  In this work, we introduce a notion of dissipative weak solution for a system
describing the evolution of a heat-conducting incompressible non-Newtonian
fluid. This concept of solution is based on the balance of entropy instead of
the balance of energy and has the advantage that it admits a weak-strong
uniqueness principle, justifying the proposed formulation. We provide a proof
of existence of solutions based on finite element approximations, thus
obtaining the first convergence result of a numerical scheme for the full
evolutionary system including temperature dependent coefficients and viscous
dissipation terms. Then we proceed to prove the weak-strong uniqueness property
of the system by means of a relative energy inequality.
