Stokes flows in three-dimensional fluids with odd and parity-violating viscosities

Abstract

The Stokes equation describes the motion of fluids when inertial forces are negligible compared with viscous forces. In this article, we explore the consequence of parity-violating and non-dissipative (i.e. odd) viscosities on Stokes flows in three dimensions. Parity-violating viscosities are coefficients of the viscosity tensor that are not invariant under mirror reflections of space, while odd viscosities are those which do not contribute to dissipation of mechanical energy. These viscosities can occur in systems ranging from synthetic and biological active fluids to magnetized and rotating fluids. We first systematically enumerate all possible parity-violating viscosities compatible with cylindrical symmetry, highlighting their connection to potential microscopic realizations. Then, using a combination of analytical and numerical methods, we analyse the effects of parity-violating viscosities on the Stokeslet solution, on the flow past a sphere or a bubble and on many-particle sedimentation. In all the cases that we analyse, parity-violating viscosities give rise to an azimuthal flow even when the driving force is parallel to the axis of cylindrical symmetry. For a few sedimenting particles, the azimuthal flow bends the trajectories compared with a traditional Stokes flow. For a cloud of particles, the azimuthal flow impedes the transformation of the spherical cloud into a torus and the subsequent breakup into smaller parts that would otherwise occur. The presence of azimuthal flows in cylindrically symmetric systems (sphere, bubble, cloud of particles) can serve as a probe for parity-violating viscosities in experimental systems.

1. Introduction

An incompressible fluid is described by the Navier–Stokes equations

(1.1a,b)\begin{equation} \rho D_t \boldsymbol{v} = \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\sigma} + \boldsymbol{f} \quad\text{and}\quad\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{v} = 0, \end{equation}

in which $\boldsymbol {v}$ is the velocity field, $\rho$ is the density of the fluid and $D_t = \partial _t + \boldsymbol {v} \boldsymbol {\cdot } \boldsymbol {\nabla }$ is the convective derivative. Surface forces in the fluid are contained in the stress tensor $\boldsymbol {\sigma }$ and body forces such as gravity are contained in $\boldsymbol {f}$. In a Newtonian fluid, the stress tensor

(1.2)\begin{equation} \sigma_{ij} = \sigma_{ij}^{h}+\eta_{ijk\ell} \, \partial_\ell v_k \end{equation}

is composed of a hydrostatic stress $\sigma _{ij}^{h}$ present even in the undisturbed fluid (in standard fluids, $\sigma ^{h}_{ij} = - P \delta _{ij}$, where $P$ is the pressure) and of a viscous stress $\eta _{ijk\ell } \partial _\ell v_k$ that arises in response to velocity gradients.

In Stokes flows, the advection term in the Navier–Stokes equation is small compared with the viscous term (at low Reynolds numbers) and can therefore be neglected (Kim & Karrila 1991). Then, the momentum conservation in the fluid reduces to the (transient/unsteady) Stokes equation

(1.3)\begin{equation} \rho \partial_t \boldsymbol{v} = \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\sigma} + \boldsymbol{f}. \end{equation}

Stokes flows are the setting for phenomena ranging from the locomotion of microscopic organisms (Taylor 1951; Purcell 1977; Lapa & Hughes 2014) to microfluidics (Stone, Stroock & Ajdari 2004) and sedimentation (Ramaswamy 2001; Guazzelli, Morris & Pic 2009; Goldfriend, Diamant & Witten 2017; Chajwa, Menon & Ramaswamy 2019). In usual fluids such as air and water, the viscosity tensor has only two components, the shear viscosity $\mu$ and the bulk viscosity $\zeta$, the latter of which can be ignored in incompressible flows. Hence, the Stokes equation takes the very simple form

(1.4)\begin{equation} \rho \partial_t \boldsymbol{v} ={-} \boldsymbol{\nabla} P + \mu \Delta \boldsymbol{v} + \boldsymbol{f} \end{equation}

along with $\boldsymbol {\nabla } \boldsymbol {\cdot }{\boldsymbol {v} } = 0$ ($\varDelta$ is the Laplacian). As the Stokes equation is linear, the flow $\boldsymbol {v}$ due to an arbitrary force field $\boldsymbol {f}$ can be obtained from the Green function of (1.4), called the Oseen tensor, or Stokeslet (see below for precise definitions). This point response can be leveraged to describe the flow due to a disturbance in the fluid or to describe the hydrodynamic interactions between colloidal particles.

In this article, we consider a class of fluids called parity-violating fluids. In these fluids, parity (i.e. mirror reflection) is broken at the microscopic level, either by the presence of external fields (e.g. a magnetic field) or by internal activity (e.g. microscopic torques). Parity-violating fluids include fluids under rotation (Nakagawa 1956), magnetized plasma (Chapman 1939) and neutral polyatomic gases under a magnetic field (Korving et al. 1967), but also artificial and biological fluids composed of active elements (Condiff & Dahler 1964; Tsai et al. 2005; Soni et al. 2019; Yamauchi et al. 2020) or vortices (Wiegmann & Abanov 2014) as well as quantum fluids describing the flow of electrons in solids under a magnetic field (Bandurin et al. 2016; Berdyugin et al. 2019). As a consequence of parity violation, the viscous response (summarized by the viscosity tensor) is richer than in usual fluids. In three-dimensional polyatomic gases subject to a magnetic field (Beenakker & McCourt 1970), two non-dissipative parity-violating viscosities have been measured (Korving et al. 1967; Beenakker & McCourt 1970) (called $\eta _4$ and $\eta _5$ in those papers). In general, even more parity-violating viscosities can exist. In § 2, we classify all possible viscous coefficients of three-dimensional fluids with cylindrical symmetry. Our classification is based on two criteria: whether the viscosities violate parity and whether they contribute to energy dissipation in the fluid. We provide a summary of the results that can be used without extensive knowledge of group theory, as well as the underlying group-theoretical analysis. In § 3, we discuss the effects of an anti-symmetric hydrodynamic stress. In § 4, we analyse in detail how the Stokeslet is affected by the presence of the additional parity-violating viscous coefficients. Qualitatively, the most important change is the presence of an azimuthal velocity in the Stokeslet, which normally vanishes. These results allow us to describe the flow past an obstacle in § 5, in which we again find the presence of azimuthal flows, even past a sphere and a spherically symmetric bubble. Finally, in § 6, we illustrate the large-scale consequences of parity-violating viscosities in the example of the sedimentation of a cloud of particles under gravity.

2. The viscosity tensor of a parity-violating fluid

2.1. Constraints from spatial symmetries

In three dimensions, the rank-four viscosity tensor $\eta _{ijk\ell }$ has 81 possible elements. However, the form of the viscosity tensor is constrained by the symmetries of the fluid it describes. For example, the most general form of the viscosity tensor for an isotropic fluid is given by

(2.1)\begin{equation} \eta_{i j k \ell} = \zeta \delta_{ij} \delta_{k \ell} + \mu \big( \delta_{ik} \delta_{j \ell } + \delta_{i\ell } \delta_{jk} - \tfrac{2}{3} \delta_{ij} \delta_{k \ell } \big) + \eta_{\text{R}} (\delta_{ik} \delta_{j \ell} - \delta_{i \ell} \delta_{jk}), \end{equation}

which contains just three independent coefficients: the shear viscosity $\mu$, the bulk viscosity $\zeta$ and the rotational viscosity $\eta _{{R}}$ (de Groot 1962). These three coefficients are invariant under parity: the exact same coefficients describe the evolution of a fluid and the image of the fluid in a mirror. In an anisotropic fluid, however, this need not be the case.

To systematically classify all the viscosity coefficients compatible with a given set of symmetries, we use the language of group theory. A general introduction to group theory in the context of fluid mechanics and applied mathematics is given in Cantwell (2002) and Hydon, Hydon & Crighton (2000). Readers unfamiliar with this formalism can skip directly to (2.7), which generalizes the expression in (2.1). Figure 1 and table 2 provide a visual summary of the possible symmetries of the fluid illustrated by microscopic examples, along with the allowed entries in the viscosity tensor for each symmetry class. In general, the less symmetry the fluid has (moving down figure 1), the larger the number of independent viscosity coefficients. Our symmetry analysis can also be read as a guide on how to build parity-violating fluids from microscopic constituents. The symmetry of the fluid can be designed using the interplay between the symmetries of the microscopic constituents and the way these constituents are collectively arranged in the fluid (for instance, whether they are aligned); see figure 1(a–g) and accompanying caption for concrete examples.

Figure 1. Axial symmetry groups, examples of their microscopic realizations and their constraints on the viscosity tensor. (a–g) Examples of microscopic systems for each axial point group (with cylindrical symmetry about the $\boldsymbol {\hat {z}}$ axis) in (h). Each example is distinguished from the others by the presence of or absence of additional spatial symmetries. (a) A fluid of spherical particles is invariant under all rotations and reflections. (b) A fluid of randomly oriented helices (with fixed chirality) is invariant under all rotations, but no reflections. (c) A fluid of elongated (nematic) particles that align with each other is invariant under reflections across all planes parallel and perpendicular to the $\boldsymbol {\hat {z}}$ axis. (d) A fluid of chiral particles that align is invariant under ${\rm \pi} /2$ rotations about any axis perpendicular to the $\boldsymbol {\hat {z}}$ axis, but not any reflections. (e) A fluid of electric dipoles under an electric field is invariant under reflections across all planes parallel, but not perpendicular, to the $\boldsymbol {\hat {z}}$ axis. (f) A fluid of charged particles under a magnetic field (or a fluid of active particles rotating about a fixed axis) is invariant under reflections across all planes perpendicular, but not parallel, to the $\boldsymbol {\hat {z}}$ axis. (g) A fluid of chiral particles that rotate about a fixed axis has no additional symmetry beyond cylindrical. The group–subgroup relations between axial point groups are shown by arrows in (h). Groups drawn in identical colour place identical constraints on the viscosity tensor. The groups $K_h \equiv O(3)$ and $K\equiv SO(3)$ (in black) give rise to the viscosity tensor of an isotropic fluid in (2.1). The groups $D_{\infty h}$, $C_{\infty v}$ and $D_\infty$ (in blue) allow all the coefficients in black in (2.7) and table 2. Some of these coefficients are anisotropic, and all are invariant under reflections parallel and perpendicular to the $\boldsymbol {\hat {z}}$ axis (even though the microscopic components are not necessarily invariant under such reflections). The groups $C_{\infty h}$ and $C_\infty$ allow for additional coefficients that change sign under reflection across planes containing the $\boldsymbol {\hat {z}}$ axis. These coefficients are shown in red in (2.7) and table 2. For more details of the symmetry groups, see Shubnikov (1988) and Hahn (2005) (in particular table § 10.1.4.2, p. 799; and figure § 10.1.4.3, p. 803).

