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arXiv physics/0511056physics.flu-dyncond-mat.soft
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Hagen-Poiffille流开始时的普遍动力学

Universal dynamics in the onset of a Hagen-Poiseuille flow

Niels Asger Mortensen, Henrik Bruus

备注：
4 pages including 1 figure
发表信息：
Phys. Rev. E 74, 017301 (2006)
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摘要：

不可压缩液体在圆形截面通道中的Hagen-Poietille流动的起始动力学已从理论上得到了很好的研究。我们使用的Hilbert空间形式主义的本征函数扩展推广的结果，以通道的任意截面。我们发现，在特征时间尺度τ =（A/P）^2（1/nu）之后达到稳态，其中A和P分别是横截面积和周长，$ν$是液体的运动粘度。对于流量Q的初始动力学，当t<<tau时，我们发现了一个普遍的线性关系，Q（t）= Q_oo（alpha/C）（t/tau），其中Q_oo是渐近稳态流量，alpha是几何校正因子，C=P^2/A是紧致性参数。对于长时间动力学，Q（t）在时间尺度$τ$上以指数方式逼近Q_0，但具有弱几何依赖的单位阶前因子，由Helmholz方程的最低本征值决定。

英文摘要：

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 $ν$ 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 $τ$, but with a weakly geometry-dependent prefactor of order unity, determined by the lowest eigenvalue of the Helmholz equation.

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