SOME DEVELOPMENTS IN VISCOUS FLUID DYNAMICS AND ITS NUMERICAL METHODS

Viscous fluid flows contain abundant "physical phenomena and the viscous fluid dynamics is of wide applications in the fields of natural and engineering sciences. After the basic equations of viscousfluiddynamics (i.e., the Navier-Stokes equations) came out, one of the most important contributions to the discipline was the boundary layer (BL) theory and the BL equations presented by Prandtl

On numerical earthquake prediction

Can earthquakes be predicted？ How should people overcome the difficulties encountered in the study of earthquake prediction？ This issue can take inspiration from the experiences of weather forecast. Although weather forecasting took a period of about half a century to advance from empirical to numerical forecast, it has achieved significant success. A consensus has been reached among the Chinese seismological community that earth- quake prediction must also develop from empirical fore- casting to physical prediction. However, it is seldom mentioned that physical prediction is characterized by quantitatively numerical predictions based on physical laws. This article discusses five key components for numerical earthquake prediction and their current status. We conclude that numerical earthquake prediction should now be put on the planning agenda and its roadmap designed, seismic stations should be deployed and observations made according to the needs of numerical prediction, and theoretical research should be carried out.

Modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies

We present an efficient, robust computational method for modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies. With appropriate mathematical treatments, the apparent numerical difficulties associated with singularities in computing elliptic integrals are completely removed. Using a boundary element discretization procedure, the governing equations are transformed into a linear algebra matrix equation that can be solved by straightforward Gauss elimination in one step without further iterations. The numerical code implemented according to our algorithm can accurately determine the surface mass density distribution in a disk galaxy from a measured rotation curve （or vice versa）. For a disk galaxy with a typical flat rotation curve, our modeling results show that the surface mass density monotonically decreases from the galactic center toward the periphery, according to Newtonian dynamics. In a large portion of the galaxy, the surface mass density follows an approximately exponential law of decay with respect to the galactic radial coordinate. Yet the radial scale length for the surface mass density seems to be generally larger than that of the measured brightness distribution, suggesting an increasing mass-tolight ratio with the radial distance in a disk galaxy. In a nondimensionalized form, our mathematical system contains a dimensionless parameter which we call the ＂galactic rotation number＂ that represents the gross ratio of centrifugal force and gravitational force. The value of this galactic rotation number is determined as part of the numerial solution. Through a systematic computational analysis, we have illustrated that the galactic rotation number remains within 4-10% of 1.70 for a wide variety of rotation curves. This implies that the total mass in a disk galaxy is proportional to V02 Rg, with V0 denoting the characteristic rotation velocity （such as the ＂flat＂ value in a typical ro- tation curve） and Rg the radius of the galactic disk. The predicted total galactic mass of the Milky Way is in good agreement with the star-count data.

Made-to-measure galaxy modelling utilising absorption line strength data

We enhance the Syer ＆ Tremaine made-to-measure （M2M） particle method of stellar dynamical modelling to model simultaneously both kinematic data and absorption line strength data, thus creating a ＇chemo-M2M＇ modelling scheme. We apply the enhanced method to four galaxies （NGC 1248, NGC 3838, NGC 4452, NGC 4551） observed using the SAURON integral-field spectrograph as part of the ATLAS3D programme. We are able to reproduce successfully the 2D line strength data achieving mean X2 per bin values of ≈ 1 with 〉 95% of particles having converged weights. Because M2M uses a 3D particle system, we are also able to examine the underlying 3D line strength distributions. The extent to which these dis- tributions are plausible representations of real galaxies requires further consideration. Overall, we consider the modelling exercise to be a promising first step in developing a ＇chemo-M2M＇ modelling system and in understanding some of the issues to be addressed. While the made-to-measure techniques developed have been applied to absorption line strength data, they are in fact general and may be of value in modelling other aspects of galaxies.