## Introduction

Theoretical and experimental studies have revealed that electrons in condensed matter can behave hydrodynamically, exhibiting fluid phenomena such as Stokes flow and vortices1,2,3,4,5,6,7,8,9. Unlike classical fluids, preferred directions inside crystals lift isotropic restrictions, necessitating a generalized treatment of electron hydrodynamics. While anisotropic viscous flows have been studied in geophysics10, their prominence in condensed matter has yet to be explored. This is of particular importance, given the recent demonstration of hydrodynamic behavior in three-dimensional materials such as Weyl semimetals11,12. Electron hydrodynamics is observed when microscopic scattering processes conserve momentum over time- and length scales that are large compared to those of the experimental probe. However, even as momentum is conserved, free energy may be dissipated from the electronic system, giving rise to a measurable viscosity in the electron flow12,13,14,15,16,17,18.

When momentum is conserved, a fluid obeys Cauchy’s laws of motion19

$$\rho \dot{{u}_{i}}={\partial }_{j}{\tau }_{ji}+\rho {f}_{i}$$
(1)
$$\rho \dot{{\sigma }_{i}}={\partial }_{j}{m}_{ji}+\rho {l}_{i}+{\epsilon }_{ijk}{\tau }_{jk},$$
(2)

where u and ρ are the fluid velocity and density, f and l are body forces and couples, τ and m are the fluid stress and couple stress, and σ is the intrinsic angular momentum density (internal spin). The superscript dot denotes the material derivative, $$\dot{x}={\partial }_{t}x \, +{u}_{j}{\partial }_{j}x$$, and ϵ is the rank-3 alternating tensor. We assume couple stresses and body couples to be zero, but allow for body forces of the form ρfi = −Rijuj, where R is a rank-2, positive–semidefinite tensor that is inversely proportional to a microscopic momentum-relaxing lifetime. In steady state and at experimentally accessible Reynolds numbers17,20, this implies that the stress tensor is symmetric19. In this limit, electron fluids obey the modified Navier–Stokes equation

$$\rho {u}_{j}{\partial }_{j}{u}_{i}=-{\partial }_{i}p+{\partial }_{j}{\tau }_{ji}-{R}_{ij}{u}_{j},$$
(3)

where τ is symmetric. Note that in electron fluids, current density is analogous to the fluid velocity, and voltage drops are analogous to changes in pressure. Assuming that the fluid velocity is much smaller than the electronic speed of sound, ucs, the electron fluids are nearly incompressible, thus

$${\partial }_{i}{u}_{i}=0.$$
(4)

In this limit, ρ is a constant, which we take to be unity. Since the fluid stress appears in a divergence, it is defined only up to a constant, which we choose to make τ vanish when u is uniformly zero21,22. We further assume that the fluid stress vanishes for uniform flow, so that it is only a function of the velocity gradient.

Without further loss of generality, the constitutive relation is written to the first order as21

$${\tau }_{ij}={A}_{ijkl}{\partial }_{l}{u}_{k},$$
(5)

where A is the fluid viscosity, a rank-4 tensor relating the fluid velocity gradient (∂jui) and the fluid stress. Since we take τ to be symmetric, A is invariant under permutation of its first two indices, i.e., Aijkl = Ajikl21,22. Viscosity is represented as the sum of three rank-4 tensor basis elements23, summarized in Table 1

$${A}_{(ij)kl}={\alpha }_{((ij)(kl))}+{\beta }_{[(ij)(kl)]}+{\gamma }_{(ij)[kl]}.$$
(6)

Tensor α describes dissipative behavior respecting both stress symmetry and objectivity, i.e., αijkl = αjikl = αklij. Tensor β on the other hand, describes nondissipative Hall viscosity7,23,24,25,26,27, i.e., βijkl = −βklij, and is nonzero only when time-reversal symmetry is broken. Finally, γ breaks stress objectivity, i.e., γijkl = −γijlk, coupling fluid stress to the vorticity. The fifth column in Table 1 specifies whether the tensor is defined according to a handedness convention.

In classical fluids, the added consideration of rotational invariance requires A to be isotropic, reducing it to the form

$${A}_{ijkl}= \, \lambda {\delta }_{ij}{\delta }_{kl}+\mu \left({\delta }_{il}{\delta }_{jk}+{\delta }_{jl}{\delta }_{ik}\right)\\ \, +{{\mathcal{B}}}_{1}\left({\epsilon }_{ik}{\delta }_{jl}+{\delta }_{ik}{\epsilon }_{jl}\right)+{\Gamma }_{1}{\delta }_{ij}{\epsilon }_{kl},$$
(7)

where δ is the Kronecker delta, ϵ is the rank-2 alternating tensor, and the Lamé parameters λ and μ can be identified as the two independent components of the proper tensor α. In the incompressible case, λ does not contribute to the stress21. $${{\mathcal{B}}}_{1}$$ and Γ1 are constants parameterizing terms with the symmetry of β and γ, respectively. Since β and γ are pseudotensors, the last three terms in Eq. (7) are only nonzero in two dimensions23,24.

