Why does pressure need to be constant in all directions to maintain equilibrium?

[anonymous99]

Homework Statement

I was reading this wikipedia article on hydrostatics and came across this:

Relevant Equations

"However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe"

I don't understand how pressure must be constant in all directions to balance out the force? Arent the forces in each direction independent, so that pressure forces in the x direction and y direction and z direction can all be different to each other, as long as they are balanced in that direction? E.g if P was 10Pa in the x direction, there would be 10N of force on the left and right sides of a cube of unit area. If the pressure was 20N in the z-direction, it would be 20N above and below. These forces cancel out so the fluid is still in equilibrium.

 

[Kyouran]

To get a relationship between the different faces, you'll have to use a tetrahedron instead as there is no way to prove such a relationship with an infinitesimal cube.

 

[BvU]

Collisions redistribute the pressure in all directions.
Think of a bicycle pump, a pressure gauge, etc.

 

[Chestermiller]

Homework Statement:: I was reading this wikipedia article on hydrostatics and came across this:
Relevant Equations:: "However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe"

I don't understand how pressure must be constant in all directions to balance out the force? Arent the forces in each direction independent, so that pressure forces in the x direction and y direction and z direction can all be different to each other, as long as they are balanced in that direction? E.g if P was 10Pa in the x direction, there would be 10N of force on the left and right sides of a cube of unit area. If the pressure was 20N in the z-direction, it would be 20N above and below. These forces cancel out so the fluid is still in equilibrium.

I agree. The Wiki explanation proves nothing.

 

[anonymous99]

I agree. The Wiki explanation proves nothing.

Do you know where I might be able to find a more complete explanation then? Thank you.

 

[Chestermiller]

As far as I know, Pascal's law that, at any given location in a fluid at static equilibrium, pressure acts equally in all directions is an observational law that has stood the test of time. I suppose that, for certain cases like ideal gases, it can be proven using, say, molecular dynamics.

 

[Kyouran]

Do you know where I might be able to find a more complete explanation then? Thank you.

I haven't read through the article, but the figure in the link below looks like it is what I meant. A proof that pressure is isotropic comes from force balance on that tetrahedron

https://www.brighthubengineering.co...c-pressure-and-pascals-law-for-static-fluids/

 

[ishika17]

I don't understand how pressure must be constant in all directions to balance out the force?

the condition is not for the pressure to be constant,Infact,the pressure is not CONSTANT ,it is EQUAL(or same) on every face of the cube .the pressure is equal because the a liquid exerts equal perpendicular force on every face of the cube and since ,According to pascal's law,
P=F/A;(Force is exerted equally on all surfaces and obviously, area of each face of the cube will be equal).hence,the presuure on each surface of the cube is same.

Arent the forces in each direction independent,

If a point in the fluid is thought of as an infinitesimally small cube,

here,we ga point as infinitesimally small cube in the fluid.so,it will be like a body which is completely immeresed in the liquid.therefore,pressure on every face of the cube is same.

These forces cancel out so the fluid is still in equilibrium.

you seem to be confused betweeen isotropy and equilibrium.
Here,we're talking of isotropy i.e.,the magnitude of pressure is same in all directions.
whereas,for a fluid,to be in equilibrium,the total pressure must be constant.

 

[jbriggs444]

Homework Statement:: I was reading this wikipedia article on hydrostatics and came across this:
Relevant Equations:: "However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe"

I don't understand how pressure must be constant in all directions to balance out the force? Arent the forces in each direction independent, so that pressure forces in the x direction and y direction and z direction can all be different to each other, as long as they are balanced in that direction? E.g if P was 10Pa in the x direction, there would be 10N of force on the left and right sides of a cube of unit area. If the pressure was 20N in the z-direction, it would be 20N above and below. These forces cancel out so the fluid is still in equilibrium.

Keeping the center of mass of the cube stationary requires that opposite faces have pair-wise equal forces from pressure. But you are correct that this, by itself, does not say that the three orthogonal pairs (top-bottom, left-right, front-back) must all three be equal to one another.

However, it is pretty intuitively clear that if top-bottom pressures were different from left-right and front-back that the cube would get squished. It would not maintain a fixed equilibrium shape.