Exact results for dissipation and steady creeping flow in three-dimensional chiral active fluids

Technical Summary: Exact Results for Dissipation and Steady Creeping Flow in Three-Dimensional Chiral Active Fluids

Problem Statement
Chiral active fluids, composed of self-spinning particles driven by microscopic energy injection, exhibit hydrodynamic behaviors governed by antisymmetric contributions to the viscosity tensor, known as "odd viscosity." While significant progress has been made in understanding two-dimensional odd systems and specific three-dimensional (3D) scenarios (e.g., microswimmers, point forces), a comprehensive theoretical framework for steady, incompressible 3D creeping flows remains limited compared to classical Stokesian fluids. Specifically, there is a lack of exact results regarding the uniqueness of solutions, the role of odd viscosity in viscous dissipation, and the explicit flow fields around suspended particles with arbitrary orientations relative to the fluid's spin angular momentum.

Methodology
The authors employ a combination of theoretical generalization and analytical derivation:

1. Generalization of Variational Principles: The Helmholtz minimum dissipation theorem is extended to systems with odd viscosity. This involves decomposing the viscosity tensor into symmetric (even) and antisymmetric (odd) parts and analyzing the dissipation rate for incompressible flows under stick boundary conditions.
2. Model Definition: A specific 3D model fluid is defined using one shear viscosity ($\eta_s$) and one odd viscosity coefficient ($\eta_o$), satisfying Onsager-Casimir reciprocity relations. The governing equations are derived for the velocity field $\mathbf{v}$ and effective pressure $\tilde{p}$.
3. Green's Function and Singularity Representations: The fundamental solution (Green's tensor) for a point force in an unbounded odd fluid is utilized. The authors apply the Lorentz reciprocal theorem for odd fluids to derive singularity representations for the flow and pressure fields around a sphere. This involves expanding the Green's tensor in a multipole series to satisfy boundary conditions for both translating and rotating spheres.
4. Analytical Integration: Exact closed-form expressions for velocity and pressure fields are computed by evaluating the multipole expansions, utilizing Fourier methods and Bessel function identities to handle the non-local effects of odd viscosity.

Key Contributions and Results

• Uniqueness and Dissipation Theorems:

  • The authors prove that steady incompressible 3D odd fluids admit a unique solution for flow fields at low Reynolds numbers, generalizing previous 2D results.
  • A generalized Helmholtz theorem is established, demonstrating that for a given set of boundary conditions, the presence of odd viscosity always increases viscous dissipation compared to a fluid with identical even viscosity but zero odd viscosity, provided the velocity field is altered by the odd viscosity. If the velocity field remains unchanged, the dissipation remains the same, though pressure fields may differ.

• Stress Response to Point Forces:

  • The paper explicitly computes the stress tensor $\Sigma(r)$ for a fluid subjected to a point force density. This result provides the necessary stress response for computational techniques like the boundary element method in odd fluids.

• Translating Sphere:

  • Exact closed-form expressions for the velocity and pressure fields around a translating sphere are derived for arbitrary orientations of the sphere's velocity $\mathbf{U}$ relative to the fluid's spin angular momentum density $\hat{\ell}$.
  • The analysis reveals that odd viscosity induces azimuthal flow components, breaking the cylindrical symmetry seen in ordinary Stokes flow when $\mathbf{U}$ is not parallel to $\hat{\ell}$.
  • Consistent with the generalized Helmholtz theorem, the authors confirm that a translating sphere dissipates more energy in an odd fluid than in a Newtonian fluid.

• Rotating Sphere:

  • For a sphere rotating with angular velocity $\mathbf{\Omega}$, the authors demonstrate that the velocity field remains identical to the classical rotlet solution of ordinary Stokes flow ($\mathbf{v} \propto \mathbf{\Omega} \times \nabla(1/r)$), regardless of the odd viscosity magnitude.
  • Consequently, the viscous dissipation for a rotating sphere is unchanged compared to a Newtonian fluid. The odd viscosity manifests only as a modification to the pressure field.
  • The authors note that this result is specific to the single odd-viscosity model (Eq. 3.1). In a generalized model with multiple odd viscosity coefficients, the velocity field would differ from the Stokes solution, leading to increased dissipation.

Significance
The paper claims to bridge the gap between extensive results for ordinary Stokesian fluids and the emerging field of odd microhydrodynamics. By providing exact analytical solutions for fundamental boundary value problems (translating and rotating spheres), the work establishes a rigorous baseline for understanding 3D chiral active fluids. The derivation of the stress tensor for point forces and the singularity representations enables the calculation of mobility and friction tensors for arbitrarily shaped particles. These results are presented as foundational for future studies on odd Brownian motion, microswimmers in chiral fluids, and the development of odd microhydrodynamics.