Abstract
It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states - the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 679-702 |
| Number of pages | 24 |
| Journal | SIAM Journal on Applied Mathematics |
| Volume | 60 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jan 1 2000 |
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ASJC Scopus subject areas
- Applied Mathematics
Cite this
Stability of pole solutions for planar propagating flames : I. Exact eigenvalues and eigenfunctions. / Vaynblat, Dimitri; Matalon, Moshe.
In: SIAM Journal on Applied Mathematics, Vol. 60, No. 2, 01.01.2000, p. 679-702.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Stability of pole solutions for planar propagating flames
T2 - I. Exact eigenvalues and eigenfunctions
AU - Vaynblat, Dimitri
AU - Matalon, Moshe
PY - 2000/1/1
Y1 - 2000/1/1
N2 - It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states - the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.
AB - It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states - the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.
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UR - http://www.scopus.com/inward/citedby.url?scp=0034565121&partnerID=8YFLogxK
U2 - 10.1137/S0036139998346439
DO - 10.1137/S0036139998346439
M3 - Article
AN - SCOPUS:0034565121
VL - 60
SP - 679
EP - 702
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
SN - 0036-1399
IS - 2
ER -