A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero.This is particularly important when boundaries are present since vorticitv is typica...A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero.This is particularly important when boundaries are present since vorticitv is typically generated at the boundary as a result of boundary layer separation.The boundary laver theory,developed by Prandtl about a hundred years ago,has become a standard tool in addressing these questions.Yet at the mathematical level,there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory.In this article,we review recent progresses on the analysis of Prandtl’s equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation.We also discuss some directions where progress is expected in the near future.展开更多
We are concerned with the shock regular reflection configurations of unsteady global solutions for a plane shock hitting a symmetric straight wedge.It has been known that patterns of the shock reflection are various a...We are concerned with the shock regular reflection configurations of unsteady global solutions for a plane shock hitting a symmetric straight wedge.It has been known that patterns of the shock reflection are various and complicated,including the regular and the Mach reflection.Most of the fundamental issues for the shock reflection have not been understood.Recently,there are great progress on the mathematical theory of the shock regular reflection problem,especially for the global existence,uniqueness,and structural stability of solutions.In this paper,we show that there are two more possible configurations of the shock regular reflection besides known four configurations.We also give a brief proof of the global existence of solutions.展开更多
We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function s...We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2.The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.展开更多
The boundary layer flow of viscous incompressible fluid over a stretching cylinder has been considered to study flow field and temperature field. Due to non-linearity, a numerical approach called Keller-box technique ...The boundary layer flow of viscous incompressible fluid over a stretching cylinder has been considered to study flow field and temperature field. Due to non-linearity, a numerical approach called Keller-box technique has been used to compute the values of velocity function f and temperature field at different points of dynamic region. The expressions for skin friction and Nusselt number have also been obtained. The dependence of velocity profile and temperature profile on the dimensionless parameter of practical interest has been analyzed in detail by graphs. The dependence of Skin friction and Nusselt number has been seen through tables.展开更多
文摘A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero.This is particularly important when boundaries are present since vorticitv is typically generated at the boundary as a result of boundary layer separation.The boundary laver theory,developed by Prandtl about a hundred years ago,has become a standard tool in addressing these questions.Yet at the mathematical level,there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory.In this article,we review recent progresses on the analysis of Prandtl’s equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation.We also discuss some directions where progress is expected in the near future.
基金supported by the National Natural Science Foundation of China(Grant no.11761077)the NSF of Yunnan province of China(2019FY003007)the Program for Innovative Research Team in Universities of Yunnan Province of China.
文摘We are concerned with the shock regular reflection configurations of unsteady global solutions for a plane shock hitting a symmetric straight wedge.It has been known that patterns of the shock reflection are various and complicated,including the regular and the Mach reflection.Most of the fundamental issues for the shock reflection have not been understood.Recently,there are great progress on the mathematical theory of the shock regular reflection problem,especially for the global existence,uniqueness,and structural stability of solutions.In this paper,we show that there are two more possible configurations of the shock regular reflection besides known four configurations.We also give a brief proof of the global existence of solutions.
基金W.-X.Li's research was supported by NSF of China(11871054,11961160716,12131017)the Natural Science Foundation of Hubei Province(2019CFA007)T.Yang's research was supported by the General Research Fund of Hong Kong CityU(11304419).
文摘We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2.The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.
文摘The boundary layer flow of viscous incompressible fluid over a stretching cylinder has been considered to study flow field and temperature field. Due to non-linearity, a numerical approach called Keller-box technique has been used to compute the values of velocity function f and temperature field at different points of dynamic region. The expressions for skin friction and Nusselt number have also been obtained. The dependence of velocity profile and temperature profile on the dimensionless parameter of practical interest has been analyzed in detail by graphs. The dependence of Skin friction and Nusselt number has been seen through tables.
