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

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:

$$\eta_{ijk\ell}=\zeta\delta_{ij}\delta_{k\ell}+\mu\left(\delta_{ik}\delta_{j\ell}+\delta_{i\ell}\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{k\ell}\right)+\eta_{\text{R}}(\delta_{ik}\delta_{j\ell}-\delta_{i\ell}\delta_{jk})$$

(5)

which contains just three independent coefficients: the shear viscosity $\mu$, the bulk viscosity $\zeta$, and the rotational viscosity $\eta_{\text{R}}$ (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 et al. (2000). Readers unfamiliar with this formalism can skip directly to Eq. 10, which generalizes the expression in Eq. 5. 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 Fig. 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 Fig. 1A-G and accompanying caption for concrete examples.

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

$$\eta_{ijk\ell}=R_{ii^{\prime}}R_{jj^{\prime}}R_{kk^{\prime}}R_{\ell\ell^{\prime}}\eta_{i^{\prime}j^{\prime}k^{\prime}\ell^{\prime}}$$

(6)

where $R$ 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 $R\in O(3)$ with $\det(R)=1$] 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 Eq. 6.

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 $\bm{\hat{z}}$). 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 Fig. 1 place identical constraints on the viscosity tensor. They break into two classes, drawn in blue and in red in Fig. 1. Fluids with the symmetry groups $D_{\infty\text{h}}$, $C_{\infty\text{v}}$, or $D_{\infty}$ (in blue) have an anisotropic viscosity tensor that is invariant under all reflections parallel and perpendicular to the ${\bm{\hat{}}z}$ axis. We call these fluids parity-preserving cylindrical, and examples include the aligned nematic particles ($D_{\infty\text{h}}$), aligned helices ($C_{\infty v}$), and dipolar molecules in an electric field ($C_{\infty v}$) shown in Fig. 1C-E. In contrast, fluids with the symmetry groups $C_{\infty\text{h}}$ or $C_{\infty}$ (in red) allow additional terms in their viscosity tensor. Examples of such fluids shown in Fig. 1F-G include spherical charged particles ($C_{\infty\text{h}}$) 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 ${\bm{\hat{}}z}$ 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 et al. (2020a, b), in which this notation is also used to describe elastic and viscoelastic media). The basis consists of

• •

  a diagonal matrix $\tau_{ij}^{1}=C_{ij}=\sqrt{\frac{2}{3}}\delta_{ij}$ corresponding to pressure and dilation,

• •

  three anti-symmetric matrices $\tau_{ij}^{A+1}=R_{ij}^{A}=\epsilon_{Aij}$ corresponding to torques and vorticity,

• •

  five traceless symmetric matrices $\tau_{ij}^{A+5}=S^{A}_{ij}$ corresponding to shear stresses and shear strain rates, whose expressions are

$S^{1}=\begin{bmatrix}1&0&0\\ 0&-1&0\\ 0&0&0\\ \end{bmatrix}S^{2}=\begin{bmatrix}0&1&0\\ 1&0&0\\ 0&0&0\\ \end{bmatrix}S^{3}=\begin{bmatrix}\frac{-1}{\sqrt{3}}&0&0\\ 0&\frac{-1}{\sqrt{3}}&0\\ 0&0&\frac{2}{\sqrt{3}}\\ \end{bmatrix}S^{4}=\begin{bmatrix}0&0&0\\ 0&0&1\\ 0&1&0\\ \end{bmatrix}S^{5}=\begin{bmatrix}0&0&1\\ 0&0&0\\ 1&0&0\\ \end{bmatrix}$

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

$$\sigma^{A}\equiv\sigma_{ij}\;\tau_{ij}^{A}\qquad\dot{e}^{A}\equiv\dot{e}_{ij}\;\tau_{ij}^{A}\qquad\eta^{AB}=\frac{1}{2}\;\tau_{ij}^{A}\,\eta_{ijk\ell}\,\tau_{k\ell}^{B}$$

(7)

we may write

$$\sigma^{A}=\eta^{AB}\,\dot{e}^{B}$$

(8)

We can transform back to Cartesian tensors via

$$\sigma_{ij}=\frac{1}{2}\;\sigma^{A}\tau_{ij}^{A}\qquad\dot{e}_{ij}=\frac{1}{2}\;\dot{e}^{A}\tau_{ij}^{A}\qquad\eta_{ijk\ell}=\frac{1}{2}\;\tau_{ij}^{A}\,\eta^{AB}\,\tau_{k\ell}^{B}.$$

(9)

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

(10)

in which the parity-violating viscosities are written in red (these are only allowed in the groups drawn in red in Fig. 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 Eq. 10 under the reflection $P_{y}$). Finally, we have restricted our attention to fluids invariant under continuous rotations about the $\bm{\hat{z}}$ 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 (Rao & Bradlyn, 2020; Varnavides et al., 2020; Toshio et al., 2020; Cook & Lucas, 2019).

In addition to the decomposition based on spatial symmetries discussed in Section 2.1, the viscosity tensor can be decomposed into symmetric and antisymmetric parts

$\displaystyle\eta_{ijk\ell}=\eta^{\text{e}}_{ijk\ell}+\eta^{\text{o}}_{ijk\ell}$

(11)

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

$$\dot{w}=\sigma_{ij}\,\partial_{j}v_{i}=\eta_{ijk\ell}\;(\partial_{j}v_{i})\;(\partial_{\ell}v_{k})=\frac{1}{2}\,\eta^{AB}\dot{e}^{A}\dot{e}^{B}.$$

(12)

Hence, the antisymmetric part $\eta^{\text{o}}_{ijk\ell}$ is purely non-dissipative, because $\eta^{\text{o}}_{ijk\ell}(\partial_{j}v_{i})\;(\partial_{\ell}v_{k})=0$. 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 $\eta^{\text{o}}_{ijk\ell}=0$ when microscopic reversibility is satisfied (Onsager, 1931; Groot, 1962; de Groot & Mazur, 1954).

The dissipative part $\eta^{\text{e}}_{ijk\ell}$ of the viscosity tensor corresponds to the symmetric part of the matrix $\eta^{AB}$ in Eq. (8), while the non-dissipative part $\eta^{\text{o}}_{ijk\ell}$ corresponds to its antisymmetric part. Hence, we have split all off-diagonal terms in Eq. (10) into odd and even parts (except when one of these is already ruled out by spatial symmetry). The non-dissipative viscosities all have a “o” superscript. In Table 2, we classify the viscosity coefficients in Eq. 10 based on whether they are dissipative or not, and on the symmetry groups in which they can occur.

The (transient) Stokes equation for an incompressible fluid found in Eq. (3) can be written as

$$\rho\partial_{t}v_{i}=-\partial_{i}P+\partial_{j}[\eta_{ijk\ell}\partial_{\ell}v_{k}]+f_{i}\quad\text{with}\quad\partial_{i}v_{i}=0.$$

(15)

in which we have used the expression (2) of the viscous stress. In reciprocal space (see Appendix A for Fourier transform conventions),

$$-\text{i}\omega\rho v_{i}=-\text{i}q_{i}P-q_{j}q_{\ell}\eta_{ijk\ell}v_{k}+f_{i}\quad\text{with}\quad\text{i}q_{i}v_{i}=0.$$

(16)

These equations can be written as

$$M(\bm{q},\omega)\bm{v}=-\text{i}P\bm{q}+\bm{f}\quad\text{with}\quad\bm{q}\cdot\bm{v}=0$$

(17)

in which we have defined the matrix

$$M_{ik}(\bm{q},\omega)=q_{j}q_{\ell}\eta_{ijk\ell}-\text{i}\omega\rho\,\delta_{ik}.$$

(18)

The matrix $M({\bm{q}},\omega)$ is always invertible at finite ${\bm{q}}$ provided that the dissipation rate $\dot{w}$ in Eq. 12 is strictly positive (see Appendix C.1). Under this hypothesis, we apply $M^{-1}(\bm{q},\omega)$ to Eq. (17). We then take the scalar product with $\bm{q}$ to obtain the pressure $P$, and then replace $P$ with its expression to obtain the velocity, giving

$$\text{i}P=\frac{\bm{q}\cdot(M^{-1}\bm{f})}{{\bm{q}\cdot(M^{-1}\bm{q})}}\quad\text{and}\quad\bm{v}=-\frac{{\bm{q}\cdot(M^{-1}\bm{f})}}{{\bm{q}\cdot(M^{-1}\bm{q})}}\;M^{-1}\bm{q}+M^{-1}\bm{f}.$$

(19)

