In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is pre...In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.展开更多
The solutions of Green’s function are significant for simplification of problem on a two-phase saturated medium.Using transformation of axisymmetric cylindrical coordinate and Sommerfeld’s integral,superposition of ...The solutions of Green’s function are significant for simplification of problem on a two-phase saturated medium.Using transformation of axisymmetric cylindrical coordinate and Sommerfeld’s integral,superposition of the influence field on a free surface,authors obtained the solutions of a two-phase saturated medium subjected to a concentrated force on the semi-space.展开更多
One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise line...One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.展开更多
A one-phase Stefan problem for the nonhomogeneous heat equation with the source term depending on an unknown parameter p(t) is considered. The existence and uniqueness of the solution (p, s, u) are also demonstrated.
In this paper, we prove the existence theorems of locbal or global classical solutions to Stefan problems with various kinetic conditions at the free boundary.
The outflow problem for the viscous two-phase flow model in a half line is investigated in the present paper.The existence and uniqueness of the stationary solution is shown for both supersonic state and sonic state a...The outflow problem for the viscous two-phase flow model in a half line is investigated in the present paper.The existence and uniqueness of the stationary solution is shown for both supersonic state and sonic state at spatial far field,and the nonlinear time stability of the stationary solution is also established in the weighted Sobolev space with either the exponential time decay rate for supersonic flow or the algebraic time decay rate for sonic flow.展开更多
In this paper we consider an evolutionary continuous casting problem with convection partial derivative + b(y) chi - Delta kappa(u) + (v) over right arrow .del u = 0 coupled with Stokes equation in the liquid phase pa...In this paper we consider an evolutionary continuous casting problem with convection partial derivative + b(y) chi - Delta kappa(u) + (v) over right arrow .del u = 0 coupled with Stokes equation in the liquid phase partial derivative(t) (v) over right arrow - v Delta (v) over right arrow + del p = (f) over right arrow(u) The mixed boundary condition is put on temprature. The existence of a weak solution is obtained by using methods of regularization, temperature dependent penalty form in the Stokes equation and compact arguments.展开更多
We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé...We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815;2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .展开更多
According to the boundary condition with the zero,negative,or positive velocity,the initial boundary problem for compressible multi-phase flow with the Dirichlet-type boundary condition can be classified into three ca...According to the boundary condition with the zero,negative,or positive velocity,the initial boundary problem for compressible multi-phase flow with the Dirichlet-type boundary condition can be classified into three cases:impermeable problem,inflow problem,or outflow problem.In this paper,we review the recent progress on the existence and nonlinear stability of the stationary solution to the outflow/inflow problems for viscous multi-phase flow.展开更多
基金supported by the National Natural Science Foundation (10871179) of China
文摘In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.
基金supported by the National Natural Science Foundation of China (10572129)
文摘The solutions of Green’s function are significant for simplification of problem on a two-phase saturated medium.Using transformation of axisymmetric cylindrical coordinate and Sommerfeld’s integral,superposition of the influence field on a free surface,authors obtained the solutions of a two-phase saturated medium subjected to a concentrated force on the semi-space.
基金supported by the Fundamental Research Funds for the Central Universities(Grant 2015XKMS014)
文摘One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.
文摘A one-phase Stefan problem for the nonhomogeneous heat equation with the source term depending on an unknown parameter p(t) is considered. The existence and uniqueness of the solution (p, s, u) are also demonstrated.
文摘In this paper, we prove the existence theorems of locbal or global classical solutions to Stefan problems with various kinetic conditions at the free boundary.
基金the paper is supported by the National Natural Science Foundation of China(Nos.11871047,11671384,11931010)the key research project of Academy for Multidisciplinary Studies,Capital Normal University,and by the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds(No.007/20530290068).
文摘The outflow problem for the viscous two-phase flow model in a half line is investigated in the present paper.The existence and uniqueness of the stationary solution is shown for both supersonic state and sonic state at spatial far field,and the nonlinear time stability of the stationary solution is also established in the weighted Sobolev space with either the exponential time decay rate for supersonic flow or the algebraic time decay rate for sonic flow.
文摘In this paper we consider an evolutionary continuous casting problem with convection partial derivative + b(y) chi - Delta kappa(u) + (v) over right arrow .del u = 0 coupled with Stokes equation in the liquid phase partial derivative(t) (v) over right arrow - v Delta (v) over right arrow + del p = (f) over right arrow(u) The mixed boundary condition is put on temprature. The existence of a weak solution is obtained by using methods of regularization, temperature dependent penalty form in the Stokes equation and compact arguments.
文摘We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815;2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .
基金supported by the National Natural Science Foundation of China(nos.11931010,11871047)the key research project of Academy for Multidisciplinary Studies,Capital Normal University,and by the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds(no.007/20530290068).
文摘According to the boundary condition with the zero,negative,or positive velocity,the initial boundary problem for compressible multi-phase flow with the Dirichlet-type boundary condition can be classified into three cases:impermeable problem,inflow problem,or outflow problem.In this paper,we review the recent progress on the existence and nonlinear stability of the stationary solution to the outflow/inflow problems for viscous multi-phase flow.
