# 2.25 Problem Sets - Section 5

## Problem 5.19: Spark ignited spherical combustion

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A combustible mixture of air and fuel is
initially at rest at a density *r*_{1}* *and uniform pressure
*p _{∞}.* At

*t*= 0, the mixture is ignited at the origin by a spark and a flame front begins to move radially outward from the origin at a constant speed

*V*

_{f}. As the flame front overtakes a gas particle, it converts the combustible mixture of density

*r*

*to a hotter combustion product of lower density*

_{1}*r*

*. This occurs so fast that the flame front can be modeled as an infinitesimally thin density discontinuity. Because the volume of a fluid particle increases as it is enveloped by the flame front, the combustible mixture ahead of the expanding flame front is pushed radially outward by the front. The combustion products of density*

_{2}*r*

*that are left behind the flame must, however, be stationary relative to the reference frame fixed in the origin.*

_{2}(a) In a reference frame fixed in the origin,
determine the radial outflow velocity *v(r,t)* of the combustible
mixture. There will be different expressions for the unburned gas
ahead of the flame front (*r > V**f**t*) and the
burned gas left behind (*r < V**f**t*). See also
Prob. 3.17. Show that the velocity distribution can be expressed
in the dimensionless form , where
is
either a dimensionless radial distance at fixed time or a dimensionless
inverse time at fixed *r*, and is
a dimensionless density difference between the unburned and burned
gases.