## Abstract

REPLACE WITH "GAS-CUSHIONED DROPLET IMPACTS WITH A THIN LAYER OF POROUS MEDIA" FOLLOWING EMAIL CONVERSATION WITH DUBRAVKA (17/11/2020)

The unsteady viscous flow induced by streamwise-travelling waves of spanwise wall velocity in an incompressible laminar channel flow is investigated. Wall waves belonging to this category have found important practical applications, such as microfluidic flow manipulation via electro-osmosis and surface acoustic forcing and reduction of wall friction in turbulent wall-bounded flows. An analytical solution composed of the classical streamwise Poiseuille flow and a spanwise velocity profile described by the parabolic cylinder function is found. The solution depends on the bulk Reynolds number R, the scaled streamwise wavelength λ

, and the scaled wave phase speed U. Numerical solutions are discussed for various combinations of these parameters. The flow is studied by the boundary-layer theory, thereby revealing the dominant physical balances and quantifying the thickness of the near-wall spanwise flow. The Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) theory is also employed to obtain an analytical solution, which is valid across the whole channel. For positive wave speeds which are smaller than or equal to the maximum streamwise velocity, a turning-point behaviour emerges through the WKBJ analysis. Between the wall and the turning point, the wall-normal viscous effects are balanced solely by the convection driven by the wall forcing, while between the turning point and the centreline, the Poiseuille convection balances the wall-normal diffusion. At the turning point, the Poiseuille convection and the convection from the wall forcing cancel each other out, which leads to a constant viscous stress and to the break down of the WKBJ solution. This flow regime is analysed through a WKBJ composite expansion and the Langer method. The Langer solution is simpler and more accurate than the WKBJ composite solution, while the latter quantifies the thickness of the turning-point region. We also discuss how these waves can be generated via surface acoustic forcing and electro-osmosis and propose their use as microfluidic flow mixing devices. For the electro-osmosis case, the Helmholtz–Smoluchowski velocity at the edge of the Debye–Hückel layer, which drives the bulk electrically neutral flow, is obtained by matched asymptotic expansion.

The unsteady viscous flow induced by streamwise-travelling waves of spanwise wall velocity in an incompressible laminar channel flow is investigated. Wall waves belonging to this category have found important practical applications, such as microfluidic flow manipulation via electro-osmosis and surface acoustic forcing and reduction of wall friction in turbulent wall-bounded flows. An analytical solution composed of the classical streamwise Poiseuille flow and a spanwise velocity profile described by the parabolic cylinder function is found. The solution depends on the bulk Reynolds number R, the scaled streamwise wavelength λ

, and the scaled wave phase speed U. Numerical solutions are discussed for various combinations of these parameters. The flow is studied by the boundary-layer theory, thereby revealing the dominant physical balances and quantifying the thickness of the near-wall spanwise flow. The Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) theory is also employed to obtain an analytical solution, which is valid across the whole channel. For positive wave speeds which are smaller than or equal to the maximum streamwise velocity, a turning-point behaviour emerges through the WKBJ analysis. Between the wall and the turning point, the wall-normal viscous effects are balanced solely by the convection driven by the wall forcing, while between the turning point and the centreline, the Poiseuille convection balances the wall-normal diffusion. At the turning point, the Poiseuille convection and the convection from the wall forcing cancel each other out, which leads to a constant viscous stress and to the break down of the WKBJ solution. This flow regime is analysed through a WKBJ composite expansion and the Langer method. The Langer solution is simpler and more accurate than the WKBJ composite solution, while the latter quantifies the thickness of the turning-point region. We also discuss how these waves can be generated via surface acoustic forcing and electro-osmosis and propose their use as microfluidic flow mixing devices. For the electro-osmosis case, the Helmholtz–Smoluchowski velocity at the edge of the Debye–Hückel layer, which drives the bulk electrically neutral flow, is obtained by matched asymptotic expansion.

Original language | English |
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Pages (from-to) | 23-49 |

Number of pages | 27 |

Journal | Journal of Engineering Mathematics |

Volume | 111 |

Issue number | 1 |

Early online date | 23 Feb 2018 |

DOIs | |

Publication status | Published - Aug 2018 |

### Bibliographical note

This work was partially supported by EPSRC First Grant EP/I033173/1. This research used computing resources at the University of Aberdeen and the University of Sheffield. We would like to thank Samuele Viaro, Eva Zincone, Claudia Alvarenga, Michele Schirru, Dr Cecile Perrault and Dr Elena Marensi for their helpful comments. We are also indebted to Georgios Tyreas and Prof. Robert Dwyer-Joyce for pointing out to us several interesting technological applications of the shear-horizontal waves. We thanks Prof. David Wagg for the encouragement and suggestions on therevision of the paper.

## Keywords

- microfluidics
- mixing
- Love waves
- biosensors
- shear-horizontal surface acoustic waves
- electro-osmosis
- electro-osmotic waves
- turbulent drag reduction