dorsal/arxiv
View SchemaUniversal dynamics in the onset of a Hagen-Poiseuille flow
| Authors | Niels Asger Mortensen, Henrik Bruus |
|---|---|
| Categories | |
| ArXiv ID | physics/0511056 |
| URL | https://arxiv.org/abs/physics/0511056 |
| DOI | 10.1103/PhysRevE.74.017301 |
| Journal | Phys. Rev. E 74, 017301 (2006) |
Abstract
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 arbitrary cross section. We find that the steady state is reached after a characteristic time scale tau = (A/P)^2 (1/nu) where A and P are the cross-sectional area and perimeter, respectively, and $\nu$ is the kinematic viscosity of the liquid. For the initial dynamics of the flow rate Q for t<<tau we find a universal linear dependence, Q(t)= Q_oo(alpha/C)(t/tau), where Q_oo is the asymptotic steady-state flow rate, alpha is the geometrical correction factor, and C=P^2/A is the compactness parameter. For the long-time dynamics Q(t) approaches Q_oo exponentially on the timescale $\tau$, but with a weakly geometry-dependent prefactor of order unity, determined by the lowest eigenvalue of the Helmholz equation.
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"abstract": "The dynamics in the onset of a Hagen-Poiseuille flow of an incompressible\nliquid in a channel of circular cross section is well-studied theoretically. We\nuse an eigenfunction expansion in a Hilbert space formalism to generalize the\nresults to channels of arbitrary cross section. We find that the steady state\nis reached after a characteristic time scale tau = (A/P)^2 (1/nu) where A and P\nare the cross-sectional area and perimeter, respectively, and $\\nu$ is the\nkinematic viscosity of the liquid. For the initial dynamics of the flow rate Q\nfor t\u003c\u003ctau we find a universal linear dependence, Q(t)= Q_oo(alpha/C)(t/tau),\nwhere Q_oo is the asymptotic steady-state flow rate, alpha is the geometrical\ncorrection factor, and C=P^2/A is the compactness parameter. For the long-time\ndynamics Q(t) approaches Q_oo exponentially on the timescale $\\tau$, but with a\nweakly geometry-dependent prefactor of order unity, determined by the lowest\neigenvalue of the Helmholz equation.",
"arxiv_id": "physics/0511056",
"authors": [
"Niels Asger Mortensen",
"Henrik Bruus"
],
"categories": [
"physics.flu-dyn",
"cond-mat.soft"
],
"doi": "10.1103/PhysRevE.74.017301",
"journal_ref": "Phys. Rev. E 74, 017301 (2006)",
"title": "Universal dynamics in the onset of a Hagen-Poiseuille flow",
"url": "https://arxiv.org/abs/physics/0511056"
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