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To my knowledge, the continuum hypothesis basically says two things:

(1) We want to consider fluid elements whose volume is large enough to contain enough molecules such that average value of its properties is not affected by molecular fluctuations.

(2) The fluid element must be small enough in comparison to the dimensions of the body of fluid under study such that a fluid element can essentially be modeled as a point in the fluid body.

I do not see how these ideas relate to the idea of treating fluid as a continuum. Why does satisfying these two conditions equate to treating a fluid as a continuum?

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If you divide a fluid up into blocks, you are making an approximation of how the fluid behaves. The make the approximation indistinguishable from reality, the blocks would have to be

  • So small enough that they don't affect anything. That is, you don't see a pixelated version of the fluid. $(2)$ guarantees this. You can look at any point in the fluid and find a block there.
  • Big enough that each block behaves like the fluid. $(1)$ guarantees this.
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  • $\begingroup$ Thanks for the answer. I got your first point -- the fluid blocks are small enough that, on a macroscopic scale, you can look at any point and find a block centered there. But could you elaborate on your second point? What do you mean by "behaving like the fluid"? Thanks. $\endgroup$
    – silverbackgorilla
    Commented Oct 23, 2021 at 14:04
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    $\begingroup$ It's the point you made. There have to be enough molecules so the block has the average properties of the fluid. For example, density of a compressible fluid is proportional to the number of molecules in the block. That breaks down if the block has room for $2 \frac{1}{2}$ molecules. $\endgroup$
    – mmesser314
    Commented Oct 23, 2021 at 15:16
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These two hypotheses do not automatically amount to a continuum treatment. What we want from a continuum theory - as opposed to a microscopic theory - is to describe an object by continuous/smooth functions, which could be made to obey a set of partial differential equations.

E.g., the microscopic density of a liquid is substantially non-zero only at the points where the atoms/molecules are located, and nearly zero between them. For many purposes such a description is unnecessary, and we can do well with considering density over some microscopically small volume, containing millions of atoms - as we do in everyday life.

Yet, this microscopically small volume should be sufficiently small, as to make the description meaningful, i.e., for the quantities to change on the scale of interest.

Finally, to add a few terms:

  • Coarse graining is the terms used to describe mathematically transition from microscopic to macroscopic description.
  • Some of the disciplines concerned are: hydrodynamics=fluid mechanics, elasticity theory, macroscopic electrodynamics, etc.
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  • $\begingroup$ Thanks for the answer. So basically, if we model a fluid as being made up of very small objects called fluid elements (instead of molecules), each with assigned values for properties like temperature, pressure, etc., we can model a fluid as a continuous function in space, to which methods of calculus can be applied. And also, by making them large enough such that their properties are unaffected by molecular fluctuations guarantees that properties will vary smoothly in space. Is that a correct interpretation of your answer? Thanks $\endgroup$
    – silverbackgorilla
    Commented Oct 23, 2021 at 14:12

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