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

$$\bm{v}=G(\bm{q},\omega)\bm{f}$$

(20)

in which

$\displaystyle G_{ij}(\bm{q},\omega)\equiv$

$\displaystyle\bigg{(}[M^{-1}]_{ij}-\frac{[M^{-1}]_{im}q_{m}q_{n}[M^{-1}]_{nj}}{q_{k}[M^{-1}]_{k\ell}q_{\ell}}\bigg{)}$

(21)

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 $v_{i}=G_{ij}$ is a solution of Eq. (15) with $\bm{f}=\delta(\bm{x})\bm{e}_{j}$, where $\bm{e}_{j}$ is the unit vector in direction $j$. For an isotropic incompressible fluid, we recover the usual (reciprocal space) Oseen tensor

$$G_{ij}^{\text{iso}}(\bm{q},\omega=0)=\frac{1}{\mu q^{2}}\left(\delta_{ij}-\frac{q_{i}q_{j}}{q^{2}}\right).$$

(22)

When the symmetric part of $\eta_{ijk\ell}$ (corresponding to dissipation) vanishes, the second term of Eq. 21 diverges at $\omega=0$ (but finite $\bm{q}$) because $M^{-1}_{k\ell}$ 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 timescale associated with viscous relaxation: in this case, the Stokes approximation is not valid. In the following, we will assume that viscous relaxation is fast enough, and focus on steady solutions that correspond to the steady Oseen tensor $G(\bm{q})\equiv G(\bm{q},\omega=0)$.

The real space Oseen tensor is then

$$G_{ij}(\bm{x})=\frac{1}{(2\pi)^{3}}\int e^{i\bm{q}\cdot\bm{x}}\,G_{ij}(\bm{q})\,\differential^{3}q$$

(23)

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

$$\bm{v}(\bm{x})=G(\bm{x})\bm{f}$$

(24)

When the antisymmetric (non-dissipative) part of the viscosity tensor vanishes ($\eta_{ijk\ell}^{o}=0$), the matrix $M$ defined by Eq. 18 is symmetric. Hence, $M^{-1}$ and the Green function $G(\bm{q})$ are symmetric as well. The symmetry of the reciprocal-space Green function $G_{ij}(\bm{q})=G_{ji}(\bm{q})$ is equivalent to $G_{ij}(\bm{x})=G_{ji}(\bm{x})$ in physical space. This is the expression of Lorentz reciprocity (Masoud & Stone, 2019, § 4.2, Eq. 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 analyze 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 will compute the real-space Oseen tensor or Stokeslet in different cases, using both numerical and analytical methods. The qualitative changes compared to 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).

Since the Stokeslet is a response to a point perturbation, multipolar responses can be computed by taking derivatives of the Green function in Eq. (23), see Kim & Karrila (1991). For example, consider a force dipole defined by a point force $\mathbf{f}$ at $\frac{1}{2}\delta\mathbf{r}$ and a point force $-\mathbf{f}$ at $-\frac{1}{2}\delta\mathbf{r}$. The corresponding fluid velocity is given by

$\displaystyle v_{i}(\mathbf{r})=G_{ik}\quantity(\mathbf{r}-\frac{1}{2}\delta\mathbf{r})f_{k}-G_{ik}\quantity(\mathbf{r}+\frac{1}{2}\delta\mathbf{r})f_{k}\approx-\partial_{j}G_{ik}(\mathbf{r})\delta r_{j}f_{k}\equiv H_{ijk}(\mathbf{r})\delta r_{j}f_{k}$

(25)

The tensor $H_{ijk}$ is often decomposed into two contributions: the symmetric part $S_{ijk}=\frac{1}{2}(H_{ijk}+H_{ikj})$, which represents the response to point shears (also known as stresslet), and the antisymmetric part $A_{i\ell}=\frac{1}{2}\epsilon_{jk\ell}H_{ijk}$ which represents the response to point torques $T_{\ell}$ (also known as rotlet). As discussed in Sec. 3, such point torques can arise from an hydrostatic torque in the fluid. An explicit expression of the Oseen tensor $G_{ik}$ is given by (107) of Appendix H in a perturbative case, from which the stresslet and rotlet can be deduced using Eq. (25).

