fluid dynamicspressure
I'd like answers both in the more intuitive side an on the more precise side.
Thinking of water as "cubes" of water, for example, would allow pressure in the z axis to be independent of the y or x axis. Choosing other "shapes", like a triangular prism, yield different results. Will this fact, then, be dependent on the format of the molecules of the liquid ?
I've heard of the rotational symmetry of liquids. What is its precise statement ? Why is it true ?
The difference between rain and water in the sink is that rain is simply falling, while water in the sink is being drawn into a center from a distance away, and the water in the sink is not perfectly still. It is rotating, if only a little bit. As it is drawn to the center, the rotation becomes more rapid.
The principle is Conservation of Angular Momentum.
Here, Ms. Kawaguti speeds up her rotation by pulling in her arms:
The fastest point of sail depends on the boat (both its hull shape and its sail plan), the wind strength, and the sea state. In general, a beam reach is not the fastest point of sail.
For instance, in very light wind some boats will go fastest on a close reach due to the increased apparent wind from going toward the wind. For boats that sail faster than the wind, the limiting factor for speed is how close they can sail to the apparent wind; when they are going faster than the wind speed the apparent wind is always forward of the beam. A broader angle to the true wind allows them to go faster before their sails are sheeted all the way in, so a broad reach is the fastest.
Check out the polar diagrams on this site:
https://76trombones.wordpress.com/2009/10/17/polar-diagrams-vmg/
90 degrees to the true wind is not usually the fastest.
As for why some angles are faster than others, that's a bit complicated and beyond the scope of this answer. Suffice it to say that as you sail closer to the wind the component of lift in the direction of travel is less, plus there's increased drag. So you slow down as you approach a close-hauled course. As you bear away to a run, the sails eventually become stalled and less efficient. Somewhere in between a close-hauled course and a run you'll find the angle of maximum speed, which is sometimes approximately 90 degrees to the true wind but not always.
BTW, I should add that the term "beam reach" doesn't have a precise definition. Some sources say a beam reach is 90 degrees to the true wind, others say its 90 degrees to the apparent wind. Most sources introduce the points of sail before the concept of apparent wind and elide over the difference entirely. For boats that only go a fraction of the wind speed (i.e. the vast majority of them) its not that important a distinction. In the answer above, I've used the true wind definition, but even if you choose the apparent wind definition a beam reach is not always fastest. Boats that go faster than the wind have their sails trimmed to close hauled regardless of their point of sail.
Best Answer
The pressure in three different directions is indeed independent for materials that are composed of cubes or other fixed shapes. But these materials are called solids, not liquids.
By definition, a liquid is a material without any regular crystallic or otherwise periodic structure. A liquid is composed of randomly arranged molecules that are as close to each other as the repulsive forces allow – the latter property distinguishes liquids from gases. Liquids and gases are two subgroups of a larger group called fluids.
When a milliliter of liquid is at rest, it means that in this milliliter of material, the molecules are randomly ordered and randomly moving so that their center-of-mass remains at rest. But when it's so, the only "force-related" quantity by which one milliliter of this liquid differs from another is the density or any function of it, such as the pressure. Because there's only one density, there's only one pressure.
Because liquids are not composed of fixed cubes but of chaotic molecules. a new molecule added to a volume of liquid with excessive pressure may escape in any direction. Whatever direction it chooses, the density (in molecules per unit volume) will be reduced to the appropriate value so the pressure may drop, too.
The independence of the fluids' pressure on the direction is known as Pascal's law and you may read an independent explanation at Wikipedia: