A Stefan problem for a non-classical heat equation with a convective condition

We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a convective boundary condition at the fixed face x = 0. Here the heat source depends on the temperature at the fixed face x = 0 that provides a heating or cooling effect depending on the properties of the source term. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations. We also obtain a comparison result of the solution (the temperature and the free boundary) with respect to the one corresponding with null source term.     

Numerical investigation of natural convection in a rectangular enclosure due to partial heating and cooling at vertical walls

A comprehensive numerical investigation on the natural convection in a rectangular enclosure is presented. The flow is induced due to the constant partial heating at lower half of the left vertical wall and partial cooling at upper half of the right vertical wall along with rest walls are adiabatic. In this investigation the Special attention is given to understand the effect of aspect ratio and heat source intensity i.e. Rayleigh number, Ra, on the fluid flow configuration as well as on the local and average heat transfer rates. The range of Rayleigh (Ra) and aspect ratio (A) is taken [10³, 10⁶] and [0.5, 4] respectively. The results are presented in terms of stream function (ψ), temperature (θ) and heat transfer rates (local Nusselt numbers Nu_L, and average Nusselt numbers Nu). The numerical experiments show that increasing of Ra implies the enhancement of thermal buoyancy force, which in turn increases the thermal convection in the cavity. As a result, the local as well as average heat transfer rate is expected to increase. The local transfer rate (Nu_L) is increases in the small region near the left vertical wall of the left wall of the cavity and after that it is decreases in the middle portion of heated region. And, it start to increase near to the middle point of left wall. It is also observed that the local heat transfer is increases as increases the aspect ratio. The average heat transfer rate (Nu) is increases as the aspect ratio A increases from 0.5 to 1 and beyond that it is decreases smoothly. It is also found that the heat transfer rate attains its maximum value at aspect ratio one.     

Determination of heat transfer properties of media with a single-needle probe

Needle probes with a line heater inside are often used in studying the heat transfer properties of loose rocks. The key problem of contact methods of measuring thermal properties of various media consists in finding thermal contact resistance at the probe/medium interface which must be taken into account in determining the thermal diffusivity of the medium. We describe a mathematical model of heating of a long needle probe in the medium under study, taking into account dimensions and thermal properties of the needle source and assuming that thermal contact between the source and the medium is not ideal. Based on the proposed model, we formulate and solve the inverse problem of finding the thermal diffusivity coefficient of the medium and the heat exchange coefficient at the probe/medium interface. The purpose of the article is to create methodology for determining thermal properties of various media in the field.     

Mathematical framework for predicting solar thermal build-up of spectrally selective coatings at the Earth’s surface

A transient finite element thermal model is formulated valid for surface coatings on any substrate material and based on the continuum conduction equations with solar loading as a heat source. The model allows cooling to be applied at outer surfaces of the body, by natural convection and accounts for ambient radiative heat loss. Hemispherical spectral reflectivities are obtained for various polymer-based coatings on a steel substrate using spectrophotometers in the 0.1 μm to 25 μm wavelengths. A time-dependent solar irradiation energy source (blackbody equivalent) is applied to an object with spectrally diffuse outer surfaces, and the incoming heat flux is split by a band approximation into reflected and absorbed energy and finally integrated over the complete spectrum to provide thermal source terms for the finite element model.     

Cooling of a composite slab in a two-fluid medium

A mixed boundary value problem associated with the diffusion equation that involves the physical problem of cooling of an infinite parallel-sided composite slab in a two-fluid medium, is solved completely by using the Wiener-Hopf technique. An analytical solution is derived for the temperature distribution at the quench fronts being created by two different layers of cold fluids having different cooling abilities moving on the upper surface of the slab at constant speedv. Simple expressions are derived for the values of the sputtering temperatures of the slab at the points of contact with the respective layers, assuming the front layer of the fluid to be of finite width and the back layer of infinite extent. The main problem is solved through a three-part Wiener-Hopf problem of a special type and the numerical results under certain special circumstances are obtained and presented in the form of a table.     

Global existence for a nonlocal model for adhesive contact

Abstract

In this paper, we address the analytical investigation into a model for adhesive contact introduced in a paper by Freddi and Fremond, which includes nonlocal sources of damage on the contact surface, such as the elongation. The resulting PDE system features various nonlinearities rendering the unilateral contact conditions, the physical constraints on the internal variables, as well as the contributions related to the nonlocal forces. For the associated initial-boundary value problem, we obtain a global-in-time existence result by proving the existence of a local solution via a suitable approximation procedure and then by extending the local solution to a global one by a nonstandard prolongation argument.     

