Induced flow due to blowing and suction flow control: an analysis of transpiration.

*(English)*Zbl 1241.76168Summary: It has previously been demonstrated that the drag experienced by a Poiseuille flow in a channel can be reduced by subjecting the flow to a dynamic regime of blowing and suction at the walls of the channel (also known as ‘transpiration’). Furthermore, it has been found to be possible to induce a ‘bulk flow’, or steady motion through the channel, via transpiration alone. In this work, we derive explicit asymptotic expressions for the induced bulk flow via a perturbation analysis. From this we gain insight into the physical mechanisms at work within the flow. The boundary conditions used are of travelling sine waves at either wall, which may differ in amplitude and phase. Here it is demonstrated that the induced bulk flow results from the effect of convection. We find that the most effective arrangement for inducing a bulk flow is that in which the boundary conditions at either wall are equal in magnitude and opposite in sign. We also show that, for the bulk flow induced to be non-negligible, the wavelength of the boundary condition should be comparable to, or greater than, the height of the channel. Moreover, we derive the optimal frequency of oscillation, for maximising the induced bulk flow, under such boundary conditions. The asymptotic behaviour of the bulk flow is detailed within the conclusion. It is found, under certain caveats, that if the amplitude of the boundary condition is too great, the bulk flow induced will become dependent only upon the speed at which the boundary condition travels along the walls of the channel. We propose the conjecture that for all similar flows, if the magnitude of the transpiration is sufficiently great, the bulk flow will depend only upon the speed of the boundary condition.

##### MSC:

76D55 | Flow control and optimization for incompressible viscous fluids |

PDF
BibTeX
XML
Cite

\textit{J. D. Woodcock} et al., J. Fluid Mech. 690, 366--398 (2012; Zbl 1241.76168)

Full Text:
DOI

##### References:

[1] | DOI: 10.1017/S0022112008004886 · Zbl 1183.76702 |

[2] | DOI: 10.1017/S0022112094000431 · Zbl 0800.76191 |

[3] | DOI: 10.1016/S0376-0421(00)00016-6 |

[4] | Batchelor, An Introduction to Fluid Dynamics (1967) · Zbl 0152.44402 |

[5] | DOI: 10.1146/annurev.fluid.35.101101.161213 · Zbl 1039.76028 |

[6] | DOI: 10.1016/S0005-1098(03)00140-7 · Zbl 1055.93039 |

[7] | DOI: 10.1017/S0022112010003083 · Zbl 1205.76097 |

[8] | DOI: 10.1017/S0022112009992904 · Zbl 1189.76133 |

[9] | DOI: 10.1017/S0022112009992916 · Zbl 1189.76132 |

[10] | DOI: 10.1146/annurev.fluid.40.111406.102156 · Zbl 1229.76043 |

[11] | DOI: 10.1017/S0022112099004498 · Zbl 0939.76506 |

[12] | Toms, Proceedings of the 1st International Congress on Rheology, 2 pp 135– (1948) |

[13] | Stokes, Trans. Camb. Phil. Soc. 9 pp 8– (1851) |

[14] | DOI: 10.1146/annurev.fluid.33.1.43 |

[15] | DOI: 10.1017/S0022112010003393 · Zbl 1205.76129 |

[16] | DOI: 10.1017/S0022112006000206 · Zbl 1094.76033 |

[17] | DOI: 10.1017/S0022112006003247 · Zbl 1105.76022 |

[18] | von Helmholtz, Verh. des naturh.-med. Vereins zu Heidelberg 5 pp 1– (1868) |

[19] | DOI: 10.1103/PhysRevE.81.046304 |

[20] | DOI: 10.1016/j.physd.2009.03.008 · Zbl 1166.76016 |

[21] | DOI: 10.1017/S002211201000340X · Zbl 1205.76127 |

[22] | DOI: 10.1063/1.1516779 · Zbl 1185.76134 |

[23] | DOI: 10.1063/1.3006057 · Zbl 1182.76439 |

[24] | DOI: 10.1080/10618569808940866 · Zbl 0941.76024 |

[25] | Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows (1963) · Zbl 0121.42701 |

[26] | Ladyzhenskaya, Dokl. Akad. Nauk SSSR 123 pp 427– (1958) |

[27] | DOI: 10.1017/S0022112009007629 · Zbl 1183.76703 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.