On the need for compatibility of thermal and mechanical data in flow problems

In this short paper, we discuss an important compatibility condition which usually goes unmentioned when discussing classical flow problems in fluid mechanics. While results are presented from a supposedly purely mechanical perspective, in reality the problems need to be cast within a fully thermodynamic framework for them to make sense. This subtle issue warrants discussion and forms the subject matter of this short paper. It is shown that many classical solutions would not be possible if the problem is not cast within a thermodynamic framework with critical boundary conditions being specified for appropriate thermal quantities. While that this is so from a physical standpoint is quite obvious, we show here for the first time, that there has to be a certain compatibility between the rate at which work is done and the heat flux at the boundary.