To determine the physical-space Stokeslet or Oseen tensor $G(\bm{x})$, one must compute the inverse Fourier transform (23). This can be done numerically in the general case, in which analytical solutions are not easily accessible. To do so, we evaluate Eq. (21) on a discrete grid in reciprocal space (each component of ${\bm{q}}$ ranges from $-Q$ to $+Q$ with increments $\delta q$). This allows us to resolve length scales larger than a few $\pi/Q$ but smaller than $\pi/\delta q$. We then use the fast Fourier transform (FFT) 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 Eq. (23) with a Gaussian kernel $e^{-\pi q^{2}/4Q^{2}}$ (Gómez-González & del Álamo, 2013; Cortez, 2001). 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 ${\bm{\hat{}}{z}}$ 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 Fig. 8 of Appendix F, in which we also validate the numerical method using the exact solution discussed in the next section, see Fig. 10. We find that the viscosity coefficients that give rise to an azimuthal flow are

$$\eta_{R}^{o}\quad\eta^{e}_{Q,2}\quad\eta^{o}_{Q,2}\quad\eta^{e}_{Q,3}\quad\eta^{o}_{Q,3}\quad\eta_{1}^{o}\quad\eta_{2}^{o}$$

(26)

The list of viscosity coefficients that we found to generate $v_{\phi}$ are a subset of the parity-violating viscosities (printed in red in Eq. (10) 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 $(\eta_{A}^{e}-\eta_{A}^{o})\nabla\cdot\bm{v}$ vanishes because $\nabla\cdot\bm{v}=0$. 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.

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 $M(\omega=0)$ defined by Eq. 18 takes the form

$\displaystyle M(\omega=0)=\begin{bmatrix}\mu q^{2}&\eta_{1}^{o}(q_{x}^{2}+q_{y}^{2})-\eta_{2}^{o}q_{z}^{2}&-\eta_{2}^{o}q_{y}q_{z}\\ -\eta_{1}^{o}(q_{x}^{2}+q_{y}^{2})+\eta_{2}^{o}q_{z}^{2}&\mu q^{2}&\eta_{2}^{o}q_{x}q_{z}\\ \eta_{2}^{o}q_{y}q_{z}&-\eta_{2}^{o}q_{x}q_{z}&\mu q^{2}\\ \end{bmatrix}$

(27)

Taking ${\bm{f}}=-\bm{\hat{z}}F_{z}\delta^{3}({\bm{x}})$, 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 Eq. 19,

	$\displaystyle\bm{\hat{v}(\bm{q})}$	$\displaystyle=\frac{F_{z}}{N(\bm{q})}\begin{bmatrix}q_{z}(q_{y}(-(\eta_{1}^{o}+\eta_{2}^{o})q_{\perp}^{2}+\eta_{2}^{o}q_{z}^{2})+\mu q_{x}(q_{\perp}^{2}+q_{z}^{2}))\\ \\ q_{z}(q_{x}((\eta_{1}^{o}+\eta_{2}^{o})q_{\perp}^{2}-\eta_{2}^{o}q_{z}^{2})+\mu q_{y}(q_{\perp}^{2}+q_{z}^{2}))\\ \\ -\mu q_{\perp}^{2}(q_{\perp}^{2}+q_{z}^{2})\\ \end{bmatrix}$		(28)
	$\displaystyle\hat{p}(\bm{q})$	$\displaystyle=i\frac{F_{z}}{N(\bm{q})}\;q_{z}[(\eta_{1}^{o}q_{\perp}^{2}-\eta_{2}^{o}q_{z}^{2})((\eta_{1}^{o}+\eta_{2}^{o})q_{\perp}^{2}-\eta_{2}^{o}q_{z}^{2})+\mu^{2}(q_{\perp}^{2}+q_{z}^{2})^{2}]$		(29)

in which $N(\bm{q})=\mu^{2}(q_{\perp}^{2}+q_{z}^{2})^{3}+q_{z}^{2}((\eta_{1}^{o}+\eta_{2}^{o})q_{\perp}^{2}-\eta_{2}^{o}q_{z}^{2})^{2}$. Second, we assume that $\eta_{1}^{o}=-2\eta_{2}^{o}$, for which simplifications occur in Eqs. 28-29. This particular case occurs in the limit of low magnetic field regime in experiments on polyatomic gases (see Eq. (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 $\bm{\hat{z}}$ axis in the space of shears (see Appendix G).

