The compositional distribution within aggregates of a given size is essential to the functionality of com- posite aggregates that are usually enlarged by rapid Brownian coagulation, There is no analytical solution for...The compositional distribution within aggregates of a given size is essential to the functionality of com- posite aggregates that are usually enlarged by rapid Brownian coagulation, There is no analytical solution for the process of such two-component systems, Monte Carlo method is an effective numerical approach for two-component coagulation, In this paper, the differentially weighted Monte Carlo method is used to investigate two-component Brownian coagulation, respectively, in the continuum regime, the free-molecular regime and the transition regime. It is found that （1） for Brownian coagulation in the continuum regime and in the free-molecular regime, the mono-variate compositional distribution, i.e., the number density distribution function of one component amount （in the form of volume of the component in aggregates） satisfies self-preserving form the same as particle size distribution in mono-component Brownian coagulation; （2） however, for Brownian coagulation in the transition regime the mono-variate compositional distribution cannot reach self-similarity; and （3） the bivariate compositional distribution, i.e., the combined number density distribution function of two component amounts in the three regimes satisfies a semi self-preserving form. Moreover, other new features inherent to aggregative mixing are also demonstrated; e.g., the degree of mixing between components, which is largely controlled by the initial compositional mass fraction, improves as aggregate size increases.展开更多
In this paper, we consider the higher divided difference of a composite function f（g（t）） in which g（t） is an s-dimensional vector. By exploiting some properties from mixed partial divided differences and multiva...In this paper, we consider the higher divided difference of a composite function f（g（t）） in which g（t） is an s-dimensional vector. By exploiting some properties from mixed partial divided differences and multivariate Newton interpolation, we generalize the divided difference form of Faà di Bruno＇s formula with a scalar argument. Moreover, a generalized Faà di Bruno＇s formula with a vector argument is derived.展开更多
基金H.Zhao was supported by funds from"The National Natural Science Foundation of China"(50876037 and 50721005)"Program for New Century Excellent Talents in University"(NCET-10-0395)"National Key Basic Research and Development Program"(2010CB227004)
文摘The compositional distribution within aggregates of a given size is essential to the functionality of com- posite aggregates that are usually enlarged by rapid Brownian coagulation, There is no analytical solution for the process of such two-component systems, Monte Carlo method is an effective numerical approach for two-component coagulation, In this paper, the differentially weighted Monte Carlo method is used to investigate two-component Brownian coagulation, respectively, in the continuum regime, the free-molecular regime and the transition regime. It is found that （1） for Brownian coagulation in the continuum regime and in the free-molecular regime, the mono-variate compositional distribution, i.e., the number density distribution function of one component amount （in the form of volume of the component in aggregates） satisfies self-preserving form the same as particle size distribution in mono-component Brownian coagulation; （2） however, for Brownian coagulation in the transition regime the mono-variate compositional distribution cannot reach self-similarity; and （3） the bivariate compositional distribution, i.e., the combined number density distribution function of two component amounts in the three regimes satisfies a semi self-preserving form. Moreover, other new features inherent to aggregative mixing are also demonstrated; e.g., the degree of mixing between components, which is largely controlled by the initial compositional mass fraction, improves as aggregate size increases.
基金Acknowledgments. This work was supported by the National Science Foundation of China （Grant Nos. 10471128, 10731060）.
文摘In this paper, we consider the higher divided difference of a composite function f（g（t）） in which g（t） is an s-dimensional vector. By exploiting some properties from mixed partial divided differences and multivariate Newton interpolation, we generalize the divided difference form of Faà di Bruno＇s formula with a scalar argument. Moreover, a generalized Faà di Bruno＇s formula with a vector argument is derived.
