KPP fronts in shear flows with cut-off reaction rates
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917703438041088 |
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| author | Needham, D. J. Tzella, A. |
| author_facet | Needham, D. J. Tzella, A. |
| contents | We consider the effect of a shear flow which has, without loss of generality, a zero mean flow rate, on a Kolmogorov--Petrovskii--Piscounov (KPP) type model in the presence of a discontinuous cut-off at concentration $u = u_c$. Its structure and speed of propagation depends on $A$ (the strength of the flow relative to the propagation speed in the absence of advection) and $B$ (the square of the front thickness relative to the channel width). We use matched asymptotic expansions to approximate the propagation speed in the three natural cases $A\to \infty$, $A\to 0$ and $A=O(1)$, with particular associated orderings on $B$, whilst $u_c\in(0,1)$ remains fixed. In all the cases that we consider, the shear flow enhances the speed of propagation in a manner that is similar to the case without cut-off ($u_c=0$). We illustrate the theory by evaluating expressions (either directly or through numerical integration) for the particular cases of the plane Couette and Poiseuille flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16617 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | KPP fronts in shear flows with cut-off reaction rates Needham, D. J. Tzella, A. Analysis of PDEs Fluid Dynamics We consider the effect of a shear flow which has, without loss of generality, a zero mean flow rate, on a Kolmogorov--Petrovskii--Piscounov (KPP) type model in the presence of a discontinuous cut-off at concentration $u = u_c$. Its structure and speed of propagation depends on $A$ (the strength of the flow relative to the propagation speed in the absence of advection) and $B$ (the square of the front thickness relative to the channel width). We use matched asymptotic expansions to approximate the propagation speed in the three natural cases $A\to \infty$, $A\to 0$ and $A=O(1)$, with particular associated orderings on $B$, whilst $u_c\in(0,1)$ remains fixed. In all the cases that we consider, the shear flow enhances the speed of propagation in a manner that is similar to the case without cut-off ($u_c=0$). We illustrate the theory by evaluating expressions (either directly or through numerical integration) for the particular cases of the plane Couette and Poiseuille flows. |
| title | KPP fronts in shear flows with cut-off reaction rates |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2406.16617 |