Simulation of planar ionization wave front propagation on an unstructured adaptive grid

The paper deals with the numerical solution of a basic 2D model of the propagation of an ionization wave. The system of equations describing this propagation consists of a coupled set of reaction–diffusion-convection equations and a Poissons equation. The transport equations are solved by a finite volume method on an unstructured triangular adaptive grid. The upwind scheme and the diamond scheme are used for the discretization of the convection and diffusion fluxes, respectively. The Poisson equation is also discretized by the diamond scheme. Numerical results are presented. We deal in more detail with numerical tests of the grid adaptation technique and its influence on the numerical results. An original behavior is observed. The grid refinement is not sufficient to obtain accurate results for this particular phenomenon. Using a second order scheme for convection is necessary.     

Three‐dimensional solidification with two possible crystallization states: Existence of solutions with flow in the melt

In this paper we discuss the existence of solutions of a system of nonlinear and singular partial differential equations constituting a phase field model with convection for non‐isothermal solidification/melting of certain metallic alloys in the case where two different kinds of crystallization are possible. Each one of these crystallization states is described by its own phase field, while the liquid phase is described by another one. The model also allows the occurrence of fluid flow in non‐solid regions, which are a priori unknown, and then we have a free‐boundary value problem. Thus, the model relates the evolutions of these three phase fields, the temperature of the solidification/melting process and the fluid flow in non‐solid regions. Copyright © 2009 John Wiley & Sons, Ltd.     

The effect of a high-frequency progressive vibration on the convective instability of a two-layer fluid

The influence of a high-frequency progressive vibration on the onset of thermal convection in a two-layer system of viscous immiscible fluids is investigated. The interface is deformable, the outer walls are rigid, and heat-transfer conditions of a general form are assigned on them. The starting equations are taken in the generalized Oberbeck–Boussinesq approximation. An averaging method is employed. It is shown that the averaged problem contains a vibrogenic external force and vibrogenic stresses that are proportional to the square of the amplitude of the vibration rate. A quasi-equilibrium solution that satisfies the closure condition is found, and its stability is investigated. It is established that, unlike the case of a single-layer fluid, the horizontal component of the vibration influences the onset of convection and have a destabilizing effect. The vertical component stabilizes the two-layer system by increasing the surface tension. The long-wavelength asymptotic is investigated. Calculations are performed for the silicone oil–Fluorinert and acetonitrile–n-hexane systems.     

Global existence of weak solution to the heat and moisture transport system in fibrous porous media

This paper is concerned with theoretical analysis of a heat and moisture transfer model arising from textile industries, which is described by a degenerate and strongly coupled parabolic system. We prove the global (in time) existence of weak solution by constructing an approximate solution with some standard smoothing. The proof is based on the physical nature of gas convection, in which the heat (energy) flux in convection is determined by the mass (vapor) flux in convection.     

Prise en compte d'une sous-couche de dissipation dans les phénomènes de convection libre

Résumé À partir du modèle de Boussinesq hydrostatique, régissant un phénomène de convection libre au voisinage d'un sol thermiquement non homogène, on construit un modèle asymptotique en trois couches lié à la variation des coefficients d'échange avec l'altitude. On donne la formulation du problème de couche limite de second approximation qui en résulte et prenant en compte l'influence de la sous-couche de dissipation qui apparaît au voisinage immédiat du sol. Un cas simple est considéré et il permet d'obtenir une solution explicite pour la perturbation de la température absolue créee en régime périodique en temps.

---

Summary From the Boussinesq hydrostatic model, governing a free convection phenomenon in the vicinity of the thermally non homogeneous flat ground, we exhibit an asymptotic model with three layers connected at the variation of exchange coefficients with the altitude. We give the formulation of the corresponding second approximation boundary layer problem which takes into account the influence of the dissipation sublayer appearing in vicinity of the flat ground. A simple case is considered and it allows to obtain an explicit solution for the perturbation of the temperature in a periodic convection with the time.

     

A cellular automaton technique for modelling of a binary dendritic growth with convection

A two-dimensional model for the simulation of a binary dendritic growth with convection has been developed in order to investigate the effects of convection on dendritic morphologies. The model is based on a cellular automaton (CA) technique for the calculation of the evolution of solid/liquid (s/l) interface. The dynamics of the interface controlled by temperature, solute diffusion and Gibbs–Thomson effects, is coupled with the continuum model for energy, solute and momentum transfer with liquid convection. The solid fraction is calculated by a governing equation, instead of some approximate methods such as lever rule method [A. Jacot, M. Rappaz, Acta Mater. 50 (2002) 1909–1926.] or interface velocity method [L. Nastac, Acta Mater. 47 (1999) 4253; L. Beltran-Sanchez, D.M. Stefanescu, Mat. and Mat. Trans. A 26 (2003) 367.]. For the dendritic growth without convection, mesh independency of simulation results is achieved. The simulated steady-state tip velocity are compared with the predicted values of LGK theory [Lipton, M.E. Glicksmanm, W. Kurz, Metall. Trans. 18(A) (1987) 341.] as a function of melt undercooling, which shows good agreement. The growth of dendrite arms in a forced convection has been investigated. It was found that the dendritic growth in the upstream direction was amplified, due to larger solute gradient in the liquid ahead of the s/l interface caused by melt convection. In the isothermal environment, the calculated results under very fine mesh are in good agreement with the Oseen–Ivanstov solution for the concentration-driven growth in a forced flow.     