We begin by noting that under a rotation or reflection of space, the viscosity tensor transforms as

(2.2)\begin{equation} \eta_{i j k \ell} = {\mathsf{R}}_{i i'} {\mathsf{R}}_{j j'} {\mathsf{R}}_{k k'} {\mathsf{R}}_{\ell \ell'} \eta_{i' j' k' \ell'}, \end{equation}

where $\boldsymbol{\mathsf{R}}$ is an orthogonal matrix that implements the transformation. We say a fluid is parity-violating if its properties are not invariant under some improper rotation, i.e. a rotation combined with a reflection. In three dimensions, the most general viscosity tensor invariant under all proper rotations (i.e. under the group $SO(3)$, consisting of the transformations $\boldsymbol{\mathsf{R}} \in O(3)$ with $\det (\boldsymbol{\mathsf{R}}) = 1$) is automatically invariant under all improper rotations as well (i.e. under the whole group $O(3)$). This happens because any improper rotation can be written as a proper rotation times $-\mathbb {1} = \operatorname {diag}(-1,-1,-1)$: the four copies of $- \mathbb {1}$ always cancel out of (2.2).

Hence, we have to consider anisotropic fluids in order to see the effects of parity violation. Here, we focus on systems with cylindrical symmetry (i.e. those invariant under rotation about a fixed axis $\hat {\boldsymbol {z}}$). The set of all reflections and rotations that leave a fluid globally unchanged forms a group $G$. It turns out that there are just nine possible symmetry groups that respect cylindrical symmetry (Shubnikov 1988; Hahn 2005). These groups, known as the axial point groups, are shown in figure 1 and differ from each other by which combinations of horizontal and/or vertical reflections are present (see Appendix B, in particular figure 7). Just as invariance under $O(3)$ and $SO(3)$ placed identical constraints on the viscosity tensor, some of the anisotropic symmetry groups in figure 1 place identical constraints on the viscosity tensor. They break into two classes, drawn in blue and in red in figure 1. Fluids with the symmetry groups $D_{\infty {h}}$, $C_{\infty {v}}$ or $D_{\infty }$ (in blue) have an anisotropic viscosity tensor that is invariant under all reflections parallel and perpendicular to the $\hat {\boldsymbol {z}}$ axis. We call these fluids parity-preserving cylindrical, and examples include aligned nematic particles ($D_{\infty {h}}$), aligned helices ($C_{\infty v}$) and dipolar molecules in an electric field ($C_{\infty v}$) shown in figure 1(c–e). In contrast, fluids with the symmetry groups $C_{\infty h}$ or $C_{\infty }$ (in red) allow additional terms in their viscosity tensor. Examples of such fluids shown in figure 1(f,g) include spherical charged particles ($C_{\infty {h}}$) and chiral charged particles ($C_{\infty }$) in a magnetic field. The additional allowed viscosity coefficients acquire a minus sign when reflected across any plane containing the $\hat {\boldsymbol {z}}$ axis. We call these fluids parity-violating cylindrical.

It is useful to organize the components of the viscosity tensor by decomposing the stress $\sigma _{ij}$ and velocity gradient $\dot {e}_{k \ell } \equiv \partial _\ell v_k$ tensors on a basis of $3 \times 3$ matrices $\tau _{ij}^A$ ($A = 1 \dots 9$) corresponding to a decomposition into irreducible representations of the orthogonal group $O(3)$ (see Appendix B). In this notation, the viscosity tensor $\eta _{ijk\ell }$ is expressed as a $9 \times 9$ matrix (see Scheibner, Irvine & Vitelli (2020a) and Scheibner et al. (2020b), in which this notation is also used to describe elastic and viscoelastic media). The basis consists of

1. (i) a diagonal matrix $\tau _{i j}^1 = {\mathsf{C}}_{ij} = \sqrt {\frac {2}{3}}\delta _{ij}$ corresponding to pressure and dilation,

2. (ii) three anti-symmetric matrices $\tau _{i j}^{A+1} = {\mathsf{R}}_{ij}^A = \epsilon _{Aij}$ corresponding to torques and vorticity, and

3. (iii) five traceless symmetric matrices $\tau _{i j}^{A+5} = {\mathsf{S}}^A_{ij}$ corresponding to shear stresses and shear strain rates, whose expressions are

(2.3)\begin{equation} \left.\begin{gathered} \boldsymbol{\mathsf{S}}^1 =\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix},\quad \boldsymbol{\mathsf{S}}^2 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix},\quad \boldsymbol{\mathsf{S}}^3 = \begin{bmatrix} \frac{-1}{\sqrt{3}} & 0 & 0 \\ 0 & \frac{-1}{\sqrt{3}} & 0 \\ 0 & 0 & \frac{2}{\sqrt{3}} \\ \end{bmatrix},\\ \boldsymbol{\mathsf{S}}^4 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix},\quad \boldsymbol{\mathsf{S}}^5 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ \end{bmatrix}. \end{gathered}\right\} \end{equation}

Note that $\tau _{ij}^A \tau _{ij}^B = 2 \delta ^{AB}$. Defining

(2.4a–c)\begin{equation} \sigma^A \equiv \sigma_{ij} \, \tau_{ij}^A, \quad \dot{e}^A \equiv \dot{e}_{i j} \, \tau_{ij}^A,\quad \eta^{AB} = \tfrac{1}{2} \tau_{ij}^A \, \eta_{ijk\ell} \, \tau_{k\ell}^B, \end{equation}

we may write

(2.5)\begin{equation} \sigma^A = \eta^{AB} \, \dot{e}^B. \end{equation}

We can transform back to Cartesian tensors via

(2.6a–c)\begin{equation} \sigma_{ij} = \tfrac{1}{2} \,\sigma^A \tau_{ij}^A, \quad \dot{e}_{ij} = \tfrac{1}{2} \, \dot{e}^A \tau_{ij}^A, \quad \eta_{ijk\ell} = \tfrac{1}{2} \, \tau_{ij}^A \, \eta^{A B} \, \tau_{k\ell}^B. \end{equation}

The most general form of $\eta ^{A B}$ satisfying cylindrical symmetry about the $\boldsymbol {\hat {z}}$ axis is

(2.7)

in which the parity-violating viscosities are written in red (these are only allowed in the groups drawn in red in figure 1). An explicit list of parity-violating viscosities is also given in the caption of table 2. Concretely, these entries of the viscosity tensor relate components of the strain rate and stress tensors with different parities under a reflection by a mirror plane containing the $\hat {z}$ axis (see table 1 for the parities of the basis tensors used in (

2.7

) under the reflection $P_y$). Finally, we have restricted our attention to fluids invariant under continuous rotations about the $\hat {\boldsymbol {z}}$ axis, because they arise when an originally isotropic fluid is submitted to a single external field. In general, the fluid can be even less symmetric, for example when the fluid is invariant under a discrete point group. This can happen when multiple external fields that are not parallel to each other are applied, or in electron fluids in crystals (Cook & Lucas 2019; Rao & Bradlyn 2020; Toshio, Takasan & Kawakami 2020; Varnavides et al. 2020).

Table 1. Effect of the reflection $P_y$ on the components of the stress and strain rate used in (2.7). The components with a $1$ are invariant under $P_y$, while those with a $-1$ change sign. The action of $P_y$ on Cartesian coordinates is $\operatorname {diag}(1,-1,1)$.

Table 2. Classes of viscosity tensors and allowed viscosity coefficients. The coefficients refer to (2.7). Parity-violating viscosities (those only present in the last column) are highlighted in red. Explicitly, these are ${\eta _A^{e}}$, ${\eta _{Q,2}^e}$, ${\eta _{Q,3}^e}$, ${\eta _{R}^o}$, ${\eta _1^o}$, ${\eta _2^o}$, ${\eta _A^{o}}$, ${\eta _{Q,2}^o}$, ${\eta _{Q,3}^o}$. See Hahn (2005) for more details of the symmetry groups.

2.2. Dissipative and non-dissipative viscosities

In addition to the decomposition based on spatial symmetries discussed in § 2.1, the viscosity tensor can be decomposed into symmetric and anti-symmetric parts:

(2.8)\begin{equation} \eta_{ijk\ell} = \eta^{{e}}_{ijk\ell} + \eta^{{o}}_{ijk\ell}, \end{equation}

in which $e/o$ (standing for even/odd) label the symmetric and anti-symmetric parts of the tensor, satisfying $\eta ^{{o}}_{ijk\ell } = - \eta ^{{o}}_{k\ell ij}$ and $\eta ^{{e}}_{ijk\ell } = \eta ^{{e}}_{k\ell ij}$. The rate of mechanical energy lost by the fluid due to viscous dissipation is (see Appendix C.1)

(2.9)\begin{equation} \dot{w} = \sigma_{ij} \, \partial_j v_i = \eta_{i j k \ell} \, (\partial_j v_i) \, (\partial_\ell v_k) = \tfrac{1}{2} \, \eta^{A B} \dot{e}^A \dot{e}^B. \end{equation}

Hence, the anti-symmetric part $\eta ^{{o}}_{ijk\ell }$ is purely non-dissipative, because $\eta ^{{o}}_{ijk\ell } (\partial _j v_i) \, (\partial _\ell v_k) = 0$. In contrast, the symmetric part $\eta ^{\text {e}}_{ijk\ell }$ does indeed contribute to viscous dissipation. In a standard fluid, the viscous dissipation corresponds to a rate of entropy production $\dot s = (1/T) \, \sigma _{ij} \partial _j v_i$, where $T$ is temperature. The symmetry of the viscosity tensor has also been related to Onsager reciprocity relations in equilibrium fluids, in which one expects $\eta ^{{o}}_{ijk\ell } = 0$ when microscopic reversibility is satisfied (Onsager 1931; de Groot & Mazur 1954; de Groot 1962).