In crystals, however, there exist preferred directions and we cannot assume rotational invariance. Instead, we must consider the effect of the crystal symmetry given by Neumann’s principle28,29, which requires that physical properties described by rank-4 tensors, such as viscosity, remain invariant under the transformation law

$${A}_{ijkl}^{\prime}=| s{| }^{\eta }{s}_{im}{s}_{jn}{s}_{ko}{s}_{lp}{A}_{mnop},$$
(8)

where s is the space representation of any given point group symmetry of the crystal, s = ±1 is the determinant of the symmetry operation, and η = 0 for proper tensors and η = 1 for pseudotensors.

Although Eq. (8) relates different components of the viscosity tensor, further constrains must be imposed to ensure that the viscosity tensor never does positive work in Eq. (3), so that for any velocity field u in d dimensions

$$\int {u}_{i}{\partial }_{j}({A}_{ijkl}{\partial }_{k}{u}_{l}){d}^{d}{\bf{r}}\le 0.$$
(9)

Letting the Fourier transform of u be

$$\tilde{{\bf{u}}}({\bf{q}})=\int {e}^{i{\bf{q}}\cdot {\bf{r}}}u({\bf{r}}){d}^{d}{\bf{r}}$$
(10)

in d dimensions, we find

$$\int {q}_{j}{q}_{k}{\tilde{u}}_{i}^{* }({\bf{q}}){\tilde{u}}_{l}({\bf{q}}){A}_{ijkl}{d}^{d}{\bf{q}}\ge 0.$$
(11)

This is satisfied when Aijkl has a positive definite biquadratic form in il and jk, so we impose this constraint in addition to ij symmetry and crystal symmetry.

Viscosity tensors are then randomly generated to satisfy the aforementioned constraints, allowing for normalized numerical deviations from isotropy lower than order unity. The viscosity tensor is assumed to be spatially uniform in all cases. To demonstrate the differences between these general viscosity tensors and those more strongly constrained by symmetry, we solve for the velocity and pressure of low Reynolds number flows in several geometries. The parameterization of the viscosity tensor in Eq. (6) allows us to explore the effects of breaking stress objectivity and time-reversal symmetry. We highlight the effects of symmetry in the last two indices (kl) because it implies that the stress only couples to the strain rate (∂kul + ∂luk) and not to the vorticity (∂kul − ∂luk). This is a property of classical fluids, which means that rigid–rotational flows are stress-free, and hence are only sensitive to rotation via weaker effects like the Coriolis force. Below, we demonstrate that with more general viscosity tensors, this is not the case, and that the resulting rotational stresses can be probed in experimentally accessible geometries.

## Results

### Effect of anisotropy

We first consider rotational flow in an annulus with inner radius Rinner = 1 and outer radius Router = 2 (Fig. 1). We apply a no-slip condition to the outer boundary, allow the inner boundary to rotate with unit angular velocity ω = 1, and solve for the steady-state flow at Reynolds number

$${\rm{Re}}\equiv \frac{\omega {R}_{{\rm{inner}}}^{2}}{\left|A\right|}=0.3,$$
(12)

where

$${\left|A\right|}^{2}={A}_{ijkl}{A}_{ijkl}.$$
(13)

The zero-pressure point is fixed at the bottom of the annulus. Experimentally, such rotational flows can be achieved by threading a time-varying magnetic flux through a Corbino disk geometry30,31 (Fig. 1a). For a fluid with an isotropic viscosity, the steady-state velocity field rotates rigidly with the angular velocity set by the inner boundary condition (Fig. 1b).

To investigate the effects of anisotropy in two-dimensional materials, we consider materials with D6 (hexagonal) and D4 (square) symmetry. Notably, D6 materials do not deviate from isotropic behavior (Fig. 1(c)), consistent with experimental observations for graphene9,17. We note that 2D materials with C3 (threefold), C6 (sixfold), and D3 (triangular) symmetry also exhibit isotropic viscosity tensors (see Supplementary Methods). By contrast, the flow deviates considerably from isotropic behavior in D4 materials (Fig. 1d). We repeat the calculation, allowing for a momentum-relaxing body force equal to RL2/A = 0.1, illustrating that the deviation from isotropy remains observable (Fig. 1e). We assume R → 0 for the rest of the paper, and investigate its effects and symmetry in Supplemenentary Figs. 1 and 2. Figure 1f shows the steady-state velocity flow difference between the isotropic case and D4 materials. We observe steady-state vortices emerging at  ~15% of the bulk flow rate overlaid onto the isotropic velocity field. While the steady-state pressure field in D6 materials mirrors that of an isotropic fluid (Fig. 1g), the pressure field in D4 materials also exhibits four vortices (Fig. 1h), with orientation set by the underlying crystal axes.

