On the importance of the pressure dependence of viscosity in steady non-isothermal shearing flows of compressible and incompressible fluids and in the isothermal fountain flow

Four steady non-isothermal viscometric flows of a class of compressible and incompressible fluids with a viscosity depending on the shear rate, pressure and temperature are considered and consistency relations between the pressure gradient in the flow direction and that in the direction of the velocity gradient are derived. In particular, the dependence of the viscosity on the pressure plays the most significant role in these relations. Examples of pressure fields obeying these compatibility conditions are given and their significance is examined. In particular, it is shown that unidirectional flows in compressible and incompressible fluids are the exception rather than the rule when the viscosity depends on pressure. Motivated by this, we examine the fountain flow effect for an incompressible fluid with a pressure dependent viscosity under isothermal conditions using the finite element method.