The dissipative part $\eta ^{{e}}_{ijk\ell }$ of the viscosity tensor corresponds to the symmetric part of the matrix $\eta ^{A B}$ in (2.5), while the non-dissipative part $\eta ^{{o}}_{ijk\ell }$ corresponds to its anti-symmetric part. Hence, we have split all off-diagonal terms in (2.7) into odd and even parts (except when one of these is already ruled out by spatial symmetry). The non-dissipative viscosities all have an ‘o’ superscript. In table 2, we classify the viscosity coefficients in (2.7) based on whether they are dissipative or not, and on the symmetry groups in which they can occur.

3. The stress tensor of a parity-violating fluid

In parity-violating fluids, it is possible that the stress tensor is asymmetric. An asymmetric stress tensor means that the fluid experiences torques. While this is not possible for classical particles interacting through central pairwise interactions, non-central pairwise interactions are sufficient to contribute an anti-symmetric part to the stress tensor (Condiff & Dahler 1964). This occurs, for instance, in polyatomic gases since the particles are not spherical (Condiff & Dahler 1964). In general, anisotropic fluids and fluids with non-symmetric stress require additional hydrodynamic fields, such as the average alignment or angular velocity of the constituents (Ariman, Turk & Sylvester 1973; Ramkissoon 1976; Hayakawa 2000). Here, we assume that all other order parameters relax much faster than the velocity field, so that their dynamics can safely be ignored. When the stress tensor is constrained to be symmetric, the viscosity has the additional symmetry $\eta _{ijk\ell } = \eta _{j i k \ell }$. (Similarly, we have $\eta _{ijk\ell } = \eta _{i j \ell k}$ when vorticity does not affect the viscous response.)

In addition to the viscous stresses discussed in the previous section, the stress tensor also contains a hydrostatic part $\sigma _{ij}^{h}$ present even when there is no velocity gradient. Under the assumption of cylindrical symmetry, the hydrostatic stress takes the form

(3.1)\begin{equation} \sigma_{ij}^{h} ={-} P \delta_{ij} + \gamma {\mathsf{S}}^3_{ij} - \tau_z {\mathsf{R}}^3_{ij}, \end{equation}

in which $P$ is the pressure, $\gamma$ is a hydrostatic shear stress and $\tau _z$ is a hydrostatic torque. In this paper, we assume that $\gamma$ and $\tau _z$ are frozen (i.e. they relax to a constant value on very short time scales), like in Banerjee et al. (2017), Markovich & Lubensky (2021) and Han et al. (2021). In addition, we assume that $\tau _z$ and $\gamma$ are spatially uniform. In this case, they do not contribute to the term $\partial _j \sigma _{ij}$ in the Stokes equation (1.3), and therefore do not affect the form of the Stokeslet, which we discuss in the next section. However, a constant hydrostatic torque $\sigma _{ij}^{h} = - \epsilon _{ijk} \tau _k$ can induce a net torque $T_k$ on an object immersed in the fluid:

(3.2)\begin{equation} T_k = \oint_{\partial \mathscr{V}} \hat{n}_i \, \sigma_{ji}^h \, \epsilon_{jk\ell} \, x_\ell \, \text{d}^2x = 2 \tau_k \int_{\mathscr{V}} \text{d}^3 \,x =2 \tau_k V, \end{equation}

where $V$ is the volume of the object $\mathscr {V}$, in which we have assumed that $\hat {n}_i \sigma _{ki}$ is the force on a unit area with normal $\hat {n}_i$ (this boundary condition might not always hold true, depending on the microscopic interactions and on the definition of the stress). The effect of the hydrostatic torque $\tau _z {\mathsf{R}}^3_{ij}$ on a sphere is further discussed in § 5.3. Similarly, the effect of a homogeneous shear stress $\gamma {\mathsf{S}}_{ij}^3$ is to shear a soft deformable body, although it has no effect on rigid bodies.

4. The Stokeslet of a parity-violating fluid

4.1. Oseen tensor and Stokeslet

The (transient) Stokes equation for an incompressible fluid found in (1.3) can be written as

(4.1)\begin{equation} \rho \partial_t v_i ={-} \partial_i P + \partial_j [\eta_{i j k \ell} \partial_\ell v_k] + f_i \quad \text{with} \ \partial_i v_i = 0, \end{equation}

in which we have used the expression (1.2) of the viscous stress. In reciprocal space (see Appendix A for Fourier transform conventions):

(4.2)\begin{equation} -\text{i} \omega \rho v_i ={-} \text{i} q_i P - q_j q_\ell \eta_{i j k \ell} v_k + f_i \quad\text{with}\ \text{i} q_i v_i = 0. \end{equation}

These equations can be written as

(4.3)\begin{equation} \boldsymbol{\mathsf{M}}(\boldsymbol{q}, \omega) \boldsymbol{v} ={-} \text{i} P \boldsymbol{q} + \boldsymbol{f} \quad\text{with}\ \boldsymbol{q} \boldsymbol{\cdot} \boldsymbol{v} = 0, \end{equation}

in which we have defined the matrix

(4.4)\begin{equation} {\mathsf{M}}_{i k}(\boldsymbol{q}, \omega) = q_j q_\ell \eta_{i j k \ell} - \text{i} \omega \rho \, \delta_{i k}. \end{equation}

The matrix $\boldsymbol{\mathsf{M}}(\boldsymbol {q}, \omega )$ is always invertible at finite $\boldsymbol {q}$ provided that the dissipation rate $\dot {w}$ in (2.9) is strictly positive (see Appendix C.1). Under this hypothesis, we apply $M^{-1}(\boldsymbol {q}, \omega )$ to (4.3). We then take the scalar product with $\boldsymbol {q}$ to obtain the pressure $P$, and then replace $P$ with its expression to obtain the velocity, giving

(4.5a,b)\begin{equation} \text{i} P = \frac{\boldsymbol{q} \boldsymbol{\cdot} (\boldsymbol{\mathsf{M}}^{{-}1} \boldsymbol{f})}{{\boldsymbol{q} \boldsymbol{\cdot} (\boldsymbol{\mathsf{M}}^{{-}1} \boldsymbol{q})}}\quad\text{and}\quad \boldsymbol{v} ={-} \frac{{\boldsymbol{q} \boldsymbol{\cdot} (\boldsymbol{\mathsf{M}}^{{-}1} \boldsymbol{f})}}{{\boldsymbol{q} \boldsymbol{\cdot} (\boldsymbol{\mathsf{M}}^{{-}1} \boldsymbol{q})}} \, \boldsymbol{\mathsf{M}}^{{-}1} \boldsymbol{q} + \boldsymbol{\mathsf{M}}^{{-}1} \boldsymbol{f}. \end{equation}

The expression of the velocity in terms of the force is then

(4.6)\begin{equation} \boldsymbol{v} = \boldsymbol{\mathsf{G}}(\boldsymbol{q}, \omega) \boldsymbol{f}, \end{equation}

in which

(4.7)\begin{equation} {\mathsf{G}}_{ij} (\boldsymbol{q}, \omega) \equiv \left( [{\mathsf{M}}^{{-}1}]_{ij} - \frac{ [{\mathsf{M}}^{{-}1}]_{im}q_m q_n [{\mathsf{M}}^{{-}1}]_{nj} }{q_k [{\mathsf{M}}^{{-}1}]_{k\ell} q_\ell } \right)\end{equation}

is the Green function of the Stokes equation, which is usually called the (reciprocal space) Oseen tensor (Kim & Karrila 1991; Kuiken 1996). Formally, it is defined so that $v_{i} = {\mathsf{G}}_{i j}$ is a solution of (4.1) with $\boldsymbol {f} = \delta (\boldsymbol {x}) \boldsymbol {e}_j$, where $\boldsymbol {e}_j$ is the unit vector in direction $j$. For an isotropic incompressible fluid, we recover the usual (reciprocal space) Oseen tensor

(4.8)\begin{equation} {\mathsf{G}}_{ij}^{{iso}}(\boldsymbol{q}, \omega=0) = \frac{1}{\mu q^2} \left(\delta_{ij} - \frac{q_i q_j}{q^2} \right). \end{equation}

When the symmetric part of $\eta _{ijk\ell }$ (corresponding to dissipation) vanishes, the second term of (4.7) diverges at $\omega = 0$ (but finite $\boldsymbol {q}$) because ${\mathsf{M}}^{-1}_{k \ell }$ is strictly anti-symmetric under exchange of $k$ and $\ell$, while the product $q_k q_\ell$ is symmetric (so the denominator $q_k [M^{-1}]_{k\ell } q_\ell$ vanishes). This corresponds to a divergence of the characteristic time scale associated with viscous relaxation: in this case, the Stokes approximation is not valid. In the following, we assume that viscous relaxation is fast enough, and focus on steady solutions that correspond to the steady Oseen tensor $\boldsymbol{\mathsf{G}}(\boldsymbol {q}) \equiv \boldsymbol{\mathsf{G}}(\boldsymbol {q}, \omega = 0)$.

