A note on the meaning of mixture viscosity using the classical continuum theories of mixtures

In this paper we provide a brief review of the basic equations for the flow of two linearly viscous fluids using the mixture theory equations given in Atkin and Craine [R.J. Atkin, R.E. Craine, Continuum theories of mixtures: applications, J. Inst. Math. Appl. 17 (1976) 153; R.J. Atkin, R.E. Craine, Continuum theories of mixtures: basic theory and historical development, Quart. J. Mech. Appl. Math. 29 (1976) 290]. We then look at certain principles (or more accurately assumptions) due to Truesdell [C. Truesdell, Sulle basi della thermomeccanica, Rand Lincei, Series 8 22 (1957) 33–38, and 158–166] and Adkins [J.E. Adkins, Non-linear diffusion, 1. Diffusion and flow of mixtures of fluids, Philos. Trans. Roy. Soc. London A 255 (1963) 607–633; J.E. Adkins, Non-linear diffusion, 2. Constitutive equations for mixtures of isotropic fluids, Philos. Trans. Roy. Soc. London A 255 (1963) 635–648] and show that if the ‘assumption of the limiting cases’ of Adkins is to hold, then a very specific structure on the material properties of the two fluids has to be imposed. This new hypothesis provides one such condition for this requirement. An attempt is made to derive a relationship for the mixture viscosity using these ideas.