Modelo de un proceso de cristalización continua de un sorbete por medio de la metodología de momentos

Freezing is an important step in the manufacturing process of ice-cream and sorbet, since the operating conditions have a strong influence on the micro-structure, and consequently on the sensorial attributes of the final product. This steep of freezing is carried out by a scraped surface heat exchanger (SSHE) where the product quality is conditioned by process conditions as the evaporation temperature of a refrigerant fluid, the mix flow rate, the dasher speed and the cylinder pressure due to the air introduction. In order to study the relevance of a control system based on the influence of process variables on product quality, this paper presents a model for a continuous crystallization of a sorbet using the method of moments, which is validated by experimental data.The model created by this methodology has been able to represent the influence of the process conditions during the crystallization of the sorbet on the final product characteristics such as crystal size and the draw temperature in the outlet of the SSHE in absence of air. The model based in moments is studied as a reduced model of the population balance equation and includes the phenomena of heterogeneous nucleation and growth. This model developed represents minimal computational requirements and is highly adapted for optimization and/or process control tasks.     

Modeling of stratified two-phase flows with non-uniform evaporation based on the exact solution of convection equations

Within the framework of the Oberbeck–Boussinesq approximation, the exact solution of equations of thermoconcentration convection is studied, which has the group origin. The issue of applicability of the exact solution for describing steady-state convective flows of a liquid and a co-current gas flux under the conditions of inhomogeneous evaporation of the diffusive type in a flat horizontal channel is discussed. Algorithms for obtaining analytical representations of the required functions for various types of the boundary conditions for the temperature function on the outer channel wall are proposed. The influence of the external thermal load and boundary thermal conditions on the structure of the velocity and temperature fields, evaporation mass flow rate, and vapor content in the gas layer is investigated at an example of the HFE-7100–nitrogen system. The solution correctly predicts hydrodynamical, temperature, and concentration parameters of convective regimes appearing in the two-phase system. The main characteristics are compared with the characteristics of the system in the case of uniform evaporation with a constant intensity.     

Asymptotic solution of wave front of the telegraph model of dispersive variability

A number of nonlinear phenomena in physical, chemical, economical and biological processes are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion. Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper we use the technique of asymptotic solution to find travelling wave solution for the telegraph model of dispersive variability which is a generalization of the Julian Cook model. Our model tackled special values of the time delay and the probability that an individual is disperser in a population of dispersers and nondispersers. The solution of our model is reduced to telegraph Fisher–Kolmogoroff invasion and Julian Cook models. Also, the effect of the time delay on the propagation speed is presented.     

A note on an accelerated high‐accuracy multigrid solution of the convection‐diffusion equation with high Reynolds number

We present a new strategy to accelerate the convergence rate of a high‐accuracy multigrid method for the numerical solution of the convection‐diffusion equation at the high Reynolds number limit. We propose a scaled residual injection operator with a scaling factor proportional to the magnitude of the convection coefficients, an alternating line Gauss‐Seidel relaxation, and a minimal residual smoothing acceleration technique for the multigrid solution method. The new implementation strategy is tested to show an improved convergence rate with three problems, including one with a stagnation point in the computational domain. The effect of residual scaling and the algebraic properties of the coefficient matrix arising from the fourth‐order compact discretization are investigated numerically. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 1–10, 2000     

Analytical and numerical investigations of a batch crystallization model

This article is concerned with the analytical and numerical investigations of a one-dimensional population balance model for batch crystallization processes. We start with a one-dimensional batch crystallization model and prove the local existence and uniqueness of the solution of this model. For this purpose Laplace transformation is used as a basic tool. A semi-discrete high resolution finite volume scheme is proposed for the numerical solution of the current model. The issues of positivity (monotonicity), consistency, stability and convergence of the proposed scheme for the current model are analyzed and proved. Finally, we give a numerical test problem. The numerical results of the proposed high resolution scheme are compared with the solution of the reduced four-moments model and the first-order upwind scheme.     

Influence of eddies on heat transfer in Couette flow over undulated substrates

In Couette flows over undulated substrates eddies can be generated under creeping flow conditions. In contrast to free surface flows on undulated substrates even smooth bottom undulations allow for eddy generation due to the kinematical constraints. The subject of our paper is how these flow patterns interact with the temperature field in non–isothermal flows. Our analysis of the thermo–mechanical coupling is focused on the two dominant effects, namely convection and thermoviscosity, whereas dissipation heat, buoyancy and temperature–dependence of the remaining material parameters are neglected. We solve the problem in two steps: First, the influence of the eddies on the convective heat transfer is considered by solving the heat conduction equation with convection. For the velocity field we take the solution resulting analytically from Reynolds' lubrication approximation for the isothermal flow. The thermoviscous feedback of the resulting temperature field to the flow is considered in forthcoming papers. For the construction of the solution an analytical approach based on a nonorthogonal series representation of the fundamental fields and a variational formulation of the field equations is used. The results are visualised and the physical effects they reveal are discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A characteristics-mixed covolume method for a convection-dominated transport problem

In this paper, we propose a characteristics-mixed covolume method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term. The velocity and press are approximated by the lowest order Raviart-Thomas mixed finite element space on rectangles. The projection of a mixed covolume element is introduced. We prove its first order optimal rate of convergence for the approximate velocities in the L² norm as well as for the approximate pressures in the L² norm.     

