Low-Frequency Velocity Correlation Spectrum of Fluid in a Rectangular Microcapillary
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In addition to the fast correlation for local stochastic motion, the velocity correlation function in a fluid enclosedwithin the pore boundaries features a slow long time-tail decay. At late times, the flow approaches that of an incompressiblefluid. Here, we consider the motion of a viscous fluid, at constant temperature, in a rectangular semipermeable channel.The fluid is driven through the rectangular capillary by a uniform main pressure gradient. Tiny pressure gradientsare allowed perpendicular to the main flux. We solve numerically the three-dimensional Navier-Stokes equationsfor the velocity field to obtain the steady solution. We then set and solve the Langevin equation for the fluid velocity.We report hydrodynamic fluctuations for the center-line velocity together with the corresponding relaxation times asa function of the size of the observing region and the Reynolds number. The effective diffusion coefficient for thefluid in the microchannel is also estimated (Deff = 1.43 × 10-10 m2·s-1 for Re = 2), which is in accordance withmeasurements reported for a similar system (Stepinik, J.; Callaghan, P. T. Physica B 2000, 292, 296-301; Stepinik,J.; Callaghan, P. T. Magn. Reson. Imaging 2001, 19, 469-472).