# Universal dynamics in the onset of a Hagen-Poiseuille flow

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Universal dynamics in the onset of a Hagen-Poiseuille flow Niels Asger Mortensen and Henrik Bruus MIC-Department of Micro and Nanotechnology, NanoDTU, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark �Received 8 February 2006; published 13 July 2006� The dynamics in the onset of a Hagen-Poiseuille flow of an incompressible liquid in a channel of circular cross section is well-studied theoretically. We use an eigenfunction expansion in a Hilbert space formalism to generalize the results to channels of an arbitrary cross section. We find that the steady state is reached after a characteristic time scale �= �A /P�2�1/��, where A and P are the cross-sectional area and perimeter, respec- tively, and � is the kinematic viscosity of the liquid. For the initial dynamics of the flow rate Q for t�� we find a universal linear dependence, Q�t�=Q��� /C��t /��, where Q� is the asymptotic steady-state flow rate, � is the geometrical correction factor, and C=P2 /A is the compactness parameter. For the long-time dynamics Q�t� approaches Q� exponentially on the time scale �, but with a weakly geometry-dependent prefactor of order unity, determined by the lowest eigenvalue of the Helmholz equation. DOI: 10.1103/PhysRevE.74.017301 PACS number�s�: 47.10.A�, 47.15.Rq, 47.27.nd, 47.27.nf I. INTRODUCTION Hagen-Poiseuille flow �or simply Poiseuille flow� is im- portant to a variety of applications ranging from macroscopic pipes in chemical plants to the flow of blood in veins. How- ever, the rapid development in the field of lab-on-a-chip sys- tems during the past decade has put even more emphasis on pressure driven laminar flow. Traditionally, capillary tubes would have circular cross sections, but today microfabricated channels come with a variety of shapes depending on the fabrication technique in use. The list of examples includes rectangular channels obtained by hot embossing in polymer wafers, semicircular channels in isotropically etched surfaces, triangular channels in pota

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