Free convection from a point heat source in a stratified fluid

A non-stationary problem of free convection from a point heat source in a stratified fluid is considered. The system of equations is reduced to a single equation for a special scalar function which determinos the velocity field, and the temperature and salinity distribution. Relations are found connecting the spatial and temporal scales of the phenomenon with the parameters of the medium and the intensity of the heat source. The magnitude of the critical source intensity at which the fluid begins to move in a jet-flow mode is established.The structure of convective flows above the heat sources depends, in the stratified media, essentially on the nature of the stratification /1/ which may be caused by a change in the temperature of the medium /2, 3/ or its salinity /4–7/, and by the form of the heat source. When a temperature gradient exists within the medium, an ascending jet forms above the point source, mushrooming outwards near the horizon of the hydrostatic equilibrium. In the case of a fluid with salinity gradient, the jet is surrounded by a sheet of descending salty fluid, and a regular system of annular convective cells is formed around it /1/.The height of the stationary jet computed in /2, 3/ on the basis of conservative laws agrees with experiment. However, this approach does not enable the temperature and velocity distribution over the whole space to be found and does not enable the problem of determining the flow to be investigated. A stationary solution of the linearized convection equations /8/ does not correspond to detail to the observed flow pattern /1, 5–7/. In this connection the study of the non-linear, non-stationary convection equations is of interest.The purpose of this paper is to construct a non-linear, non-stationary free convection equation above a point heat source, and to analyse the scales of the resulting structure and the critical conditions under which the flow pattern changes.     

Linear stability of a thin non-isothermal droplet spreading on a rotating disk

Previous works show that the linear stability of the contact line of an isothermal spreading droplet depends on the base state curvature of the free interface at the position of the unperturbed contact line [1]. Here the linear stability of a thin liquid droplet on a rotating disk is investigated, where the disk has a certain radial temperature profile and the ambient passive gas a constant temperature. It is shown that two different Marangoni effects influence the spreading beside centrifugal, gravitational, and capillary effects. The stability analysis prevails, that temperature gradients in the disk amplify some and damp other modes, while cooling the entire disk (or heating the gas) damps the growth rate of all modes and therefore seems to be an appropriate way to suppress the instability. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

公众参与视角下的央地环境规制博弈分析

本文首先构建环境规制中中央政府和地方政府的两方演化博弈模型,并在此基础上将公众作为第三方参与主体,构建中央政府、地方政府和公众三方演化博弈模型,详细比较两方和三方博弈模型的区别,探究各个主体策略行为的影响因素。研究发现:(1)未有公众参与下,地方政府策略选择主要受地方政府积极执行成本、环境收益、经济损失,消极执行的环境政绩损失,中央政府监管力度、治理补贴和对地方政府的处罚等因素影响;中央政府的监管策略主要受到严格监管的成本以及对地方政府的治理补贴和处罚等因素影响。引入公众参与后,在央地两方博弈的基础上,地方政府环境规制执行策略的影响因素还增加了地方政府被举报后所受到的追加处罚,中央政府监管策略的影响因素还增加了中央政府监管力度、对地方政府的追加处罚以及中央政府的公信力损失。(2)未有公众参与下,中央政府严格监管的概率随地方政府积极执行概率的增大而减小。引入公众参与后,中央政府严格监管率随地方政府积极执行概率的增大而增大。说明在公众参与下,地方政府积极执行环境规制对中央政府严格监管产生的抑制作用转变成了促进作用。(3)地方政府积极执行的概率、中央政府严格监管的概率都随公众举报概率的增大而增大。说明公众参与不仅对地方政府承担环保责任具有促进作用,而且有利于促使中央政府落实环境治理政策。     

Dynamical Symmetry Breaking in Geometrodynamics

Using a first-order perturbative formulation, we analyze the local loss of symmetry when a source of electromagnetic and gravitational fields interacts with an agent that perturbs the original geometry associated with the source. We had proved that the local gauge groups are isomorphic to local groups of transformations of special tetrads. These tetrads define two orthogonal planes at every point in space–time such that every vector in these local planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. Because the local gauge symmetry in Abelian or even non-Abelian field structures in four-dimensional Lorentzian space–times is manifested by the existence of local planes of symmetry, the loss of symmetry is manifested by a tilt of these planes under the influence of an external agent. In this strict sense, the original local symmetry is lost. We thus prove that the new planes at the same point after the tilting generated by the perturbation correspond to a new symmetry. Our goal here is to show that the geometric manifestation of local gauge symmetries is dynamical. Although the original local symmetries are lost, new symmetries arise. This is evidence for a dynamical evolution of local symmetries. We formulate a new theorem on dynamical symmetry evolution. The proposed new classical model can be useful for better understanding anomalies in quantum field theories.     

ON LOCAL STRUCTURAL STABILITY OF ONE-DIMENSIONAL SHOCKS IN RADIATION HYDRODYNAMICS

In this paper, we are concerned with the local structural stability of one-dimensional shock waves in radiation hydrodynamics described by the isentropic Euler-Boltzmann equations. Even though in this radiation hydrodynamics model, the radiative effects can be understood as source terms to the isentropic Euler equations of hydrodynamics, in general the radiation field has singularities propagated in an angular domain issuing from the initial point across which the density is discontinuous. This is the major difficulty in the stability analysis of shocks. Under certain assumptions on the radiation parameters, we show there exists a local weak solution to the initial value problem of the one dimensional Euler-Boltzmann equations, in which the radiation intensity is continuous, while the density and velocity are piecewise Lipschitz continuous with a strong discontinuity representing the shock-front. The existence of such a solution indicates that shock waves are structurally stable, at least local in time, in radiation hydrodynamics.     

