Flow Through a Rectangular Duct

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Consider the fluid flow through a rectangular duct. The flow velocity in the cross section obeys the following governing equation:

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.

The associated boundary conditions are (no slip condition) on the four sides of the duct.

The analytical solution of the partial differential equation obtained using the separation of variables technique [1] is given by:

,

where .

The Demonstration uses the finite element method embedded in Mathematica's NDSolve to display the velocity distribution at the nodes of the user-set discretization of the domain. A value of the root-mean-square deviation (RMSD) is reported on the plot as an indication of the accuracy of the numerical solution when compared to the exact analytic solution. For the smaller values of the cell measure parameter, one finds excellent agreement between our numerical solution and the analytic solution (i.e. the RMSD is quite small).

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Contributed by: Housam Binous, Ahmed Bellagi, and Brian G. Higgins (August 2015)
Open content licensed under CC BY-NC-SA


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Reference

[1] T. Bennett, Transport by Advection and Diffusion, New York: Wiley, 2012.



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