Time-Dependent Dispersion of Small Particles in Rectangular Conduits

Abstract

The mean velocity and dispersion coefficient of a Brownian particle suspended in a fluid flowing in a rectangular conduit are evaluated, obtaining analytical results over the whole time domain and for any initial position of the particle. These new results are obtained by exploiting the symmetry of the problem, showing that the particle longitudinal motion within the rectangular conduit is equivalent to the one occurring in an unbounded velocity field, which is the even periodic extension of the original velocity profile. Two main parameters are encountered: the aspect ratio $\eta $ of the conduit cross section, and the nondimensional time $\tau $, scaled by the characteristic time associated with the particle molecular diffusion. Simple asymptotic expressions are obtained for the particle velocity and displacement, the velocity auto-correlation, and the dispersion coefficient. The limits addressed are for small and large times $\tau $ and small $\eta $ ratios. The main results are the following. In the limit of long times the mean displacement strongly depends on the initial position of the particle, while the dispersion coefficient is independent of it and, for a fixed cross section area, it attains a maximum value at $\eta = 0.65$. In the limit of vanishingly small $\eta $ two distinct timescales $\tau _y $ and $\tau _z = \tau _y /\eta ^2 $ are defined, characterizing the particle molecular diffusion along the short and the long side of the conduit cross section, respectively. It is found that for short times $t < \tau _y $, the dispersivity increases either linearly or quadratically, depending on the initial conditions, for intermediate times $\tau _y < t < \tau _z $ it reaches a plateau, and for long times $t < \tau _z $ it attains a new higher limit value. While the long-time limit value is independent of the initial location of the Brownian particle and is about 7.95 times higher than the two-dimensional dispersivity, the intermediate-time plateau strongly depends on the initial conditions. Namely, when the particle initial location is not too close to the short lateral wall, the intermediate-time dispersion coefficient coincides with the two-dimensional dispersivity, indicating that the particle does not “know” of the existence of the lateral boundary. However, when the particle is initially introduced near the lateral wall, the intermediate-time dispersion coefficient is higher and is equal to up to four times the two-dimensional value.