On a nonlocal problem modelling Ohmic heating in planar domains

In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

The application of time-domain numerical simulation methods to the microwave heating of foods

Simulation of the microwave processing of food and other materialsin general involves two distinct but interdependent processes.These are the determination of the electromagnetic field profileand the resulting temperature profile as a function of timeinside the load. Both these processes are described in thispaper from the point of view of transmission-line modelling.The techniques described are related to other methods such asthe finite-difference time-domain method. Typical results ofsimulations are presented and outstanding problems are identified.     

Response of moist paperboard during rapid compression and heating

The response of moist paperboard exposed to extensive compression and heating in short periods of time is investigated. A generic framework describing this response, in a thermodynamically consistent manner, has been derived previously. The present paper provides explicit formats of the necessary constitutive relations specific to moist paperboard exposed to extensive compression and heating in short periods of time. The transient transports of mass, momentum and energy, as well as specific interaction terms are considered for orthotropic paperboard. The elasto-plastic response is taken into account in a large strain setting. The exchange of mass between the water bound to the fibers and the water vapor during the sealing is also considered. Simulations of an idealized sealing of two sheets of paperboard are performed and the predicted distributions of temperature, vapor pressure, out-of-plane stress and Forchheimer number are studied. The discussion related to the results from the simulations provides a deeper insight to how the different transport processes will affect the paperboard and how these are coupled. The closed system of equations, including the explicit formats of constitutive relations, provided in this paper makes it possible to set up suitable experiments for validation of the model.     

Buoyancy and chemical reaction effects on MHD mixed convection heat and mass transfer in a porous medium with thermal radiation and Ohmic heating

The combined effect of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid past a vertical permeable surface embedded in a porous medium is analyzed. The heat equation includes the terms involving the radiative heat flux, Ohmic dissipation, viscous dissipation and the internal absorption whereas the mass transfer equation includes the effects of chemically reactive species of first-order. The non-linear coupled differential equations are solved analytically by perturbation technique. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of chemical reaction, thermal stratification and magnetic field. It is observed that the effect of thermal radiation and magnetic field is to decrease the velocity, temperature and concentration profiles in the boundary layer. There is also considerable effect of magnetic field and chemical reaction on skin-friction coefficient and Nusselt number.     

Stress-optimal local heating during welding of unlike plates with variable thicknesses

A method is proposed and is used to solve the problem of determining the stress-optimal temperature fields of the local heating during welding of unlike plates with variable thicknesses. Computational modelling was carried out for plates of M-40 material and St. 3 steel, with thickened edges, depending on the width of the heating zone and the beginning of the region of elastic deformation.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 53–56, 1987.     

Mixed convection magnetohydrodynamic heat and mass transfer past a stretching surface in a micropolar fluid-saturated porous medium under the influence of Ohmic heating,Soret and Dufour effects

A numerical model is developed to examine the combined effects of Soret and Dufour on mixed convection magnetohydrodynamic heat and mass transfer in micropolar fluid-saturated Darcian porous medium in the presence of thermal radiation, non-uniform heat source/sink and Ohmic dissipation. The governing boundary layer equations for momentum, angular momentum (microrotation), energy and species transfer are transformed to a set of non-linear ordinary differential equations by using similarity solutions which are then solved numerically based on shooting algorithm with Runge–Kutta–Fehlberg integration scheme over the entire range of physical parameters with appropriate boundary conditions. The influence of Darcy number, Prandtl number, Schmidt number, Soret number and Dufour number, magnetic parameter, local thermal Grashof number and local solutal Grashof number on velocity, temperature and concentration fields are studied graphically. Finally, the effects of related physical parameters on local Skin-friction, local Nusselt number and local Sherwood number are also studied. Results showed that the fields were influenced appreciably by the Soret and Dufour effects, thermal radiation and magnetic field, etc.     

基于欧姆加热模型的一类非局部双曲方程解的渐近性态

In this paper,the asymptotic behavior of a non-local hyperbolic problem modelling Ohmic heating is studied.It is found that the behavior of the solution of the hyperbolic problem only has three cases:the solution is globally bounded and the unique steady state is globally asymptotically stable;the solution is infinite when t→∞;the solution blows up.If the solution blows up,the blow-up is uniform on any compact subsets of(0,1] and the blow-up rate is lim t → T*-u(x,t)(T*-t)1/α+βp-1=(α+βp-1/1-α)1/1-α-βp,where T* is the blow-up time.     

Change in the properties of impact polystyrene during multiple processing

It has been shown experimentally that the multiple extrusion of grade UP-1E impact polystyrene (5–7 cycles) has almost no effect on most of the physicomechanical properties, the content of volatiles and helium or the residual monomer content. Multiple processing of the same polystyrene by injection molding (10 cycles) causes partial degradation of the material and the destruction of its crosslinked fraction, which results in increased fluidity, reduced molecular weight, and a partial improvement in mechanical properties (approximately up to the fifth or sixth cycle).Mekhanika Polimerov, Vol. 3, No. 1, pp. 156–160, 1967     

Influence of heating on the stability of fiberglass reinforced plastic shells during axial compression

Results are described of the experimental investigation of the patterns of change of carrying capacity and deformation of smooth thin-walled round cylindrical shells made from fiberglass reinforced plastic based on phenol-formaldehyde binder, loaded by an axial compressing force in a temperature field uniform with respect to the thickness of the wall and the surface. The experimental data are compared with calculated ones obtained according to the formulae of the theory of orthotropic shells. The influence of circumferential compressing stresses arising in the junction zone of the heated wall with less heated thicker parts at the edges (ribs) on the carrying capacity of the shells is evaluated.Professor N. E. Zhukovskii Central Aerodynamics Institute, Moscow Oblast. Translated from Mekhanika Polimerov, No. 5, pp. 838–845, September–October, 1972.     

