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Thread: Motion between points tending to zero dimensions

  1. #1 Motion between points tending to zero dimensions 
    Administrator KALSTER's Avatar
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    Logic dictates that everything is infinitely devisable, for when you choose to stop dividing after any number of divisions you could still in principle divide to yet another level. So if you take it to the limit (pun intended) mathematically, you will be left with a particle with dimensions that tend toward zero. An infinite number of these particles together? How could motion relative to each other be expressed?


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  3. #2 Re: Motion between points tending to zero dimensions 
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    Quote Originally Posted by KALSTER
    Logic dictates that everything is infinitely devisable
    Logic? Well, then it’s the logic on which Euclidean geometry is based – which leads to Zeno’s paradoxes.


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  4. #3  
    Administrator KALSTER's Avatar
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    Cool link! I also looked further into Incommensurable magnitudes as well. Calculus has provided a means to deal with such paradoxes, but that still does not answer my question really. A sea of particles with dimensions tending towards zero would mean that the volume of such a system would also tend toward zero. But what if the number of particles was assumed to be infinite? How could the fluid dynamics be expressed mathematically with these givens?
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

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    "It is the mark of an educated mind to be able to entertain a thought without accepting it." - Aristotle
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    Forum Professor sunshinewarrior's Avatar
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    Quote Originally Posted by KALSTER
    Cool link! I also looked further into Incommensurable magnitudes as well. Calculus has provided a means to deal with such paradoxes, but that still does not answer my question really. A sea of particles with dimensions tending towards zero would mean that the volume of such a system would also tend toward zero. But what if the number of particles was assumed to be infinite? How could the fluid dynamics be expressed mathematically with these givens?
    Only by calculus and the concept of limit, as far as I'm aware.
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  6. #5  
    Forum Masters Degree bit4bit's Avatar
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    Quote Originally Posted by KALSTER
    Cool link! I also looked further into Incommensurable magnitudes as well. Calculus has provided a means to deal with such paradoxes, but that still does not answer my question really. A sea of particles with dimensions tending towards zero would mean that the volume of such a system would also tend toward zero. But what if the number of particles was assumed to be infinite? How could the fluid dynamics be expressed mathematically with these givens?
    Sounds to me like continuum mechanics is what you're looking for...fluids are considered to be continuums, that is, all properties (pressure, velocity, etc) vary continuously across the fluid. This is just like considering pressure as a statistical mean of kinetic energies of particles...but when we talk about pressure in a container we don't necessarily consider the individual particles, but assign a property called pressure to the fluid.

    In fluid dynamics, it's the same kind of thing. The size of the particles is negligible, and any interactions between them is described as a property of the continuum. Viscosity for examples is transfer of momentum from particles colliding.

    As sunshinewarrior said, it's based on limits and calculus...the fluid is flowing, so we are considering rates of change/integrals of fluid properties. The derivations are based on picking an abitrary control volume, whose volume tends to 0 in the limit, and looking at the flux through it.

    As another note, there are different ways to analyse a fluid. The flowing fluid can be expressed as a system of equations (one for each particle) parameterized by time, where the progress of each particle is 'tracked' (Systems analysis)....or alternatively you can consider an arbitrary, fixed 'control volume', and look at the changes of fluid properties within the volume over time (Control volume analysis). You can flip between them with 'Reynolds transport theorem'.
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    Administrator KALSTER's Avatar
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    Thanks, I'll take a look at those :wink: . The fluid I am takjing about is a bit different than the standard, in that the size of the individual particles might vary and the seperations between them might be zero. Also, there is no friction between them, but there is resistence to the enlarging of the particles, the enlarging enabling fluidity. Would these mathematical models be able to deal with that do you think? Maybe considering each particle as a control volume, but then treating the interactions between the particles with system analysis? Also, the scale involved in the calculations would need to cover many orders of magnitude with the end result of quasi-chaotic movement on small scale having to be predictable on larger scales?
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

    "Gullibility kills" - Carl Sagan
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  8. #7  
    Forum Masters Degree bit4bit's Avatar
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    If the sizes of the particles vary, then the co-efficient of viscosity for the fluid would change accordingly....but then you say there is no 'friction' between particles? If by this you mean that the fluid is inviscid, then you are saying that the particles cannot exchange momentum with each other through collisions, cause that's what viscosity is.

    I think you need to state more clearly what these particles are, and exactly what properties they have. Then from that you can derive all the properties the fluid consisting of those particles will have, and then analyse it with systems or control volume analysis. I'm talking about classical fluids, made up of atoms or molecules, but i get the impression you are talking about something other than that?

    Whatever it is, calculus is a very powerful tool nonetheless.
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  9. #8  
    Administrator KALSTER's Avatar
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    Superfluidity is the word I was looking for! I was reading through the wiki article on Superfluidity and came across this gob-smacking passage:

    "A more fundamental property than the disappearance of viscosity becomes visible if superfluid is placed in a rotating container. Instead of rotating uniformly with the container, the rotating state consists of quantized vortices. That is, when the container is rotated at speed below the first critical velocity (related to the quantum numbers for the element in question) the liquid remains perfectly stationary. Once the first critical velocity is reached, the superfluid will very quickly begin spinning at the critical speed. The speed is quantized - i.e. it can only spin at certain speeds."

    This is exactly the type of stuff I was thinking and wondering about. My excitement is overflowing!

    Is there some kind of simulation engine where I can feed it variables and see what happens anywhere available?
    Disclaimer: I do not declare myself to be an expert on ANY subject. If I state something as fact that is obviously wrong, please don't hesitate to correct me. I welcome such corrections in an attempt to be as truthful and accurate as possible.

    "Gullibility kills" - Carl Sagan
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    "It is the mark of an educated mind to be able to entertain a thought without accepting it." - Aristotle
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  10. #9  
    Forum Masters Degree bit4bit's Avatar
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    Glad you found it...I don't know much about superfluidity, but I know some CFD software/NSE solvers. If you use Solidworks (excellent mechanical CAD software), you can get various add-ons from the COSMOS suite, including CFD packages. Also ANSYS FLUENT is meant to be really good (never used it though). A word of warning though, CFD software like these are very expensive. For more CFD softwares check this page...some are free/open-source, but mainly for US citizens only.

    The drawback is that I don't think any of these codes/software will allow superfluidity simulations. If you're a good programmer, and know about CFD, then you could modify a code yourself to allow for it, but generally that's done by professional CFD experts working for organisations like NASA and other big aerospace companies.
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