Stokes flows in three-dimensional fluids with odd viscosity
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1
Stokes flows in three-dimensional fluids with
arXiv:2011.07681v1 [physics.flu-dyn] 16 Nov 2020
odd viscosity
Tali Khain1,2 , Colin Scheibner1,2 , and Vincenzo Vitelli1,2,3
1
James Franck Institute, The University of Chicago, Chicago, IL 60637, USA
2
Department of Physics, The University of Chicago, Chicago, IL 60637, USA
3
Kadanoff Center for Theoretical Physics, The University of Chicago, Chicago, IL 60637, USA
(Received xx; revised xx; accepted xx)
The Stokeslet is the fundamental Green’s function associated with point forces in viscous
flows. It prescribes how the work done by external forces is balanced by the energy
dissipated through velocity gradients. In ordinary fluids, viscosity is synonymous with
energy dissipation. Yet, in fluids with broken microscopic time-reversal symmetry, the
viscosity tensor can acquire a dissipationless contribution called odd viscosity. As the
ratio between odd and dissipative viscosity diverges, energy balance requires that the
resulting flow gradients become singular. Here, we find that these singularities give rise
to additional contributions to the Stokeslet flow that persist even when the odd viscosity
is small. In this limit, we solve for the flow past a sphere and illustrate the distinct effects
of odd shear and rotational viscosities. When applied to many-body sedimentation, our
analysis reveals the emergence of non-reciprocal hydrodynamic interactions and chiral
modifications to particle trajectories.
1. Introduction
At low Reynolds numbers, fluid flow is controlled by viscosity (Purcell 1977). Viscosity
itself is the ability of a fluid to exert stresses in response to velocity gradients, as expressed
by the phenomenological equation
σij (x) = ηijkl ∂k vl (x) (1.1)
where σij (x) is the viscous stress, vl (x) is the velocity, and ηijkl is the viscosity tensor.
For systems with microscopic time-reversal symmetry, the Onsager reciprocity relations
imply that the viscosity tensor must be symmetric ηijkl = ηklij , and hence contribute to
the entropy production ṡ ∝ σij ∂i vj (Groot 1962). However, for systems with broken
microscopic time reversal symmetry, the viscosity tensor can acquire an additional
antisymmetric part
o o
ηijkl = −ηklij (1.2)
o o
The antisymmetric tensor ηijkl is known as odd (or Hall) viscosity. Crucially, since ηijkl is
antisymmetric, odd viscosity does not contribute to entropy production, i.e. dissipation,
in a fluid.
Fluids with odd viscosity arise in various domains. Experimental measurements of
odd viscosity have been performed in electron fluids (Berdyugin et al. 2019), gases of
polyatomic molecules subject to magnetic fields (Korving et al. 1967; Hulsman et al.
1970), and colloids composed of spinning parts (Soni et al. 2019). Moreover, odd viscosity
has received theoretical attention in contexts inlcuding active matter (Banerjee et al.
2017; Han et al. 2020; Souslov et al. 2020; Markovich & Lubensky 2020; Banerjee et al.
2
2020), plasmas (Chapman 1939), quantum fluids (Avron et al. 1995; Vollhardt 1990;
Hoyos & Son 2012), and fluids of vortices (Wiegmann & Abanov 2014; Bogatskiy 2019).
As odd viscosity implies parity violation in isotropic two-dimensional fluids, it serves as
an exciting platform for exotic wave phenomena (Souslov et al. 2019; Tauber et al. 2020;
Baardink et al. 2020; Abanov et al. 2018; Avron et al. 1995; Avron 1998; Monteiro &
Ganeshan 2020). Since odd viscosity itself does not entail entropy production, extensive
attention has been paid to the ideal Bernoulli limit that is amenable to Hamiltonian and
action principles (Abanov & Monteiro 2019; Markovich & Lubensky 2020; Banerjee et al.
2017).
Here, we examine a different facet of odd viscosity. Working in two and three dimen-
sions, we investigate the odd viscous response to local probes in the limit that inertial
forces are negligible. Viscosity-dominated flows are the setting for phenomena ranging
from many-body sedimentation (Purcell 1977; Goldfriend et al. 2017; Chajwa et al. 2019)
to the locomotion of microswimmers (Huang et al. 2019; Elfring & Lauga 2020; Lapa
& Hughes 2014). Due to the linearity of the Stokes equation, general solutions can be
constructed from Green’s functions, known as Stokeslets (Happel 1983). Qualitatively, the
interplay between the Stokeslet and odd viscosity can be viewed from the perspective
of energy balance: the work done by an external force must be balanced by the heat
generated by velocity gradients. As the dissipative viscosity is removed, flow gradients
must become increasingly intense in order to offset the work done by the external agent.
Consequently, in the formal limit that the dissipative viscosity vanishes and only the
odd viscosity remains, the velocity gradient field must exhibit singularities. Our study
illustrates how such singularities in the dissipationless limit belie the qualitative character
of the odd viscous flow even when dissipation is present.