In order to find the real space solution, we compute the inverse Fourier transform in Eq. 23 (see Appendix G for the detailed calculation). Parameterizing the final flow field by $\gamma=\eta_{2}^{o}/\mu$, we obtain the velocity field

$\displaystyle v_{r}(\mathcalligra{r},\theta)$

$\displaystyle=-\frac{F_{z}}{4\pi\eta_{2}^{o}}\frac{\cot{\theta}}{\gamma\mathcalligra{r}}\left(1-\frac{1}{\sqrt{1+\gamma^{2}\sin^{2}\theta}}\right)$

(30)

	$\displaystyle v_{\phi}(\mathcalligra{r},\theta)$	$\displaystyle=\frac{F_{z}}{4\pi\eta_{2}^{o}}\frac{\cot{\theta}}{\mathcalligra{r}}\left(1-\frac{1}{\sqrt{1+\gamma^{2}\sin^{2}{\theta}}}\right)$		(31)
	$\displaystyle v_{z}(\mathcalligra{r},\theta)$	$\displaystyle=\frac{F_{z}}{4\pi\eta_{2}^{o}}\frac{1}{\gamma\mathcalligra{r}}\left(1-\frac{\gamma^{2}+1}{\sqrt{1+\gamma^{2}\sin^{2}\theta}}\right)$		(32)

as well as the pressure field

$\displaystyle p(\mathcalligra{r},\theta)$

$\displaystyle=\frac{F_{z}}{4\pi}\frac{\cos{\theta}}{\mathcalligra{r}^{2}}\left(1-\frac{2(\gamma^{2}+1)}{(1+\gamma^{2}\sin^{2}{\theta})^{3/2}}\right)$

(33)

Here, $\mathcalligra{r}$ is the radius in spherical coordinates (see schematic in Figure 2A and Appendix A). Streamlines of the velocity field are visualized for a range of $\gamma$ in Fig. 2 and Supplementary Movie 1. 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 Fig. 2A and in three dimensions in Fig. 2B. Notably, as the blue and red arrows in Fig. 2A indicate, the flow develops an azimuthal component for $\gamma\neq 0$ (Fig. 2C-D), consistent with the fact that $\eta_{1}^{o}$ and $\eta_{2}^{o}$ are parity-violating (see Table 2 and Eq. 10).

As $\gamma$ is increased, the magnitude of the azimuthal component grows, while the radial component diminishes. When $\gamma\gg 1$, the $\bm{\hat{r}}$-component of the velocity field goes to zero, while $v_{\phi}$ and $v_{z}$ approach $(\mathcalligra{r}\sin{\theta})^{-1}$. (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.

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 $\bm{v}_{\text{Stokes},1}$ to the standard Stokeslet (given in Eq. 94 of Appendix G) due to the parity-violating coefficients $\eta_{1}^{o},\eta_{2}^{o}$ and $\eta_{R}^{o}$ is of the form

$\displaystyle\bm{v}_{\text{Stokes},1}=v_{\phi,1}\bm{\hat{\phi}}=\left[v_{\phi,1}^{(\eta_{1}^{\text{o}})}+v_{\phi,1}^{(\eta_{2}^{\text{o}})}+v_{\phi,1}^{(\eta_{R}^{o})}\right]\bm{\hat{\phi}}$

(34)

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

Starting back from Eqs. 28-29 (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 Section 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 $\bm{\hat{\phi}}$-component of the velocity field. The contributions of the two viscosities are

	$\displaystyle v_{\phi,1}^{(\eta_{1}^{\text{o}})}(\mathcalligra{r},\theta)$	$\displaystyle=-\epsilon_{1}\frac{F_{z}}{128\pi\mu}\frac{(5+3\cos{2\theta})\sin{2\theta}}{\mathcalligra{r}}+\order{\epsilon_{1}^{2}}$		(35)
	$\displaystyle v_{\phi,1}^{(\eta_{2}^{\text{o}})}(\mathcalligra{r},\theta)$	$\displaystyle=-\epsilon_{2}\frac{F_{z}}{64\pi\mu}\frac{(1+3\cos{2\theta})\sin{2\theta}}{\mathcalligra{r}}+\order{\epsilon_{2}^{2}}$		(36)

while the presssure is not modified at first order. The azimuthal component is visualized in the vertical $r$-$z$ plane in Fig. 3. In the absence of odd viscosity (Fig. 3A), $v_{\phi}=0$. The non-dimensionalized $v_{\phi}$ profiles for $\eta_{1}^{o}$ and $\eta_{2}^{o}$ given by Eq. 35-36 are shown in Fig. 3B-C. While both velocity fields decay as $1/\mathcalligra{r}$, they differ appreciably in their angular dependence: $\eta^{o}_{2}$ includes an additional sign change.

