I'm a bit confused by the continuum hypothesis stating that fluid are continuous objects rather than made out of discrete objects.
Say for $\rho (x,t)$ (density) is there more than one fluid particle at $x$ or less than one.
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asked Mar 28, 2015 at 18:40
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$\begingroup$ It'd depend on the fluid you are modeling and the density at $x$. If you have a hydrogen plasma ($m_H\approx1.67\times10^{-24}\,\rm g$) and the density is $\rho\leq10^{-25}\,\rm g/cm^3$ then you have less than one particle per cubic centimeter, but if you have $\rho\geq10^{-24}\,\rm g/cm^3$, then you have more than one. $\endgroup$– Kyle KanosCommented Mar 28, 2015 at 18:45
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The continuum hypothesis means the following: at each point of the region of the fluid it is possible to construct one volume small enough compared to the region of the fluid and still big enough compared to the molecular mean free path.
Why is that important? Because of two things. First, since the volume you can build at each point is very small compared to the size of the region of the fluid, you can think of the volume as located at a point instead of considering it as a collection of points. Imagine Earth for instance. If you build a small volume of $1 m^3$ somewhere on the surface of the Earth it is so small compared to Earth's size it can be considered to be associated with a particular point.
The secont thing is that since the volume is big enough compared to the molecular mean free path, this means it contains a large enough number of molecules. Why is that something you would want? Because containing a reasonable number of molecules allows you to take means on the volume and those means will make sense.
So, for instance, you can go there, compute the mass of each molecule, sum them up and divide by the volume. Given this hypothesis, this mean makes sense. And given the first hypothesis, you can think about this mean as associated with the point.
Because of that it makes sense of talking about fields defined on the region of the fluid. The mass density for example or the velocity field. They are in truth, means of quantities associated with the molecules, but that on the macroscopic point of view, can be considered just as fields associating quantities to points of the region.
On the density case, it really means: if $x$ is a point on the region of the fluid and $t$ an instant of time, $\rho(x,t)$ is the mean value of the mass of molecules contained inside one such small volume associated at $x$ at time $t$. From the macroscopic point of view, it is just a density that allows you to get mass through integration.
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$\begingroup$ Fantastic. Do these individual volumes that have been constructed around each point neccesarily represent individual fluid particles? $\endgroup$ Commented Mar 28, 2015 at 19:31
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$\begingroup$ We usually call this volume containing a number of molecules a fluid particle. That is a terminology used, where one might think "well a fluid particle is at a point", but what one really means is "a very small volume, which we think as located at a point with many molecules inside". So yes, they are fluid particles, but a fluid particle contains many true molecules of the fluid. $\endgroup$– GoldCommented Mar 28, 2015 at 19:38
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$\begingroup$ I'm not sure what is meant by 'taking means on the volume'. Do you mean taking the mean of the molecular masses inside the volume around a point and requiring there to be a large number of molecules so this mean is representative of the fluid? $\endgroup$ Commented Mar 28, 2015 at 20:40
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$\begingroup$ Yes that is the idea. If such small volume according to the hypothesis exists at each point, you simply take the mean of the mass of the molecules in the volume. Then intuitively for this mean to make sense you need a big enough number of molecules on the volume. $\endgroup$– GoldCommented Mar 28, 2015 at 20:50
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$\begingroup$ Would you not take the mean and then multiply it by the number of molecules inside to get the mass of that volume as opposed to just taking the mean of the masses? $\endgroup$ Commented Mar 28, 2015 at 20:56