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U-decay chain + diffusion

  1. Feb 8, 2012 #1
    Ok guys, I've got a problem which I'd like to solve in an elegant way, but I don't how to solve it (if it can be solved analytically).

    I'm considering the decay chain of Uranium. So Uranium decays to Thorium, which decays to Protactinium, and so on. I know how to solve such a linear system, since the only variable is time, with the decay constants being constant (duh). Example:

    [itex]\frac{dU}{dt} = -{k}_{1}U[/itex]

    [itex]\frac{dTh}{dt} = -{k}_{2}Th + {λ}_{1}U[/itex]

    [itex]\frac{dPa}{dt} = -{k}_{3}Pa + {λ}_{2}Th[/itex]

    in which the k's are decay constants. Or in vector notation:

    [itex]{N}^{*} = AN[/itex]

    in which [itex]{N}^{*}[/itex] is the time derivative of species [itex]{N}_{i}[/itex], A is a matrix consisting of k's. This solves to:

    [itex]N(t) = S\Lambda{S}^{-1}N(0)[/itex]

    in which S is the matrix of eigenvectors and [itex]\Lambda[/itex] is a diagonal matrix with [itex]{e}^{{λ}_{i}t}[/itex] as the diagonal entries.

    But now I want to solve it when diffusion into or out of the system is allowed. So my initial problem becomes something like this:

    [itex]\frac{\partial U}{\partial t} = -{k}_{1}U - \nabla*(-D \nabla U)[/itex]

    [itex]\frac{\partial Th}{\partial t} = -{k}_{2}Th + {k}_{1}U - \nabla*(-D\nabla Th)[/itex]

    and so on...

    with D being a tensor consisting of diffusion constants for that species and * denoting the dot product (couldn't find a nicer dot).

    Now the concentration of species depends on both t and x, which are independent of eachother. I was wondering if it even is possible to solve this analytically and if so, how? I imagine the solution would have a position and a time dependent component, something like:

    [itex]N(t) = PQ{P}^{-1}x + ST{S}^{-1}N(0)[/itex]

    in which P is the eigenvector matrix of some position dependent matrix, Q the solution matrix in terms of position and x the initial position, and S is the eigenvector matrix of the time dependent matrix and T the solution matrix in terms of time and N(0) the initial concentrations.

    Please note that the diffusion flux depends on the concentration gradient in the system, which in turn depends on time, since species decay.

    Could anyone please help me with the above?
     
  2. jcsd
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