Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems.     

Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems. Received: July 14, 2003     

Spatial decay estimates for the Maxwell-Cattaneo equations with mixed boundary conditions

In this paper the authors derive spatial decay bounds for the temperature and heat flux as defined by the Generalized Maxwell-Cattaneo equations for heat conduction in a semi-infinite cylinder when the temperature and the tangential components of the heat flux vector vanish on the lateral surface of the cylinder. The results here supplement those previously found by the authors [5] when the heat flux vector was assumed to be zero on the lateral surface but no condition was imposed on the temperature.Received: February 7, 2002; revised: June 3, 2002     

Temporal decay bounds in generalized heat conduction

In this paper the authors derive exponential decay bounds for the temperature and heat flux as defined by the generalized Maxwell-Cattaneo equations for heat conduction in a bounded region when the temperature and the tangential components of the heat flux vanish on the boundary. They also derive bounds in for the heat flux and temperature when the heat flux is assumed to vanish on the boundary but no boundary condition is imposed on the temperature.     

Spatial decay estimates for the Maxwell-Cattaneo equations with mixed boundary conditions

In this paper the authors derive spatial decay bounds for the temperature and heat flux as defined by the Generalized Maxwell-Cattaneo equations for heat conduction in a semi-infinite cylinder when the temperature and the tangential components of the heat flux vector vanish on the lateral surface of the cylinder. The results here supplement those previously found by the authors [5] when the heat flux vector was assumed to be zero on the lateral surface but no condition was imposed on the temperature.     

Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: A Lie group analysis

In this paper, heat and mass transfer analysis for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/blowing is investigated. The governing partial differential equation and auxiliary conditions are converted to ordinary differential equations with the corresponding auxiliary conditions via Lie group analysis. The boundary layer temperature, concentration and nanoparticle volume fraction profiles are then determined numerically. The influences of various relevant parameters, namely, thermophoresis parameter Nt, Brownian motion parameter Nb, Lewis number Le, suction/injection parameter S, permeability parameter k₁, source/sink parameter λ and Prandtl parameter Pr on temperature and concentration as well as wall heat flux and wall mass flux are discussed. Comparison with published results is presented.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0, 1[^d. The error of an algorithm is defined in L₂-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of nonuniform time discretizations is crucial.     

Optimal Control of a Parabolic Equation with Dynamic Boundary Condition

We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the dynamic boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an L ^p function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.     

On the existence of weak solutions to the equations of steady flow of heat-conducting fluids with dissipative heating

We study a system of partial differential equations describing the steady flow of a heat conducting incompressible fluid in a bounded three dimensional domain, where the right-hand side of the momentum equation includes the buoyancy force. In the present work we prove the existence of a weak solution under both the smallness and a sign condition on physical parameters α₀ and α₁ which appear on the right hand side.     

General formulas for the smoothed analysis of condition numbers

We provide estimates on the volume of tubular neighborhoods around a subvariety Σ of real projective space, intersected with a disk of radius σ. The bounds are in terms of σ, the dimension of the ambient space, and the degree of equations defining Σ. We use these bounds to obtain smoothed analysis estimates for some conic condition numbers. To cite this article: P. Bürgisser et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Explicit solutions for a two-phase unidimensional Lamé-Clapeyron-Stefan problem with source terms in both phases

A two-phase Stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi-infinite phase-change material is studied. We assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x=0, a constant temperature or a heat flux of the form (q₀>0). The internal heat source functions are given by (j=1 solid phase; j=2 liquid phase) where βj=βj(η) are functions with appropriate regularity properties, ρ is the mass density, l is the fusion latent heat by unit of mass, is the diffusion coefficient, x is the spatial variable and t is the temporal variable. We obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x=0. Moreover, the equivalence of the two free boundary problems is also proved. We generalize the solution obtained in [J.L. Menaldi, D.A. Tarzia, Generalized Lamé-Clapeyron solution for a one-phase source Stefan problem, Comput. Appl. Math. 12 (2) (1993) 123-142] for the one-phase Stefan problem. Finally, a particular case where βj (j=1,2) are of exponential type given by βj(x)=exp(−²(x+dj)) with x and dj∈R is also studied in details for both boundary temperature conditions at x=0. This type of heat source terms is important through the use of microwave energy following [E.P. Scott, An analytical solution and sensitivity study of sublimation-dehydration within a porous medium with volumetric heating, J. Heat Transfer 116 (1994) 686-693]. We obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x=0 is considered; a similar result is obtained for a heat flux condition imposed on x=0 if an inequality for parameter q₀ is satisfied.     