The real-space Oseen tensor is then

(4.9)\begin{equation} {\mathsf{G}}_{i j}(\boldsymbol{x}) = \frac{1}{(2{\rm \pi})^3}\int {\text{e}}^{{\text{i}} \boldsymbol{q} \boldsymbol{\cdot} \boldsymbol{x}} \, {\mathsf{G}}_{i j}(\boldsymbol{q}) \, \text{d}^3 q \end{equation}

and the flow generated by a point force $\boldsymbol {f}(\boldsymbol {x}) = \boldsymbol {f} \delta (\boldsymbol {x})$ is the Stokeslet

(4.10)\begin{equation} \boldsymbol{v}(\boldsymbol{x}) = \boldsymbol{\mathsf{G}}(\boldsymbol{x}) \boldsymbol{f}. \end{equation}

When the anti-symmetric (non-dissipative) part of the viscosity tensor vanishes ($\eta _{ijk\ell }^o = 0$), the matrix $\boldsymbol{\mathsf{M}}$ defined by (4.4) is symmetric. Hence, ${\mathsf{M}}^{-1}$ and the Green function $\boldsymbol{\mathsf{G}}(\boldsymbol {q})$ are symmetric as well. The symmetry of the reciprocal-space Green function ${\mathsf{G}}_{i j}(\boldsymbol {q}) = {\mathsf{G}}_{j i}(\boldsymbol {q})$ is equivalent to ${\mathsf{G}}_{i j}(\boldsymbol {x}) = {\mathsf{G}}_{j i}(\boldsymbol {x})$ in physical space. This is the expression of Lorentz reciprocity (Masoud & Stone 2019, § 4.2, (4.7)), which can be interpreted as a symmetry in the exchange between the source (producing a force) and the receiver (measuring the velocity field). Conversely, Lorentz reciprocity is broken by the presence of non-dissipative (or, equivalently, odd) viscosities.

We can now analyse the effect of parity-violating viscosities on the Stokeslet. Unlike the situation in a two-dimensional, isotropic incompressible fluid (see Appendix D), in three dimensions the odd and parity-violating viscosities can modify the Stokeslet velocity field. To see this, we compute the real-space Oseen tensor or Stokeslet in different cases, using both numerical and analytical methods. The qualitative changes compared with usual isotropic fluids can be anticipated without any computation from symmetry arguments. When the driving force is along the axis of azimuthal symmetry, a fluid from the classes ‘isotropic’ and ‘parity-preserving cylindrical’ in table 2 cannot exhibit an azimuthal flow because a reflection symmetry constrains the azimuthal component of the velocity to be opposite to itself – this is indeed the case for the standard Stokeslet solution (Kim & Karrila 1991). In contrast, an azimuthal flow is allowed when parity-violating terms are introduced in the viscosity tensor (class ‘parity-violating cylindrical’ in table 2).

4.2. Stresslet, rotlet and multipolar responses

Since the Stokeslet is a response to a point perturbation, multipolar responses can be computed by taking derivatives of the Green function in (4.9) (see  Kim & Karrila 1991). For example, consider a force dipole defined by a point force $\boldsymbol {f}$ at $\frac 12 \delta \boldsymbol {r}$ and a point force $-\boldsymbol {f}$ at $-\frac 12 \delta \boldsymbol {r}$. The corresponding fluid velocity is given by

(4.11)\begin{equation} v_i( \boldsymbol{r}) = {\mathsf{G}}_{ik} \big(\boldsymbol{r} - \tfrac12 \delta \boldsymbol{r}\big) f_k - {\mathsf{G}}_{ik} \big(\boldsymbol{r} + \tfrac12\delta \boldsymbol{r}\big) f_k \approx{-}\partial_j {\mathsf{G}}_{ik} (\boldsymbol{r}) \delta r_j f_k \equiv {\mathsf{H}}_{ijk} (\boldsymbol{r}) \delta r_j f_k. \end{equation}

The tensor ${\mathsf{H}}_{ijk}$ is often decomposed into two contributions: the symmetric part ${\mathsf{S}}_{ijk} = \frac 12( {\mathsf{H}}_{ijk} + {\mathsf{H}}_{ikj} )$, which represents the response to point shears (also known as stresslet), and the anti-symmetric part $A_{i\ell } = \frac 12 \epsilon _{jk \ell } {\mathsf{H}}_{ijk}$, which represents the response to point torques $T_\ell$ (also known as rotlet). As discussed in § 3, such point torques can arise from a hydrostatic torque in the fluid. An explicit expression of the Oseen tensor ${\mathsf{G}}_{i k}$ is given by (H6) of Appendix H in a perturbative case, from which the stresslet and rotlet can be deduced using (4.11).

4.3. General numerical solution

To determine the physical-space Stokeslet or Oseen tensor $G(\boldsymbol {x})$, one must compute the inverse Fourier transform (4.9). This can be done numerically in the general case, in which analytical solutions are not easily accessible. To do so, we evaluate (4.7) on a discrete grid in reciprocal space (each component of $\boldsymbol {q}$ ranges from $-Q$ to $+Q$ with increments $\delta q$). This allows us to resolve length scales larger than a few ${\rm \pi} /Q$ but smaller than ${\rm \pi} /\delta q$. We then use the fast Fourier transform algorithm to compute the real-space Oseen tensor (or the real-space Stokeslet). To avoid numerical instabilities (Gibbs oscillations) due to the sharp cutoff in reciprocal space, we regularize the integrand in (4.9) with a Gaussian kernel $\exp ({- {\rm \pi}q^2/ 4 Q^2})$ (Cortez 2001; Gómez-González & del Álamo 2013). This procedure allows us to compute the Stokeslet for an arbitrary viscosity tensor. Our code for this computation is available at https://github.com/talikhain/StokesletFFT.

We consider an external force parallel to the $\hat {\boldsymbol {z}}$ axis and examine the perturbative effect of each coefficient separately. We set the normal shear viscosity to $\mu _1= \mu _2 = \mu _3 = 1$ and vary each of the other viscosity coefficients one by one, setting them to be $\eta _i = 0.01 \mu$. This flow is visualized for each viscosity in figure 8 of Appendix F, in which we also validate the numerical method using the exact solution discussed in the next section (see figure 10). We find that the viscosity coefficients that give rise to an azimuthal flow are

(4.12a–g) \begin{equation} \eta_R^o, \quad\eta^e_{Q,2}, \quad \eta^o_{Q,2}, \quad \eta^e_{Q,3}, \quad \eta^o_{Q,3}, \quad \eta_1^o, \quad \eta_2^o. \end{equation}

The list of viscosity coefficients that we found to generate $v_{\phi }$ are a subset of the parity-violating viscosities (in red in (2.7) and listed in the caption of table 2), as expected. In fact, the only parity-violating viscosities that do not give rise to azimuthal flow are $\eta ^e_A$ and $\eta ^o_A$. This is because we have assumed that the flow is incompressible. First, the term $(\eta _A^e - \eta _A^o) \boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {v}$ vanishes because $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {v} = 0$. Second, the term $(\eta _A^e + \eta _A^o) \omega _3$ contributes to the component $\sigma _C$ of the stress, and can therefore be absorbed in the pressure.

4.4. The Stokeslet of an odd viscous fluid: exact solution

We now consider a particular case in which the real-space Stokeslet can be computed analytically. First, we set $\mu \equiv \mu _1=\mu _2=\mu _3$ and consider only the odd shear viscosities $\eta _1^o$ and $\eta _2^o$ (all the other viscosities are assumed to vanish, except perhaps the bulk $\xi$ viscosity which drops out of the Stokes equation). In this case, the matrix $\boldsymbol{\mathsf{M}}(\omega = 0)$ defined by (4.4) takes the form

(4.13)\begin{equation} \boldsymbol{\mathsf{M}}(\omega = 0) =\begin{bmatrix} \mu q^2 & \eta_1^o(q_x^2 + q_y^2) - \eta_2^o q_z^2 & -\eta_2^o q_y q_z \\ - \eta_1^o(q_x^2 + q_y^2) + \eta_2^o q_z^2 & \mu q^2 & \eta_2^o q_x q_z \\ \eta_2^o q_y q_z & -\eta_2^o q_x q_z & \mu q^2 \end{bmatrix}. \end{equation}

Taking $\boldsymbol {f} = -\boldsymbol {\hat z} F_z \delta ^3(\boldsymbol {x})$, and defining $q_{\perp }^2 \equiv q_x^2 + q_y^2$, we find the full expressions for the velocity and pressure in Fourier space by using (4.5a,b):

(4.14) \begin{gather} \boldsymbol{\hat{v}(\boldsymbol{q})}= \frac{F_z}{N(\boldsymbol{q})} \begin{bmatrix} q_z(q_y(-(\eta_1^o + \eta_2^o)q_{{\perp}}^2 +\eta_2^o q_z^2) + \mu q_x (q_{{\perp}}^2 + q_z^2))\\ q_z(q_x((\eta_1^o+\eta_2^o)q_{{\perp}}^2 -\eta_2^o q_z^2) + \mu q_y (q_{{\perp}}^2 + q_z^2))\\ -\mu q_{{\perp}}^2(q_{{\perp}}^2 + q_z^2)\end{bmatrix}, \end{gather}

(4.15)\begin{gather}\hat{p}(\boldsymbol{q}) = {\text{i}} \frac{F_z}{N(\boldsymbol{q})} \, q_z[(\eta_1^o q_{{\perp}}^2 - \eta_2^o q_z^2)((\eta_1^o + \eta_2^o)q_{{\perp}}^2 - \eta_2^oq_z^2) + \mu^2(q_{{\perp}}^2 + q_z^2)^2]. \end{gather}

in which $N(\boldsymbol {q}) = \mu ^2(q_{\perp }^2 + q_z^2)^3+q_z^2((\eta _1^o + \eta _2^o)q_{\perp }^2 - \eta _2^o q_z^2)^2$. Second, we assume that $\eta _1^o = -2 \eta _2^o$, for which simplifications occur in (4.14)–(4.15). This particular case occurs in the limit of low magnetic field regime in experiments on polyatomic gases (see (13) in Hulsman et al. 1970), and was also obtained in a theoretical Hamiltonian description of fluids of spinning molecules (Markovich & Lubensky 2021). In this limit, the viscosity matrix can be seen as a simple combination of isotropic contractions and rotations about the $\boldsymbol {\hat {z}}$ axis in the space of shears (see Appendix G).

