Partial Differential Form Research Articles

Time-dependent probability density of statistical mechanics

Methods based on the density matrix for calculating the time-dependent probability density of a quantum system approaching equilibrium are presented. Explicit expressions are derived for the time-dependent probability density for a double-well potential. The effects of tunneling and transitions between energy levels on the probability density are discussed. For the case of closely spaced energy levels, a partial differential form of the density matrix equation is derived and used to calculate time-dependent probability densities.

Prediction of Transpired Turbulent Boundary Layers

A model for the mixing length distribution near the wall in turbulent boundary layer flow with transpiration is presented. The model is based on a new formulation of the exponential damping function originally suggested by Van Driest. The analysis used to evaluate the damping function employs the same set of assumptions successfully used by several investigators in the past to develop the law of the wall with blowing. This mixing length model is then used with the calculation method previously developed by the author to solve the governing conservation equations of mass, momentum, and energy in partial differential form. Predicted velocity profiles, skin friction coefficients, and Stanton numbers are compared with experimental results taken over a wide range of transpiration for both incompressible and compressible flows.