Solution of the boundary value problem for the Navier-Stokes equation for the flow of a gaseous medium past a heated spheroid

We find the solution of the Navier-Stokes equation linearized with respect to the velocity with regard of a power-law dependence of the molecular transport coefficients (viscosity and heat conductivity) and the gaseous medium density on the temperature. The uniqueness of the solution is proved.     

On the Analytic Solutions of a Special Boundary Value Problem for a Nonlinear Heat Equation in Polar Coordinates

The paper addresses a nonlinear heat equation (the porous medium equation) in the case of a power-law dependence of the heat conductivity coefficient on temperature. The equation is used for describing high-temperature processes, filtration of gases and fluids, groundwater infiltration, migration of biological populations, etc. The heat waves (waves of filtration) with a finite velocity of propagation over a cold background form an important class of solutions to the equation under consideration. A special boundary value problem having solutions of such type is studied. The boundary condition of the problem is given on a sufficiently smooth closed curve with variable geometry. The new theorem of existence and uniqueness of the analytic solution is proved.     

Analytical solutions for movement of cold water thermal front in a heterogeneous geothermal reservoir

In the present study an analytical model has been presented to describe the transient temperature distribution and advancement of the thermal front generated due to the reinjection of heat depleted water in a heterogeneous geothermal reservoir. One dimensional heat transport equation in porous media with advection and longitudinal heat conduction has been solved analytically using Laplace transform technique in a semi infinite medium. The heterogeneity of the porous medium is expressed by the spatial variation of the flow velocity and the longitudinal effective thermal conductivity of the medium. A simpler solution is also derived afterwards neglecting the longitudinal conduction depending on the situation where the contribution to the transient heat transport phenomenon in the porous media is negligible. Solution for a homogeneous aquifer with constant values of the rock and fluid parameters is also derived with an aim to compare the results with that of the heterogeneous one. The effect of some of the parameters involved, on the transient heat transport phenomenon is assessed by observing the variation of the results with different magnitudes of those parameters. Results prove the heterogeneity of the medium, the flow velocity and the longitudinal conductivity to have great influence and porosity to have negligible effect on the transient temperature distribution.     

Inverse determination of thermal conductivity using semi-discretization method

In this work a semi-discretization method is presented for the inverse determination of spatially- and temperature-dependent thermal conductivity in a one-dimensional heat conduction domain without internal temperature measurements. The temperature distribution is approximated as a polynomial function of position using boundary data. The derivatives of temperature in the differential heat conduction equation are taken derivative of the approximated temperature function, and the derivative of thermal conductivity is obtained by finite difference technique. The heat conduction equation is then converted into a system of discretized linear equations. The unknown thermal conductivity is estimated by directly solving the linear equations. The numerical procedures do not require prior information of functional form of thermal conductivity. The close agreement between estimated results and exact solutions of the illustrated examples shows the applicability of the proposed method in estimating spatially- and temperature-dependent thermal conductivity in inverse heat conduction problem.     

A superstatistical model for anomalous heat conduction and diffusion

We propose a superstatistical model for anomalous heat conduction and diffusion, which is formulated by the thermal conductivity distribution, overall temperature and heat flux distributions. Our model obeys Fourier's law and the continuity equation at the individual level. The evolution of the thermal conductivity distribution is described by an advection-diffusion equation. We show that the superstatistical model predict anomalous behaviors including the time-dependent effective thermal conductivity and slow long-time asymptotics. The time-dependence of the effective thermal conductivity is determined by the mean square displacement (MSD), which coincides with existing investigations. The superstatistical structure can also be extended into other non-Fourier models including the Cattaneo and fractional-order heat conduction models.     

Similarity analysis of a nonlinear fin equation

A nonlinear fin equation in which the thermal conductivity is an arbitrary function of the temperature and the heat transfer coefficient is an arbitrary function of a spatial variable is considered. Scaling, translational and spiral group symmetries of the equations are determined. Classification of the functions for which these symmetries exist is performed. In general, no useful symmetries exist for arbitrary thermal conductivity and heat transfer coefficients. However, for some restricted forms of the functions, useful symmetries exist. A similarity transformation is used to reduce the partial differential equation to an ordinary differential equation as an example.     

Thermal stress state of a bimaterial with a closed interfacial crack having rough surfaces

We have formulated the problem of thermoelasticity for a bimaterial whose components differ only in their shear moduli, with a closed interfacial crack having rough surfaces. The bimaterial is subjected to the action of compressive loads and heat flow normal to the interfacial surface. We have taken into account the dependence of thermal conductance of the defect on the contact pressure of its faces and heat conductivity of the medium that fills it. The problem is reduced to a Prandtl-type nonlinear singular integro-differential equation for temperature jump between the crack surfaces. An analytical solution of this problem has been constructed for the case of action of the heat flow only. We have analyzed the dependence of contact pressure of the defect faces, temperature jump between them, and the intensity factor of tangential interfacial stresses on the value of given heat flow, roughness of the surfaces, and ratio between the shear moduli of joined materials.     

