scholarly journals Non-classical heat equation with singular memory term

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 219-226
Author(s):  
Van Ho
Keyword(s):  
Heat Conduction ◽  
Heat Equation ◽  
Mild Solution ◽  
Solid Mechanics ◽  
Memory Term ◽  
Conduction Theory ◽  
Singular Memory

In this paper, we consider the non-classical heat equation with singular memory term. This equation has many applications in various fields, for example liquids mechanics, solid mechanics, and heat conduction theory first, we prove that the solution exists locally in time. Then we investigate the converegence of the mild solution of non-classical heat equation, and the mild solution of classical heat equation.

Approximate solution of some nonlinear problems of heat conduction theory

1969 ◽  
Vol 17 (5) ◽  
pp. 1420-1423
Author(s):  
A. N. Luppov ◽  
B. G. Ogloblin
Keyword(s):  
Approximate Solution ◽  
Heat Conduction ◽  
Nonlinear Problems ◽  
Conduction Theory

The Lumped Capacitance Model for Unsteady Heat Conduction in Regular Solid Bodies With Natural Convection to Nearby Fluids Engages the Nonlinear Bernoulli Equation

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Antonio Campo
Keyword(s):  
Heat Conduction ◽  
Natural Convection ◽  
Heat Equation ◽  
Biot Number ◽  
Bernoulli Equation ◽  
Convective Coefficient ◽  
Unsteady Heat Conduction ◽  
The Mean ◽  
Capacitance Model ◽  
Solid Bodies

For the analysis of unsteady heat conduction in solid bodies comprising heat exchange by forced convection to nearby fluids, the two feasible models are (1) the differential or distributed model and (2) the lumped capacitance model. In the latter model, the suited lumped heat equation is linear, separable, and solvable in exact, analytic form. The linear lumped heat equation is constrained by the lumped Biot number criterion Bil=h¯(V/S)/ks < 0.1, where the mean convective coefficient h¯ is affected by the imposed fluid velocity. Conversely, when the heat exchange happens by natural convection, the pertinent lumped heat equation turns nonlinear because the mean convective coefficient h¯ depends on the instantaneous mean temperature in the solid body. Undoubtedly, the nonlinear lumped heat equation must be solved with a numerical procedure, such as the classical Runge–Kutta method. Also, due to the variable mean convective coefficient h¯ (T), the lumped Biot number criterion Bil=h¯(V/S)/ks < 0.1 needs to be adjusted to Bil,max=h¯max(V/S)/ks < 0.1. Here, h¯max in natural convection cooling stands for the maximum mean convective coefficient at the initial temperature Tin and the initial time t = 0. Fortunately, by way of a temperature transformation, the nonlinear lumped heat equation can be homogenized and later channeled through a nonlinear Bernoulli equation, which admits an exact, analytic solution. This simple route paves the way to an exact, analytic mean temperature distribution T(t) applicable to a class of regular solid bodies: vertical plate, vertical cylinder, horizontal cylinder, and sphere; all solid bodies constricted by the modified lumped Biot number criterion Bil,max<0.1.

The stationary state and the heat equation for a variant of Davies' model of heat conduction

1985 ◽  
Vol 38 (5-6) ◽  
pp. 1051-1070 ◽  
Author(s):  
R. Artuso ◽  
V. Benza ◽  
A. Frigerio ◽  
V. Gorini ◽  
E. Montaldi
Keyword(s):  
Heat Conduction ◽  
Heat Equation ◽  
Stationary State

scholarly journals Universality in heat conduction theory: weakly nonlocal thermodynamics

2012 ◽  
Vol 524 (8) ◽  
pp. 470-478 ◽  
Author(s):  
P. Ván ◽  
T. Fülöp
Keyword(s):  
Heat Conduction ◽  
Conduction Theory

Investigation on the Use of a Spacetime Formalism for Modeling and Numerical Simulations of Heat Conduction Phenomena

2020 ◽  
Vol 45 (3) ◽  
pp. 223-246
Author(s):  
Roula Al Nahas ◽  
Alexandre Charles ◽  
Benoît Panicaud ◽  
Emmanuelle Rouhaud ◽  
Israa Choucair ◽  
...  
Keyword(s):  
Heat Conduction ◽  
Heat Equation ◽  
Numerical Simulations ◽  
Numerical Comparison ◽  
Thermal Models ◽  
Frame Indifference ◽  
Thermomechanical Couplings ◽  
Covariance Principle

AbstractThe question of frame-indifference of the thermomechanical models has to be addressed to deal correctly with the behavior of matter undergoing finite transformations. In this work, we propose to test a spacetime formalism to investigate the benefits of the covariance principle for application to covariant modeling and numerical simulations for finite transformations. Several models especially for heat conduction are proposed following this framework and next compared to existing models. This article also investigates numerical simulations using the heat equation with two different thermal dissipative models for heat conduction, without thermomechanical couplings. The numerical comparison between the spacetime thermal models derived in this work and the corresponding Newtonian thermal models, which adds the time as a discretized variable, is also performed through an example to investigate their advantages and drawbacks.

scholarly journals Blow up and globality of solutions for a nonautonomous semilinear heat equation with Dirichlet condition

2019 ◽  
Vol 53 (1) ◽  
pp. 57-72
Author(s):  
Marcos Josías Ceballos-Lira ◽  
Aroldo Pérez
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Mild Solution ◽  
Blow Up ◽  
Local Existence ◽  
Dirichlet Condition ◽  
Infinite Time ◽  
Diffusion Operator ◽  
Intrinsic Ultracontractivity ◽  
Semilinear Heat Equation

In this paper we prove the local existence of a nonnegative mild solution for a nonautonomous semilinear heat equation with Dirichlet condition, and give sucient conditions for the globality and for the blow up infinite time of the mild solution. Our approach for the global existence goes back to the Weissler's technique and for the nite time blow up we uses the intrinsic ultracontractivity property of the semigroup generated by the diffusion operator.

Algebraically Explicit Analytical Solutions of Unsteady Conduction With Variable Thermal Properties in Spheroidal Coordinates

2004 ◽  
Author(s):  
Ruixian Cai ◽  
Na Zhang
Keyword(s):  
Thermal Conductivity ◽  
Thermal Properties ◽  
Heat Conduction ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Spherical Coordinates ◽  
Rectangular Coordinates ◽  
Conduction Theory ◽  
Unsteady Heat Conduction ◽  
Spheroidal Coordinates

The analytical solutions of unsteady heat conduction with variable thermal properties (thermal conductivity, density and specific heat are functions of temperature or coordinates) are meaningful in theory. In addition, they are very useful to the computational heat conduction to check the numerical solutions and to develop numerical schemes, grid generation methods and so forth. Such solutions in rectangular coordinates have been derived by the authors; some other solutions for unsteady point symmetrical heat conduction in spherical coordinates are given in this paper to promote the heat conduction theory and to develop the relative computational heat conduction.

Microbeam Beam Heating Analysis of Thin Foils Using Heat Conduction Theory

2009 ◽  
Author(s):  
B. Lovelace ◽  
A. W. Haberl ◽  
H. Bakhru ◽  
J. C. Kimball ◽  
R. E. Benenson ◽  
...  
Keyword(s):  
Heat Conduction ◽  
Thin Foils ◽  
Beam Heating ◽  
Conduction Theory
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