We develop the dimension-reduced hyperbolic moment method for the Boltzmann equation, to improve solution efficiency using a numerical regularized moment method for problems with low-dimensional macroscopic variables and high-dimensional microscopic variables. In the present work, we deduce the globally hyperbolic moment equations for the dimension-reduced Boltzmann equation based on the Hermite expansion and a globally hyperbolic regularization. The numbers of Maxwell boundary condition required for well-posedness are studied. The numerical scheme is then developed and an improved projection algorithm between two different Hermite expansion spaces is developed. By solving several benchmark problems, we validate the method developed and demonstrate the significant efficiency improvement by dimension-reduction.

The advent of charge-coupled devices (CCD) has significantly extended the horizons offull-field measurement methods in many fields of fundamental and applied sciences,including fluid flow dynamics. One of the major applications of imaging sensors in fluidmechanics is the Digital Particle Image Velocimetry (DPIV), which is a non-invasive, fullfield optical measuring technique. Due to recent advances in image processing methods, andin digital imaging, laser and optics DPIV has become a dominant tool for obtainingvelocity information about fluid motion. On the other hand, over the last years,developments in micro- and nanofluidic systems have promised a shrinking of thedesktop-sized chemical and biological devices to microscale size, so a detailedinvestigation of the behaviour of flow inside these microdevices is essential for theoptimum design of microsystems, as well as for an understanding of flow dynamics in micronscales. In this work, the advances in visualization and quantitative measurements of flowvelocity measurements are reviewed and explained.