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an interesting problem of supplying boundary condition for a PDE

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92557] an interesting problem of supplying boundary condition for a PDE
  • From: pratip <pratip.official at gmail.com>
  • Date: Sun, 5 Oct 2008 06:05:25 -0400 (EDT)

Hi All,
Down is the equation system

Jc = -\[Rho] Subscript[\[GothicCapitalD], 
   k] (D[\[Omega][x, y, t], x] + D[\[Omega][x, y, t], y]) - 
  Subscript[\[GothicCapitalD], 
   k] (D[Log[T[x, y, t]], x] + D[Log[T[x, y, t]], y]);

ContEq = D[u[x, y, t], x] + D[u[x, y, t], y] == 0;
MomentumEq = \[Rho] (D[u[x, y, t]^2, x] + D[u[x, y, t]^2, y]) == 
   D[(\[Mu] (D[u[x, y, t], x] + D[u[x, y, t], y]) - 
       2/3 \[Mu] (D[u[x, y, t], x] + D[u[x, y, t], y]) ), x] + 
    D[(\[Mu] (D[u[x, y, t], x] + D[u[x, y, t], y]) - 
       2/3 \[Mu] (D[u[x, y, t], x] + D[u[x, y, t], y]) ), y] - 
    D[P[x, y, t], x] + D[P[x, y, t], y];
TranportEq = \[Rho] \[Omega][x, y, 
      t] + \[Rho] (D[\[Omega][x, y, t] u[x, y, t], x] + 
       D[\[Omega][x, y, t] u[x, y, t], y]) == D[Jc, x] + D[Jc, y];
EnergyEq = 
  Subscript[c, 
    p] (D[(\[Rho] u[x, y, t] T[x, y, t]), x] + 
      D[(\[Rho] u[x, y, t] T[x, y, t]), y]) == 
   D[(\[Lambda] (D[T[x, y, t], x] + D[T[x, y, t], y])), x] + 
    D[(\[Lambda] (D[T[x, y, t], x] + D[T[x, y, t], y])), y];
{ContEq, MomentumEq, TranportEq, EnergyEq} // 
  FullSimplify // TableForm

Now we need a boundary condition that Mathematica can handle. I have freedom to choose any initial values for the unknowns. For this kind of equation I don't know what type of BC I should take.  I just took

BC = {u[x, y, 0] == x + y, u[0, y, t] == u[1, y, t], 
  u[x, 0, t] == u[x, 1, t], P[x, y, 0] == x, P[0, y, t] == P[1, y, t],
   P[x, 0, t] == P[x, 1, t], T[x, y, 0] == 2 y + 1, 
  T[0, y, t] == T[1, y, t], 
  T[x, 0, t] == T[x, 1, t], \[Omega][x, y, 0] == 
   x - y, \[Omega][0, y, t] == \[Omega][1, y, t], \[Omega][x, 0, 
    t] == \[Omega][x, 1, t]}

Now I Finally insert values to the constants in the PDE so that I can call NDSolve.

Eq = {ContEq, MomentumEq, TranportEq, EnergyEq} /. \[Lambda] -> .1 /. 
     Subscript[c, p] -> 1.2 /. \[Rho] -> .01 /. \[Mu] -> 1 /. 
  Subscript[\[GothicCapitalD], k] -> 1;
EQN = Join[Eq, BC]

To solve the PDE I call

Sol = NDSolve[
   EQN, {u, P, T, \[Omega]}, {x, 0, 1}, {y, 0, 1}, {t, 0, 3}, 
   Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> {"TensorProductGrid"}}] // Timing

And I see the following error

*****************
*****************
NDSolve::pdord: Some of the functions have zero differential order so \
the equations will be solved as a system of differential-algebraic \
equations. >>

NDSolve::mxsst: Using maximum number of grid points 100 allowed by \
the MaxPoints or MinStepSize options for independent variable x. >>

NDSolve::mxsst: Using maximum number of grid points 100 allowed by \
the MaxPoints or MinStepSize options for independent variable y. >>

NDSolve::mxsst: Using maximum number of grid points 100 allowed by \
the MaxPoints or MinStepSize options for independent variable x. >>

General::stop: Further output of NDSolve::mxsst will be suppressed \
during this calculation. >>

NDSolve::ibcinc: Warning: Boundary and initial conditions are \
inconsistent. >>

NDSolve::ivcon: The given initial conditions were not consistent with \
the differential-algebraic equations.  NDSolve will attempt to \
correct the values. >>
*****************
*****************
It runs for long and finally tells LinearSolve out of memory.
I hope people with your expertise can give me some insight. You can take full freedom to define BC as you wish. We need just a solution. I am mainly interested to find a class of BC that makes the PDE solvable with Mathematica.
I hope for some prompt reply.


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