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Darned heat conduction problem

  1. Jun 1, 2010 #1
    1. The problem statement, all variables and given/known data


    In each of Problems 1 through 8 find the steady-state solution of the heat conduction equation ∂2uxx=ut that satisfieds that given set of boundary problems.

    .....

    3. ux(0,t)=0, u(L,t)=0

    2. Relevant equations

    Assume u(x,t)=X(x)T(t)


    3. The attempt at a solution

    2X'' T = X T'

    X'' / X = (1/∂2) T' / T = -ß

    X'' + ßX = 0,
    T + ∂2ßT' = 0.

    Since ux(0,t)=0 and u(x,t)=X(x)T(t),

    X'(0)T(t)=0 ---> X'(0)=0.

    Solving X'' + ßX = 0 like a I would any differential equation,

    X(x) = C1cos(√(ß)x) + C2sin(√(ß)x)

    --->

    X'(x) = -√(ß)C1sin(√(ß)x) + √(ß)C2(cos(√(ß)x)

    --->

    X'(0)=0= 0 + √(ß)C2 ---> C2=0

    --->

    X(x) = C1cos(√(ß)x).

    Also, u(L,t)=0 ---> X(L)T(t)=0 ---> X(L)=0.

    I'm getting somewhere, right?

    Since we're talking about a steady-state solution being reached, some function of t and possibly x will disappear, leaving us with just a function v(x) that shows the temperature at any place in the rod.

    u(x,t) = v(x) + w(x,t),

    limt-->∞ u(x,t) = v(x)


    ...............


    Anyhow, I'm sort of stuck. Problem 1 was was easy because it gave me u(0,t)=T1, u(L,t)=T2, just like the examples in the chapter; but this one is throwing me off with the ux(0,t). I'm having trouble putting the pieces together.

    Please help, geniuses.
     
  2. jcsd
  3. Jun 1, 2010 #2

    Mapes

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    At steady state, [itex]\partial T/\partial t=0[/itex]. Why not just integrate [itex]\partial^2 T/\partial x^2=0[/itex] twice and apply the boundary conditions? Or are you required to use separation of variables?
     
  4. Jun 1, 2010 #3
    Hmmmmm ..... In the textbook example it never has us integrate T''(t)=0 when solving these problems. It does say, however:

    Since v(x) must satisfy the equation of the heat equation ß2uxx=ut, we have

    v''(x)=0, 0<x<L.
     
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