Abstract
In this chapter, the reader will be introduced to a variety of non-Newtonian phenomena exhibited by real fluids, namely stress relaxation, nonlinear creep, shear-thinning and shear-thickening, thixotropy, development of normal stress differences in simple shear flows, yield, etc.
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Notes
- 1.
A thermodynamic framework has been put into place by Rajagopal and Srinivasa in [234] that appeals to the maximization of the rate of entropy production to provide a basis for fluids of grade 2 and the same framework can be used to develop the model of the grade 3 fluid and the Bingham fluid.
- 2.
This is the vector version of equation (13) in Stokes’ paper [259]. We have not yet introduced the notation that will be followed in this book. Here, \(\mu \) is the viscosity, \(\varrho \) the density, \({\varvec{v}}\) the velocity, p the pressure, \({\varvec{b}}\) the specific body force, and \(\varDelta \) denotes the Laplacian, \(\mathbf{grad}\) the gradient, div the divergence and \(\frac{d}{d t}\) the material time derivative, see Section 2.3 for more details.
- 3.
There are several phenomena associated with the response of non-Newtonian fluids such as “elastic turbulence,” “drag reduction due to turbulence,” “secondary flows in turbulent flows of non-Newtonian fluids,” etc. that are a consequence of some of the above-mentioned response characteristics of non-Newtonian fluids coupled with the effects of turbulence. As turbulence, to date, has defied proper understanding, even within the context of fluids that in the laminar range are described by the classical Navier–Stokes fluid, we shall not discuss such characteristics in the book.
- 4.
If one waits sufficiently long, then one could discern motion in nearly all bodies due to the effect of gravity. Deborah [80] remarks in the Old Testament that “Even mountains quaked in the presence of the Lord.” Usually, Deborah’s statement is translated to read “Even mountains flowed in the presence of the Lord,” but this translation does not seem to be felicitous (see Rajagopal [228] for a detailed discussion concerning this issue).
- 5.
Recently, it has been shown that the Cauchy stress in many rate type models can be expressed as an elastic response from an evolving natural configuration, i.e., the stress can be expressed in terms of a Cauchy–Green stretch tensor which obeys an evolution equation (see Rajagopal and Srinivasa [235]).
- 6.
More precisely, \({\varvec{R}}\in {\mathcal O^+}\), i.e., det\(\,{\varvec{R}}=1\). This follows from the fact that, as a consequence of the conservation of mass, det\(\,{\varvec{F}}>0\) at all times. However, some authors relax this constraint by allowing \({\varvec{F}}\) to become singular at some points in the presence of shocks.
- 7.
There is a popular misconception that by definition, a fluid is isotropic. If by fluid we mean a simple fluid in the sense of Noll and furthermore a body whose symmetry group is the unimodular group, then such fluids are isotropic. From such a narrow perspective one cannot have an anisotropic liquid, say a fluid model for a liquid crystal. On the other hand, the common definition that a fluid cannot support a shear stress indefinitely allows one to describe anisotropic fluids. For such fluids, the symmetry group is not the unimodular group (cf. Rajagopal [225] and Rajagopal and Srinivasa [235] for a discussion of the relevant issues).
- 8.
A body is said to be mechanically isolated if there is no working due to either the tractions on the boundary or due to the body forces.
- 9.
Reiner [242] derived and discussed special flows of a subclass of such fluids by using a power-series expansion. We shall discuss mathematical issues concerning these fluids as well as a more general subclass of those fluids that are referred to as Reiner–Rivlin fluids.
- 10.
If the fluid is incompressible but inhomogeneous then the density is not a constant everywhere, the density of a specific material point remains the same, i.e., \(\varrho = \varrho ({\varvec{x}})\).
- 11.
Ting [268] studied several flows of fluids of grade two, that is fluids modeled by (2.4.19). None of the problems he studied had bounded solutions when \(\alpha _1\) was negative.
- 12.
More precisely, in the second integral \(\varphi \) stands for \(\varphi ({\varvec{x}}({\varvec{X}},0),0)\).
- 13.
An interesting discussion of the difficulties inherent to the no-slip boundary condition can be found in Frehse and Málek [104].
- 14.
More generally, in the flow of a fluid of grade n, the equations of motion are partial differential equations of order \(n+1\). However, in view of thermodynamic restrictions, it is possible that the equations are of lower order as is the case of third-grade fluids.
- 15.
In a thermodynamically compatible fluid of grade two, Tanner’s comment apply to general three-dimensional flows as \(\alpha _1 + \alpha _2 = 0\). If in a plane flow, \(\alpha _1 + \alpha _2 \ne 0\), it can be shown that the terms that are multiplied by \(\alpha _1 + \alpha _2\) reduce to the gradient of a scalar and hence can be absorbed in the pressure.
- 16.
Huilgol [137] showed that the Stokes solution is the only solution for the creeping flow equations in the plane flow of a second grade fluid with \(\alpha _1 <0\).
- 17.
This evolution of material symmetry was given with regard to physical constants that appear in the constitutive equations and is different from the evolution of material symmetry of materials with multiple natural configurations (cf. Rajagopal and Srinivasa [235] for a discussion of material symmetry in anisotropic liquids).
- 18.
It is worth emphasizing that Maxwell [184] developed his original model for a viscoelastic fluid to describe the dynamic response of air, a fluid that can store as well as dissipate energy. The classical Euler fluid is a perfectly elastic fluid incapable of dissipation; it can only store energy, and usually gases are either modeled as ideal fluids or Van der Waal’s fluids. Once a choice is made for the rate of dissipation, as it is nonnegative, it serves as a Lyapunov function and decreases with time satisfying the minimum entropy production theorem of Onsager [201] (see also Prigogine [218]) that characterizes the steady states for special choices of the rate of entropy production (see [234]).
- 19.
In general, \({\varvec{F}}_{\kappa _{p(t)}}\) is a mapping transforming the vectors belonging to the tangent space at a material point of \(\kappa _{p(t)}\) into the tangent space at the same material point in the configuration \(\kappa _t\).
- 20.
The phenomenon of a large increase in the volume flow rate due to a small increase in the driving pressure is referred to as “spurt.”
- 21.
Bulíček et al. [53, 54] have provided a detailed mathematical study of the equations governing the flows of such fluids.
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Cioranescu, D., Girault, V., Rajagopal, K.R. (2016). Mechanics. 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_2
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