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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.

Qmechanic
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usainlightning
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  • 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. – Kyle Kanos Mar 28 '15 at 18:45

1 Answers1

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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.

Gold
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  • Fantastic. Do these individual volumes that have been constructed around each point neccesarily represent individual fluid particles? – usainlightning Mar 28 '15 at 19:31
  • 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. – Gold Mar 28 '15 at 19:38
  • 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? – usainlightning Mar 28 '15 at 20:40
  • 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. – Gold Mar 28 '15 at 20:50
  • 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? – usainlightning Mar 28 '15 at 20:56
  • No, you simply want to know the total mass divided by the volume which is a mean value of the mass there. It is not a mean value of the mass per molecule, but the mass per volume that you want. – Gold Mar 28 '15 at 20:59
  • Sorry for opening this thread again. When we talk about flow rate [link],(https://www.khanacademy.org/science/physics/fluids/fluid-dynamics/a/what-is-volume-flow-rate) and we take a surface A. We multiply the velocity of fluid at the surface A and we obtain the volume flow rate. My question here is the velocity at the surface is also an average over a small volume? – Tonylb1 Oct 24 '15 at 14:37
  • @Tonylb1, the idea to associate quantities to points in a continuum is exactly the idea that located at the point you may construct a very small volume and take the average of the quantity inside that volume considering the actual particles which compose the continuum. Velocity of a fluid is just one example: at any point, when you consider the velocity field at the point what you are considering is the average over a very small volume around the point of the velocities of the particles which compose the fluid. – Gold Oct 24 '15 at 14:53
  • @user1620696 great but for example if we say a flow rate of 500l/s over a surface those 500 litres is where exactly? on the surface or near it? – Tonylb1 Oct 24 '15 at 14:57
  • This is a little different. You are seeing what is the volume of fluid which crosses a small surface element around the point per unit second. Take a look at this question of mine http://physics.stackexchange.com/questions/132096/why-is-this-the-volume-flow-rate-per-unit-area, it may help you out with this. – Gold Oct 24 '15 at 15:12
  • @user1620696 I checked it. I understood the unit explanation but the other comment is like the website I mentioned earlier. If Q = V/t and V = A.d then if t=1s Q = V = 500L/s let's suppose A = 1m2 than d= 500m? so here we are taking a very big volume that has a constant velocity unlike the continuum theory. – Tonylb1 Oct 24 '15 at 15:38
  • I might be wrong or misunderstanding what you are saying. This volume is not the volume we use to take averages of quantities. To take the averages and define the quantities pointwise, we imagine we take around each point a small neighborhood and average quantities associated to the particles there. When we talk about the flow rate accross a surface we are talking about the amount of volume which crosses the surface per unit time. If a lot of volume crosses the surface it might be because the flow of the fluid is quite fast, but this has nothing to do with the continuum approximation. – Gold Oct 24 '15 at 16:01
  • @user1620696 Yes true. but we are taking the velocity at the surface and we multiply by the surface so we obtain the flow rate. Later we say it is an amount of volume that pass through the surface and this volume has the same velocity. That's why I am confused – Tonylb1 Oct 24 '15 at 16:06
  • I think I'm not understanding the point here, perhaps someone with a more "practical" understanding would help you better. I suggest, instead, that you ask this as a sepparate question, exposing your point better. – Gold Oct 24 '15 at 16:14