### Effect of asymmetry

We next examine the importance of symmetry in the last two indices of the viscosity tensor. We calculate the flow profile for the annulus in Fig. 1 scaled by a factor of two, equipped with a pressure gauge, as shown in Fig. 2a. The pressure gauge is a channel with no-slip boundary conditions, allowing us to measure the difference between the flow and a nearly stationary fluid. To isolate the effects of $${{\mathcal{B}}}_{1}$$ and Γ1 in Eq. (7), Fig. 2a, b shows the flow and pressure fields in the annulus for a material with isotropic viscosity tensor where both $${{\mathcal{B}}}_{1}$$ and Γ1 have been set to zero (SO(2){α}). These are nearly unchanged inside the annulus as compared to Fig. 1b, g, with a constant pressure in the gauge. Allowing for nonzero stress-breaking components, i.e., using a material with isotropic viscosity for $${{\mathcal{B}}}_{1}=0$$ and Γ1 = 0.25 (SO(2){γ}), we observe a significant pressure buildup near the gauge. This is due to the shear stress between the rotating and stationary fluids, while the pressure within the gauge itself is nearly uniform, as shown in Fig. 2c.

To quantify the pressure difference between SO(2){α} and SO(2){γ}, note that the pressure is fixed to zero at a point p, the bottom of the annulus domain. The pressure in the gauge may be written as the path integral

$${p}_{{\rm{gauge}}}=\int_{p}^{g}\nabla p\cdot d{\bf{s}},$$
(14)

where g is a point in the gauge. At low Reynolds numbers, we may neglect ujjui in Eq. (3), to find in the steady state

$$\nabla p={A}_{ijkl}{\partial }_{i}{\partial }_{k}{u}_{l}.$$
(15)
$${p}_{{\rm{gauge}}}=\int_{p}^{g}{A}_{ijkl}{\partial }_{i}{\partial }_{k}{u}_{l}d{s}_{j}.$$
(16)

Taking into account Eq. (7) and noting that the changes in fluid flow are negligible, we find

$$\Delta {p}_{{\rm{gauge}}}=\int_{p}^{g}\Delta {A}_{ijkl}{\partial }_{i}{\partial }_{k}{u}_{l}d{s}_{j}={\Gamma }_{1}\int_{p}^{g}{\partial }_{i}\omega d{s}_{i}={\Gamma }_{1}\Delta \omega ,$$
(17)

where ωi = ϵijkjuk is the vorticity of the flow. For the geometry used, we find Δpgauge = 0.15, vorticity in the gauge is zero, and that in the annulus is 0.6, so Δω = 0.6 (Fig. 2c). Since we chose Γ1 = 0.25, we see that in this setup, the pressure gauge (Δpgauge = Γ1Δω = 0.25 × 0.6 = 0.15) is directly sensitive to the asymmetry in the viscosity, which couples the rotation directly to the pressure field and the stress. We note that the same setup is sensitive to Hall viscosity coefficients for time-reversal broken systems, i.e., for the case where both $${{\mathcal{B}}}_{1}$$ and Γ1 are nonzero, the pressure gauge generalizes to

$$\Delta {p}_{{\rm{gauge}}}=\int_{p}^{g}\Delta {A}_{ijkl}{\partial }_{i}{\partial }_{k}{u}_{l}d{s}_{j}=({{\mathcal{B}}}_{1}+{\Gamma }_{1})\Delta \omega .$$
(18)

