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I'm studying fluid mechanics and the book says the following:

For any continuum, forces acting on a piece of material are of two types. First there are forces of stress, whereby the piece of material is acted on by forces across its surface by the rest of the continuum.

Now, I can't get the significance of this. Since he says "forces across its surface" it seems that the piece of the material must be three dimensional. Is this right? So, forces of stress are always exterted on three dimensional pieces of a material?

What I mean by this question is: suppose $D\subset \mathbb{R}^3$ is a three dimensional region filled with a fluid, then forces of stress are always exerted on three dimensional subsets of $D$?

Also, what they represent? As I understand, they represent the force that the rest of the fluid exerts on that piece of it, but why they exert such forces? Is it some sort of force exerted at molecular level that has this macroscopic effects?

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Yes, forces are exerted at the molecular level that have macroscopic effects. This is what thermodynamics is all about. If you hold a sheet of tissue paper in the middle of a gas-filled box, molecules of gas on one side are constantly bouncing off it, creating a force (pressure). Molecules on the other side are also doing so, so the tissue doesn't move. Replace the tissue by an imaginary thin membrane and the same thing happens. The molecules on one side are bouncing off the molecules on the other side, exerting pressure both ways.

If one block of gas is moving past the other, think of two trains moving past each other, and the passengers are amusing themselves by tossing bowling balls back and forth. Each time a bowling ball lands in the other train, it is not going straight in. It is going at an angle. This tends to reduce the difference in speed between the two trains - effectively it is a "drag" between the two trains. When this happens at the molecular level, it is called shear stress, and is one basis for viscosity.

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I don't know much about fluid mechanics, but I think this can be answered by remembering that everything we can touch/see is three dimensional on some scale. A two dimensional object would be infinitely thin if we could see its side. I don't know about in the higher-level physics (When you talk about superstrings etc. Which can exist in several dimensions, or so I've been told.), but I have always been tough that nothing two dimensional can exist in our three dimension universe.

I hope that helps.

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If you think of a gas under pressure, there is a force that is proportional to the surface area that pushes in a direction orthogonal to the surface.

Stress is a generalization where the force could also/instead go in a direction parallel to the surface.

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