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Differential Equations - Help with scaling / dimensionalizing

  1. Sep 11, 2012 #1
    Hi all,

    I have a homework assignment related to the conservation of energy in a fluid.
    This is given in terms of the density [itex]ρ[/itex], specific heat [itex]c_{p}[/itex], thermal conductivity [itex]k[/itex], bulk viscosity [itex]μ[/itex], dissipation function [itex]\widetilde{\phi}_{v}[/itex], velocity function [itex]\widetilde{u}[/itex], spacial derivative of the Temperature function (respect to [itex]L[/itex]) [itex]\widetilde{∇}\widetilde{T}[/itex], second spacial derivative with respect to [itex]L, \widetilde{∇}^{2}\widetilde{T} [/itex], and partial derivative of the temperature function [itex]\frac{∂\widetilde{T}}{∂\widetilde{t}}[/itex]. [itex]T[/itex] is temp, [itex]t[/itex] is time, [itex]U[/itex] is velocity, mass is [itex]M[/itex], length is [itex]L[/itex].

    All of the tilde's are supposed to be on top of that preceding variable, I am sorry if this is way too confusing, tried my best..

    That's a lot of variables, but it is OK.

    The basic part of this problem is to dimensionalize the problem and find the dimensionless parameters according to the pi theorem.

    It looks something like this [itex]ρc_{p}(\frac{∂\widetilde{T}}{∂\widetilde{t}} + \widetilde{u}\bullet\widetilde{∇}\widetilde{T}) = k\widetilde{∇}^{2}\widetilde{T} + μ\widetilde{\phi}_{v}[/itex]

    I know the units for the basic nonfunctional variables like [itex]ρ[/itex] and [itex]c_{p}[/itex].
    My issue is with the functions. My instructor said that, for example [itex]\widetilde{\phi}_{v} = \frac{U^{2}\phi_{v}}{L^{2}}[/itex], where [itex]\phi_{v} [/itex]is the value and not the function.

    I can figure out all of the terms' dimensions and come out OK.

    But, how would I convert something like [itex]\widetilde{T}[/itex] (temperature function) into dimensions? Would that be as simple as just [itex]\frac{T}{t}[/itex]? So the partial [itex]\frac{∂\widetilde{T}}{∂\widetilde{t}}[/itex] is dimensionalized as [itex]\frac{T}{t}[/itex]?
    Or am I missing something here that turns the function into a different set of terms when given initial temp [itex]T_{0}[/itex]?

    I am given initial temperature [itex]T_{0}[/itex], velocity [itex]U[/itex], and length [itex]L[/itex].

    When I find the dimensionless parameters, I am calculating that [itex]α_{1}[/itex], for example is something with insane powers like [itex]ρc_{p}^{6}\widetilde{u}^{{-14}}k(\frac{\widetilde{∂T}}{\widetilde{∂t}})^{5}[/itex]

    This just seems completely wrong, we go from nothing more than 3rd powers at the most to all of the sudden -14 powers?

    Thx in advance. I don't need direct answers but help is appreciated!

    EDIT: if it helps, I am fine with the math side of this, but I took Physics C Mechanics in highschool, that was it and we never even came close to problems of this complexity. I have to get used to this physics mumbo jumbo.

    Okay I made this into LaTeX, which I had forgotten. Hopefully I can get an answer now!
     
    Last edited: Sep 11, 2012
  2. jcsd
  3. Sep 12, 2012 #2
    Can you show a bit of your work... So for example, what are the units of [itex] \mu [/itex] and [itex]\phi_v [/itex]? What do the tildes mean?
     
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