We discuss the linear hydrodynamic response of a two-dimensional active chiral compressible fluid with odd viscosity. The viscosity coefficient represents broken time-reversal and parity symmetries in the 2D fluid and characterizes the deviation of the system from a passive fluid. Taking into account the hydrodynamic coupling to the underlying bulk fluid, we obtain the odd viscosity-dependent mobility tensor, which is responsible for the non-reciprocal hydrodynamic response to a point force. Furthermore, we consider a finite-size disk moving laterally in the 2D fluid and demonstrate that the disk experiences a non-dissipative lift force in addition to the dissipative drag one.
Stokesian Dynamics is a well-established computational method for simulating dynamics of many particles suspended in a conventional passive fluid medium. Active fluids composed of self-propelling particles with broken time reversal symmetry permit the emergence of a so-called odd viscosity. In this work, we extended the conventional Stokesian Dynamics formalism to incorporate the additional hydrodynamic effects due to odd viscosity, which enables simulating collective behaviors of many particles suspended in an active fluid medium with both even viscosity and odd viscosity.
Hydrodynamic flow field and frictional resistance coefficient of a disk rotating steadily in a compressible fluid layer with odd viscosity on a rigid substrate
We discuss hydrodynamic forces acting on a two-dimensional liquid domain that moves laterally within a supported fluid membrane in the presence of odd viscosity. Since active rotating proteins can accumulate inside the domain, we focus on the difference in odd viscosity between the inside and outside of the domain. Taking into account the momentum leakage from a two-dimensional incompressible fluid to the underlying substrate, we analytically obtain the fluid flow induced by the lateral domain motion, and calculate the drag and lift forces acting on the moving liquid domain. In contrast to the…
Analytical response functions for a compressible thin fluid layer with odd viscosity
The Open University（开放大学）；Max Planck Institute for Dynamics and Self-Organization (MPI–DS)（马克斯·普朗克动力学与自组织研究所）；Kyoto University（京都大学）；Wenzhou Institute, University of Chinese Academy of Sciences（中国科学院大学温州研究院）
We apply the geometric theory of swimming at low Reynolds number to the study of nearly circular swimmers in two-dimensional fluids with non-vanishing Hall, or "odd", viscosity. The Hall viscosity gives an off-diagonal contribution to the fluid stress-tensor, which results in a number of striking effects. In particular, we find that a swimmer whose area is changing will experience a torque proportional to the rate of change of the area, with the constant of proportionality given by the coefficient $η^o$ of odd viscosity. After working out the general theory of swimming in fluids with Hall visc…
When time reversal is broken the viscosity tensor can have a non vanishing odd part. In two dimensions, and only then, such odd viscosity is compatible with isotropy. Elementary and basic features of odd viscosity are examined by considering solutions of the wave and Navier-Stokes equations for hypothetical fluids where the stress is dominated by odd viscosity.
In common fluids, viscosity is associated with dissipation. However, when time-reversal-symmetry is broken a new type of non-dissipative `viscosity' may emerge. Recent theories and experiments on classical 2D systems with active spinning particles have heightened interest in odd viscosity, but a microscopic theory for it in active materials is still absent. Here we present such first-principles microscopic Hamiltonian theory, valid for both 2D and 3D, showing that odd viscosity is present in any system, equilibrium or not, with aligned spinning components. Our work substantially extends the ap…
Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam（阿姆斯特丹大学范德瓦尔斯-泽曼研究所物理学院）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）
In this work, we present observable consequences of parity-violating odd viscosity term in incompressible 2+1D hydrodynamics. For boundary conditions depending on the velocity field (flow) alone we show that: (i) The fluid flow quantified by the velocity field is independent of odd viscosity, (ii) The force acting on a closed contour is independent of odd viscosity, and (iii) The odd viscosity part of torque on a closed contour is proportional to the rate of change of area enclosed by the contour with the proportionality constant being twice the odd viscosity. The last statement allows us to d…
We present variational and Hamiltonian formulations of incompressible fluid dynamics with free surface and nonvanishing odd viscosity. We show that within the variational principle the odd viscosity contribution corresponds to geometric boundary terms. These boundary terms modify Zakharov's Poisson brackets and lead to a new type of boundary dynamics. The modified boundary conditions have a natural geometric interpretation describing an additional pressure at the free surface proportional to the angular velocity of the surface itself. These boundary conditions are believed to be universal sinc…
In this letter, we derive the Korteweg-de Vries (KdV) equation corresponding to the surface dynamics of a shallow depth ($h$) two-dimensional fluid with odd viscosity ($ν_o$) subject to gravity ($g$) in the long wavelength weakly nonlinear limit. In the long wavelength limit, the odd viscosity term plays the role of surface tension albeit with opposite signs for the right and left movers. We show that there exists two regimes with a sharp transition point within the applicability of the KdV dynamics, which we refer to as weak $(|ν_o|< \sqrt{gh^3}/6)$ and strong $(|ν_o|> \sqrt{gh^3}/6)$ parity-…
We study the problem of flow of a neutral gas past an infinite cylinder at right angle to its axis at low Reynolds number when the fluid is characterized by broken time-reversal invariance, and hence by odd viscosity in addition to the normal even one. We solve the Oseen approximation to Navier-Stokes equation and calculate the lift force which appears due to the odd viscosity.
