### Abstract

We analyze the behavior of solutions of steady advection-diffusion problems in bounded domains with prescribed Dirichlet data when the Péclet number Pe ≫ 1 is large. We show that the solution converges to a constant in each flow cell outside a boundary layer of width O(ε ^{1/2}), ε = Pe ^{-1}, around the flow separatrices. We construct an ε-dependent approximate "water pipe problem" purely inside the boundary layer that provides a good approximation of the solution of the full problem but has ε-independent computational cost. We also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic, ε-independent problem on this graph. Finally, we show that the Dirichlet-to-Neumann map of the water pipe problem approximates the Dirichlet-to-Neumann map of the separatrix problem with an error independent of the flow outside the boundary layers.

Original language | English (US) |
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Pages (from-to) | 867-922 |

Number of pages | 56 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 58 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*58*(7), 867-922. https://doi.org/10.1002/cpa.20058

}

*Communications on Pure and Applied Mathematics*, vol. 58, no. 7, pp. 867-922. https://doi.org/10.1002/cpa.20058

**Boundary layers for cellular flows at high Péclet numbers.** / Novikov, Alexei; Papanicolaou, George; Ryzhik, Lenya.

Research output: Contribution to journal › Review article

TY - JOUR

T1 - Boundary layers for cellular flows at high Péclet numbers

AU - Novikov, Alexei

AU - Papanicolaou, George

AU - Ryzhik, Lenya

PY - 2005/7/1

Y1 - 2005/7/1

N2 - We analyze the behavior of solutions of steady advection-diffusion problems in bounded domains with prescribed Dirichlet data when the Péclet number Pe ≫ 1 is large. We show that the solution converges to a constant in each flow cell outside a boundary layer of width O(ε 1/2), ε = Pe -1, around the flow separatrices. We construct an ε-dependent approximate "water pipe problem" purely inside the boundary layer that provides a good approximation of the solution of the full problem but has ε-independent computational cost. We also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic, ε-independent problem on this graph. Finally, we show that the Dirichlet-to-Neumann map of the water pipe problem approximates the Dirichlet-to-Neumann map of the separatrix problem with an error independent of the flow outside the boundary layers.

AB - We analyze the behavior of solutions of steady advection-diffusion problems in bounded domains with prescribed Dirichlet data when the Péclet number Pe ≫ 1 is large. We show that the solution converges to a constant in each flow cell outside a boundary layer of width O(ε 1/2), ε = Pe -1, around the flow separatrices. We construct an ε-dependent approximate "water pipe problem" purely inside the boundary layer that provides a good approximation of the solution of the full problem but has ε-independent computational cost. We also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic, ε-independent problem on this graph. Finally, we show that the Dirichlet-to-Neumann map of the water pipe problem approximates the Dirichlet-to-Neumann map of the separatrix problem with an error independent of the flow outside the boundary layers.

UR - http://www.scopus.com/inward/record.url?scp=14844311436&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=14844311436&partnerID=8YFLogxK

U2 - 10.1002/cpa.20058

DO - 10.1002/cpa.20058

M3 - Review article

AN - SCOPUS:14844311436

VL - 58

SP - 867

EP - 922

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 7

ER -