A note on the modeling of incompressible fluids with material moduli dependent on the mean normal stress

Abstract

In incompressible materials, both fluids and solids, a part of the stress is not prescribed by constitutive specification, that is, the part of the stress is not determined in terms of kinematical quantities, temperature, et cetera. This “indeterminate” part of the stress is variously referred to as the “constraint stress”, the “reaction stress” or the “Lagrange multiplier” enforcing the constraint. In the case of an incompressible Navier–Stokes fluid, the part of the stress, that is a consequence of the constraint, also happens to coincide with the mean value of the stress which is referred to as the “mechanical pressure”. However, in general non-Newtonian fluids this is not the case, and, unfortunately, in view of the widespread use of the Navier–Stokes equation, the terminology “pressure” is used interchangeably for both the part of the stress that is not constitutively specified and the mean value of the stress, leading to considerable confusion with regard to important issues concerning non-Newtonian fluids. Recognizing the distinction between the mean value of the stress and the part of the stress that is not constitutively specified becomes critical in materials whose moduli depend on the mean value of the stress. An example of the same concerns the viscosity, which depending on whether it is a function of the indeterminate part of the stress or the mean value of the stress could lead to different flow characteristics. In this short note we discuss an error that is a consequence of not recognizing the distinction between these different quantities but misidentifying them as being the same, the mechanical “pressure”.