While time-reversal and stress objectivity-breaking terms persist in two-dimensional isotropic materials, the handedness of the pseudotensor implies that mirror operations set them to zero in 3D. This can be directly observed by comparing low- and high-symmetry three-dimensional crystals. We consider the same rotational flow along the ab crystal plane of orthorhombic materials, such as the hydrodynamically reported Weyl semimetal WP211,12. Along this plane, the difference between the two viscosity tensors can be parameterized as follows:

$$\begin{array}{l}{A}_{ijkl}^{{C}_{2v}^{(001)}}={A}_{ijkl}^{{D}_{2h}^{(001)}}+{\Gamma }_{2}{\delta }_{ij}{\epsilon }_{kl}+{\Gamma }_{3}{\sigma }_{ij}^{z}{\epsilon }_{kl}\\ \hspace{1em}+{{\mathcal{B}}}_{2}\left({\delta }_{li}{\epsilon }_{jk}-{\epsilon }_{li}{\delta }_{jk}\right)+{{\mathcal{B}}}_{3}\left({\delta }_{ij}{\sigma }_{kl}^{x}-{\sigma }_{ij}^{x}{\delta }_{kl}\right),\end{array}$$
(19)

where $${{\mathcal{B}}}_{2}$$, $${{\mathcal{B}}}_{3}$$, Γ2, and Γ3 are constants parameterizing terms with the symmetry of β and γ, respectively, σx and σz are Pauli matrices. Figure 2d shows the pressure difference between a material with D2h symmetry and one with C2v symmetry (for $${{\mathcal{B}}}_{2}={{\mathcal{B}}}_{3}={\Gamma }_{3}=0$$ and Γ2 = 0.25), indicating the same pressure buildup as in Fig. 2c inside the gauge along with a nontrivial pressure structure in the annulus.

### 2D flows in 3D crystals

Finally, we consider flow through an expanding channel along high-symmetry planes in 3D. This geometry has been proposed as a diagnostic of electron hydrodynamics because it naturally generates vortices, not present in ordinary ohmic flow. The case with isotropic viscosity is shown in Fig. 3a, where the small vortices that form in the corners are clearly detached from the bulk of the flow. We consider the Td (tetrahedral) and Oh (cubic) point groups. In particular, we consider flows along the polar {111}, nonpolar {110}, and semipolar {001} family of planes (Fig. 3b)

$${A}_{ijkl}^{{T}_{{\rm{d}}}^{(111)}}={A}_{ijkl}^{{O}_{{\rm{h}}}^{(111)}}+{{\mathcal{B}}}_{4}\left({\sigma }_{ij}^{x}{\sigma }_{kl}^{z}-{\sigma }_{ij}^{z}{\sigma }_{kl}^{x}\right)+{\Gamma }_{4}{\delta }_{ij}{\epsilon }_{kl}$$
(20a)
$${A}_{ijkl}^{{T}_{{\rm{d}}}^{(110)}}={A}_{ijkl}^{{O}_{{\rm{h}}}^{(110)}}$$
(20b)
$${A}_{ijkl}^{{T}_{{\rm{d}}}^{(001)}}={A}_{ijkl}^{{O}_{{\rm{h}}}^{(001)}}+{{\mathcal{B}}}_{5}\left({\sigma }_{ij}^{z}{\delta }_{kl}^{z}-{\delta }_{ij}{\sigma }_{kl}^{z}\right)+{\Gamma }_{5}{\sigma }_{ij}^{x}{\epsilon }_{kl}.$$
(20c)

Along these planes, the difference between the two viscosity tensors can be parameterized according to Eqs. (20a), (20b), (20c). We impose fully developed (parabolic) inlet and outlet flows with constant discharge, and solve for the steady-state flow at low Reynolds number. Figure 3c shows the difference between the flow in an isotropic material and the flow in a cubic material along a {111} close-packed plane, which exhibits rotational invariance. Along the nonpolar {110} planes, terms with β and γ symmetry vanish. However, $${{A}}^{{{O}}_{{\rm{h}}}^{(110)}}$$ is anisotropic along this plane, with Fig. 3d showing the difference in flow between the isotropic case. Finally, along the semipolar {001} family of planes, the viscosity tensor is both anisotropic (Fig. 3e) and asymmetric. Figure 3f quantifies the additional vortices generated by the asymmetry at  ~10%, for $${{\mathcal{B}}}_{5}=0$$ and Γ5 = 0.25.

## Discussion

We found that electron fluids in crystals with anisotropic and asymmetric viscosity tensors can exhibit steady-state fluid behaviors not observed in classical fluids. In 3D, discrete deviations from isotropy allow the fluid stress to couple to the fluid vorticity with or without breaking time-reversal symmetry, for the case of Hall viscosity and objectivity-breaking viscosity, respectively. Recent measurements of spatially resolved flows9,17,32 suggest that these effects can be directly observed in systems beyond graphene. Our findings further hint at potential applications. For instance, the pressure gauge in Fig. 2 could be used as a magnetometer, converting a time-varying magnetic flux through a modified Corbino disk geometry into current in the annulus, and ultimately into a voltage drop between it and the gauge. Our work highlights the importance of crystal symmetry on electronic flow, and invites further exploration of time-dependent flows in systems with internal spin degrees of freedom and asymmetric stress tensors.