Nonreacting chemical transport in two-phase reservoirs — Factoring diffusive and wave properties

The vertical transport of mass, energy andn unreacting chemical species in a two-phase reservoir is analysed when capillarity can be ignored. This results in a singular system of equations, comprising mixed parabolic and hyperbolic equations. We derive a natural factorisation of these equations into diffusive and wave equations. If diffusive or conductive effects are important for onlyN–1 of the chemical species, then there areN parabolic equations, andn+2–N wave equations. Each wave equation allows for the corresponding variable to be discontinous, or equivalently, for shock propagation to occur. Steady flows were shown to allow for more than two (vapour and liquid dominated) saturations for a given mass, energy and chemical flux, but when thermal conduction and chemical diffusion are unimportant, only the vapour and liquid dominated cases appear likely to occur. For infinitesimal shocks there is a continuous flux vector for each diffusive variable. The existence of these continuous flux vectors results in significant simplifications of the corresponding wave equations. It remains an open question if continuous flux vectors exist for finite shocks. General expressions are obtained for the diffusion and wave matrices, which require no knowledge of continuous flux vectors.     

Null‐Controllability of a System of Linear Thermoelasticity

We consider a linear system of thermoelasticity in a compact, C ^infin, n-dimensional connected Riemannian manifold. This system consists of a wave equation coupled to a heat equation. When the boundary of the manifold is non‐empty, Dirichlet boundary conditions are considered. We study the controllability properties of this system when the control acts in the hyperbolic equation (and not in the parabolic one) and has its support restricted to an open subset of the manifold. We show that, if the control time and the support of the control satisfy the geometric control condition for the wave equation, this system of thermoelasticity is null-controllable. More precisely, any finite‐energy solution can be driven to zero at the control time. An analogous result is proved when the control acts on the parabolic equation. Finally, when the manifold has no boundary, the null‐controllability of the linear system of three‐dimensional thermoelastic ity is proved. (Accepted June 13, 1996)     

Thermal waves propagation due to localised heat inputs – the Laplace transforms method analysis

 Propagation of thermal waves due to localised heat inputs is analysed by the Laplace transforms method for the case of constant thermophysical parameters of the heat-conducting media and a temperature dependent internal heat source. Both the hyperbolic and parabolic models of heat conduction are used and compared. For the hyperbolic model, an energy pulse travelling along the conductor is observed. Long time hyperbolic solutions do not differ qualitatively from the related parabolic solutions, but the quantitative differences between the two solutions do not vanish with time because of the temperature dependent heat source. The cases considered in the paper describe the evolution of normal zones in technical superconductors. Received on 10 May 2000 / Published online: 29 November 2001     

Thermal waves induced by a rapidly moving line source

For the cases involving a fast moving heat source or extremely short pulses emitted by lasers or short time after the start of transients, the classical theory of heat conduction breaks down since the wave nature of heat transport dominates. In this study, the temperature field due to a fast moving line source was determined analytically using the wave concept. The results are given for different values of thermal Mach number (M=V/C). For M>1 the heat affected zone is confined in a wedge shape region behind the source. The wedge half angle is equal to sin^?1 (1/M). It was confirmed that the difference between the results of diffusion and wave models depends on the corresponding time scale and the relaxation time.     

Fast precise integration method for hyperbolic heat conduction problems

A fast precise integration method is developed for the time integral of the hyperbolic heat conduction problem. The wave nature of heat transfer is used to analyze the structure of the matrix exponential, leading to the fact that the matrix exponential is sparse. The presented method employs the sparsity of the matrix exponential to improve the original precise integration method. The merits are that the proposed method is suit- able for large hyperbolic heat equations and inherits the accuracy of the original version and the good computational efficiency, which are verified by two numerical examples.     

Non-Fourier heat conduction by axisymmetric thermal waves in an infinite solid medium

The non-stationary heat conduction in an infinite solid medium internally bounded by an infinitely long cylindrical surface is considered. A uniform and time- dependent temperature is prescribed on the boundary surface. An analytical solution of the hyperbolic heat conduction equation is obtained. The solution describes the wave nature of the temperature field in the geometry under consideration. A detailed analysis of the cases in which the temperature imposed on the boundary surface behaves as a square pulse or as an exponentially decaying pulse is provided. The evolution of the temperature field in the case of hyperbolic heat conduction is compared with that obtained by solving Fourier's equation. Received on 28 January 1998     

