Acoustic heating produced in the thermoviscous flow of a Bingham plastic

Abstract

This study is devoted to the instantaneous acoustic heating of a Bingham plastic. The model of the Bingham plastic’s viscous stress tensor includes the yield stress along with the shear viscosity, which differentiates a Bingham plastic from a viscous Newtonian fluid. A special linear combination of the conservation equations in differential form makes it possible to reduce all acoustic terms in the linear part of of the final equation governing acoustic heating, and to retain those belonging to the thermal mode. The nonlinear terms of the final equation are a result of interaction between sounds and the thermal mode. In the field of intense sound, the resulting nonlinear acoustic terms form a driving force for the heating. The final governing dynamic equation of the thermal mode is valid in a weakly nonlinear flow. It is instantaneous, and does not imply that sounds be periodic. The equations governing the dynamics of both sounds and the thermal mode depend on sign of the shear rate. An example of the propagation of a bipolar initially acoustic pulse and the evolution of the heating induced by it is illustrated and discussed.