In order to find the real-space solution, we compute the inverse Fourier transform in (4.9) (see Appendix G for the detailed calculation). Parameterizing the final flow field by $\gamma = \eta _2^o/ \mu$, we obtain the velocity field

(4.16)\begin{gather} v_r( {\Huge\unicode[Palace Script MT]{x0072}}, \theta) ={-}\frac{F_z}{4{\rm \pi} \eta_2^o}\frac{\cot{\theta}}{\gamma{\Huge\unicode[Palace Script MT]{x0072}}} \left(1 - \frac{1}{ \sqrt{1 + \gamma^2\sin^2\theta}}\right), \end{gather}

(4.17)\begin{gather}v_{\phi}({\Huge\unicode[Palace Script MT]{x0072}}, \theta) =\frac{F_z}{4{\rm \pi} \eta_2^o}\frac{\cot{\theta}}{{\Huge\unicode[Palace Script MT]{x0072}}} \left(1 -\frac{1}{ \sqrt{1+\gamma^2 \sin^2{\theta}}}\right), \end{gather}

(4.18)\begin{gather}v_z({\Huge\unicode[Palace Script MT]{x0072}}, \theta) =\frac{F_z }{4{\rm \pi} \eta_2^o } \frac{1}{ \gamma {\Huge\unicode[Palace Script MT]{x0072}}}\left(1 -\frac{\gamma^2 + 1}{ \sqrt{1+\gamma^2\sin^2 \theta}}\right), \end{gather}

as well as the pressure field

(4.19)\begin{equation} p({\Huge\unicode[Palace Script MT]{x0072}},\theta) =\frac{F_z}{4{\rm \pi}}\frac{\cos{\theta}}{{\Huge\unicode[Palace Script MT]{x0072}}^2}\left(1 - \frac{2(\gamma^2 + 1)}{(1 + \gamma^2 \sin^2{\theta})^{3/2}}\right) . \end{equation}

Here, ${\Huge\unicode[Palace Script MT]{x0072}}$ is the radius in spherical coordinates (see schematic in figure 2(a) and Appendix A). Streamlines of the velocity field are visualized for a range of $\gamma$ in figure 2 and supplementary movie 1 available at https://doi.org/10.1017/jfm.2021.1079. In the absence of odd viscosity, the Stokeslet flow only has two components, $v_r$ and $v_z$ (Appendix G), visualized in the vertical $x$–$z$ plane in figure 2(a) and in three dimensions in figure 2(b). Notably, as the blue and red arrows in figure 2(a) indicate, the flow develops an azimuthal component for $\gamma \neq 0$ (figure 2c–d), consistent with the fact that $\eta _1^o$ and $\eta _2^o$ are parity-violating (see table 2 and (2.7)).

Figure 2. A Stokeslet in an odd viscous fluid. (a) The streamlines of a standard Stokeslet flow are shown in black. The blue and red arrows indicate the appearance of an azimuthal flow once odd viscosity is introduced. A schematic of the system and the coordinate convention is shown in the inset. An external force, $\boldsymbol {F}$, is applied at the origin in the $-\hat {\boldsymbol {z}}$ direction. (b–d) A three-dimensional rendition of the Stokeslet streamlines initialized around a circle (i.e. many copies of the bold streamline in b), for a range of viscosity ratios, $\gamma = \eta ^o/\mu$. As the odd viscosity increases, the velocity field develops an azimuthal component that changes sign across the $z = 0$ plane, where the source is located. In the limit of only odd viscosity (d), the familiar radial component of the flow vanishes.

As $\gamma$ is increased, the magnitude of the azimuthal component grows, while the radial component diminishes. When $\gamma \gg 1$, the $\hat {\boldsymbol {r}}$ component of the velocity field goes to zero, while $v_{\phi }$ and $v_z$ approach $({\Huge\unicode[Palace Script MT]{x0072}} \sin {\theta })^{-1}$. (For the approximation of a steady Stokes flow to remain valid, the dissipative shear viscosity $\mu$ must remain finite in order to ensure that the relaxation time of the fluid is also finite, so the limit $\gamma = \infty$ is never actually reached.) At smaller $\gamma$, the central line splits into lobes of high azimuthal velocity that migrate away from the vertical, as illustrated in Appendix G.

4.5. Stokeslet: perturbative solution

In this section, we consider more generally the effect of the odd shear viscosities and of the rotational viscosities on the Stokeslet by treating the problem perturbatively (with respect to the small parameters characterizing the magnitude of these viscosities). We find that the first-order correction $\boldsymbol {v}_{\text {Stokes},1}$ to the standard Stokeslet (given in (G1) of Appendix G) due to the parity-violating coefficients $\eta _1^o, \eta _2^o$ and $\eta _R^o$ is of the form

(4.20)\begin{equation} \boldsymbol{v}_{\text{Stokes},1} = v_{\phi,1}\hat{\boldsymbol{\phi}} = \left[ v_{\phi, 1}^{(\eta_1^{o})} + v_{\phi, 1}^{(\eta_2^{o})} + v_{\phi, 1}^{(\eta_R^o)} \right] \hat{\boldsymbol{\phi}}. \end{equation}

Let us now discuss the explicit form of each of these terms, starting with the odd shear viscosities.

Starting back from (4.14)–(4.15) (in which $\eta _1^o$ and $\eta _2^o$ are independent), we perform a perturbative expansion in the quantities $\epsilon _{1(2)} \equiv \eta _{1(2)}^o/\mu \ll 1$. Computing the inverse Fourier transform to obtain the flow fields in real space as in § 4.4 (see Appendix H for the detailed calculation), we find that both $\eta _1^o$ and $\eta _2^o$ contribute to leading order by introducing terms contained entirely in the $\hat {\boldsymbol {\phi }}$ component of the velocity field. The contributions of the two viscosities are

(4.21)\begin{gather} v_{\phi, 1}^{(\eta_1^{o})}({\Huge\unicode[Palace Script MT]{x0072}},\theta)={-}\epsilon_1 \frac{F_z} {128 {\rm \pi}\mu}\frac{(5 + 3\cos{2\theta})\sin{2\theta}}{{\Huge\unicode[Palace Script MT]{x0072}}}+\mathscr{O}(\epsilon_1^2), \end{gather}

(4.22)\begin{gather}v_{\phi, 1}^{(\eta_2^{o})}({\Huge\unicode[Palace Script MT]{x0072}}, \theta)={-}\epsilon_2 \frac{F_z} {64 {\rm \pi}\mu } \frac{(1 + 3\cos{2\theta})\sin{2\theta}}{{\Huge\unicode[Palace Script MT]{x0072}}}+\mathscr{O}(\epsilon_2^2), \end{gather}

while the pressure is not modified at first order. The azimuthal component is visualized in the vertical $r$–$z$ plane in figure 3. In the absence of odd viscosity (figure 3a), $v_{\phi }=0$. The non-dimensionalized $v_{\phi }$ profiles for $\eta _1^o$ and $\eta _2^o$ given by (4.21)–(4.22) are shown in figure 3(b–c). While both velocity fields decay as $1/{\Huge\unicode[Palace Script MT]{x0072}}$, they differ appreciably in their angular dependence: $\eta ^o_2$ includes an additional sign change.

Figure 3. The non-dimensionalized azimuthal component of the Stokeslet flow for small shear and rotational odd viscosity coefficients. (a) In the absence of odd viscosity, the azimuthal component of the velocity field is zero. (b–d) The first-order correction of the Stokeslet due to $\eta _1^o, \eta _2^o$ and $\eta _R^o$, respectively, taking $\eta ^o/\mu = 0.1$. The origin is removed due to the singularity of the flow at $r = z = 0$. The azimuthal flow is odd with respect to $z$, and forms lobe-like regions of concentrated rotation. Blue indicates flow into the page, red corresponds to flow out of the page. Overall, the fluid flows out of the page in the upper lobe (in red) and into the page in the lower lobe (in blue). In (c), two small additional lobes have opposite velocities compared with the larger ones.

We now consider rotational viscosities, which couple vorticity and torques. These viscosities break both minor symmetries of the viscosity tensor ($\eta _{ijk\ell } \neq \eta _{jik\ell } \neq \eta _{ji\ell k}$), because the vorticity and torques are the anti-symmetric parts of the strain rate and stress tensors, and are shown in the block outlined in green in (2.7), reproduced below:

(4.23)\begin{equation} \begin{bmatrix} \sigma^1_R\\ \sigma^2_R\\ \sigma^3_R\end{bmatrix} = \begin{bmatrix} \eta_{R,1} & \eta_R^o & 0 \\ -\eta_R^o & \eta_{R,1} & 0 \\ 0 & 0 & \eta_{R,2} \end{bmatrix} \begin{bmatrix} \omega_1\\ \omega_2\\ \omega_3 \end{bmatrix} . \end{equation}

These rotational viscosities are often ignored in standard fluids because their contribution to the stress relaxes to zero over short times (de Groot 1962), but occur in the hydrodynamics of liquid crystals (Miesowicz 1946; Ericksen 1961; Leslie 1968; Parodi 1970) as well as in the hydrodynamics of electrons in materials with anisotropic Fermi surfaces (Cook & Lucas 2019).

