# Boundary layers for cellular flows at high Péclet numbers

Alexei Novikov, George Papanicolaou, Lenya Ryzhik

Research output: Contribution to journalReview article

23 Citations (Scopus)

### 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) 867-922 56 Communications on Pure and Applied Mathematics 58 7 https://doi.org/10.1002/cpa.20058 Published - Jul 1 2005

### Fingerprint

Boundary Layer
Boundary layers
Pipe
Water
Dirichlet-to-Neumann Map
Separatrix
Diffusion Problem
Streamlines
Behavior of Solutions
Graph in graph theory
Costs
Dirichlet
Computational Cost
Bounded Domain
Converge
Dependent
Cell
Approximation

### All Science Journal Classification (ASJC) codes

• Mathematics(all)
• Applied Mathematics

### Cite this

Novikov, Alexei ; Papanicolaou, George ; Ryzhik, Lenya. / Boundary layers for cellular flows at high Péclet numbers. In: Communications on Pure and Applied Mathematics. 2005 ; Vol. 58, No. 7. pp. 867-922.
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abstract = "We analyze the behavior of solutions of steady advection-diffusion problems in bounded domains with prescribed Dirichlet data when the P{\'e}clet 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.",
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Boundary layers for cellular flows at high Péclet numbers. / Novikov, Alexei; Papanicolaou, George; Ryzhik, Lenya.

In: Communications on Pure and Applied Mathematics, Vol. 58, No. 7, 01.07.2005, p. 867-922.

Research output: Contribution to journalReview 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

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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.

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