一个具有内部物质对流的两项连铸Stefan问题

考虑了一个具有内部物质对流和非线性边界热交换的多维连铸Stefan问题,并得到了这个问题整体弱解的存在性、唯一性和对初边界条件的连续依赖性。本项工作改进和推广了J.F.Fodri-gues＆F.Yi的结果,放宽了他们对内部流和边界条件的一些不太符合实际的限制。     

On a free boundary problem in ground freezing

We consider a boundary value problem describing the stationary flow of a non‐Newtonian fluid through the frozen ground, with a free interface between the liquid and the solid phases. We prove the existence of at least one weak solution of the problem. Copyright © 2000 John Wiley & Sons, Ltd.     

Unconditional non‐linear stability for a fluid of third grade

The thermal convection in a layer of a third grade fluid is investigated, with viscosity being a general function of temperature. We develop a non‐linear stability analysis and prove that unconditional non‐linear stability criterion is achieved using a natural energy approach. This shows that, in some sense, the equations for a fluid of third grade are preferable to those for a fluid of second grade or a dipolar fluid. Copyright © 2004 John Wiley & Sons, Ltd.     

On the well‐posedness of a mathematical model describing water‐mud interaction

In this paper, we consider a mathematical model describing the two‐phase interaction between water and mud in a water canal when the width of the canal is small compared with its depth. The mud is treated as a non‐Newtonian fluid, and the interface between the mud and fluid is allowed to move under the influence of gravity and surface tension. We reduce the mathematical formulation, for small boundary and initial data, to a fully nonlocal and nonlinear problem and prove its local well‐posedness by using abstract parabolic theory. Copyright © 2012 John Wiley & Sons, Ltd.     

On transport equations driven by a non‐divergence‐free force field

The paper is devoted to initial boundary value problems for transport equations with non‐divergence‐free external field. The crucial role is played by integration along characteristics and associated Green's formula for which we provide a new proof which generalizes and clarifies previous versions. The paper concludes with an application of general theory to the Spencer–Lewis equation. Copyright © 2007 John Wiley & Sons, Ltd.     

On the cauchy problem for a one‐dimensional full compressible non‐Newtonian fluids

This paper is concerned with global existence and asymptotic behavior of H¹ solutions to the Cauchy problem of one‐dimensional full non‐Newtonian fluids with the weighted small initial data. We then obtain the global existence of Hⁱ(i = 2,4) solutions and their asymptotic behavior for the system. Copyright © 2015 John Wiley & Sons, Ltd.     

Darcy's laws for non‐stationary viscous fluid flow in a thin porous medium

We consider a non‐stationary Stokes system in a thin porous medium Ω_? of thickness ? which is perforated by periodically solid cylinders of size a _? . We are interested here to give the limit behavior when ? goes to zero. To do so, we apply an adaptation of the unfolding method. Time‐dependent Darcy's laws are rigorously derived from this model depending on the comparison between a _? and ? . Copyright © 2016 John Wiley & Sons, Ltd.     

Weak solution to a one‐dimensional full compressible non‐Newtonian fluid

This paper is devoted to the existence of global‐in‐time weak solutions to a one‐dimensional full compressible non‐Newtonian fluid. A semi‐discrete finite element scheme is taken to generate approximate solutions, based on an exact projection technique. To enforce convergence of the approximate solutions, the uniform estimate is obtained using an iteration method and energy method, with the help of the weak compactness and convexity. Numerical simulations showing the existence of solutions are presented.     

On a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition

In this paper, a classical Stefan problem with a prescribed and small time-dependent temperature at the boundary is studied. By using a multiple time-scales perturbation method, it is shown analytically how the moving boundary profile is influenced by the prescribed temperature at the boundary and the initial conditions. Only a few exact solutions are available for this type of problems and it turns out that the constructed approximations agree very well with these exact solutions. In particular, approximations of solutions for this type of problems, with periodic and decaying temperatures at the boundary, are constructed. Furthermore, these approximations are valid on a long time scale, and seems to be not available in the literature.     

Heat transfer and Couette flow of a chemically reacting non‐linear fluid

In this paper we study the flow and heat transfer in a chemically reacting non‐linear fluid between two long horizontal parallel flat plates that are at different temperatures. The top plate is sheared, whereas the bottom plate is fixed. The fluid is modeled as a generalized power‐law fluid whose viscosity is also assumed to be a function of the concentration. The effects of radiation are neglected. The equations are made dimensionless and the boundary value problem is solved numerically; the velocity and temperature profiles are obtained for various dimensionless numbers. Published in 2009 by John Wiley & Sons, Ltd.     

Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation

The work presents a mathematical model describing the time fractional anomalous-diffusion process of a generalized Stefan problem which is a limit case of a shoreline problem. In this model, the governing equations include a fractional time derivative of order 0 < α ? 1 and variable latent heat. The approximate solution of the problem is obtained by homotopy perturbation method. The results thus obtained are compared graphically with the exact solutions. A brief sensitivity study is also performed.     

An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients

We consider a nonlinear heat conduction problem for a semi-infinitematerial x > 0, with phase-change temperature T₁, an initialtemperature T₂ (> T₁) and a heat flux of the type q (t) =q₀/t imposed on the fixed face x = 0. We assume that the volumetricheat capacity and the thermal conductivity are particular nonlinearfunctions of the temperature in both solid and liquid phases. We determine necessary and/or sufficient conditions on the parametersof the problem in order to obtain the existence of an explicitsolution for an instantaneous nonlinear twophase Stefan problem(solidification process).     

A mathematical model for the study of gliding motion of bacteria on a layer of non‐Newtonian slime

This paper is concerned with a mathematical hydrodynamical model of motility involving an undulating cell surface. The cell surface transmits stresses through a layer of exuded slime to the substratum. The slime is considered as a Johnson–Segalman fluid. A perturbation approach is used to find the analytic solution. Analytical expressions for the stream function, velocity, pressure gradient and pressure rise over a wavelength as well as the corresponding computational results are presented. The propulsive and lift forces and the power required for gliding propulsion have also been determined. The presented mechanism is found to generate a force for the propulsion of glider at a realistic speed and requires an output of power that is much less than the organism's metabolic rate of energy production. It is observed that unlike the Newtonian case of slime, the lift force is generated due to the Weissenberg number for non‐Newtonian slime, represented by the model of Johnson–Segalman fluid. It is also found that power required for translation in Johnson–Segalman fluid is reduced. Copyright © 2004 John Wiley & Sons, Ltd.     

The existence of positive solutions to a non‐local singular boundary value problem

We consider the non‐local singular boundary value problem (1) where q ∈ C⁰([0,1]) and f, h ∈ C⁰((0,∞)), limf(x)=?∞, limh(x)=∞. We present conditions guaranteeing the existence of a solution x ∈ C¹([0,1]) ∩ C²((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.     

The use of linear approximation scheme for solving the Stefan problem

This paper deals with the linear approximation scheme to approximate a singular parabolic problem: the two-phase Stefan problem on a domain consisting of two components with imperfect contact. The results of some numerical experiments and comparisons are presented. The method was used to determine the temperature of steel in the process of continuous casting.     

A Stefan problem modelling crystal dissolution and precipitation

A simple 1D model for crystal dissolution and precipitationis presented. The model equations resemble a one-phase Stefanproblem and involve non-linear and multivalued exchange ratesat the free boundary. The original equations are formulatedon a variable domain. By transforming the model to a fixed domainand applying a regularization, we prove the existence and uniquenessof a solution. The paper is concluded by numerical simulations.     

Gliding motion of bacteria on power‐law slime

The present investigation deals with an undulating surface model for the motility of bacteria gliding on a layer of non‐Newtonian slime. The slime being the viscoelastic material is considered as a power‐law fluid. A hydrodynamical model of motility involving an undulating cell surface which transmits stresses through a layer of exuded slime to the substratum is examined. The non‐linear differential equation resulting from the balance of momentum and mass is solved numerically by a finite difference method with an iteration technique. The manner in which the various exponent values of the power‐law flow affect the structure of the boundary layer is delineated. A comparison is made of the power‐law fluid with the Newtonian fluid. For the power‐law fluid with respect to different power‐law exponent values, shear‐thinning and shear‐thickening effects can be observed, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

Analytic solution of natural convection flow of a non‐newtonian fluid between two vertical flat plates by using decomposition method

An analysis has been performed to study the natural convection of a non‐Newtonian fluid between two infinite parallel vertical flat plates and the effects of the non‐Newtonian nature of fluid on the heat transfer are studied. The governing boundary layer and temperature equations for this problem are reduced to an ordinary form and are solved by Adomian decomposition method (ADM) and numerical method. Velocity and temperature profiles are shown graphically. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and non‐linear equations and so can be found widely applicable in engineering and science. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1384–1395, 2010