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Elliptic Solution to the Emmons Problem.

pdf icon Elliptic Solution to the Emmons Problem. (220 K)
Baum, H. R.; Atreya, A.

Session H4 - Fire; Paper H22;

Combustion Institute/Western States Section. 5th US Combustion Meeting. Fundamentals of Combustion, Air Pollution and Global Warming, Alternative Fuels. Proceedings. Session H4. March 25-28, 2007, San Diego, CA, 1-17 pp, 2007.


fire research; geometry; flammability; equations; mixture fraction; flow fields; velocity field; boundary layers; mathematical models; soot


The classical Emmons problem provides a well-defined geometry with analytical solutions that is relatively easy to establish experimentally. It has therefore been very useful for flammability assessment of materials. In this paper, the Emmons' problem is formulated in terms of an elliptic equation for the mixture fraction developing in a variable density elliptic flow field. Exact analytical solutions are developed for the mass and mixture fraction conservation equations in parabolic coordinates. The corresponding velocity field incorporates both the Emmons boundary layer result and an elliptic upstream influence that asymptotically satisfies the full Navier-Stokes equations. Thus the solution for the velocity field is exact everywhere outside the boundary layer. In the burning boundary layer, the error is small except in a small region 0(20 Stokes lengths 2mm) downstream of the leading edge where the velocity field is only qualitatively correct. However, the singularity at the leading edge is geometrical, and unlike the boundary layer solution, the singularity is confined to a point rather than the whole line x=O. This framework is used to analyze soot transport with generation and destruction. The soot model is also analytically tractable and seems to yield physically plausible results.