Holes in flames, flame isolas, and flame edges

We examine a simple model problem designed to elucidate how easily a hole in a diffusion flame can close and how easily an isolated region of burning in a mixing region (a flame isola) can grow. With the thickness of the mixing layer/diffusion flame defining the characteristic length, we find that a hole of diameter approximate to 1 closes for all but a tiny interval of Damkohler numbers above the one-dimensional quenching value; a hole of diameter approximate to 3 closes for Damkohler numbers that exceed the quenching value by more than 30%-40%, depending on the Lewis number: and a hole of infinite diameter closes for Damkohler numbers that exceed the quenching value by more than similar to 70%. Even larger Damkohler numbers are required for isolas to grow and we calculate these values for different radii. We investigate the speeds with which hole edges or isola edges advance or retreat and provide evidence that for a shrinking hole or isola, a meaningful speed can be defined that depends only on the combustion parameters and the instantaneous hole/isola radius.