The calculation of free electron density in Cassandra

Cassandra is an AWE opacity code used to model plasmas in local thermal equilibrium: there is a desire to expand its use to calculating plasma equations of state. Cassandra's self-consistent field calculation (scf) uses the local density approximation for bounds states and has a free electron contribution based upon the Thomas-Fermi model [B.J.B. Crowley et al., J. Quant. Spectro. Radiat. Trans. 71, 257(2001)]. Whilst this is applicable for very high temperature or low density plasmas; in hot and dense matter the effect of ionization will lead to discontinuities in the effective ionisation, Z^?. The electron contribution to hydrostatic pressure is associated with Z^?, thus these discontinuities produce unphysical jumps in the resulting calculated material pressure.We describe a procedure to mitigate the effect by calculating the free electron wave functions within the generalized ion-cell model [B.J.B. Crowley et al., Phys. Rev. A 41, 2179(1990)], and thus explicitly calculate free-electron resonances.     

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) =  C _γ ρ ^γ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.     

Enhanced boiling heat transfer using radial fins

A numerical bifurcation analysis is carried out in order to determine the solution structure of radial fins subjected to multi-boiling heat transfer mode. One-dimensional conduction is employed throughout the thermal analysis. The fluid heat transfer coefficient is temperature dependent on the three regimes of phase-change of the fluid. Six fin profiles, defined in the text, are considered. Multiplicity structure is obtained to determine different types of bifurcation diagrams, which describe the dependence of a state variable of the system like the temperature or the heat dissipation on the fin design parameters, conduction–convection parameter (CCP) or base temperature difference (ΔT). Specifically, the effects of ΔT, CCP and Biot number are analyzed. The results are presented graphically, showing the significant behavioral features of the heat rejection mechanism.

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MHD mixed convective heat transfer flow about an inclined plate

Mixed convection heat transfer about a semi-infinite inclined plate in the presence of magneto and thermal radiation effects is studied. The fluid is assumed to be incompressible and dense. The nonlinear coupled parabolic partial differential equations governing the flow are transformed into the non-similar boundary layer equations, which are then solved numerically using the Keller box method. The effects of the mixed convection parameter R _i, the angle of inclination α, the magnetic parameter M and the radiation–conduction parameter R _d on the velocity and temperature profiles as well as on the local skin friction and local heat transfer parameters. For some specific values of the governing parameters, the results are compared with those available in the literature and a fairly good agreement is obtained.     

Hyperbolic heat conduction in the semi-infinite body with the heat source which capacity linearly depends on temperature

It is commonly taken for granted that long-time solutions of Cattaneo's hyperbolic heat conduction equation tend to overlap the related parabolic solutions. Usually, for small times considerable qualitative as well as quantitative differences between the hyperbolic and parabolic solutions are observed which vanish completely for large times. However, in the case of a heat conducting body with the temperature dependent inner heat generation, the quantitative differences may grow with time. This arises from the feedback between the temperature and the source capacity. To illustrate this effect, Cattaneo's hyperbolic equation for the semi-infinite body, with the heat source which capacity is linearly dependent on temperature, is solved analytically by the Laplace transforms method. Received on 1 July 1997     

An explicit asymptotic model for the Bleustein–Gulyaev waveUn modèle explicite pour l'onde de Bleustein–Gulyaev

The antiplane motion of a transversely isotropic piezoelectric half-space is considered. An explicit asymptotic model is derived for the far field of the surface wave. It involves, in particular, a 1D hyperbolic equation for surface shear deformation propagating with the finite wave speed predicted for the first time by J.L. Bleustein and Yu.V. Gulyaev. Neumann and Dirichlet problems are formulated to restore interior mechanical and electric fields. The derivation utilizes asymptotic arguments combined with Lourier symbolic integration. Comparison with the exact solution is presented for surface impact loading. To cite this article: J. Kaplunov et al., C. R. Mecanique 332 (2004).     

Conditional Stability of Unstable Viscous Shock Waves in Compressible Gas Dynamics and MHD

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic–parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small L ¹ ∩ H ³ perturbations, converging time asymptotically to a translate of the unperturbed wave. That is, for a shock with p unstable eigenvalues, we establish conditional stability on a codimension-p manifold of initial data, with sharp rates of decay in all L ^p . For p = 0, we recover the result of unconditional stability obtained by Mascia and Zumbrun. The main new difficulty in the hyperbolic–parabolic case is to construct an invariant manifold in the absence of parabolic smoothing.     

On some diffusive waves in non-linear heat conduction

We address the non-linear heat conduction in the presence of absorption for the case of spherical symmetry geometry. The non-linear model is based on both a temperature-dependent thermal conductivity and a non-linear generalization of the Fourier law. The governing equation belongs to a class of degenerate parabolic equations. We obtain similarity solutions in closed form for the Cauchy problem corresponding to an instantaneous point source problem. We investigate the non-linear effects on the propagation of the temperature distrubances. We find that in certain cases the temperature distribution displays travelling wave characteristics. The solution for the Cauchy problem is recovered by considering a suitable first boundary value problem.     