Application of Laguerre Transform on heat conduction problem

Summary In this paper the author has utilised the Laguerre transform introduced by him in an earlier paper (1960), to solve the heat equation for one-dimensional linear flow of heat in a very long non-homogenous bar of prescribed thermal conductivity under certain boundary conditions, the source of heat is being taken within the medium.

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Riassunto In questa Nota l'autore si serve della trasformaziones di Laguerre [da lui stesso usata in un suo precedente lavoro (1960)] per risolvere l'equazione del calore nei caso in cui esso fluisce lungo una sola direzione lineare attraverso una lunghissima sbarra, non omogenea, di data conduttività termica, e ciò sotto certe condizioni. Si suppone che la sorgente di calore si trovi entro il mezzo considerato.

     

Determination of the coefficient in the stationary nonlinear equation of heat conduction

The article considers the problem of determining the solution-dependent coefficient of heat conductivity in a stationary nonlinear equation of heat conduction containing a parameter. Additional information for the determination of heat conductivity is provided by a function dependent on a parameter, which is obtained by solving a boundary-value problem. A uniqueness theorem is proved for the inverse problem.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 13–17, 1993.     

A local iteration scheme for nonlinear two-dimensional steady-state heat conduction: a BEM approach

An efficient algorithm is proposed to solve the steady-state nonlinear heat conduction equation using the boundary element method (BEM). Nonlinearity of the heat conduction equation arises from nonlinear boundary conditions and temperature dependence of thermal conductivity. Using Kirchhoff's transformation, the case of temperature dependence of thermal conductivity can be transformed to the nonlinear boundary conditions case. Applying the BEM technique, the resulting matrix equation becomes nonlinear. The nonlinearity, however, only involves the boundary nodes that have nonlinearboundary conditions. The proposed local iterative scheme reduces the entire BEM matrix equation to a smaller matrix equation whose rank is the same as the number of boundary nodes with nonlinear boundary conditions. The Newton-Raphson iteration scheme is used to solve the reduced nonlinear matrix equation. The local iterative scheme is first applied to two one-dimensional problems (analytical solutions are possible) with different nonlinear boundary conditions. It is then applied to a two-region problem. Finally, the local iterative scheme is applied to two cavity problems in which radiation plays a role in the heat transfer.     

Propagation of nonlinear magnetoacoustic waves in a compressible medium of finite electrical conductivity

We study the one-dimensional propagation of weakly nonlinear waves in a compressible medium of finite electrical conductivity subjected to the action of a magnetic field. We obtain evolution equations that describe the wave processes under small and finite magnetic Reynolds numbers. It is shown that in a medium of finite conductivity the evolution of perturbations in a fluid is described by the modified Bürger's equation. We find the stationary and automodel solution of this equation and use them as the basis for analyzing the influence of effects of electrical conductivity on the structure of weak shock waves.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 73–76.     

Thermoelastic contact of half-spaces with equal thermal distortivities in the presence of a heat-permeable intersurface gap

We consider the interaction of elastic half-spaces with equal thermal distortivities in the presence of a heat-permeable medium in an intercontact gap caused by a recess on the surface of one of the bodies. Outside the gap, a perfect thermal and frictionless mechanical contact takes place between the bodies. Using the method of functions of intercontact gaps, the formulated contact problem is reduced to a singular integral equation for a derivative of the height of the gap, which is solved analytically, and to a Prandtl-type singular integro-differential equation for the difference of temperature of the surfaces in the region of the gap, for the solution of which we propose an analytic-numerical approach. Plots illustrate the influence of load and the thermal conductivity of the filler on the temperature difference between the edges of the gap, contact stresses, heat flows, and longitudinal strains between the half-spaces.     

Analytic solutions to heat transfer problems on a basis of determination of a front of heat disturbance

With the use of additional boundary conditions in integral method of heat balance, we obtain analytic solution to nonstationary problem of heat conductivity for infinite plate. Relying on determination of a front of heat disturbance, we perform a division of heat conductivity process into two stages in time. The first stage comes to the end after the front of disturbance arrives the center of the plate. At the second stage the heat exchange occurs at the whole thickness of the plate, and we introduce an additional sought-for function which characterizes the temperature change in its center. Practically the assigned exactness of solutions at both stages is provided by introduction on boundaries of a domain and on the front of heat perturbation the additional boundary conditions. Their fulfillment is equivalent to the sought-for solution in differential equation therein. We show that with the increasing of number of approximations the accuracy of fulfillment of the equation increases. Note that the usage of an integral of heat balance allows the application of the given method for solving differential equations that do not admit a separation of variables (nonlinear, with variable physical properties etc.).     