2. The Stokeslet of an odd viscous fluid: exact solution
We begin our analysis by obtaining the Stokeslet solution in the presence of odd
viscosity in an incompresssible fluid. When inertial forces are negligible, fluid flow is
governed by the Stokes equation
−∂j p(x) + ∂i σij (x) =fj (x) (2.1)
where p(x) is the pressure and fi (x) is the external force density. In Fourier space, Eq. 2.1
may be written as
iqi p(q) + Mil (q)vl (q) = fi (q) (2.2)
−1
subject to the incompressiblity condition qi vi = 0, where Mil (q) = qi qk ηijkl . When Mij
exists, the general solution is given by vi (q) = Gij (q)fj (q), where
−1 −1
Mim qm qn Mnj
−1
Gij (q) ≡ Mij − −1 (2.3)
qk Mkl ql
1 2 q q
For a standard isotropic, incompressible fluid, Gij = 8π q (δij + qi 2j ). Notice that Gij (q)
o
is symmetric whenever ηijkl vanishes. Moreover, whenever the symmetric (i.e. dissipative)
component of ηijkl vanishes, the second term of Eq. 2.3 diverges, as can be anticipated
on the basis of energy arguments.
To concretely illustrate the consequences of Eq. 2.3, the form of ηijkl must be specified.
As discussed in Appendix A, the most general ηijkl has 34 = 81 independent components
in three dimensions in absence of any physical symmetries or constraints. Imposing
cylindrical symmetry about an axis ẑ restricts ηijkl to the 19 independent components
shown in Eq. A 2. We first consider only the shear viscosities which relate shear stress
3
Figure 1. A Stokeslet in an odd viscous fluid. An external force, F , is applied at the origin
in the −ẑ direction (panel D). Panels A-C visualize the streamlines of the Stokeslet solution
for a range of odd to even viscosity ratios, γ = η2o /µ. All flows are obtained analytically for
the special case η1o = −2η2o (see Eq. 2.5-2.6). As the odd viscosity is added (panels B-C), 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 (panel C), the familiar radial component
of the flow vanishes. Panels E and F show slices of the azimuthal velocity field on the r-z plane,
and correspond to panels B and C, respectively. Panel G further details the angular dependence
of vφ on θ for a variety of values of γ. The lobes in panels E and G arise from the singularity
along the z-axis at γ → ∞ in panel F and swing out as γ decreases.
to shear strain: µ1 , µ2 , µ3 , η1o , and η2o . Here, µ1 , µ2 , and µ3 respect Onsager reciprocity,
while η1o and η2o violate Onsager reciprocity. Particularly elegant solutions are available
when µ1 = µ2 = µ3 = µ and η1o = −2η2o , and we will assume this ansatz for the
rest of this section. In this case, the viscosity tensor can be viewed geometrically as
performing a rotation through the angle arctan(η2o /µ) about the ẑ axis. We note that
the choice η1o = −2η2o is consistent with viscous terms derived from Hamiltonain coarse-
graining procedures on fluids of spinning molecules (Markovich & Lubensky 2020), and
representative of the range −2 . η1o /η2o . −1/2 measured in three-dimensional gases of
polyatomic molecules (Hulsman et al. 1970). See Appendix A for further discussion of
the symmetry properties of ηijkl .
Taking f = −ẑFz δ 3 (x), we derive the resulting flow field parameterized by γ = η2o /µ,
!
Fz cot θ 1
vr (ρ, θ) = − 1− p (2.4)
4πη2o γρ 1 + γ 2 sin2 θ
4
!
Fz cot θ 1
vφ (ρ, θ) = 1− p (2.5)
4πη2o ρ 1 + γ 2 sin2 θ
!
Fz 1 γ2 + 1
vz (ρ, θ) = 1− p (2.6)
4πη2o γρ 1 + γ 2 sin2 θ
and the pressure field,
2(γ 2 + 1)
Fz cos θ
p(ρ, θ) = 1− (2.7)
4π ρ2 (1 + γ 2 sin2 θ)3/2
Streamlines of the velocity field are visualized for a range of γ in figure 1a-c and
supplementary movie 1. Notably, for γ 6= 0, the flow develops an azimuthal component
(figure 1e-f), consistent with the planar chirality of η1o and η2o . In the limit µ → 0
(or γ → ∞), the r̂-component of the velocity field vanishes, and the two remaining
components, vφ and vz , become proportional to (ρ sin θ)−1 . As anticipated on the basis
of energy arguments, gradients in the velocity have singularities, in this case arising
along the axis of symmetry, sin θ = 0. In terms of vorticity ω = ∇ × v, ωρ takes the
form ωρ (ρ, θ) ∝ [δ(sin θ) − 1]/ρ2 when γ → ∞. In this limit, ωρ is the combination of two
contributions: (i) a 1/ρ2 term that resembles the electric field emanating from a point
charge and (ii) a delta-function along the axis of cylindrical symmetry that acts to cancel
the net source of vorticity within any surface enclosing the Stokeslet. (See Appendix B
for the Stokeslet in two dimensions.)
