• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제13권1호
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• pp.21-29
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• 2009

This paper treats the conditions for the existence and stability properties of stationary solutions of a predator-prey interaction with self and cross-diffusion. We show that at a certain critical value a diffusion driven instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion) but becomes unstable with respect to the system with diffusion and that Turing instability takes place. We note that the cross-diffusion increase or decrease a Turing space (the space which the emergence of spatial patterns is holding) compared to the Turing space with self-diffusion, i.e. the cross-diffusion response is an important factor that should not be ignored when pattern emerges.

• 한국연소학회지
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• 제18권2호
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• pp.42-50
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• 2013

Linear stability analysis of radiating counterflow diffusion flames is numerically conducted to examine the instability characteristics of cellular patterns. Lewis number is assumed to be 0.5 to consider diffusional-thermal instability. Near kinetic limit extinction regime, growth rates of disturbances always have real eigen-values and neutral stability condition of planar disturbances perfectly falls into quasi-steady extinction. Cellular instability of disturbance with transverse direction occurs just before steady extinction. However, near radiative limit extinction regime, the eigenvalues are complex and pulsating instability of planar disturbances appears prior to steady extinction. Cellular instability occurs before the onset of planar pulsating instability, which means the extension of flammability.

• 대한수학회지
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• 제57권1호
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• pp.249-281
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• 2020

A reaction-diffusion model with spatiotemporal delay modeling the dynamical behavior of a single species is investigated. The parameter regions for the local stability, global stability and instability of the unique positive constant steady state solution are derived. The conditions of the occurrence of Turing (diffusion-driven) instability are obtained. The existence of time-periodic solutions, the existence and nonexistence of nonconstant positive steady state solutions are proved by bifurcation method and energy method. Numerical simulations are presented to verify and illustrate the theoretical results.

• 대한기계학회논문집B
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• 제21권9호
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• pp.1218-1229
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• 1997

The diffusional-thermal instability of diffusion flames in the premixed-flame regime is studied in a constant-density two-dimensional counterflow diffusion-flame configuration, to investigate the instability mechanism by which periodic wrinkling, travelling or pulsating of the reaction sheet can occur. Attention is focused on flames with small departures of the Lewis number from unity and with small values of the stoichiometric mixture fraction, so that the premixed-flame regime can be employed for activation-energy asymptotics. Cellular patterns will occur near quasisteady extinction when the Lewis number of the more completely consumed reactant is less than a critical value( ~ =0.7). Parametric studies for the instability onset conditions show that flames with smaller values of the Lewis number and stoichiometric mixture fraction and with larger values of the Zel'dovich number tend to be more unstable. For Lewis number greater than unity, near-extinction flame are found to exhibit either travelling instability or pulsating instability.

• 대한기계학회논문집B
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• 제28권6호
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• pp.688-696
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• 2004

Nonlinear dynamic behavior of diffusive-thermal instability in diluted CH$_4$/O$_2$ diffusion flames is numerically investigated by adopting detailed chemistry and transport. Counterflow diffusion flame is adopted as a model flamelet. Particular attention is focused on the pulsating-instability regime, which arises for Lewis numbers greater than unity, and the instability occurs at high strain rate near extinction condition in this flame configuration. Once a steady flame structure is obtained for a prescribed value of initial strain rate, transient solution of the flame is calculated after a finite amount of strain-rate perturbation is imposed on the steady flame. Transient evolution of the flame depends on the initial strain rate and the amount of perturbed strain rate. Basically, the dynamic behaviors can be classified into two types, namely non-oscillatory decaying solution and diverging solution leading to extinction. The peculiar oscillatory solution, which has been found in the previous study adopting one-step chemistry and constant Lewis numbers, is net observed in this study, which is attributed to both convective flow and preferential diffusion effects.