We consider perturbations in the quantities $\epsilon _{R,1} = \eta _{R,1}/\mu, \epsilon _{R,2} = \eta _{R,2}/\mu$ and $\epsilon _R^o = \eta _R^o/\mu$. The matrix $\boldsymbol{\mathsf{M}}$ is given by

(4.24)\begin{equation} \boldsymbol{\mathsf{M}} = \mu \begin{bmatrix} q^2 - \epsilon_{R,1} q_z^2 - \epsilon_{R,2}q_y^2 & \epsilon_{R,2} q_x q_y -\epsilon_R^o q_z^2 & \epsilon_{R,1} q_x q_z + \epsilon_R^o q_y q_z \\ \epsilon_{R,2} q_x q_y + \epsilon_R^o q_z^2 & q^2 -\epsilon_{R,1} q_z^2 - \epsilon_{R,2}q_x^2 & -\epsilon_R^o q_x q_z + \epsilon_{R,1} q_y q_z \\ \epsilon_{R,1} q_x q_z - \epsilon_R^o q_y q_z & \epsilon_R^o q_x q_z + \epsilon_{R,1} q_y q_z & q^2 - \epsilon_{R,1} q^2 \end{bmatrix}. \end{equation}

Applying (4.5a,b), we calculate the velocity and pressure in Fourier space:

(4.25)\begin{gather} \boldsymbol{v}(\boldsymbol{q}) = \frac{F_z}{\mu \, N_2(\boldsymbol{q})} \begin{bmatrix} - q_x q_z (q_{{\perp}}^2 + q_z^2) -\epsilon_R^o q_y q_z (q_{{\perp}}^2 + q_z^2) + \epsilon_R q_x q_z^3\\ - q_y q_z (q_{{\perp}}^2 + q_z^2) + \epsilon_R^o q_x q_z (q_{{\perp}}^2 + q_z^2) + \epsilon_R q_y q_z^3\\ q_{{\perp}}^2(q_{{\perp}}^2 + q_z^2) - \epsilon_R q_{{\perp}}^2 q_z^2 \end{bmatrix}, \end{gather}

(4.26)\begin{gather}p(\boldsymbol{q}) = \frac{-\text{i} Fz}{N_2(\boldsymbol{q})} \,[q_z (q_{{\perp}}^2 + q_z^2)^2 -\epsilon_R q_z (q_{{\perp}}^2 + q_z^2)(q_{{\perp}}^2 + 2q_z^2)+ (\epsilon_R^2 + (\epsilon_R^o)^2) q_z^3(q_{{\perp}}^2 + q_z^2)], \end{gather}

in which

(4.27)\begin{equation} N_2(\boldsymbol{q}) = {(q_{{\perp}}^2 + q_z^2)^3+ \epsilon_R(q_{{\perp}}^2 + q_z^2)^2({-}q_{{\perp}}^2 - 2q_z^2) + (\epsilon_R^2 + (\epsilon_R^o)^2)q_z^2 (q_{{\perp}}^2 + q_z^2)^2}. \end{equation}

Note that the coefficient $\epsilon _{R,2}$ does not affect the flow (see Appendix H for further details). For the remaining coefficients, we expand the above expressions up to first order in $\epsilon _{R,1}$ and $\epsilon _R^o$, and compute their inverse Fourier transform to find the real-space fields.

Of the three rotational viscosities, only $\eta _R^o$ violates parity (see table 2 and (2.7)) and as a consequence, gives rise to an azimuthal flow:

(4.28)\begin{equation} v_{\phi, 1}^{(\eta_R^o)}({\Huge\unicode[Palace Script MT]{x0072}},\theta)=\epsilon_R^o\frac{F_z}{16 {\rm \pi}\mu}\frac{\sin(2 \theta )}{ {\Huge\unicode[Palace Script MT]{x0072}}}+\mathscr{O}({(\epsilon_R^o)^2}) . \end{equation}

The $v_\phi$ profile due to $\eta _R^o$ is shown in figure 3(d). While the parity-violating shear and rotational viscosities generate quantitatively different azimuthal flows, their qualitative effect is the same. The pressure is again not modified to first order.

5. Odd viscous flow past an obstacle

5.1. Odd viscous flow past a sphere

Two-dimensional flows past obstacles in the presence of a non-dissipative (odd) viscosity have previously been studied experimentally in Soni et al. (2019) and theoretically in Kogan (2016). Lapa & Hughes (2014) also analysed the consequences on swimmers at low Reynolds numbers. In these two-dimensional cases, only the pressure field is modified by the additional viscous terms, while the velocity field remains unchanged. Nevertheless, Kogan (2016) reported that a lift force appears in the Oseen approximation (including inertia) of the flow past an infinite cylinder due to the non-dissipative viscosity. In this section, we consider three-dimensional flows. Even in the Stokes limit (without inertia), we find that parity-violating viscosities have a qualitative effect on the flow past a sphere: the Stokes drag is not modified at this order, but an azimuthal velocity develops despite the symmetry of the obstacle.

Let us begin by considering the viscous flow past a finite-radius sphere (Kim & Karrila 1991). We assume a uniform velocity field $\boldsymbol {v} =U \hat {\boldsymbol {z}}$ at ${\Huge\unicode[Palace Script MT]{x0072}} \to \infty$ and a no-slip boundary condition with $\boldsymbol {v}=0$ on the surface of the sphere ${\Huge\unicode[Palace Script MT]{x0072}} =a$. The streamlines of this flow in a standard fluid are shown in black in figure 4(a) on the $r$–$z$ plane. Here, we assume that the sphere cannot (or does not) rotate. In § 5.3, we discuss the case in which the sphere is allowed to rotate.

Figure 4. Odd viscous flow past (a–e) a sphere and (f–j) a bubble. (a) The streamlines of a standard flow past a sphere are shown in black. The blue and red arrows indicate the appearance of an azimuthal flow once odd viscosity is introduced. (b–e) The non-dimensionalized azimuthal velocity component of the flow visualized on the $r$–$z$ plane. If the odd viscosity is absent (b), the azimuthal component is zero. Perturbative additions of $\eta _1^o, \eta _2^o$ and $\eta _R^o$ (taking $\eta ^o/\mu = 0.1$) significantly affect the flow past a sphere by introducing a non-zero $v_{\phi }$ that is odd in $z$ (c–e). (f) The streamlines of a standard flow outside and inside a spherical bubble are shown in black. The blue and red arrows indicate the appearance of an azimuthal flow once odd viscosity is introduced. (g–j) The non-dimensionalized azimuthal velocity component of the flow visualized on the $r$–$z$ plane. Unlike the case of the sphere, the velocity field extends into the bubble, with a continuous velocity across the bubble surface. If the odd viscosity is absent (g), the azimuthal component is zero. Perturbative additions of $\eta _1^o$, $\eta _2^o$ and $\eta _R^o$ (taking $\eta ^o/\mu = 0.1$) significantly affect the flow by introducing a non-zero $v_{\phi }$ both inside and outside the bubble (h–j).

We once again look for a perturbative solution to (4.1) with $f_i = 0$ in the small parameters $\epsilon _1$, $\epsilon _2$ and $\epsilon _R^o$. To leading order in the parity-violating viscosities, the pressure field about the sphere is given simply by the pressure term due to the Stokeslet, as in a standard isotropic fluid. Since the Stokeslet pressure does not have a first-order correction (§ 4.5), (4.1) reduces to the vector Poisson equation for the first-order velocity field:

(5.1)\begin{equation} \Delta \boldsymbol{v_1} ={-} \varDelta_{\alpha} \boldsymbol{v_0}, \end{equation}

in which $\varDelta _{\alpha }$ is the second-order differential operator associated with the viscosity $\alpha$ (here, $\alpha = \eta _1^{o}, \eta _2^{o}, \eta _{R}^{o}$; see Appendix E for explicit form) and $\boldsymbol {v_0}$ is the flow past a sphere in a standard fluid (given by (I4)). The resulting vector Poisson equation for the perturbed flow is formally equivalent to the electrostatics problem of finding the electric potential due to a conducting spherical cavity enclosing a point charge. We use the corresponding Dirichlet Green function by expanding the solution in spherical harmonics (Jackson 1999). The details of this calculation are provided in Appendix I. Solving for the flow $\boldsymbol {v_1}$ to leading order in $\epsilon _1$, $\epsilon _2$ and $\epsilon _R^o$, we can express the resulting velocity field in terms of the Stokeslet solution, $\boldsymbol {v_{{Stokes}, 1}}$ in (4.21), (4.22) and (4.28), from § 4.5, as

(5.2)\begin{equation} v_{\phi, 1} ({\Huge\unicode[Palace Script MT]{x0072}}, \theta) =\left(\frac{6{\rm \pi} a U \mu}{F_z} \boldsymbol{v_{\text{Stokes}, 1}} + \frac{{\rm \pi} a^3 U \mu}{F_z}\Delta \boldsymbol{v_{\text{Stokes}, 1}} + \frac{{\rm \pi} a^5 U \mu}{20 F_z}\varDelta^2 \boldsymbol{v_{\text{Stokes}, 1}} \right)\boldsymbol{\cdot} \hat{\boldsymbol{\phi}} \end{equation}

with no modifications to $v_r$ and $v_z$ at leading order.

In standard isotropic fluids, a superposition of the Stokeslet ($v \propto 1/{\Huge\unicode[Palace Script MT]{x0072}}$) and its second derivative (a source dipole $v \propto 1/{\Huge\unicode[Palace Script MT]{x0072}} ^3$) is sufficient to satisfy the boundary conditions. In the presence of odd viscosity, we find that higher-order gradients are necessary, as can be seen from (5.2). Even so, by equating the far field of the flow and the Stokeslet solution, we find $F_z = 6{\rm \pi} a U\mu$. Hence, the Stokes drag experienced by the sphere remains unchanged to first order in $\eta _R^{o}$, $\eta _1^{o}$ and $\eta _2^{o}$ compared with a standard fluid.

Rewriting (5.2) more explicitly, we have

(5.3)\begin{equation} v_{\phi, 1} ({\Huge\unicode[Palace Script MT]{x0072}}, \theta) =\frac{3U}{64}\left[ g^1(\theta) \frac{a}{{\Huge\unicode[Palace Script MT]{x0072}}} + g^3(\theta) \left(\frac{a}{{\Huge\unicode[Palace Script MT]{x0072}}} \right)^3 + g^5(\theta) \left(\frac{a}{{\Huge\unicode[Palace Script MT]{x0072}}} \right)^5 \right] \sin{2\theta}, \end{equation}

where

(5.4)\begin{gather} g^1(\theta)=8\epsilon_R^o - ( 5 + 3\cos{2\theta} ) \epsilon_1 - (2 + 6\cos{2\theta}) \epsilon_2 +\mathscr{O}({\epsilon^2}) , \end{gather}

(5.5)\begin{gather}g^3(\theta)={-}8\epsilon_R^o + (6 +10\cos{2\theta}) \epsilon_1 + (4 +20\cos{2\theta}) \epsilon_2+\mathscr{O}({\epsilon^2}) , \end{gather}

(5.6)\begin{gather}g^5(\theta)={-}( 1 + 7\cos{2\theta} ) \epsilon_1 - (2 + 14\cos{2\theta}) \epsilon_2 +\mathscr{O}({\epsilon^2}). \end{gather}

These velocity fields are shown in figure 4(c–e) on the $r$–$z$ plane. As in the Stokeslet case, each odd viscosity coefficient results in an azimuthal flow, but the quantitative features of the velocity field vary depending on the exact viscosity chosen.

