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Nondimensinalize PDE

  1. Sep 22, 2012 #1
    $$
    \frac{1}{\alpha}T_t = T_{xx}
    $$
    B.C are
    $$
    T(0,t) = T(L,t) = T_{\infty}
    $$
    I.C is
    $$
    T(x,0) = T_i.
    $$
    By recasting this problem in terms of non-dimensional variables, the diffusion equation along with its boundary conditions can be put into a canonical form.
    Suppose that we introduce variable scaling defined by
    $$
    x_* = \frac{x}{L}\quad\quad t_* = \frac{\alpha t}{L^2}\quad\quad \theta = \frac{T - T_{\infty}}{T_i - T_{\infty}}
    $$
    With this change of variables, show that the problem definition becomes
    \begin{alignat*}{5}
    \theta_{t_*} & = & \theta_{x_*x_*} & & \\
    \theta(0,t_*) & = & \theta(1,t_*) & = & 0\\
    \theta(x_*,0) & = & 1
    \end{alignat*}

    $$
    \frac{1}{L^2}\frac{\partial T}{\partial t_*} = \frac{1}{L^2}\frac{\partial^2 T}{\partial x_*^2}
    $$
    Do I make the [itex]T[/itex] substitution just as [itex]T = \theta(T_i - T_{\infty}) + T_{\infty}[/itex]?
     
  2. jcsd
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