• 한국연소학회:학술대회논문집
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• 한국연소학회 2004년도 제28회 KOSCO SYMPOSIUM 논문집
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• pp.95-101
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• 2004

Dynamic behavior of diffusive-thermal instability in diluted $CH_4/O_2$ diffusion flames is numerically investigated by adopting detailed chemistry and transport. Counterflow diffusion flame is adopted as a model flamelet. Particular attention is focused on the pulsating-instability regime, which arises for Lewis numbers greater than unity, and the instability occurs at high strain rate near extinction condition in this flame configuration. Once a steady flame structure is obtained for a prescribed value of initial strain rate. transient solution of the flame is calculated after a finite amount of strain-rate perturbation is imposed Oil the steady flame. Transient evolution of the flame depends on the initial strain rate and the amount of perturbed strain rate. Basically, the dynamic behaviors can be classified into two types, namely non-oscillatory decaying solution and diverging solution leading to extinction. The peculiar oscillatory solution. which has been found in the previous study adopting one-step chemistry and constant Lewis numbers, is not observed in this study, which is attributed to both convective flow and preferential diffusion effects.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제23권2호
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• pp.139-155
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• 2019

In this paper, we study the asymptotic behavior and Hopf bifurcation of the modified Holling-Tanner models for the predator-prey interactions in the absence of diffusion. Further the direction of Hopf bifurcation and stability of bifurcating periodic solutions are investigated. Diffusion driven instability of the positive equilibrium solutions and Turing instability region regarding the parameters are established. Finally we illustrate the theoretical results with some numerical examples.

• 한국진공학회지
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• 제21권3호
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• pp.130-135
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• 2012

드리프트-확산 근사식을 이용한 1차원 유체 방정식으로부터 선형적 안정성 이론을 전개하여 Tosend 방전에서 전자 확산이 불안정성에 미치는 영향을 관찰하였다. 본 연구에서 관찰된 바에 따르면 Townsend 불안정성은 전자 확산과 공간 전하에 의해 형성된 전기장의 효과가 결합되어 발생하며, 공간전하에 의한 효과가 작은 영역, 즉 방전 전류가 낮은 영역에서는 전자 확산 효과가 커질수록 불안정이 더 빨리 진행된다는 것이 발견되었다.

• 한국연소학회지
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• 제10권3호
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• pp.25-33
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• 2005

Fast-time instability is investigated for diffusion flames with Lewis numbers greater than unity by employing the numerical technique called the Evans function method. Since the time and length scales are those of the inner reactive-diffusive layer, the problem is equivalent to the instability problem for the $Li\tilde{n}\acute{a}n#s$ diffusion flame regime. The instability is primarily oscillatory, as seen from complex solution branches and can emerge prior to reaching the upper turning point of the S-curve, known as the $Li\tilde{n}\acute{a}n#s$ extinction condition. Depending on the Lewis number, the instability characteristics is found to be somewhat different. Below the critical Lewis number, $L_C$, the instability possesses primarily a pulsating nature in that the two real solution branches, existing for small wave numbers, merges at a finite wave number, at which a pair of complex conjugate solution branches bifurcate. For Lewis numbers greater than $L_C$, the solution branch for small reactant leakage is found to be purely complex with the maximum growth rate found at a finite wave number, thereby exhibiting a traveling nature. As the reactant leakage parameter is further increased, the instability characteristics turns into a pulsating type, similar to that for L < $L_C$.

• 한국연소학회지
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• 제15권3호
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• pp.74-81
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• 2010

In this study, soot formation characteristics on the instability of laminar diffusion flames were investigated experimentally using a concentric co-flow burner. When a small amount of air was supplied through an inner nozzle, a stable propane laminar diffusion flame became unstable and began to oscillate mainly due to the dilution effect. The increase of air flow rate transformed an oscillating non-sooting flame into a stable nonsooting flame. When the air flow rate was continuously increased an inner flame was formed and the flame was changed to an oscillating sooting flame, an oscillating non-sooting flame and finally a stable non-sooting hollow flame. When the air flow rate was decreased, a non-sooting hollow flame was eventually changed back to a stable non-sooting flame. The presence of an inner flame, however, altered the soot formation characteristics of a flame. More soot production was observed with the presence of an inner flame. The increased or decreased soot formation/oxidation rates, the radiation heat loss, and the heating effect of inner flames are most likely to be responsible for the observed instability of laminar diffusion flames.