Evanescent and inertial-like waves in rigidly-rotating odd viscous liquids

This paper demonstrates that rigidly rotating odd viscous liquids support a diverse spectrum of non-axisymmetric oscillatory, evanescent, and mixed-type inertial-like waves, the classification and precession characteristics of which offer a pathway to experimentally determine odd viscosity coefficients while establishing a formal equivalence between two- and three-dimensional formulations.

The Gist

Imagine a special kind of liquid that behaves like a spinning top with a mind of its own. This isn't your everyday water or oil; it's an "odd viscous liquid." Unlike normal fluids that get hot when you stir them (dissipation), this liquid doesn't heat up. Instead, it has a built-in "twist" that makes it react to motion in a way that feels almost magical.

This paper explores what happens when you take this special liquid, put it in a spinning container, and watch how waves move through it. Here is the breakdown of their findings using simple analogies:

1. The Spinning Dance Floor

Think of the liquid as a dance floor spinning at a constant speed. In normal physics, if you drop a pebble in a spinning pool, you get ripples that travel in predictable circles. But because this liquid has "odd viscosity," it creates two very different types of waves that behave like distinct dancers:

• The "Wall Huggers" (Evanescent Waves): Imagine a dancer who is terrified of the center of the room. They stay glued to the edge, shivering and vibrating right against the wall, but their energy dies out instantly as you look toward the middle of the room. In the paper, these are called Wall Modes. They are "evanescent," meaning they fade away exponentially as you move away from the solid boundary.
• The "Center Stage" Dancers (Oscillatory Waves): Now imagine a dancer who loves the middle of the floor. They bounce and ripple all the way across the room, filling the entire space with movement. These are the Body Modes. They are "oscillatory," meaning they travel through the liquid like a standard wave, not fading away immediately.
• The "Hybrid" Dancers (Mixed Modes): Sometimes, the liquid does both at once. Some parts of the wave hug the wall, while others dance in the center. The paper calls these Mixed Modes.

2. The Secret Code (The Wavenumber)

How do the scientists know which dancer will show up? They use a mathematical "secret code" called a wavenumber (represented by the Greek letter kappa, $\kappa$).

• If the code is a real number, you get the "Center Stage" dancers (waves that travel through the middle).
• If the code is an imaginary number (a concept from math that acts like a decay factor), you get the "Wall Huggers" (waves that fade away).
• If the code is a complex mix of both, you get the "Hybrid" dancers.

The paper maps out exactly when each type of dancer appears based on how fast the container spins and how "twisty" the liquid is.

3. The "Ghost" Column

In normal spinning fluids, if you poke a hole in the liquid, the disturbance travels straight up and down, forming a rigid column (like a ghostly pillar). In this odd liquid, the authors found that the liquid still forms these columns, but the "twist" of the odd viscosity changes how the waves move inside them. It's as if the ghostly column has a slight lean or a different rhythm depending on the liquid's properties.

4. Why This Matters (The "Speed Trap")

The most exciting practical takeaway the authors suggest is a way to measure the "twistiness" of this liquid.

Currently, scientists don't know the exact values of the "odd viscosity" coefficients for many of these materials. It's like knowing a car has an engine but not knowing its horsepower.

• The Solution: If you spin this liquid and watch the patterns of waves (the dancers) precess (rotate) around the container, the speed at which they rotate tells you the exact value of the odd viscosity.
• The Analogy: It's like listening to the pitch of a siren. If you know the speed of the siren, you can figure out how fast the car is moving. Here, by watching how fast the wave patterns spin, you can calculate the liquid's hidden "twist" coefficient.

5. The 2D vs. 3D Connection

The paper also points out a fascinating trick: The math for a flat, 2D spinning disk is almost identical to the math for a 3D spinning cylinder.

• In the 2D disk, the "density" of the liquid acts like the main character.
• In the 3D cylinder, the "vertical speed" (how fast the liquid moves up and down) plays the exact same role as the density did in the 2D version.
  It's as if the 3D problem is just the 2D problem wearing a different hat, but the underlying dance steps are the same.

Summary

The authors have built a mathematical map showing that spinning "odd" liquids create three distinct types of waves: those that hide near the walls, those that fill the room, and those that do both. By observing how fast these wave patterns spin, scientists can finally measure the mysterious "odd viscosity" of these materials, turning a theoretical curiosity into a measurable physical property.

Technical Summary

Problem Statement
This paper investigates the wave dynamics of odd viscous liquids subjected to rigid-body rotation. While previous studies have established that non-rotating three-dimensional odd viscous liquids support Taylor columns and axisymmetric inertial-like waves, the behavior of these fluids under rotation requires further clarification. Specifically, the authors aim to characterize the non-axisymmetric wave modes that arise in both two-dimensional compressible and three-dimensional incompressible odd viscous liquids rotating with a constant angular velocity $\Omega$. The central problem is to determine the nature of these waves—whether they are oscillatory, evanescent, or a combination—and to classify them based on the mathematical character of their planar wavenumbers.