We develop three asymptotic models of surface waves in a non-newtonian fluid with odd viscosity. This viscosity is also known as Hall viscosity and appears in a number of applications such as quantum Hall fluids or chiral active fluids. Besides the odd viscosity effects, these models capture both gravity and capillary forces up to quadratic interactions and take the form of nonlinear and nonlocal wave equations. Two of these models describe bidirectional waves while the third PDE studies the case of unidirectional propagation. We also prove the well-posedness of these asymptotic models in spac…
Ferrofluids are a synthetic magnetic colloid consisting of magnetized nanoparticles surrounded by a repulsive surfactant layer. When subjected to an external magnetic field the ferrofluid acquires a macroscopic magnetization density which leads to magnetic behavior that is intricately coupled to the ambient fluid dynamics. Ferrofluids share several features with the chiral active fluids composed of unidirectionally spinning hematite cubes, which have been shown to possess a 2D non-dissipative odd viscosity term (Nature Physics, 15, 1188-1194(2019)). In standard ferrofluid dynamics, 3D versions…
The flow of momentum and energy in a fluid is typically associated with dissipative transport coefficients: viscosity and thermal conductivity. Fluids that break certain symmetries such as mirror symmetry and time-reversal invariance can display non-dissipative transport coefficients called odd (or Hall) viscosities and thermal conductivities. The goal of this paper is to elucidate the microscopic origin of these dissipationless transport coefficients using kinetic theory. We show that odd viscosity and odd thermal conductivity arise when the linearized collision operator is not Hermitian. Thi…
In the paper [Chaos, Solitons & Fractals, 2006, vol. 30, pp. 851-858]the authors treat the boundary layer flow of a micropolar fluid along a horizontal flat plate with blowing or suction. The fluid viscosity and thermal conductivity are assumed functions of temperature. The boundary layer equations are transformed into ordinary ones and subsequently are solved using the Chebyshev finite difference method. However, there are some deficiencies and errors in this paper.
Wave-crests around obstacles in odd viscous liquids
Center for Computation and Theory of Soft Materials, Robert R. McCormick School of Engineering and Applied Science, Northwestern University（西北大学罗伯特·R·麦考密克工程与应用科学学院软物质计算与理论中心）；Department of Physics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校物理系）
We consider free surface dynamics of a two-dimensional incompressible fluid with odd viscosity. The odd viscosity is a peculiar part of the viscosity tensor which does not result in dissipation and is allowed when parity symmetry is broken. For the case of incompressible fluids, the odd viscosity manifests itself through the free surface (no stress) boundary conditions. We first find the free surface wave solutions of hydrodynamics in the linear approximation and study the dispersion of such waves. As expected, the surface waves are chiral and even exist in the absence of gravity and vanishing…
In simple fluids, such as water, invariance under parity and time-reversal symmetry imposes that the rotation of constituent 'atoms' are determined by the flow and that viscous stresses damp motion. Activation of the rotational degrees of freedom of a fluid by spinning its atomic building blocks breaks these constraints and has thus been the subject of fundamental theoretical interest across classical and quantum fluids. However, the creation of a model liquid which isolates chiral hydrodynamic phenomena has remained experimentally elusive. Here we report the creation of a cohesive two-dimensi…
Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）；Dutch Institute for Emergent Phenomena (DIEP), University of Amsterdam（阿姆斯特丹大学荷兰涌现现象研究所）
Wave turbulence theory of odd fluids and solids: kinetic equations and solutions
Eindhoven University of Technology（埃因霍温理工大学）；The University of Chicago（芝加哥大学）；Université Paris-Saclay（巴黎萨克雷大学）；École polytechnique（巴黎综合理工学院）；ESPCI Paris（巴黎高等物理化工国立学院）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；CNR-IAC（意大利国家研究委员会应用研究所）
We present a versatile perturbative calculation scheme to determine the mobility matrix for two and more particles in a low Reynolds number fluid with spatially variant viscosity. Assuming an asymptotic non-constant viscosity perturbation superimposed on a constant viscosity background, we exploit the Lorentz reciprocal theorem and a reflection method to obtain the leading order correction to the mobility matrix. We apply our approach firstly to an interface-like linear viscosity gradient with an extension much larger than the length scale of the particle separation. We find that the viscosity…
We investigate the bulk hydrodynamics of the chiral vortex matter on an arbitrary closed surface, extending the ideas of [20, 41]. Placing this important example of a chiral medium onto a curved geometry reveals the geometric nature of odd viscosity. The anomalous odd viscosity of the vortex matter is associated with a special interaction of point vortices with curvature.
City College of the City University of New York（纽约市立大学城市学院）；Western Washington University（西华盛顿大学）；Graduate Center of City University of New York（纽约市立大学研究生院）
Particle motion in non-Newtonian fluids can be markedly different than in Newtonian fluids. Here we look at the change in dynamics for a few problems involving rigid spherical particles in shear-thinning fluids in the absence of inertia. We give analytical formulas for sedimenting spheres, obtained by means of the reciprocal theorem, and demonstrate quantitively differences in comparison to a Newtonian fluid. We also calculate the first correction to the suspension viscosity, the Einstein viscosity, for a dilute suspension of spheres in a weakly shear-thinning fluid.
We consider an isotropic compressible non-dissipative fluid with broken parity subject to free surface boundary conditions in two spatial dimensions. The hydrodynamic equations describing the bulk dynamics of the fluid as well as the free surface boundary conditions depend explicitly on the parity breaking non-dissipative odd viscosity term. We construct a variational principle in the form of an effective action which gives both bulk hydrodynamic equations and free surface boundary conditions. The free surface boundary conditions require an additional boundary term in the action which resemble…
