Exploitation of the entropy principle: Proof of Liu theorem if the gradients of the governing equations are considered as constraints

The exploitation of the entropy principle in thermodynamics with Lagrange multipliers (the so called Liu procedure) is based on the celebrated Liu theorem. In some recent papers, where either rigid heat conductors or Korteweg fluids have been considered, the Liu procedure has been generalized by using the gradients of the governing equations as additional constraints. Here, a rigorous proof of the validity of the extended procedure to arbitrary first-order nonlocal continua is provided, thus explicitly generalizing the Liu theorem. As an application, heat conducting first-order Korteweg-type fluids with a scalar internal variable are analyzed. The thermodynamic restrictions are determined by the extended Liu procedure. Moreover, a comparison with the thermodynamic analysis obtained by the application of an extended Coleman-Noll procedure is performed.