An experimental investigation into the influence of Brownian motion on shear-induced particle migration of monodisperse suspensions of micrometre-sized colloidal particles is presented. The suspension is pumped through a 50 µm × 500 µm rectangular crosssection glass channel. The experiments are characterized chiefly by the sample volume fraction (φ = 0.1 − 0.4), and the flow rate expressed as the Péclet number (Pe = 10 − 400). For each experiment we measure the entrance length, which is the distance from the inlet of the channel required for the concentration profile to develop to its non-uniform steady state. The entrance length increases strongly with increasing Pe for Pe 100, in marked contrast to non-Brownian flows for which the entrance length is flow-rate independent. For larger Pe, the entrance length reaches a constant value which depends on the other experimental parameters. Additionally, the entrance length decreases with increasing φ; this effect is strongest for low φ. Modelling of the migration based on spatial variation of the normal stresses due to the particles captures the primary features observed in the axial evolution over a range of Pe and φ.

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We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent PN and SPN equations, of radiative transfer. The method, which works for arbitrary moment order N , makes use of the specific coupling between the moments in the PN equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.2000 Mathematics Subject Classification. 65M06; 35L50; 65M12; 35Q20.

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We study mixed-moment models (full zeroth moment, half higher moments) for a Fokker-Planck equation in one space dimension. Mixed-moment minimum-entropy models are known to overcome the zero net-flux problem of full-moment minimum entropy Mn models. Realizability theory for these mixed moments of arbitrary order is derived, as well as a new closure, which we refer to as Kershaw closures. They provide non-negative distribution functions combined with an analytical closure. Numerical tests are performed with standard first-order finite volume schemes and compared with a finite-difference Fokker-Planck scheme.