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Fluid dynamics - find distribution of a conserved variable

  1. Sep 9, 2014 #1
    I accidentally posted this to the "Calculus & Beyond" forum when I meant to post it to the physics forum. If someone can tell me how to move this post, I will get rid of it here!

    1. The problem statement, all variables and given/known data
    Consider a property, for example temperature θ, that is conserved during advection (i.e. Dθ/Dt = 0 where D/Dt represents the total derivative). Suppose the velocity of the field advecting the fluid is:

    u = ax, v = -ay, w = 0

    Find the form of the temperature for t > 0 when at t=0, θ(t=0) = θ0y/L.

    2. Relevant equations

    Dθ/Dt = ∂θ/∂t + u*(∂θ/∂x) + v*(∂θ/∂y)

    3. The attempt at a solution

    So far, I've calculated the trajectories:
    x(t) = C1e^(at)
    y(t) = C2e^(-at)

    I also know that since θ is conserved, θ is a function of where in space the parcel originated. In other words, θ=θ(Y) where Y is the Lagrangian variable.

    I've expanded the total derivative and set it equal to zero since θ is conserved:
    (1) Dθ/Dt = ∂θ/∂t + u*(∂θ/∂x) + v*(∂θ/∂y) = 0

    Then, I thought to say that θ(t)=(θ0y(t)/L)*f(t), where f(t) is some function of time that we want to find out so we can see what the function θ(t) looks like.

    I took this θ(t)=(θ0y(t)/L)*f(t) and plugged it back into equation (1), but don't think I'm getting anywhere. Can anyone tell me if I'm on the right track or if I'm going about this completely the wrong way?
     
  2. jcsd
  3. Sep 9, 2014 #2
    You started off OK, but then you started going in a fruitless direction. Let's focus on time zero. Let x(0) and y(0) be the coordinates of a material particle at time zero. So, at time zero, the "temperature" of the material particle is θ(x(0),y(0))=θ0y(0)/L. What do the lines of constant θ look like at time t = 0? When the fluid stretches the way it is described, does θ at later times vary with x? What do the lines of constant θ look like at later times t? If a material particle is at x(0) and y(0) at time zero, what are the coordinates of this same material point x(t) and y(t) at time t? What is the value of θ for the material point x(t) and y(t) at time t?

    Chet
     
  4. Sep 9, 2014 #3
    Hi Chet, thank you very much for your reply - it helped get me out of my rut and thinking in a new way. Here's what I've come up with:

    We've already found that
    x(t)=C2eat
    y(t)=C3e-at

    Let's say that the particle is at (x0,y0) at t=0. Then
    x(t)=x0eat
    y(t)=y0e-at

    Therefore we can find out where the particle started by inverting the above equations:
    x0=x(t)e-at
    y0=y(t)eat

    Now, let's think about a particle that has travelled to the point (x(t),y(t)). Since θ is conserved, in order to get the particle's temperature θ(t) all we need to know is its original y-coordinate (and we are lucky here because the equations for x(t) and y(t) are not coupled). In other words,
    θ(x,y,t)=θ0*y(t=0)/L=θ0*y0/L

    Thus the final answer is
    θ(x,y,t)=θ0*y(t)*exp(at)/L

    Do you think this looks better now? I think before I was getting caught up in trying to plug things into the total derivative instead of thinking about the picture of an individual parcel.
     
    Last edited: Sep 9, 2014
  5. Sep 9, 2014 #4
    Excellent job. Very nice.

    Chet
     
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