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I'm trying to solve the equation:

$$ \frac{\partial^2 v_z(x,y)}{\partial x^2} + \frac{\partial^2 v_z(x,y)}{\partial y^2} = \frac{\Delta P}{\mu \Delta X} $$

This flow is supposed to be flowing in a square channel (laminar,steady,fully developed).

I know I need to solve it numerically and if there was no source term I could just take the average of all four points around each node and get my answer. The problem is that I need to find the appropriate source term and I honestly don't know how to start. I also know I need to make the equation dimensionless. I know the source term can't be found analytically and I honestly am lost there. Can someone explain what the source term is and how I can start my quest to find it?

tpg2114
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Adam Marciniak
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    What do you think the terms are in the equation you're trying to solve? Isn't one of those the source term? – tpg2114 Feb 06 '14 at 04:52
  • I did not do the calculation, but I think this question and the answer by Ron Maimon may help you:

    http://physics.stackexchange.com/questions/31260/is-there-an-analytical-solution-for-fluid-flow-in-a-square-duct

     – Ján Lalinský Feb 06 '14 at 11:21
  • Why do you want to solve it numerically? An analytical solution (in terms of an infinite series, but you can easily cut it off) exists for it. Check out the Shah and London, 1978 – Michiel Feb 13 '14 at 21:59

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