In the Flow of dry water chapter of Feynmann lecture, this following point is written (see here):
The law of hydrostatics, therefore, is that the stresses are always normal to any surface inside the fluid. The normal force per unit area is called pressure. From the fact that there is no shear in a static fluid, it follows that the pressure stress is the same in all directions (Fig. 40-1). We will let you entertain yourself by proving that if there is no shear on any plane in a fluid, the pressure must be the same in any direction.
I am trying to figure out how to prove the statement "if there is no shear on any plane in a fluid, the pressure must be the same in any direction."
What I've found so far:
The premise he puts itself is confusing for me because according to Wikipedia, the pressure is a scalar field rather than a vector field and hence it should have no associated direction.
Anyways, I assumed maybe that since pressure’s direction is determined by the area which it may act on and hence any area element below a given height from the top of the container would have the same pressure acting throughout it.
After some searching on stack exchange, I found that the explanation of pressure being the same in all direction is given by Pascal's law (See Here), but on seeing the Wikipedia page for pascal's law (here), I formed the impression that Pascal's law may be related to pressure transmission rather than what direction pressure acts in.
This led me to search for proof for pascals law.
I found this answer in which the author proves that pressure change in any point in a fluid is transmitted throughout the whole fluid undiminished(pascal's law) but I think that the statement doesn't really explain the isotropic nature of pressure.
Hence this leads me to two main questions:
- How is pascal's law related to the isotropic nature of pressure?
- How do you prove the isotropic nature of pressure using pascal's law or however another way?