5.2. Odd viscous flow past a bubble

Closely related to the flow past a solid sphere is the flow past a spherical bubble without surface tension, in which the bubble itself is filled with a fluid (Hadamard 1911; Rybczynski 1911; Lamb 1924; Batchelor 1967). Here we assume that the inner and outer fluid have the same viscosities. As with the sphere, we solve for a steady velocity field configuration satisfying $\boldsymbol {v} = U \hat {\boldsymbol {z}}$ as ${\Huge\unicode[Palace Script MT]{x0072}} \to \infty$, but now we require that the velocity field be continuous throughout all space (even across the nominal boundary of the bubble). In a standard fluid, the flow outside the bubble resembles that of the flow past a sphere, while the flow inside is described by Hill's spherical vortex (Hill 1894), with the boundary condition imposing continuous velocity at the surface. The streamlines of this velocity field are visualized in black in figure 4(f). Following the set-up above, let us consider the effect of the odd viscosities $\eta _1^o, \eta _2^o, \eta _R^o$ in the perturbative limit.

Like in the case of the sphere, the first-order correction to the pressure vanishes, and the flow outside the bubble reduces to (5.1). To solve this equation, we again employ Green function methods. Unlike the sphere problem, however, the boundary condition no longer requires no-slip velocity on the bubble surface, so we do not need to use the Dirichlet Green function. The details of this calculation are provided in Appendix J. Solving for the flow to leading order in $\epsilon _1$, $\epsilon _2$ and $\epsilon _R^o$, we can write it in terms of the Stokeslet solution, $\boldsymbol {v_{\text {Stokes}, 1}}$:

(5.7)\begin{equation} v_{\phi, 1}^{{out}} ({\Huge\unicode[Palace Script MT]{x0072}}, \theta) =\left(\frac{5{\rm \pi} a U \mu}{F_z} \boldsymbol{v_{\text{Stokes}, 1}} + \frac{{\rm \pi} a^3 U \mu}{2 F_z}\Delta \boldsymbol{v_{\text{Stokes}, 1}} + \frac{{\rm \pi} a^5 U \mu}{56 F_z}\varDelta^2 \boldsymbol{v_{\text{Stokes}, 1}} \right)\boldsymbol{\cdot} \hat{\boldsymbol{\phi}}, \end{equation}

with no modifications to $v_r$ and $v_z$ at leading order. As in the case of the sphere, the higher-order $1/{\Huge\unicode[Palace Script MT]{x0072}} ^5$ term is necessary to satisfy (5.1) and the boundary condition. By equating the far-field flow and the Stokeslet solution, we find $F_z = 5{\rm \pi} aU \mu$, which corresponds to the Stokesian drag on a bubble in a standard fluid; that is, the drag force is again unaffected at first order in odd viscosity. Note that the general form of the drag on a spherical bubble in a standard fluid is given by $F_z = (4{\rm \pi} a U \mu _{{out}} )(({\mu _{{out}} + ({3}/{2})\mu _{{in}}})/({\mu _{{out}} + \mu _{{in}}}))$, where $\mu _{{out}}$ and $\mu _{{in}}$ are the even shear viscosities outside and inside the bubble, respectively. In the case we are considering, $\mu _{{in}} = \mu _{{out}}$, so $F_z$ reduces to the expression above (Batchelor 1967).

Rewriting (5.7) more explicitly, we have

(5.8)\begin{equation} v_{\phi, 1}^{{out}} ({\Huge\unicode[Palace Script MT]{x0072}}, \theta) =\frac{U}{896}\left[ g^1_{{out}}(\theta) \frac{a}{{\Huge\unicode[Palace Script MT]{x0072}}} + g^3_{{out}}(\theta) \left(\frac{a}{{\Huge\unicode[Palace Script MT]{x0072}}} \right)^3 + g^5_{{out}}(\theta) \left(\frac{a}{{\Huge\unicode[Palace Script MT]{x0072}}} \right)^5 \right] \sin{2\theta}, \end{equation}

where

(5.9)\begin{gather} g^1_{{out}}(\theta)= 280 \epsilon_R^o - (175 + 210 \cos{2\theta}) \epsilon_1 - (70 + 105\cos{2\theta}) \epsilon_2 +\mathscr{O}({\epsilon^2}), \end{gather}

(5.10)\begin{gather}g^3_{{out}}(\theta)={-}168 \epsilon_R^o + ( 126 + 210 \cos{2\theta}) \epsilon_1 + (84 + 420\cos{2\theta}) \epsilon_2+\mathscr{O}({\epsilon^2}), \end{gather}

(5.11)\begin{gather}g^5_{{out}}(\theta)={-}(15 + 105 \cos{2\theta}) \epsilon_1 - (30 + 210\cos{2\theta}) \epsilon_2 +\mathscr{O}({\epsilon^2}). \end{gather}

Next, we consider the flow inside the bubble. In this case, $\varDelta _{\alpha } \boldsymbol {v}_{\boldsymbol {0}}^{{in}} = 0$, so (4.1) reduces to the vector Laplace equation:

(5.12)\begin{equation} \Delta \boldsymbol{v}_{\boldsymbol{1}}^{{in}} = 0, \end{equation}

with the boundary condition $\boldsymbol {v}_{\boldsymbol {1}}^{{out}}(a, \theta ) = \boldsymbol {v}_{\boldsymbol {1}}^{{in}}(a, \theta )$. The solution to this Dirichlet problem involves the Dirichlet Green function used in the sphere computation (Jackson 1999) (see details in Appendix J). Solving for $\boldsymbol {v}_{\boldsymbol {1}}^{{in}}$, we find

(5.13)\begin{equation} v_{\phi, 1}^{{in}} ({\Huge\unicode[Palace Script MT]{x0072}}, \theta) = \frac{U}{56}\left[ g^1_{{in}}(\theta) \left(\frac{{\Huge\unicode[Palace Script MT]{x0072}}}{a}\right)^2 \right] \sin{2\theta}, \end{equation}

where

(5.14)\begin{equation} g^1_{{in}}(\theta)= 7\epsilon_R^o - 4\epsilon_1 - \epsilon_2 +\mathscr{O}({\epsilon^2}), \end{equation}

again with no modifications to $v_r$ and $v_z$ at leading order. We plot the velocity fields outside and inside the bubble in figure 4(h–j).

5.3. Effect of hydrostatic torque

As we have mentioned in § 3, parity-violating fluids, such as fluids made of spinning particles, can exhibit a hydrostatic torque $\sigma ^{h}_{ij} = -\epsilon _{ijz} \tau _z$ and a hydrostatic shear stress density in their hydrostatic stress. Let us illustrate the effect of the hydrostatic torque on a finite sphere. We assume a no-slip boundary condition at the surface of the sphere. In contrast to the situation of § 5.1, where the velocity at the surface of the sphere was assumed to vanish, here this velocity is determined by the balance between the torques due to the viscous stress and to the hydrostatic stress. Note that other boundary conditions could be appropriate, depending on the microscopic interactions between the constituents of the fluid and the boundary.

In the absence of odd viscosity, the hydrostatic torque leads to a total torque $T_z = \frac 83 {\rm \pi}a^3 \tau _z$ on the sphere (see (3.2)). Hence, the sphere rotates at a steady angular velocity $\varOmega = {T_z}/{8 {\rm \pi}a^3 \mu } = {\tau _z}/{3 \mu }$ (see Hobbie & Roth 2007) and introduces an additional azimuthal component

(5.15)\begin{equation} v_\phi ({\Huge\unicode[Palace Script MT]{x0072}}, \theta) = \frac{\varOmega a^3 \sin \theta}{{\Huge\unicode[Palace Script MT]{x0072}}^2} \end{equation}

to the flow. This expression is valid for all ${\Huge\unicode[Palace Script MT]{x0072}}$ (in both near and far fields), as it satisfies the boundary condition. This azimuthal flow is even in $z$, unlike the flow due to the parity-violating viscosities, which is odd in $z$.

When odd viscosity is present, we can still compute the far-field flow using the perturbative Oseen tensor (computed in (H6) of Appendix H). The far-field flow is given by $v_i = A_{i z} T_z$ using the rotlet $A_{i \ell }$ (see § 4.2), and we find that

(5.16)\begin{equation} \boldsymbol{v} = \frac{T_z}{8{\rm \pi}\mu} \frac{\sin{\theta}}{{\Huge\unicode[Palace Script MT]{x0072}}^2}\boldsymbol{\hat{\phi}} + \epsilon \frac{T_z}{16{\rm \pi}\mu}\frac{1 + 3\cos{2\theta}}{{\Huge\unicode[Palace Script MT]{x0072}}^2} \hat{\boldsymbol{{\Huge\unicode[Palace Script MT]{x0072}}}}, \end{equation}

in which $\epsilon \equiv \eta ^o/\mu \ll 1$ and $\eta _2^o = - \eta _1^o /2 \equiv \eta ^o$. Again, we find an azimuthal flow even in $z$ as a consequence of the hydrostatic torque. In addition, we see that the presence of odd viscosity combined with a hydrostatic torque generates flow in the radial direction that is absent in the zeroth-order case.

6. Sedimentation in a parity-violating fluid

6.1. Few particles: mechanisms

We now examine the role of parity-violating viscosities and of the corresponding azimuthal flows in the problem of sedimentation, in which particles driven by an external field (e.g. gravity) interact hydrodynamically at low Reynolds number. We assume the particles to be small identical spheres without inertia, which sediment under gravity and are advected by the flow due to the other particles.