Cattaneo–Christov heat flux model for three-dimensional magnetohydrodynamic flow of an Eyring Powell fluid over an exponentially stretching surface with convective boundary condition

The present model concentrates on three-dimensional steady incompressible flow of an Eyring-Powell nanofluid past an exponentially stretching sheet with magnetic field. The Cattaneo–Christov heat flux with convective boundary condition is accounted for. Shooting method is the instrumental for obtaining numerical solution of the transformed-converted system of the mathematical models. Behavior of the determining thermo-physical parameters on the velocity, temperature, skin friction, heat transfer rate, and finally isotherms are considered. The major relevant outcomes of the current investigation are that increment in Eyring-Powell parameter uplifts flow velocity, while that peters out the fluid temperature. Enhanced values of the mixed convection parameter weakened the skin friction coefficient while it slightly strengthened the rate of heat transfer.     

Continuous dependence on boundary and Soret coefficients in double diffusive bidispersive convection

We develop a theory for double diffusive convection in a double porosity material along the Brinkman scheme. The Soret effect is included whereby a temperature gradient may directly influence salt concentration. The boundary conditions on the temperature and salt fields are of general Robin type. A number of a priori estimates are established whereby, through energy arguments, we prove continuous dependence of the solution on the Soret coefficient and on the coefficients in the boundary conditions in the L²- norm.     

Kronecker product approximation preconditioners for convection–diffusion model problems

We consider the iterative solution of linear systems arising from four convection–diffusion model problems: scalar convection–diffusion problem, Stokes problem, Oseen problem and Navier–Stokes problem. We design preconditioners for these model problems that are based on Kronecker product approximations (KPAs). For this we first identify explicit Kronecker product structure of the coefficient matrices, in particular for the convection term. For the latter three model cases, the coefficient matrices have a 2 × 2 block structure, where each block is a Kronecker product or a summation of several Kronecker products. We then use this structure to design a block diagonal preconditioner, a block triangular preconditioner and a constraint preconditioner. Numerical experiments show the efficiency of the three KPA preconditioners, and in particular of the constraint preconditioner that usually outperforms the other two. This can be explained by the relationship that exists between these three preconditioners: the constraint preconditioner can be regarded as a modification of the block triangular preconditioner, which at its turn is a modification of the block diagonal preconditioner based on the cell Reynolds number. Copyright © 2009 John Wiley & Sons, Ltd.     

Modeling of three dimensional thermocapillary flows with evaporation at the interface based on the solutions of a special type of the convection equations

Theoretical and numerical study of the convection processes, which are accompanied by evaporation/condensation, in the framework of new non-standard problem is largely motivated by new physical experiments. One of the principal questions is to understand the character and to evaluate the degree of influence of particular factors or their combined action on the structure of the joint flows of liquid and gas-vapor mixture. The flow topology is determined by four main mechanisms: natural and thermocapillary convection, tangential stresses and mass transfer due to evaporation at the interface. The mathematical modeling of the fluid flows in an infinite channel with a rectangular cross section is carried out on the basis of the solution of a special type of the convection equations. The effects of thermodiffusion and diffusive thermal conductivity in the gas phase and evaporation at the thermocapillary interface are taken into consideration. Numerical investigations are performed for the liquid – gas (ethanol – nitrogen) system under normal and low gravity. The fluid flows are characterized as translational and progressively rotational motions and can be realized in various forms.     

Crystallization waves in thin amorphous layers on heat conducting substrates

A crystallization process in thin films is considered, where, driven by the release of the latent heat of fusion, the transformation from an amorphous state to the crystalline state takes place in a progressing wave of invariant shape. The crystallization rate is determined by a rate equation. The influence of the heat loss due to heat conduction into the substrate is taken into account. The resulting system of an ordinary differential equation and an integro-differential equation is solved numerically using a collocation method. The propagation speed of the wave in dependence on a non-dimensional heat loss parameter is determined. It turns out that the existence of a self-sustaining crystallization wave requires the heat loss parameter to be smaller than a certain critical value. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic solution of wave front of the telegraph model of dispersive variability

A number of nonlinear phenomena in physical, chemical, economical and biological processes are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion. Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper we use the technique of asymptotic solution to find travelling wave solution for the telegraph model of dispersive variability which is a generalization of the Julian Cook model. Our model tackled special values of the time delay and the probability that an individual is disperser in a population of dispersers and nondispersers. The solution of our model is reduced to telegraph Fisher–Kolmogoroff invasion and Julian Cook models. Also, the effect of the time delay on the propagation speed is presented.