Anomalous Reactive Transport in the Framework of the Theory of Chromatography

The anomalous reactive transport considered here is the migration of contaminants through strongly sorbing permeable media without significant retardation. It has been observed in the case of heavy metals, organic compounds, and radionuclides, and it has critical implications on the spreading of contaminant plumes and on the design of remediation strategies. Even in the absence of the well-known fast migration pathways, associated with fractures and colloids, anomalous reactive transport arises in numerical simulations of reactive flow. It is due to the presence of highly pH-dependent adsorption and the broadening of the concentration front by hydrodynamic dispersion. This leads to the emergence of an isolated pulse or wave of a contaminant traveling at the average flow velocity ahead of the retarded main contamination front. This wave is considered anomalous because it is not predicted by the classical theory of chromatography, unlike the retardation of the main contamination front. In this study, we use the theory of chromatography to study a simple pH-dependent surface complexation model to derive the mathematical framework for the anomalous transport. We analyze the particular case of strontium (Sr²⁺) transport and define the conditions under which the anomalous transport arises. We model incompressible one-dimensional (1D) flow through a reactive porous medium for a fluid containing four aqueous species: H⁺, Sr²⁺, Na⁺, and Cl⁻. The mathematical problem reduces to a strictly hyperbolic 2 × 2 system of conservation laws for effective anions and Sr²⁺, coupled through a competitive Langmuir isotherm. One characteristic field is linearly degenerate while the other is not genuinely nonlinear due to an inflection point in the pH-dependent isotherm. We present the complete set of analytical solutions to the Riemann problem, consisting of only three combinations of a slow wave comprising either a rarefaction, a shock, or a shock–rarefaction with fast wave comprising only a contact discontinuity. Highly resolved numerical solutions at large Péclet numbers show excellent agreement with the analytic solutions in the hyperbolic limit. In the Riemann problem, the anomalous wave forms only if: hydrodynamic dispersion is present, the slow wave crosses the inflection locus, and the effective anion concentration increases along the fast path.     

The generation and quantitative visualization of breaking internal waves

New techniques for the generation and quantitative visualization of breaking progressive internal waves are presented. Laboratory techniques applicable to general stratified flow experiments are also demonstrated. The planar laser-induced fluorescence (PLIF) technique is used to produce calibrated images of the wave breaking process, and the details of the PLIF measurements are described in terms of the necessary corrections and considerations for the application of PLIF to stratified flows. Results of the flow visualization and wave generation techniques are presented, which show that the nature of internal wave breaking is strongly dependent on the type of breaking internal wave considered.

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Hyperbolic heat conduction in the semi-infinite body with a time-dependent laser heat source

 The Cattaneo hyperbolic and classical parabolic models of heat conduction in the laser irradiated materials are compared. Laser heating is modelled as an internal heat source, whose capacity is given by g(x,t)= I(t)(1−R)μexp(−μx). Analytical solution for the one-dimensional, semi-infinite body with the insulated boundary is obtained using Laplace transforms and the discussion of solutions for different time characteristics of the heat source capacity (constant, instantaneous, exponential, pulsed and periodic) is presented. Received on 18 May 1999     

Heat transfer from convecting-radiating fins of different profile shapes

A finite difference method is used to predict the performance of convecting-radiating fins of rectangular, trapezoidal, triangular, and concave parabolic shapes. The analysis assumes one-dimensional, steady conduction in the fin and neglects radiative exchange between adjacent fins and between the fin and its primary surface. For the range of thermal and geometrical parameters investigated, the variation of heat transfer rate and the fin efficiency with other profile shapes was found to be within 11 percent of the rectangular shape. The effect of profile shape is most pronounced when the Biot number,Bi, and radiation number,N _r, are small compared to unity. Because of several limiting assumptions, the results would be used only for preliminary analysis and design particularly when a fin assembly is involved rather than an individual fin.     

On hyperbolic heat conduction in solids: minimum mean free path of energy carriers and a method of estimating thermal diffusivity and heat propagation characteristics

The paper gives the analytical solution to the one dimensional hyperbolic heat conduction equation in an insulated slab-shaped sample that is heated uniformly on the front face with δ or laser impulse. The solution results in a formula that enables to estimate the minimum mean free path of energy carriers in the sample to detect the second sound (i.e. the thermal wave) at the sample rear face. A method of experimental data evaluation at the second sound effect is proposed, which gives the thermal diffusivity of the sample and the parameters of heat propagation.