On controllable heat flux fields and the determination of temperature-dependent thermal conductivities
Quarterly of Applied Mathematics
Introduction. It is known that there is no single non-uniform temperature field that is possible in all heat conductors which have a temperature-dependent thermal conductivity  . We say then that there are no controllable states of temperature associated with such materials. We shall show below, however, that controllable heat flux fields do exist for this class of materials, and we shall demonstrate how the knowledge of such fields enables one to design experimental programs which are
... ams which are universally applicable to the determination of nonlinear thermal conductivities. The controllable states will be seen to be derivable from a potential, but, unlike the classical linear case, the potential will be a scalar field distinct from the temperature field in the heat conductor. 2. Nonlinear heat conduction. In the typical boundary-value problem in continuum physics one wishes to determine states of stress, temperature, etc., within a body when conditions of loading, temperature, etc., are specified on the surface of the body. Physical principles such as balance of momentum and conservation of energy provide the differential field equations to be solved, while experimental measurements, experiential inferences, or assumptions provide the boundary data. It is often the case, however, that the physical quantities which appear as dependent variables in the field equations are not the same as those which are available as boundary data. For example, in steadystate heat flow the balance-of-energy equation involves the divergence of the heat flux vector, while the most readily available boundary data may be temperature values.