Rheology of Complex Fluids
by H. Pleiner
(external collaboration with H.R. Brand, M. Liu and J.L. Harden)
The flow behavior of complex fluids is rather involved. Viscoelasticity, shear thinning and thickening, normal stress differences are well known for a long time. What is missing is a coherent derivation of the appropriate hydrodynamic-like equations especially in the nonlinear domain. Frequently ad-hoc models are used, which have some success in describing the main features, but lack a solid physical foundation. Our goal is to lay a framework for such nonlinear equations that is in accordance with basic physical principles, like thermodynamics and irreversible thermodynamics.
In order to describe visco-elasticity we start with a decent hydrodynamic description of nonlinear elasticity [B]. This requires the use of a bi-vectorial field, ∇i aα, in Eulerian description, where aα(r,t) is the coordinate of a (translated and rotated) body point measured in the laboratory frame ri at time t. Since the elasticity of a body must not depend on the orientation and position of the frame, in which its initial state is measured (subcript α), compared to the laboratory frame (subsript i), ∇i aα is not a (symmetric) tensor, but
contains deformations as well as rotations. The dynamic equation for that field is uniquely determined by general physical laws. From it implicit hydrodynamic equations for strains and rotations can be derived, which can be made explicit in the form of an expansion in the strains.
Allowing for strain relaxation viscoelasticity is obtained [B]. For the Eulerian strain tensor the reversible part of the strain evolution equation resembles that of an lower convected model, while for other strain tensors different forms are obtained. This is a clear signal that the reversible nonlinearities cannot be postulated by ad-hoc principles like the so-called frame indifference principle, but have to be derived from first principles.
Other non-Newtonian features, like shear thinning or thickening and normal stress differences, are treated at least partially by this approach, but further relaxing fields may also be important.
Meanwhile, we have shown [E] that the hydrodynamically derived model for non-Newtonian fluids in terms of the Eulerian strain tensor contains most of, and is more general than, the standard rheological models (expressed as "constitutive" equations for the stress tensor) as special cases and discards a few of them. The hydrodynamic method allows to discriminate those parts of the dynamics that are due to general principles from the unavoidable phenomenological part. The latter is given in the form of truncated power series in the strain tensor that can systematically be generalized when necessary. For this part we stick to the well-established 'linear irreversible thermodynamics', which, being linear in the generalized forces, nevertheless leads to equations highly nonlinear in the variables like the strain tensor.
A different approach to describe viscoelastic effects in semi-flexible and
stiff polymeric systems [C] is based on the use of a nematic-like orientational order parameter tensor (Doi-Edwards). For the true nematic phase
we combine the director hydrodynamics in a nematic phase
with the relaxing dynamics of the scalar degree of order into
an effective dynamic equation for the 2nd rank order parameter tensor
(including an external field). The nonlinear convective terms are
definitely of the Jaumann type describing the coupling to
rotational flow, but there are in addition phenomenological (linear and
nonlinear) couplings to elongational flow, which can be of equal
importance. Jaumann-type convective nonlinearities are also found in the
isotropic case, however, the phenomenological couplings have a very different
structure from those in the nematic phase due to the
underlying director hydrodynamics. Thus, it is not possible to use the same set of
equations for the orientational dynamics in the nematic and the isotropic
Furthermore we give the form of the appropriate orientational-elastic
stresses in the stress tensor. They are completely
fixed by the orientation dynamics and no choices are left. Thus, there
are again different expressions for the isotropic and the nematic phase. In
particular, a simple stress-optical law is valid for the isotropic phase
only -- and only in linear approximation.
As in the case of a relaxing elastic strain tensor, the hydrodynamic model of non-Newtonian fluids described in terms of a transient orientational order can be put approximately into the form of a "constitutive" equation with an effective dynamic equation for the stress tensor [F].
Of the traditional constitutive models some forms of the Johnson-Segalman and the Jeffreys model are compatible with this origin of non-Newtonian behavior, while alternatively, some forms of the Maxwell and Oldroyd models are compatible with a relaxing elasticity [E]. Giesekus-type models can be compatible with both types.
In the two-fluid description [D], each component or phase is treated as a continuum
described by local thermodynamic variables (e.g. temperature, density, and
relevant order parameters), and dynamical quantities (e.g. velocity or momentum).
In general, these variables for the constituents are coupled. For instance, the
effective friction between components in a binary fluid mixture leads to a drag
in the macroscopic description that is proportional to the local velocity
difference. Such a description is important, e.g. for manifestly heterogeneous systems, in particular for their phase separation dynamics or any other flow-induced structural evolution phenomena.
We discuss general 2-fluid hydrodynamic equations for complex
fluids, where one kind is a simple Newtonian fluid, while the other is either
liquid-crystalline or polymeric/elastomeric, thus being applicable to
lyotropic liquid crystals, polymer solutions, and swollen elastomers. The
procedure can easily be generalized to other
complex fluid solutions.