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 antisymmetric parts of the strain rate and stress tensors, and are shown in the block outlined in green in Eq. 10, reproduced below:

$\displaystyle\begin{bmatrix}\sigma^{1}_{R}\\ \sigma^{2}_{R}\\ \sigma^{3}_{R}\\ \end{bmatrix}=\begin{bmatrix}\eta_{R,1}&&\eta_{R}^{o}&&0\\ -\eta_{R}^{o}&&\eta_{R,1}&&0\\ 0&&0&&\eta_{R,2}\\ \end{bmatrix}\begin{bmatrix}\omega_{1}\\ \omega_{2}\\ \omega_{3}\\ \end{bmatrix}$

(37)

These rotational viscosities are often ignored in standard fluids because their contribution to the stress relaxes to zero over short times (Groot, 1962), but occur in the hydrodynamics of liquid crystals (Miesowicz, 1946; Leslie, 1968; Ericksen, 1961; 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 ${M}$ is given by

$\displaystyle{M}=\mu\begin{bmatrix}q^{2}-\epsilon_{R,1}q_{z}^{2}-\epsilon_{R,2}q_{y}^{2}&\epsilon_{R,2}q_{x}q_{y}-\epsilon_{R}^{o}q_{z}^{2}&\epsilon_{R,1}q_{x}q_{z}+\epsilon_{R}^{o}q_{y}q_{z}\\ \epsilon_{R,2}q_{x}q_{y}+\epsilon_{R}^{o}q_{z}^{2}&q^{2}-\epsilon_{R,1}q_{z}^{2}-\epsilon_{R,2}q_{x}^{2}&-\epsilon_{R}^{o}q_{x}q_{z}+\epsilon_{R,1}q_{y}q_{z}\\ \epsilon_{R,1}q_{x}q_{z}-\epsilon_{R}^{o}q_{y}q_{z}&\epsilon_{R}^{o}q_{x}q_{z}+\epsilon_{R,1}q_{y}q_{z}&q^{2}-\epsilon_{R,1}q^{2}\\ \end{bmatrix}$

(38)

Applying Eq. 19, we calculate the velocity and pressure in Fourier space,

	$\displaystyle\bm{v}(\bm{q})$	$\displaystyle=\frac{F_{z}}{\mu\;N_{2}(\bm{q})}\begin{bmatrix}-q_{x}q_{z}(q_{\perp}^{2}+q_{z}^{2})-\epsilon_{R}^{o}q_{y}q_{z}(q_{\perp}^{2}+q_{z}^{2})+\epsilon_{R}q_{x}q_{z}^{3}\\ \\ -q_{y}q_{z}(q_{\perp}^{2}+q_{z}^{2})+\epsilon_{R}^{o}q_{x}q_{z}(q_{\perp}^{2}+q_{z}^{2})+\epsilon_{R}q_{y}q_{z}^{3}\\ \\ q_{\perp}^{2}(q_{\perp}^{2}+q_{z}^{2})-\epsilon_{R}q_{\perp}^{2}q_{z}^{2}\\ \end{bmatrix}$		(39)
	$\displaystyle p(\bm{q})$	$\displaystyle=\frac{-\text{i}Fz}{N_{2}(\bm{q})}\;[q_{z}(q_{\perp}^{2}+q_{z}^{2})^{2}-\epsilon_{R}q_{z}(q_{\perp}^{2}+q_{z}^{2})(q_{\perp}^{2}+2q_{z}^{2})+(\epsilon_{R}^{2}+(\epsilon_{R}^{o})^{2})q_{z}^{3}(q_{\perp}^{2}+q_{z}^{2})]$		(40)

in which

$\displaystyle N_{2}(\bm{q})={(q_{\perp}^{2}+q_{z}^{2})^{3}+\epsilon_{R}(q_{\perp}^{2}+q_{z}^{2})^{2}(-q_{\perp}^{2}-2q_{z}^{2})+(\epsilon_{R}^{2}+(\epsilon_{R}^{o})^{2})q_{z}^{2}(q_{\perp}^{2}+q_{z}^{2})^{2}}.$

(41)

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 Eq. 10) and as a consequence, gives rise to an azimuthal flow,

$\displaystyle v_{\phi,1}^{(\eta_{R}^{o})}(\mathcalligra{r},\theta)$

$\displaystyle=\epsilon_{R}^{o}\frac{F_{z}}{16\pi\mu}\frac{\sin(2\theta)}{\mathcalligra{r}}+\order{(\epsilon_{R}^{o})^{2}}$

(42)

The $v_{\phi}$ profile due to $\eta_{R}^{o}$ is shown in Fig. 3D. 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.

