gdz) is integrated as shown in Figure 3, we can get:
Fig. 3 Static pressure of static fluid
This equation is called the basic equation of incompressible hydrostatics.[1]