Lyapunov, Opial and Beesack inequalities for one-dimensional p(t)-Laplacian equations

We generalize the classical Lyapunov, Opial and Beesack inequalities for one-dimensional differential equations to nonstandard growth p(t)-Laplacian.     

Analysis of dual-phase-lag heat conduction in cylindrical system with a hybrid method

The dual-phase-lag heat transfer model is applied to investigate the transient heat conduction in an infinitely long solid cylinder for an exponentially decaying pulse boundary heat flux and for a short-pulse boundary heat flux. A hybrid application of the Laplace transform method and the control volume scheme is used to obtain the numerical solutions. Comparison between the numerical results and the analytic solution for an exponentially decaying heat flux pulse evidences the accuracy of the present numerical results. Results further show that the present numerical scheme can overcome the mathematical difficulties to analyze such problems. Effects of the thermal lag ratio τ_q/τ_T, the shift time τ_q–τ_T, the function form of heating pulse, and geometry of medium on the behavior of heat transfer are investigated.     

Optimal paths for minimizing entransy dissipation during heat transfer processes with generalized radiative heat transfer law

A common of finite-time heat transfer processes between high- and low-temperature sides with generalized radiative heat transfer law [q ∝ Δ(Tⁿ)] is studied in this paper. In general, the minimization of entropy generation in heat transfer processes is taken as the optimization objective. A new physical quantity, entransy, has been identified as a basis for optimizing heat transfer processes in terms of the analogy between heat and electrical conduction recently. Heat transfer analyses show that the entransy of an object describes its heat transfer ability, as the electrical energy in a capacitor describes its charge transfer ability. Entransy dissipation occurs during heat transfer processes, as a measure of the heat transfer irreversibility with the dissipation related thermal resistance. Under the condition of fixed heat load, the optimal configurations of hot and cold fluid temperatures for minimizing entransy dissipation are derived by using optimal control theory. The condition corresponding to the minimum entransy dissipation strategy with Newtonian heat transfer law (n = 1) is that corresponding to a constant heat flux rate, while the condition corresponding to the minimum entransy dissipation strategy with the linear phenomenological heat transfer law (n = −1) is that corresponding to a constant ratio of hot to cold fluid temperatures. Numerical examples for special cases with Newtonian, linear phenomenological and radiative heat transfer law (n = 4) are provided, and the obtained results are also compared with the conventional strategies of constant heat flux rate and constant hot fluid (reservoir) temperature operations and optimal strategies for minimizing entropy generation. Moreover, the effects of heat load changes on the optimal hot and fluid temperature configurations are also analyzed.     

Heat transfer in granular materials: effects of nonlinear heat conduction and viscous dissipation

In this paper, we study the heat transfer in a one‐dimensional fully developed flow of granular materials down a heated inclined plane. For the heat flux vector, we use a recently derived constitutive equation that reflects the dependence of the heat flux vector on the temperature gradient, the density gradient, and the velocity gradient in an appropriate frame invariant formulation. We use two different boundary conditions at the inclined surface: a constant temperature boundary condition and an adiabatic condition. A parametric study is performed to examine the effects of the material dimensionless parameters. The derived governing equations are coupled nonlinear second‐order ordinary differential equations, which are solved numerically, and the results are shown for the temperature, volume fraction, and velocity profiles. Copyright © 2013 John Wiley & Sons, Ltd.     

Heat kernel bounds and desingularizing weights

We study the parabolic operator ∂_t−Δ_x+V(t,x), in d?1, with a potential V=V+−V−,V±?0 assumed to be from a parabolic Kato class, and obtain two-sided Gaussian bounds on the associated heat kernel. The constraints on the Kato norms of V⁺ and V⁻ are completely asymmetric, as they should be. Further improvements to our heat kernel bounds are obtained in the case of time-independent potentials.If V has singularities of the type ±c|x|⁻² with a suitably small constant c, we obtain new lower and (sharp) upper weighted heat kernel bounds. The rate of growth of the weights depends (explicitly) on the constant c. The standard bounds and methods (estimates in L^p-spaces without desingularizing weights) fail for singular potentials.