We can decompose gas into several cuboid volume elements. When only one element is considered, the situation that all cuboids are a whole can be solved.
There are three kinds of forces: the downward force acting on the top of the micro-cube caused by the pressure p on the fluid. According to the definition of pressure,
Ftop=Ptop A
Similarly, the force acting on the volume element pushed upward from the fluid pressure is
Bottom =-bottom a
In this equation, the negative sign comes from the direction-this force supports the volume elements instead of pulling them down (we assume that the positive force acts downward, and if you think of "down" as "up", the result is the same as that of hydrostatic pressure).
Finally, the weight of the volume element leads to a downward force. If the density is expressed by rho and the volume is v, then g is the standard gravity, then:
fw weight =ρgV
The volume of a micro cube is equal to the area of the top or bottom multiplied by the height, which is the formula for calculating the cube.
fw weight = pgAH
Balancing these forces, the resultant force acting on the gas is
f total = Ftop+f bottom+fw weight = Ftop A-f bottom A+pgAH
If the gas does not move, it will be zero. If we divide by a,
0=Ftop-Fbottom+pgAh
Or,
Ftop-Fbottom=-pgAH
Ptop？ The change of Pbottom pressure, H is the height of volume element-the change of distance on the ground. Considering that these changes are infinitesimal, the equation can be expressed in the form of derivative.
dP=-pg dH
Density varies with pressure and gravity varies with height, so the equation becomes:
dP=-p(p) g(h) dH
Note that this last equation can be solved according to the three-dimensional Naville-Stokes equation under the condition of hydrostatic pressure balance.
The only common equation is the equation, which is pronounced as
This hydrostatic equilibrium can be consider as a special case of Naville-Stokes equation.