In this paper,we revisit the global well-posedness of the classical viscous surface waves in the absence of surface tension effect with the reference domain being the horizontal infinite slab,for which the first compl...In this paper,we revisit the global well-posedness of the classical viscous surface waves in the absence of surface tension effect with the reference domain being the horizontal infinite slab,for which the first complete proof was given in Guo-Tice[Anal.PDE 6,1429-1533(2013)]via a hybrid of Eulerian and Lagrangian schemes.The fluid dynamics are governed by the gravity-driven incompressible Navier-Stokes equations.Even though Lagrangian formulation is most natural to study free boundary value problems for incompressible flows,few mathematical works for global existence are based on such an approach in the absence of surface ten-sion effect,due to breakdown of Beale’s transformation.We develop a mathematical approach to establish global well-posedness based on the Lagrangian framework by analyzing suitable“good unknowns”associated with the problem,which requires no nonlinear compatibility conditions on the initial data.展开更多
The magnetohydrodynamic（MHD） flow induced by a stretching or shrinking sheet under slip conditions is studied.Analytical solutions based on the boundary layer assumption are obtained in a closed form and can be appl...The magnetohydrodynamic（MHD） flow induced by a stretching or shrinking sheet under slip conditions is studied.Analytical solutions based on the boundary layer assumption are obtained in a closed form and can be applied to a flow configuration with any arbitrary velocity distributions. Seven typical sheet velocity profiles are employed as illustrating examples. The solutions to the slip MHD flow are derived from the general solution and discussed in detail. Different from self-similar boundary layer flows, the flows studied in this work have solutions in explicit analytical forms. However, the current flows require special mass transfer at the wall, which is determined by the moving velocity of the sheet. The effects of the slip parameter, the mass transfer at the wall, and the magnetic field on the flow are also demonstrated.展开更多
文摘In this paper,we revisit the global well-posedness of the classical viscous surface waves in the absence of surface tension effect with the reference domain being the horizontal infinite slab,for which the first complete proof was given in Guo-Tice[Anal.PDE 6,1429-1533(2013)]via a hybrid of Eulerian and Lagrangian schemes.The fluid dynamics are governed by the gravity-driven incompressible Navier-Stokes equations.Even though Lagrangian formulation is most natural to study free boundary value problems for incompressible flows,few mathematical works for global existence are based on such an approach in the absence of surface ten-sion effect,due to breakdown of Beale’s transformation.We develop a mathematical approach to establish global well-posedness based on the Lagrangian framework by analyzing suitable“good unknowns”associated with the problem,which requires no nonlinear compatibility conditions on the initial data.
文摘The magnetohydrodynamic（MHD） flow induced by a stretching or shrinking sheet under slip conditions is studied.Analytical solutions based on the boundary layer assumption are obtained in a closed form and can be applied to a flow configuration with any arbitrary velocity distributions. Seven typical sheet velocity profiles are employed as illustrating examples. The solutions to the slip MHD flow are derived from the general solution and discussed in detail. Different from self-similar boundary layer flows, the flows studied in this work have solutions in explicit analytical forms. However, the current flows require special mass transfer at the wall, which is determined by the moving velocity of the sheet. The effects of the slip parameter, the mass transfer at the wall, and the magnetic field on the flow are also demonstrated.