A solution method for the sub-surface stresses and local deflection of a semi-infinite inhomogeneous elastic medium

This paper proposes analytical Fourier series solutions (based on the Airy stress function) for the local deflection and subsurface stress field of a two-dimensional graded elastic solid loaded by a pre-determined pressure distribution. We present a selection of numerical results for a simple sinusoidal pressure which indicates how the inhomogeneity of the solid affects its behaviour. The model is then adapted and used to derive an iterative algorithm which may be used to solve for the contact half width and pressure induced from contact with a rigid punch. Finally, the contact of a rigid cylindrical stamp is studied and our results compared to those predicted by Hertzian theory. It is found that solids with a slowly varying elastic modulus produce results in good agreement with those of Hertz whilst more quickly varying elastic moduli which correspond to solids that become stiffer below the surface give rise to larger maximum pressures and stresses whilst the contact pressure is found to act over a smaller area.     

On final motions of a Chaplygin ball on a rough plane

A heavy balanced nonhomogeneous ball moving on a rough horizontal plane is considered. The classical model of a “marble” body means a single point of contact, where sliding is impossible. We suggest that the contact forces be described by Coulomb’s law and show that in the final motion there is no sliding. Another, relatively new, contact model is the “rubber” ball: there is no sliding and no spinning. We treat this situation by applying a local Coulomb law within a small contact area. It is proved that the final motion of a ball with such friction is the motion of the “rubber” ball.     

Recovering unknown terms in a nonlinear boundary condition for Laplace's equation

^** Email: gabriele{at}fi.iac.cnr.it We consider a thin metallic plate whose top side is inaccessibleand in contact with a corroding fluid. Heat exchange betweenmetal and fluid follows linear Newton's cooling law as longas the inaccessible side is not damaged. We assume that theeffects of corrosion are modelled by means of a nonlinear perturbationin the exchange law. On the other hand, we are able to heatthe conductor and take temperature maps of the accessible side.Our goal is to recover the nonlinear perturbation of the exchangelaw on the top side from thermal data collected on the oppositeone (thermal imaging). In this paper, we use a stationary model,i.e. the temperature inside the plate is assumed to fulfil Laplace'sequation. Hence, our problem is stated as an inverse ill-posedproblem for Laplace's equation with nonlinear boundary conditions.We study identifiability and local Lipschitz stability. In particular,we prove that the nonlinear term is identified by one Cauchydata set. Moreover, we produce approximated solutions by meansof an optimizational method.     

Heat transfer in sliding of a plane-parallel layer over a base

An analytic solution of the thermal problem of friction for a plane-parallel layer–base tribosystem under conditions of incomplete thermal contact between contacting bodies is obtained. Asymptotics of the obtained solution for small and large values of time are determined. For the materials of a cermet layer–iron base friction pair, we investigate the influence of the thermal conductivity coefficient of the contact on the temperature distribution and intensity of a heat fluxes.     

Determination of coherence length from speckle contrast on a rough surface

Conclusions The speckle pattern in the image of a diffusely scattering plane surface illuminated by two mutually inclined quasi plane waves split from a common laser source gives a direct display of the coherence properties of that light source. This can be used as a simple device to check the coherence of a laser source for holographic or interferometric work. The speckle contrast is a linear function of the square modulus of the degree of coherence. This relation has been proved experimentally for coherent and for incoherent laser radiation. The main difference of the intensity distributions for coherent and incoherent illumination occurs at low intensities, which have maximum probability in the coherent and minimum probability in the incoherent case. The intensity probability distributions have been determined experimentally for two limiting cases. Within the experimental limitations they show very good agreement with the theoretical predictions.     

Numerical analysis of a bilateral frictional contact problem for linearly elastic materials

We consider a mathematical model which describes the contactbetween a linearly elastic body and an obstacle, the so-calledfoundation. The process is quasistatic and the contact is bilateral,i.e. there is no loss of contact during the process. The frictionis modelled with Tresca's law. The variational formulation ofthe problem is a nonlinear evolutionary inequality for the displacementfield which has a unique solution under certain assumptionson the given data. We study spatially semi-discrete and fullydiscrete schemes for the problem with finite-difference discretizationin time and finite-element discretization in space. The numericalschemes have unique solutions. We show the convergence of thescheme under the basic solution regularity. Under appropriateregularity assumptions on the solution, we derive optimal ordererror estimates. Finally, we present numerical results in thestudy of two-dimensional test problems.