Nonlinear hyperbolic theory of thermal waves in metals

Abstract

A closed-form solution for cylindrical thermal waves in metals is given based on the nonlinear hyperbolic system of energy conservation and heat flux relaxation equations. It is shown that heat released from a line source propagates radially outward with finite speed in the form of a thermal wave which exhibits a discontinuous wave front. Unique nonlinear thermal wave solutions exist up to a critical amount of driving energy, i.e., for larger energy releases the thermal flow becomes multivalued (occurrence of shock waves). By comparison, it is demonstrated that the parabolic thermal wave theory gives in general a misleading picture of the profile and propagation of thermal waves, and leads to physical (infinite speed of heat propagation) and mathematical (divergent energy integrals) difficulties. Attention is drawn to the importance of temporal heat flux relaxation for the physical understanding of fast, transient processes, such as thermal waves, and more general explosions and implosions.