Emphasis is placed on the rigorous
derivation of the dynamic equations within the framework of hydrodynamics as
contrasted to ad-hoc treatments. The resulting equations are rather general
and complicated. They can and have to be simplified for special applications
or systems by appropriate and well-defined approximations. One of the
advantages of starting from the general theory is the possibility to identify
and characterize the approximations made.
Of special interest are
nonlinearities that originate from the 2-fluid description, like the transport
part of the total time derivatives.
It is shown that the proper velocities, with which the hydrodynamic quantities
are convected, cannot be chosen at will, since there are subtle relations
among them. Within allowed combinations the convective velocities are
generally material dependent. The so-called stress
division problem, i.e. how the nematic or elastic stresses are distributed
between the two fluids, is shown to depend partially on the choice of the
convected velocities, but is otherwise also material dependent. A set of
reasonably simplified equations is given as well as a linearized version of an
effective concentration dynamics that may be used for comparison with
H. Pleiner and H.R. Brand, "Macroscopic dynamics and hydrodynamic Maxwell equations",
Phys. Rev. Lett. 74, 1883 (1995), [56 kB pdf-file]
H. Pleiner and H.R. Brand, "Slow relaxational dynamics and rheology of complex fluids"[A],
in Slow Dynamics in Complex Systems: 8th Tohwa University International Conference eds. M. Tokuyama and I. Oppenheim, p.160 (1999), [461 kB pdf-file]
H. Temmen, H. Pleiner, M. Liu and H.R. Brand, "Convective Nonlinearity in Non-Newtonian Fluids"[B1],
Phys. Rev. Lett., 84, 3228 (2000), [518 kB pdf-file] and Proc. 13th Int. Congr. on Rheology, 4, 96 (2000), [106 kB pdf-file];
"Temmen at al. Reply",
Phys. Rev. Lett., 86, 745 (2001), [282 kB
pdf-file] DOI: 10.1103/PhysRevLett.86.745
see also M. Grmela, Phys.
Lett. A, 296, 97 (2002).
H. Pleiner, M. Liu and H.R. Brand, "The Structure of Convective Nonlinearities in Polymer Rheology"[B2],
Rheologica Acta, 39, 560 (2000), [131 kB pdf-file]
H. Pleiner, M. Liu and H.R. Brand, "Convective Nonlinearities for the Orientational Tensor Order
Parameter in Polymeric Systems"[C],
Rheologica Acta, 41, 375 (2002), [150 kB pdf-file]
and "Dynamics of the Orientational Tensor Order Parameter in Polymeric Systems", in
Progress in Rheology: Theory and
Applications, eds. F.J.M. Boza, A.G. Conejo, P.P López, J.M.F. Gómez, and
J.M. García (GER, Sevilla), p.7 (2002), [124 kB pdf-file]
H. Pleiner and J.L. Harden, "General Nonlinear 2-Fluid Hydrodynamics of Complex Fluids and Soft Matter"[D], in Nonlinear Problems of Continuum Mechanics, Special issue of
Notices of Universities. South of Russia. Natural sciences, p.46 - 61 (2003) [335 kB pdf-file] and arXiv:cond-mat/0404134,
Macromol. Chem. Phys., 204, F33 (2003) [378 kB pdf-file] (short version) and
Slow Dynamics - Sendai 2003, AIP Conference Proceedings 708, 46 (2004) [57 kB pdf-file] - contains simplified 2-fluid equations for gels
H. Pleiner, M. Liu and H.R. Brand, "Nonlinear Fluid Dynamics Description of non-Newtonian Fluids"[E],
Rheologica Acta, 43, 502 (2004) [158 kB pdf-file] DOI 10.1007/s00397-004-0365-8 and arXiv:cond-mat/0404137
- contains a comparison with standard rheological models
and "A Physicists' View on Constitutive Equations",
Proc. XIVth Int. Congr. on Rheology, 168 (2004) [229 kB pdf-file]
H. Pleiner, M. Liu and H.R. Brand, "Non-Newtonian constitutive equations using the orientational order parameter"[F],
IMA Volume in Mathematics
and its Applications, Volume 141: Modeling of Soft
Matter, edited by M.-C. Calderer and E. Terentjev, (Springer 2005) p. 99 [164 kB pdf-file]
O. Müller, M. Liu, H. Pleiner and H.R. Brand, "Transient elasticity and polymeric fluids: Small-amplitude deformations",
Phys. Rev. E 93, 023113 (2016) DOI: 10.1103/PhysRevE.93.023113 [943 kB pdf-file]
O. Müller, M. Liu, H. Pleiner and H.R. Brand, "Transient elasticity and the rheology of polymeric fluids with large amplitude deformations",
Phys. Rev. E 93, 023114 (2016) DOI: 10.1103/PhysRevE.93.023114 [773 kB pdf-file]
preprints of the theory group
Other research topics:
Instabilities in Complex Fluids and Active Soft Matter
Banana and Tetrahedral Phases
Liquid Crystalline Polymers and Elastomers
Membranes, Films and Surface Waves
Last modified: March 5th, 2004