Nonlinear Hyperbolic Rigid Heat Conductor of the Coleman Type
Titel:
Nonlinear Hyperbolic Rigid Heat Conductor of the Coleman Type
Auteur:
Ignaczak, Jozef Domanski, Wlodzimierz
Verschenen in:
Journal of thermal stresses
Paginering:
Jaargang 31 (2008) nr. 5 pagina's 416-437
Jaar:
2008-05
Inhoud:
A one-dimensional nonlinear hyperbolic homogeneous isotropic rigid heat conductor proposed by Coleman is analyzed using the method of weakly nonlinear geometric optics. For such a model the law of conservation of energy, the dissipation inequality, the Cattaneo's equation, and a generalized energy-entropy relation with a parabolic variation of the energy and entropy along the heat-flux axis, are postulated. First, it is shown that the model can be described by a non-homogeneous quasi-linear hyperbolic matrix partial differential equation of the first order for an unknown vector u = (θ, Q)T, where θ and Q are the dimensionless absolute temperature and heat-flux fields, respectively. Next, the Cauchy problem for the matrix equation with a weakly perturbed initial condition is formulated, and an asymptotic solution to the problem in terms of the amplitudes σα (α = 1, 2) that satisfy a pair of nonlinear first order partial differential equations, is obtained. The Cauchy problem is then solved in a closed form when the initial data are suitably restricted. Numerical examples are included.
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