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 analyzed 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 $\bm{v}=U\hat{\bm{z}}$ at $\mathcalligra{r}\to\infty$ and a no-slip boundary condition with $\bm{v}=0$ on the surface of the sphere $\mathcalligra{r}=a$. The streamlines of this flow in a standard fluid are shown in black in Fig. 4A on the $r$-$z$ plane. Here, we assume that the sphere cannot (or does not) rotate. In Sec. 5.3, we will discuss the case in which the sphere is allowed to rotate.

We once again look for a perturbative solution to Eq. 15 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 (Section 4.5), Eq. 15 reduces to the vector Poisson equation for the first order velocity field,

$\displaystyle\Delta\bm{v_{1}}=-\Delta_{\alpha}\bm{v_{0}}$

(43)

in which $\Delta_{\alpha}$ is the second-order differential operator associated to the viscosity $\alpha$ (here, $\alpha=\eta_{1}^{\text{o}},\eta_{2}^{\text{o}},\eta_{\text{R}}^{\text{o}}$, see Appendix E for explicit form), and $\bm{v_{0}}$ is the flow past a sphere in a standard fluid (given by Eq. 111). 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 $\bm{v_{1}}$ 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, $\bm{v_{\text{Stokes},1}}$ in Eqs. 35, 36, and 42, from Section 4.5, as

$\displaystyle v_{\phi,1}(\mathcalligra{r},\theta)$

$\displaystyle=\left(\frac{6\pi aU\mu}{F_{z}}\bm{v_{\text{Stokes},1}}+\frac{\pi a^{3}U\mu}{F_{z}}\Delta\bm{v_{\text{Stokes},1}}+\frac{\pi a^{5}U\mu}{20F_{z}}\Delta^{2}\bm{v_{\text{Stokes},1}}\right)\cdot\bm{\hat{\phi}}$

(44)

with no modifications to $v_{r}$ and $v_{z}$ at leading order.

In standard isotropic fluids, a superposition of the Stokeslet ($v\propto 1/\mathcalligra{r}$) and its second derivative (a source dipole $v\propto 1/\mathcalligra{r}^{3}$) is sufficient to satisfy the boundary conditions. In the presence of odd viscosity, we find that higher order in gradients are necessary, as can be seen from Eq. 44. Even so, by equating the far field of the flow and the Stokeslet solution, we find $F_{z}=6\pi aU\mu$. 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 to a standard fluid.

Rewriting Eq. 44 more explicitly, we have

$\displaystyle v_{\phi,1}(\mathcalligra{r},\theta)$

$\displaystyle=\frac{3U}{64}\left[g^{1}(\theta)\frac{a}{\mathcalligra{r}}+g^{3}(\theta)\left(\frac{a}{\mathcalligra{r}}\right)^{3}+g^{5}(\theta)\left(\frac{a}{\mathcalligra{r}}\right)^{5}\right]\sin{2\theta}$

(45)

where

	$\displaystyle g^{1}(\theta)=$	$\displaystyle\ 8\epsilon_{R}^{o}-(5+3\cos{2\theta})\epsilon_{1}-(2+6\cos{2\theta})\epsilon_{2}+\order{\epsilon^{2}}$		(46)
	$\displaystyle g^{3}(\theta)=$	$\displaystyle-8\epsilon_{R}^{o}+(6+10\cos{2\theta})\epsilon_{1}+(4+20\cos{2\theta})\epsilon_{2}+\order{\epsilon^{2}}$		(47)
	$\displaystyle g^{5}(\theta)=$	$\displaystyle-(1+7\cos{2\theta})\epsilon_{1}-(2+14\cos{2\theta})\epsilon_{2}+\order{\epsilon^{2}}$		(48)

These velocity fields are shown in Fig. 4C-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.