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Elliptic Solution to the Emmons Problem.

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.

### Keywords:

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

### Abstract:

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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.
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