In a standard isotropic fluid, an isolated sedimenting particle experiences a Stokesian drag and thus sinks at a velocity $U = F_z/(6{\rm \pi} \mu a)$. In a co-moving reference frame, the velocity field generated by a single sedimenting sphere $\beta$ is simply given by the Stokes flow past a sphere (I4). In the dilute limit, we can neglect the near-field terms that fall off faster than $1/{\Huge\unicode[Palace Script MT]{x0072}}$ (Happel 1983). In particular, we neglect the higher-order $1/{\Huge\unicode[Palace Script MT]{x0072}} ^2$ velocity field contribution associated with particle rotation (see § 5.3), which may occur in a fluid with hydrostatic torques. As a result, the velocity field generated by each particle simply reduces to the Stokeslet (G1). If all the sedimenting particles experience the same force $\boldsymbol {f} = -\hat {\boldsymbol {z}}F_z$, the equation of motion in the co-moving frame for particle $\alpha$ becomes (Hocking 1964; Guazzelli et al. 2009)

(6.1)\begin{equation} \frac{{\text{d}}\kern0.7pt\boldsymbol{x}^{\alpha}}{{\text{d}}t} = \sum_{\alpha \neq \beta} G(\boldsymbol{x}^{\alpha} - \boldsymbol{x}^{\beta})\boldsymbol{f} , \end{equation}

where $G(\boldsymbol {x})$ is the Green function of the Stokes equation (Oseen tensor) from (4.9), $\alpha$ and $\beta$ are particle indices and $\boldsymbol {x}^\alpha$ is the position of particle $\alpha$ in the co-moving reference frame.

In a parity-violating fluid, we replace the standard Stokeslet field on the right-hand side of (6.1) with the odd viscous Stokeslet from (4.16)–(4.18). For simplicity, here we will consider $\eta _2^o = - \eta _1^o /2 \equiv \eta ^o$ and $\mu _1 = \mu _2 = \mu _3 \equiv \mu$ (all other viscosities in (2.7) are set to zero). Since each of the sedimenting particles experiences an identical vertical force, we can move into their co-moving reference frame. We then numerically integrate (6.1) over time with a standard fourth-order Runge–Kutta algorithm to obtain the trajectories of the particles.

Figure 5(a,b) shows the trajectories of three particles in the $x$–$z$ plane with $\eta ^o=0$ and $\eta ^o >0$. While the sedimentation of as few as three Stokeslets in a standard isotropic fluid is already chaotic (Hocking 1964; Jánosi et al. 1997), the parity-violating flow introduces simple and well-defined modifications to the trajectories. For example, the trajectories of two particles interacting through the standard Stokeslet are confined to the vertical plane (figure 5c) containing initial positions. In the presence of odd viscosity, the particle trajectories are deflected out of this plane due to the azimuthal flow present in the odd Stokeslet (figure 5d). Similarly, the dynamics of a three-particle system initialized along a horizontal line is constrained to the initial vertical plane, as shown in figure 5(e) (Hocking 1964). The azimuthal flow in the odd Stokeslet allows the trajectories to escape out of this plane and follow three-dimensional trajectories (figure 5f). More generally, the area, $\boldsymbol {\varDelta } \boldsymbol {\cdot } \hat {\boldsymbol {z}}$, of the projection on the $x$–$y$ plane of the triangle formed by three sedimenting particles is constant in time for standard fluids (Hocking 1964), even though its shape will generally change (figure 5g). This conservation law is broken when azimuthal flow is present, as illustrated in figure 5(h). In the figure, we have used a high $\eta ^o = 1$ for ease of visualization, but the effect is also present in the perturbative regime.

Figure 5. Few-particle sedimentation in an odd viscous fluid. (a,b) The trajectories of three particles, initially positioned along a horizontal line, without and with odd viscosity, respectively. Although the change to the trajectories is small, the addition of $\eta ^o$ is sufficient to qualitatively change the long-time dynamics of the system, as the red particle gets left behind in (b), while the black particle is lost in (a). (c,d) The trajectories of two particles initialized at different heights without and with odd viscosity, respectively. Rather than moving along the line connecting the two particles (c), the trajectories are rotated by an angle in the presence of $\eta ^o$ (d). (e,f) Three particle trajectories without and with odd viscosity, respectively. In (e), the three particles remain in the same vertical plane as they sediment, but the addition of $\eta ^o$ in (f) allows the particles to follow three-dimensional trajectories. (g,h) The projected triangle area of a three-particle system remains constant with time without odd viscosity (g), but can change with the addition of $\eta ^o$ (h). Panels (b), (d) and (f) are computed with $\eta ^o = 0.1$, panel (h) with $\eta ^o = 1$.

6.2. Sedimentation of clouds

We now consider a cloud composed of many sedimenting particles. We start with $N = 2000$ particles uniformly distributed within a spherical volume of radius $a = 0.5$, and evolve the system by integrating (6.1). Figure 6(a–c) shows snapshots of the evolution for different values of odd viscosity (see also supplementary movie 2).

Figure 6. Sedimentation of a cloud in an odd viscous fluid. (a–c) Snapshots of the falling cloud from simulations with $N = 2000$ particles for different values of $\gamma = \eta ^o/\mu$. In the absence of odd viscosity, the initially spherical cloud deforms into a torus, and subsequently breaks apart into smaller clouds. As the odd viscosity is increased, the breakup event only occurs in a fraction of the runs, and for even higher values of $\gamma$, the cloud no longer forms a torus, instead deforming into an ellipsoid. (d,e) Streamlines of the fluid flow in the $x$–$z$ plane with $y=0$, computed at $t=0$ in the instantaneous reference frame of the cloud. When $\gamma = 0$, the flow field corresponds to Hill's spherical vortex. In contrast, for a large odd viscosity, the initially spherical cloud immediately deforms to an ellipsoid due to the stretched vortices. (f–h) The velocity field in the $x$–$y$ plane with $z = 0.33$, computed at $t = 0$, for various values of $\gamma$. As the odd viscosity is increased, the radial component of the velocity decreases while the azimuthal component increases. (i–k) Sample particle trajectories for varying $\gamma$, with supplementary schematics to highlight the main features. The colour map indicates distance from the viewer, with dark blue closest.

The case of a standard fluid (figure 6a) was analysed theoretically and experimentally by Batchelor (1972), Nitsche & Batchelor (1997), Ekiel-Jeżewska, Metzger & Guazzelli (2006) and Metzger, Nicolas & Guazzelli (2007) (see also references therein). In this case, the cloud develops a vertical tail of particles that are lost from the outside layer of the cloud. Then, the circulating motion of the flow inside the cloud (figure 6d) depletes the number of particles along the central vertical axis of the cloud, leading to the formation of a torus. If the initial cloud is sufficiently large, the torus undergoes a breakup event into smaller clouds. As shown in figure 6(d), the streamlines of the fluid velocity field due to the particles (or, equivalently, the trajectories of the particles themselves, since inertia is neglected) initially coincide with the flow past a bubble from § 5.2 (Kojima, Hinch & Acrivos 1984; Pozrikidis 1990; Ekiel-Jeżewska et al. 2006; Shimokawa et al. 2016).

We now analyse the effect of a non-zero odd viscosity. In the regime of small $\gamma = \eta ^o/\mu$, the streamlines inside the cloud still agree with the velocity field in a bubble from (5.13). Particles develop a small in-plane tangential velocity component that points in opposite directions below and above the equator. This perturbative modification due to odd viscosity does not yet affect the qualitative features of the cloud dynamics (i.e. the formation of a torus and its breakup). As odd viscosity is increased further, the strength of the azimuthal flow increases in comparison with the radial component, and qualitative features of the evolution begin to change. At $\gamma = 1$, for instance, the breakup event does not always happen. At $\gamma = 5$, we find that the cloud no longer deforms into a torus. Instead, the cloud adopts an ellipsoidal shape which persists until all particles have leaked into the trailing tail (figure 6c). The formation of the ellipsoid is visible from the initial flow within the cloud; unlike the Hill spherical vortex shown in figure 6(d), the initial streamlines in the high odd viscosity case form a stretched vortex flow (figure 6e). The radial and azimuthal flows, as seen from above, are shown in figure 6(f–h).

In all cases, the particles within the cloud follow approximately closed trajectories (figure 6i–k). In the absence of odd viscosity, the closed-loop trajectories are angled radially inward to the central axis of the cloud (figure 6i). When odd viscosity is non-zero, the loop deforms due to the azimuthal flow. Since the azimuthal component changes sign above and below the equator of the cloud, the particles change rotation direction along the trajectories, now creating curved closed loops (see figure 6j). For values of $\gamma$ that correspond to ellipsoidal clouds, the trajectories wrap around layers of the cloud with little radial motion, again rotating in opposite directions above and below the equator (figure 6h).

7. Conclusion

In this article, we have explored the effects of parity violation on the viscous response of a fluid in three dimensions. The broken mirror symmetry gives rise to azimuthal flows even when the external forcing is aligned with the axis of azimuthal symmetry. The changes in a single Stokeslet lead to qualitative changes in the sedimentation of both few and many particles. The situations we have analysed theoretically and numerically are within experimental reach. In the context of soft matter, this could be done in multiple-scale colloids: a colloidal suspension of rotating particles can produce effective fluids with parity-violating viscosities (as has already been demonstrated in two dimensions; Soni et al. 2019), while larger particles can act as colloidal particles for the effective fluid. In these systems, the presence of parity-violating coefficients could also affect hydrodynamic instabilities such as the fingering instabilities observed in colloidal rollers in suspension (Wysocki et al. 2009; Driscoll et al. 2016). In the context of hard condensed matter, recent experimental and theoretical works (Hoyos & Son 2012; Levitov & Falkovich 2016; Holder, Queiroz & Stern 2019) have focused on the hydrodynamic behaviour of electrons in solids. There, sizeable parity-violating viscosities can occur and have been observed when the sample is under a magnetic field (Berdyugin et al. 2019) and Stokes flow can be realized by introducing holes in the sample (Gusev et al. 2020).