Stability of fronts in the Kuramoto-Sivashinsky equation advected by a Poiseuille flow

We study reaction fronts described by the Kuramoto-Sivashinsky equation subject to a Poiseuille flow. The fronts propagate with or against the flow located inside a two-dimensional slab. Steady front profiles can be flat, axisymmetric, or nonaxisymmetric, depending on the gap between the plates and the average flow speed. We first obtain the steady front solutions, later executing a linear stability analysis to determine the stability of the fronts. Applying fluid flow can turn initially unstable fronts into stable fronts. Stable steady fronts propagating in the adverse direction of the Poiseuille flow are axisymmetric for slow fluid flows. However, for higher speeds an adverse flow can lead to stable nonaxisymmetric fronts. We also show regions of bistability where stable nonaxisymmetric and axisymmetric fronts can coexist.

Figure 1

Front profiles for different average velocities of the Poiseuille flow. For supportive flows the fronts are axisymmetric. Small adverse flows ($\overline{\mathrm{v}}=-2.0$) allow stable axisymmetric fronts, but for higher speeds ($\overline{\mathrm{v}}=-2.5$) the front becomes nonaxisymmetric. The fronts are shifted for comparison.

Figure 2

Front velocities for different distances between plates (L). The thick solid line (A) corresponds to stable nonaxisymmetric fronts. The solid line (B) corresponds to stable axisymmetric fronts. The dashed line (E) corresponds to unstable axisymmetric fronts. Broken lines (C, D, and F) correspond to unstable nonaxisymmetric fronts.

Figure 3

The largest real part of the eigenvalues $\sigma$ for different distances L between plates. The front is unstable when the largest real part of the eigenvalues $\sigma$ is positive. The thickest solid line (A) corresponds to nonaxisymmetric fronts. Solid line ${\mathrm{B}}_1$ corresponds to axisymmetric fronts and solid line ${\mathrm{B}}_2$ corresponds to other axisymmetric fronts. Broken lines (C, D, and F) correspond to nonaxisymmetric fronts. The dashed line (E) corresponds to other axisymmetric fronts. Solid line G corresponds to flat fronts.

Figure 4

Front velocities subject to a Poiseuille flow in the same direction as the propagating front. The average velocity of the flow is $\overline{\mathrm{v}}=0.1$. Solid lines correspond to stable fronts; all broken lines correspond to unstable fronts. The thick solid line (A) corresponds to nonaxisymmetric fronts. Branch ${\mathrm{B}}_1$ corresponds to axisymmetric fronts. Branch ${\mathrm{B}}_2$ corresponds to other axisymmetric fronts. Broken lines (C, D, and F) correspond to nonaxisymmetric fronts.

Figure 5

The largest real part of the eigenvalues $\sigma$ for different distances L between plates. The Poiseuille flow is in the same direction as the propagating front. The average velocity of the flow is $\overline{\mathrm{v}}=0.1$. The front is unstable when the largest real part of the eigenvalues $\sigma$ is positive. The thickest solid line (A) corresponds to nonaxisymmetric fronts. Solid line ${\mathrm{B}}_1$ corresponds to axisymmetric fronts. Solid line ${\mathrm{B}}_2$ corresponds to other axisymmetric fronts. Broken lines (C, D, and F) correspond to nonaxisymmetric fronts.

Figure 6

Front velocities subject to a Poiseuille flow in the adverse direction to the propagating front. The average velocity of the flow is $\overline{\mathrm{v}}=-0.1$. Solid lines correspond to stable fronts; all broken lines correspond to unstable fronts. The thick solid line (A) corresponds to nonaxisymmetric fronts. Branch ${\mathrm{B}}_1$ corresponds to axisymmetric fronts. Branch ${\mathrm{B}}_2$ corresponds to other axisymmetric fronts. Broken lines correspond to nonaxisymmetric fronts. Line D also corresponds to other nonaxisymmetric fronts.

Figure 7

The largest real part of the eigenvalues $\sigma$ for different distances L between plates. The Poiseuille flow is in the adverse direction to the propagating front. The average velocity of the flow is $\overline{\mathrm{v}}=-0.1$. The front is unstable when the largest real part of the eigenvalues $\sigma$ is positive. The thickest solid line (A) corresponds to nonaxisymmetric fronts. Solid line ${\mathrm{B}}_1$ corresponds to axisymmetric fronts. Solid line ${\mathrm{B}}_2$ also corresponds to other axisymmetric fronts. Broken lines (C and F) correspond to nonaxisymmetric fronts. Solid line D also corresponds to other nonaxisymmetric fronts.

Figure 8

Front velocities as a function of the distance L between plates for different Poiseuille flows. All fronts propagate in the same direction as the Poiseuille flow. The average flow speed ($\overline{\mathrm{v}}$) affects the shape of the fronts. For $\overline{\mathrm{v}}=0$ the curve has two maxima. The first one corresponds to stable nonaxisymmetric fronts; the second corresponds to axisymmetric fronts. As $\overline{\mathrm{v}}$ increases, nonaxisymmetric fronts disappear, as is the case of $\overline{\mathrm{v}}=2.5$.

Figure 9

Regions for the existence of nonaxisymmetric and axisymmetric fronts for different values of the average flow velocity and plate separation. Only axisymmetric fronts can exit for $\overline{\mathrm{v}}>2.46$.

Figure 10

Front velocities as a function of the width L between plates for adverse Poiseuille flows. The figure only shows stable fronts. (a) The solid line corresponds to no Poiseuille flow ($\overline{\mathrm{v}}=0$), showing two maxima. The first one corresponds to stable nonaxisymmetric fronts; the second corresponds to axisymmetric fronts. The dashed line corresponds to $\overline{\mathrm{v}}=-0.3$. Branches ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ represent stable axisymmetric fronts. (b) The solid line corresponds to $\overline{\mathrm{v}}=-0.5$, with branches ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ representing axisymmetric fronts. The dashed line corresponds to $\overline{\mathrm{v}}=-0.7$. Here there is only one branch B, corresponding to concave upward axisymmetric fronts.

Figure 11

Front propagation velocity as a function of the average velocity of Poiseuille flow. The distance L between plates is 3.5. Branch A corresponds to axisymmetric fronts, both stable (solid lines) and unstable (dashed line). Branch B (dot-dashed line) corresponds to unstable nonaxisymmetric fronts. The thick solid line (NA) corresponds to stable nonaxisymmetric fronts.