Analytical and numerical analysis of time fractional dual-phase-lag heat conduction during short-pulse laser heating

Numerical Algorithms - In this study, we analytically and numerically investigate the non-Fourier heat conduction behavior within a finite medium based on the time fractional dual-phase-lag model....     

A hybrid technique for computing the power distribution generated in a lossy medium during microwave heating

Over the years researchers in the field of computational electromagnetics (CEM) have investigated and explored a number of different techniques to resolve electromagnetic fields inside waveguide and cavity structures. The equations that govern the fundamental behaviour of electromagnetic wave propagation in such structures are Maxwell's equations. In the literature, a number of different techniques have been employed to solve these equations and out of these methods, the classical finite-difference time-domain (FD-TD) scheme, which uses a staggered time and space discretisation, is the most well-known and widely used. However, this scheme is complicated to implement on an irregular computational domain using unstructured meshes.This research work builds upon previous work undertaken for a waveguide, where a coupled method was introduced for the solution of the governing electromagnetic equations. In that work, the free-space component of the solution was computed in the time-domain, whilst the power distribution in the load was resolved using the frequency dependent electric field Helmholtz equation. This methodology resulted in a time-frequency domain hybrid scheme. In this paper, the hybrid method has been tested further for both waveguide and cavity configurations that are loaded with a lossy dielectric material. Numerical tests highlight both the accuracy and computational efficiency of the proposed hybrid strategy for predicting the power distribution generated during microwave heating processes. The accuracy of the hybrid scheme is gauged by direct comparison with the FD-TD numerical solutions and previously published thermal images.     

Microwave heating of ceramic composites

The microwave heating of a ceramic composite is modelled andanalysed. The composite consists of many small ceramic particlesembedded in a ceramic cement. The composite is assumed to bewell insulated, and each particle is assumed to be in imperfectthermal contact with the surrounding cement. Based on thesetwo assumptions an asymptotic theory exploiting the small Biotnumber and small non-dimensional contact conductance is developed.Our asymptotic theory yields a set of nonlinear partial differentialequations which govern the temperature in the composite. Theseare reduced to a set of coupled nonlinear ordinary differentialequations in which the surface area of each particle entersas a parameter. Recent experiments with such composites haveshown that the steady-state temperature of the composite isstrongly dependent upon the radii of the embedded particles.Our model captures this effect. In fact, our analysis showsthat the assumption of imperfect thermal contact between theparticles and the ceramic cement is essential for this trendto be established.     

Geometric network simulation of high porosity foods

Solid foods are formed of a network of air cells distributed within a solid matrix. Size of air cells and how they are interconnected influence moisture sorption, thermal conductivity, aroma retention and texture of cellular foods. Available geometric pore structure models widely used in petroleum reservoir engineering, soil science, and catalysis are for low porosity (?0.4) materials. However, food materials generally have higher porosities (0.5–0.9). Microstructure of cookie (porosity = 0.54) and bread (porosity = 0.90) samples were simulated using the one-dimensional (1-D) corrugated pore model. Model parameters were: Unit cell dimensions; number of capillary segments, number of diameter sub-ranges, geometric placement of capillary segments in the unit cell; capillary segment length, and the seed number in random number generation. Accurate simulation of porosity depended on number of corrugated pores, and thus the number of diameter sub-ranges in the unit cell. For the cookie sample, a cubic unit cell that was able to accommodate more than a single capillary segment in a unit square and 3 diameter sub-ranges gave realistic results. Simulation of the bread crumb required a similar cubic unit cell arrangement with 12 diameter sub-ranges.     

Optimization of the thermostressed state of piecewise homogeneous glass shells of revolution during thermal processing

We propose a physico-mathematical model and state a problem of stress-optimization of regimes for heating thin piecewise homogeneous glass shells of revolution with given ranges of admissible temperature and stress values by means of an optimal choice of temperature for the surrounding medium. To solve the problem we apply the method of local variations with a specially chosen initial value of the control function and numerical analytic solution of the corresponding direct problem of thermomechanics for the shell in question.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 52–56.     

On the aerodynamic heating of blunt bodies

Zusammenfassung Die Berechnung der Erw?rmung stumpfer Flugk?rper wird für grosse Geschwindigkeiten in Berücksichtigung der übergangserscheinungen er?rtert. Im besondern wird auf das Problem der Abkühlung durch Massentranspiration eingegangen.

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This research was supported by the United States Air Force, through the Air Force Office of Scientific Research of the Air Research and Development Command, under contract AF 18(600)-786.