[Paper Review] General Nonlinear 2-Fluid Hydrodynamics of Complex Fluids and Soft Matter

Pleiner, H., Harden, J. L.|arXiv (Cornell University)|Apr 6, 2004

Rheology and Fluid Dynamics Studies17 citations

TL;DR

This paper develops a rigorous, general nonlinear two-fluid hydrodynamic framework for complex fluids, treating one component as a Newtonian fluid and the other as a structured fluid (e.g., polymer or liquid crystal). It derives consistent hydrodynamic equations by enforcing symmetries, conservation laws, and thermodynamics, showing that convective velocities and stress partitioning are material-dependent and constrained by nonlinear couplings. The key contribution is a systematic derivation of viscous dissipation terms and the identification of consistent approximations for experimental comparison, including a linearized effective concentration dynamics.

ABSTRACT

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. Special emphasis is laid on such 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 experiments.

Motivation & Objective

• To develop a general, thermodynamically consistent two-fluid hydrodynamic model for complex fluids, including polymeric and liquid crystalline systems.
• To clarify the role of convective velocities in nonlinear two-fluid dynamics and show they are not arbitrary but material-dependent.
• To resolve the stress division problem by demonstrating its dependence on both convected velocities and material-specific properties.
• To derive a simplified, experimentally relevant set of equations, including a linearized effective concentration dynamics.

Proposed method

• Derives hydrodynamic equations using conservation laws, symmetries, and thermodynamics as a foundation.
• Introduces two distinct fluid phases with individual mass densities, velocities, and momentum densities.
• Incorporates nonlinear couplings through generalized viscous dissipation tensors involving velocity gradients and relative motion.
• Uses a 2-fluid description with separate momentum and stress balances, including relative velocity w = v₁ − v₂.
• Derives the most general form of viscous dissipation, identifying distinct viscosity tensors for different physical mechanisms.
• Applies harmonic approximation to relate the 2-fluid dissipation tensors to standard 1-fluid limits and validates consistency across approximations.

Experimental results

Research questions

• RQ1 How do convective velocities in a two-fluid model affect the hydrodynamic equations and their nonlinear structure?
• RQ2 What determines the distribution of nematic or elastic stresses between the two fluid components (the stress division problem)?
• RQ3 How do different approximations for viscous dissipation (e.g., neglecting certain viscosity tensors) affect the consistency and physical validity of the model?
• RQ4 What is the correct 1-fluid limit of the two-fluid model, and how do the viscous tensors transform in this limit?
• RQ5 Can a simplified, linearized effective concentration dynamics be derived that is suitable for experimental comparison?

Key findings

• The convective velocities in the two-fluid model are not arbitrary; they are constrained by consistency conditions, and their choice affects stress partitioning.
• The stress division between the two fluids is not unique and depends on both the choice of convected velocities and intrinsic material properties.
• The most general viscous dissipation tensor includes multiple terms involving velocity gradients, relative velocity, and curl of momentum, with distinct tensor symmetries that must be preserved.
• The 1-fluid limit is consistently recovered only when specific combinations of viscosity tensors vanish, particularly ν(w), ν(c), ν(r), ν(d), and ν(e).
• Neglecting certain viscous terms (e.g., ν(c) = 0) is inconsistent with other approximations (e.g., ν(12) = 0), showing that common simplifications in the literature are not always compatible.
• A linearized effective concentration dynamics is derived that can be used to compare theoretical predictions with experimental rheological data.