This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covarianc...This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.展开更多
Natural convection in air-filled rectangular cavities inclined with respect to gravity, so that the heated wall is facing upwards, is studied numerically under the assumption of two-dimensional laminar flow. A computa...Natural convection in air-filled rectangular cavities inclined with respect to gravity, so that the heated wall is facing upwards, is studied numerically under the assumption of two-dimensional laminar flow. A computational code based on the SIMPLE-C algorithm is used for the solution of the system of the mass, momentum and energy transfer governing equations. Simulations are performed for height-to-width aspect ratios of the enclosure from 0.25 to 8, Rayleigh numbers based on the length of the heated and cooled walls from 10~2 to 10~7, and tilting angles of the enclosure from 0° to 75°. The existence of an optimal tilting angle is confirmed for any investigated configuration, at a location that increases as the Rayleigh number is decreased, and the height-to-width aspect ratio of the cavity are increased, unless the value of the Rayleigh number is that corresponding to the onset of convection or just higher. Dimensionless correlating equations are developed to predict the optimal tilting angle and the heat transfer performance of the enclosure.展开更多
基金supported by an NSERC granta startup fund of University of Alberta
文摘This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
文摘Natural convection in air-filled rectangular cavities inclined with respect to gravity, so that the heated wall is facing upwards, is studied numerically under the assumption of two-dimensional laminar flow. A computational code based on the SIMPLE-C algorithm is used for the solution of the system of the mass, momentum and energy transfer governing equations. Simulations are performed for height-to-width aspect ratios of the enclosure from 0.25 to 8, Rayleigh numbers based on the length of the heated and cooled walls from 10~2 to 10~7, and tilting angles of the enclosure from 0° to 75°. The existence of an optimal tilting angle is confirmed for any investigated configuration, at a location that increases as the Rayleigh number is decreased, and the height-to-width aspect ratio of the cavity are increased, unless the value of the Rayleigh number is that corresponding to the onset of convection or just higher. Dimensionless correlating equations are developed to predict the optimal tilting angle and the heat transfer performance of the enclosure.
基金supported by National Natural Science Foundation of China(Grant Nos.11025107,10831008 and 10901165)the Fundamental Research Funds for Central Universities(Grant No.201034000-3162643)+2 种基金High Level Talent Project in High Schools in Guangdong Province(Grant No.34000-5221001)the Fundamental Research Funds for the Central Universities(Grant No.101gpy25)China Post-doctoral Science Foundation(Grant No.201003382)
文摘We construct the canonical solitons,in terms of Cabezas-Rivas and Topping,associated with some generalized Ricci flows.