CFD predictions for cement kilns including flame modelling,heat transfer and clinker chemistry

Clinker formation in coal-fired rotary cement kilns under realistic operation conditions has been modelled with a commercial axisymmetric CFD code for the gaseous phase including a Monte Carlo method for radiation, a finite-volume code for the energy equation in the kiln walls, and a novel code for the species and energy conservation equations, including chemical reactions, for the clinker. An iterative procedure between the predictions for the temperature field of the gaseous phase, the radiative heat flux to the walls, and the kiln and clinker temperature is used to predict the distribution of the inner wall temperature explicitly, including the calculation of heat flow to the clinker. It was found that the dominant mode of heat transfer between the gas and the kiln walls is by radiation and that the heat lost through the refractories to the environment is about 10% of the heat input and a further 40% is used for charge heating and clinker formation. The predictions are consistent with trends based on experience and limited measurements in a full-scale cement kiln.     

Effective Boundary Conditions for the Heat Equation with Interior Inclusion

Of concern is the scenario of a heat equation on a domain that contains a thin layer, on which the thermal conductivity is drastically different from that in the bulk. The multi-scales in the spatial variable and the thermal conductivity lead to computational difficulties, so we may think of the thin layer as a thickless surface, on which we impose "effective boundary conditions"(EBCs). These boundary conditions not only ease the computational burden, but also reveal the effect of the inclusion. In this paper, by considering the asymptotic behavior of the heat equation with interior inclusion subject to Dirichlet boundary condition, as the thickness of the thin layer shrinks, we derive, on a closed curve inside a two-dimensional domain, EBCs which include a Poisson equation on the curve, and a non-local one. It turns out that the EBCs depend on the magnitude of the thermal conductivity in the thin layer,compared to the reciprocal of its thickness.     

An analytical solution for the unidirectional solidification problem

The extraction of heat from a molten casting is resisted by an imperfect thermal contact at the mold-casting interface. The nature of the contact varies throughout the casting process and has the effect of increasing the thermal resistance at the interface. This can be modelled by incorporating a gaseous gap at the mold-casting interface that grows with increasing time.

This paper is concerned with an analytical solution of the unidirectional solidification problem, which incorporates movement of the casting at the interface. The derivation of the analytical solution requires the simultaneous solution of the transient heat equations, for the mold, gaseous gap, and solid and liquid parts of the melt. The analytical solution is extended so that contamination layers on the mold and casting can be incorporated as well as an initial gap. This is achieved by introducing virtual layers of mold, gas, and casting. Using the extended solution, the effects of interfacial resistance, air conductivity, and gap variation on solidification rates are examined.     

Calculation of deformative and thermal properties of multiphase composite materials filled with composite or hollow spherical inclusions by the effective medium method

A theoretical investigation was carried out to examine the possibilities of a structural approach to prediction of elastic constants, creep functions, and thermal properties of multiphase polymer composite materials filled with composite or hollow spherical Inclusions of several types. The problem of determining effective properties of the composite was solved by generalizing the effective medium method, a variant of the self-consistent method, for the case of a four-phase kernel-shell-matrix-equivalent homogeneous medium model. Exact analytical expressions for the bulk modulus thermal expansion coefficient, thermal conductivity coefficient, and specific heat were obtained. The solution for the shear modulus is given in the form of a nonlinear equation whose coefficients are the solution of a system of 12 linear equations.To be presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 462–472, July–August, 1995.     

Symmetry analysis of a heat conduction model for heat transfer in a longitudinal rectangular fin of a heterogeneous material

We consider a heat conduction model arising in transient heat transfer through longitudinal fins of a heterogeneous (functionally graded) material. In this case, the thermal conductivity depends on the spatial variable. The heat transfer coefficient depends on the temperature and is given by the power law function. The resulting nonlinear partial differential equation is analyzed using both classical and nonclassical symmetry techniques. Both the transient state and the steady state result in a number of exotic symmetries being admitted by the governing equation. Furthermore, nonclassical symmetries are also admitted. Both classical and nonclassical symmetry analysis results in some useful reductions and some remarkable exact solutions are constructed.     

Analytical solutions to nonlinear equations arising in heat transfer by variational iteration,homotopy perturbation,and Adomian decomposition methods

As thermal conductivity plays an important role on fin efficiency, we tried to solve heat transfer equation with thermal conductivity as a function of temperature. In this research, some new analytical methods called homotopy perturbation method, variational iteration method, and Adomian decomposition method are introduced to be applied to solve the nonlinear heat transfer equations, and also the comparison of the applied methods (together) is shown graphically. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum

In this paper, we study the well-posedness of the thermal boundary layer equation in two-dimensional incompressible heat conducting flow. The thermal boundary layer equation describes the behavior of thermal layer and viscous layer for the two-dimensional incompressible viscous flow with heat conduction in the small viscosity and heat conductivity limit. When the initial datum are analytic, with respect to the tangential variable of the boundary, and without the monotonicity condition of the tangential velocity, by using the Littlewood-Paley theory, we obtain the local-in-time existence and uniqueness of solution to this thermal boundary layer problem.