Abstract

Analytic solutions for cylindrical thermal waves in solid medium are given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation.We consider the original non-linear hyperbolic system itself with the self-similar Ansatz for the temperature distribution and for the heat flux. As results continuous.

Highlights

  • Analytic solutions for cylindrical thermal waves in solid medium are given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations

  • The desired local character can be restored with the Taylor expansion of q by time which is usually truncated at the first order namely q(x,t) +τ ∂q(x,t) = −κ∇T (x,t). ∂ t. This is the well-known Cattaneo heat conduction law [6] the second term on the left hand side is known as the "thermal intertia"

  • Our model is presented to describe the heat conduction of any kind of solid state without additional restrictions, room or even higher temperature can be considered with large negative ω exponents. Even from these examples we can see that it has a need to investigate the general heat conduction problem, where the coefficients have general power law dependence

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Summary

Introduction

Analytic solutions for cylindrical thermal waves in solid medium are given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence From such laws one cannot form a second order parabolic or telegraph-type equation. In contemporary heat transport theory (ever since Maxwell’s paper [1]) it is widely accepted in the literature that only for stationary and weakly non-stationary temperature fields the constitutive equation assumes that a temperature gradient ∇T instantaneously produces heat flux q according to the Fourier law Combining this equation with the energy conservation law the usual parabolic heat conduction equation is given. This is the well-known Cattaneo heat conduction law [6] the second term on the left hand side is known as the "thermal intertia" Combining this constitutive equation with the energy conservation yields the hyperbolic telegraph heat conduction equation where τ and are constants. J Generalized Lie Theory Appl S2: 010. doi:10.4172/1736-4337.S2-010

Our starting point is the following qt
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