科研之友
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摘要
We study diffusion in point-source-driven shear flows of generalized second-grade fluids. We start by obtaining exact solutions of shear flows triggered by point sources under various boundary conditions. For unrestricted flows, we demonstrate that the velocity distribution is the probability density function of a coupled or uncoupled continuous-time random walk. In the first instance, the motion is described by a compound Poisson process with an explicit probability density function corresponding to the velocity distribution. The average waiting time in this situation is finite and is identified with the structural relaxation time. In the second case, we obtain an explicit formula for the probability density function in terms of special functions. In both cases, the probability density functions of the associated stochastic processes are leptokurtic at all finite times with variances linear in time. By using the method of images, we infer velocity fields for restricted flows from those of unrestricted flows. Equipped with some exact expressions of the velocity field, we analyze advection-diffusion via the Feynman-Kac formula, which lends itself naturally to Monte Carlo simulation.
- 出版日期2010-11
