Stability and wave motion problems in continuous media with second sound

Abstract

In this thesis we investigate thermal convection and wave motion in models of second sound such as the Cattaneo model, Green and Laws model, Batra model, and Green and Naghdi model. For the Green and Laws and Batra models we also investigate questions of stability and uniqueness. The term second sound means the transport of heat as a thermal wave. The models are all presented within the framework of continuum mechanics. We use a mathematical technique involving an acceleration wave to solve some problems. Furthermore, in one of the chapters we use a numerical method, namely a D(^2) Chebyshevtau method to find eigenvalues of a thermal convection problem. This technique is a highly accurate method. In Chapter two we study thermal convection with the Cattaneo model. The model is about thermal convection in a layer of fluid heated from below. We also employ D(^2) Chebyshev tau method to obtain numerical results for the model. In Chapter three we study various properties such as instability and uniqueness of the model of second sound which is derived by Green and Laws. We investigate the model of Green and Laws for which the generalized temperature Φ depends on θ and θ. We also show differences between the results when the boundary and initial conditions have been changed. In Chapter four we study uniqueness, instability and wave motion of a Batra model. In Chapter five we investigate thermal waves in a rigid heat conductor. This is a more recent model of heat transport in a rigid body, namely that derived by Green and Naghdi. In the final chapter, Chapter six, we consider a generalization of the theory of Chapter five, to include fluid mechanical behaviour. We adopt a special relation for the Helmholtz free energy in the model of Green and Naghdi. We analyse behaviour of an acceleration wave for the model.