Methodology
The authors employ a linearized theoretical framework, assuming small-amplitude motions and neglecting nonlinear terms in the Navier-Stokes equations.

• Constitutive Laws: The study utilizes specific constitutive laws for odd viscosity. In two dimensions, a single odd viscosity coefficient ($\eta_o$) is considered. In three dimensions, two coefficients ($\eta_o$ and $\eta_4$) are included to account for the full tensor structure in cylindrical coordinates.
• Governing Equations: The analysis is conducted in a frame rotating with the fluid. The momentum and continuity equations are reduced to a single scalar equation for the variable density (in 2D) or a modified pressure (in 3D). These equations take the form of Poincaré-Cartan equations.
• Solution Strategy: The authors seek solutions in the form of Bessel functions ($J_m(\kappa r)$) multiplied by azimuthal and temporal phase factors ($e^{i(m\phi - \omega t)}$). This leads to a dispersion relation linking the planar wavenumber $\kappa$, the precession frequency $\omega$, the rotation rate $\Omega$, and the odd viscosity coefficients.
• Boundary Conditions: No-slip boundary conditions are imposed at the solid sidewalls of the domain (a disk in 2D, a cylinder in 3D). This results in a secular equation (a determinant condition) that must be satisfied for non-trivial solutions to exist.
• Classification: The admissible wavenumbers $\kappa$ are analyzed in the parameter space defined by $\omega$ and the odd viscosity coefficients. The roots of the dispersion relation are categorized as real, imaginary, or complex.

Key Contributions and Results

1. Classification of Wave Modes: The paper identifies three distinct types of wave modes based on the character of the planar wavenumber $\kappa$:

   • Body Modes: Correspond to real $\kappa$, resulting in oscillatory inertial-like waves that fill the interior of the cylinder.
   • Wall Modes: Correspond to imaginary $\kappa$, resulting in evanescent waves that decay exponentially away from the solid boundary.
   • Mixed Modes: Correspond to complex $\kappa$ (a combination of real and imaginary parts), exhibiting characteristics of both oscillatory and evanescent behavior.
     These modes are shown to precess either prograde or retrograde with respect to the rotating frame.

2. Formal Equivalence of 2D and 3D Problems: A significant theoretical finding is the formal equivalence between the two-dimensional compressible problem and the three-dimensional incompressible problem. The variable density $\rho'$ in the 2D case plays the same mathematical role as the axial velocity component $v_z$ in the 3D case. The three-dimensional problem with non-zero $\eta_4$ can be mapped to the $\eta_4=0$ case by renormalizing the angular velocity and the odd viscosity coefficient.

3. Recovery of Known Phenomena: The formulation recovers recent results regarding equatorial and topological waves in two-dimensional odd viscous liquids as special cases, specifically establishing the presence of such waves at frequencies $\omega < 2\Omega$.

4. Connection to Rayleigh-Bénard Convection: The wall modes (evanescent waves) and body modes (oscillatory waves) are shown to resemble the wall and body modes observed in non-odd, rapidly-rotating Rayleigh-Bénard convection, despite the different physical origins (odd viscosity vs. thermal forcing and shear viscosity).

5. Experimental Implications: The authors demonstrate that the admissible pairs of precession frequency $\omega$ and odd viscosity coefficients ($\nu_o, \nu_4$) form specific curves in parameter space derived from the secular equation. They suggest that by experimentally observing the precession rate of these patterns, one could theoretically determine the largely unknown values of the odd viscosity coefficients. Additionally, the paper adapts experimental conditions from Nosan et al. (2021) to show that fluid particle paths in these evanescent waves form ellipses in the $r-z$ plane, offering another potential method for determining viscosity coefficients.

Significance
The paper claims that its primary contribution is the rigorous derivation and classification of wall, body, and mixed modes in rigidly-rotating odd viscous liquids. By establishing the link between the mathematical character of the wavenumber $\kappa$ and the physical behavior of the wave (evanescent vs. oscillatory), the work provides a theoretical basis for interpreting experimental data. The authors posit that this framework offers a viable pathway to experimentally quantify odd viscosity coefficients, which have remained largely unknown. Furthermore, the demonstration of the formal equivalence between 2D compressible and 3D incompressible formulations unifies the understanding of wave propagation in these systems, while the analysis of particle trajectories and the inclusion of shear viscosity effects (perturbatively) bridge the gap between idealized theory and realistic experimental conditions.