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A Kinetic Theory Approach to Ordered Fluids

José A. Carrillo, Patrick E. Farrell, Andrea Medaglia, Umberto Zerbinati

Abstract

We develop a unified kinetic theory for ordered fluids, which systematically extends the phase space with the appropriate generalized angular momenta. Our theory yields a uniquely determined mesoscopic model for any continuum with microstructure that is characterized by Capriz's order parameter manifold. We illustrate our theory with three running examples: liquids saturated with non-diffusive gas bubbles, liquids composed of calamitic (rodlike) molecules, and liquids composed of calamitic molecules with additional head-to-tail symmetry. We discuss the symmetries of the microscopic interactions via Noether's theorem, and use them to characterize the conserved quantities mesoscopic dynamics. We derive the mesoscopic model for ordered fluids from a kinetic point of view assuming that the microscopic interactions are of weak nature when it comes to the ordering of the fluid. Lastly, we discuss under which conditions an H-theorem result holds at the mesoscopic scale and for which Vlasov potential we can expect the emergence of collective behavior.

A Kinetic Theory Approach to Ordered Fluids

Abstract

We develop a unified kinetic theory for ordered fluids, which systematically extends the phase space with the appropriate generalized angular momenta. Our theory yields a uniquely determined mesoscopic model for any continuum with microstructure that is characterized by Capriz's order parameter manifold. We illustrate our theory with three running examples: liquids saturated with non-diffusive gas bubbles, liquids composed of calamitic (rodlike) molecules, and liquids composed of calamitic molecules with additional head-to-tail symmetry. We discuss the symmetries of the microscopic interactions via Noether's theorem, and use them to characterize the conserved quantities mesoscopic dynamics. We derive the mesoscopic model for ordered fluids from a kinetic point of view assuming that the microscopic interactions are of weak nature when it comes to the ordering of the fluid. Lastly, we discuss under which conditions an H-theorem result holds at the mesoscopic scale and for which Vlasov potential we can expect the emergence of collective behavior.

Paper Structure

This paper contains 14 sections, 11 theorems, 160 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Order parameter and Lagrangian formulation
  3. Order parameter manifold
  4. Symmetries and conserved quantities
  5. Microscopic collision
  6. The BBGKY hierarchy for ordered fluids
  7. Related literature
  8. The hierarchy
  9. Weak-order interaction
  10. Collision invariants and steady state
  11. Mean-field potential
  12. Conclusion
  13. Frame indifference in terms of group actions
  14. Mathematical aspects of the collision operator

Key Result

Proposition 2.17

If the group action $\mathcal{A}$ is transitive and $\mathcal{L}_1$ in eq:decomposition is frame indifferent, then the matrix ${ { = } }(\nu)$ is a constant. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Collision of two hard spheres, moving with velocities $v_1$ and $v_2$, respectively, and with centers of mass $\mathbf{x}_1$ and $\mathbf{x}_2$.
  • Figure 1: Mean (top) and standard deviation (bottom) of a system of particles evolving according to \ref{['eq:ODEs']} for $\beta=0$ and $\alpha=1$ (saddle point, left), $\beta=1$ and $\alpha=1$ (stable spiral, middle), and $\beta=1$ and $\alpha=0.25$ (stable node, right). The initial conditions for $\underline{\nu}$ and $\underline{\varsigma}$ are drawn from a uniform distribution in $[-\pi,\pi]$ and $[-1,1]$ respectively. Lastly we fixed $\underline{\hat{\nu}}=\left(\sin(\hat{\theta}),\cos(\hat{\theta})\right)$ with $\hat{\theta}=0.4$.
  • Figure 2: Collision of two gas bubbles. As a result of the collision the volume fraction of each bubble is exchanged, with $\nu_1^\prime + \nu_2^\prime = \nu_1 + \nu_2$.
  • Figure 3: Collision of calamitic molecules in two dimensions.

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.7: Landau--de Gennes Q-tensor
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • Definition 2.14
  • Remark 2.15
  • Example 2.16
  • Proposition 2.17
  • ...and 37 more