Abstract
Many real fluids exhibit response characteristics that cannot be satisfactorily described by the classical Navier–Stokes fluid model and such fluids are referred to as non-Newtonian fluids.
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Notes
- 1.
It would have been more appropriate to have used the terminology non-Navier–Stokesian fluids rather than non-Newtonian fluids as the theories of fluid advanced by Newton are quite far removed from the Navier–Stokes theory. This is not surprising as the notion of a partial derivative had not yet been invented. Also, Newton advanced more than one theory to describe the motion of fluids. As Dugas [86] points out, Newton proposed two theories for fluids, the first “a schematic theory of fluids, which he considered to be formed of an aggregate of elastic particles, which repelled each other, were arranged at equal distances from each other, and were free”, and “In the second theory the particles of the fluid are contiguous”. The two theories led to results to the resistance due to the translation of a solid cylinder in a fluid body of infinite extent to differ by a factor of four. In [272], Truesdell goes as far as to aver that “Newton’s theories of fluids are largely false”. We shall use the terminology non-Newtonian fluid as this has become the terminology adopted by the practitioners in the field. We shall also use the terminology “Newtonian fluid” and “Navier–Stokes fluid” interchangeably.
- 2.
We are interested in departures from the behavior exhibited by “Newtonian fluids” in what one understands as laminar flows. Turbulent flows of fluids described in the laminar flow regime by the Navier–Stokes model remain shrouded in mystery with numerous models being used to describe different categories of turbulent flows.
- 3.
Some fluid models can be cast as rate type or integral type models, and example of the same being the eponymous model due to Maxwell [184]. Recently, Rajagopal in [227] (see also Prusa and Rajagopal [219]) has introduced a new class of fluid models wherein the history of the deformation gradient and the history of the stress are implicitly related. The class introduced by Rajagopal in [227] includes most of the fluid models of the differential, rate, and integral type and introduces new models that can describe phenomena that are not possible to characterize within the context of models currently in use, an example being the dependence of the material moduli on the invariants of both the stress, kinematical variables and mixed invariants of the stress and the appropriate kinematical tensors.
- 4.
If by a fluid one means a body that cannot resist a shear stress, however small the time interval, then the terminology “Bingham fluid” is an oxymoron as the appellation is supposed to describe a body that can resist a shear force until a particular threshold is achieved for the shear stress. Since the material flows like a fluid once the threshold is exceeded, it is considered a fluid.
- 5.
It would be fair to say that the readers are introduced to a particular thermodynamic perspective as there is considerable disagreement concerning in what form the second law of thermodynamics is to be enforced. We shall not get into a discussion of the various perspectives, their merits and demerits. We shall merely present the consequences of the Clausius-Duhem inequality being used as the surrogate for the second law of thermodynamics.
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© 2016 Springer International Publishing Switzerland
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Cioranescu, D., Girault, V., Rajagopal, K.R. (2016). Introduction. In: Mechanics and Mathematics of Fluids of the Differential Type. Advances in Mechanics and Mathematics, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-319-39330-8_1
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DOI: https://doi.org/10.1007/978-3-319-39330-8_1
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