While the singular line dominates the flow field at |γ| = ∞, a finite dissipative viscosity
µ regularizes the singularity. As γ decreases, the singular line splits into lobes of high
azimuthal velocity that migrate away from the vertical, illustrated by the angular profile
of vφ in figure 1g. We note that the singular solution is purely a formal limiting case, as
the diverging velocity is physically inconsistent with the assumption of negligible inertial
forces. In the next sections, we examine flows regularized by large dissipative viscosity
and objects of finite radius.
3. Stokeslet: perturbative solution
In the section above, we considered the special case of η1o = −2η2o , which enables
an exact solution of the Stokeslet problem for all values of γ = η2o /µ. We now take
η1o and η2o to be independent and perform a perturbative expansion of Eq. 2.1 in the
o
quantities 1(2) ≡ η1(2) /µ 1. We find that both η1o and η2o contribute to leading order
by introducing terms contained entirely in the φ̂-component of the velocity
Fz (5 + 3 cos 2θ) sin 2θ
+ O 21
vφ (ρ, θ) = −1 (3.1)
128πµ ρ
Fz (1 + 3 cos 2θ) sin 2θ
+ O 22
vφ (ρ, θ) = −2 (3.2)
64πµ ρ
Figure 2a-b depicts the vφ profiles for η1o and η2o separately. While both velocity fields de-
cay as 1/ρ, they differ appreciably in their angular dependence: η2o includes an additional
sign change. As demonstrated in the previous section, this lobe structure originates from
the singularities in the limit of µ → 0, yet persists for small η1o , η2o and evolves into the
shape shown in figure 2.
Fluids exhibiting odd shear viscosities often have sources of angular momentum that
may additionally couple vorticity and anti-symmetric stress σij 6= σji . Such “rotational”
viscosity coefficients, shown in the green top left block of Eq. A 2, break both minor5
Figure 2. Contour plots of the first-order correction to Stokeslet flow due to a variety of
distinct shear and rotational odd viscosity coefficients. Assuming that the odd viscosity is small
as compared to the even viscosity, panels A-C visualize the azimuthal flow for a slice at constant
φ as functions of the coordinates r, z, for odd viscosities η1o , η2o , and ηR , respectively. Blue denotes
fluid flow into the page, and red denotes flow out of the page. The zeroth order streamlines of
the normal Stokeslet are plotted in panel A. The inset in panel B demonstrates the complex
near-field of the odd viscous flow past a sphere, for η2o /µ 1; an additional higher order term
(1/ρ5 ) is needed to satisfy the no-slip boundary condition. The far-field is described by the
Stokeslet.
symmetries of the viscosity tensor, ηijkl , as they are odd under exchange of i and j as
well as k and l. Here, we once again consider a system with cylindrical symmetry and
investigate the role of the rotational viscosities Γ1 , Γ3 , and η R on the Stokeslet solution.
Both Γ1 and Γ3 respect Onsager reciprocity, while η R violates Onsager reciprocity. Of
the three coefficients, only η R violates planar chirality and, as a consequence, gives rise
to an azimuthal flow:
Fz sin(2θ)
+ O 2R
vφ (ρ, θ) = R (3.3)
16πµ ρ
While both the shear and the rotational viscosities generate azimuthal flow, the vφ profile
due to η R shown in figure 2c differs subtly in its θ dependence from the effects of the
shear viscosities.
4. Odd viscous flow past a sphere
We now consider the odd viscous flow past a finite radius sphere. We assume a uniform
velocity field v = U ẑ at ρ → ∞ and no-slip boundary conditions v = 0 on the surface of
the sphere ρ = a. In standard isotropic fluids, a superposition of the Stokeslet (v ∝ 1/ρ)
and its second derivative (v ∝ 1/ρ3 ) are sufficient to satisfy the boundary conditions and
provide the force that holds the sphere in place. However, in the presence of odd viscosity,
we find that higher order gradients are necessary. Solving for the flow to leading order
in 1 , 2 , and R , we obtain
" 3 5 #
3U 1 a 3 a 5 a
vφ (ρ, θ) = f (θ) + f (θ) + f (θ) sin(2θ) (4.1)
64 ρ ρ ρ6
Figure 3. The trajectories of three inertia-less particles sedimenting in each others’ Stokeslet
flows. Panel A visualizes the three-dimensional nature of the particle paths in the presence
of odd viscosity, η o = η1o = η2o . Without odd viscosity (panels B and C), the trajectories are
purely two-dimensional: if the particles are initialized with y = 0, they remain so throughout
the dynamics. Panel B shows a top-down view (x-y plane), and panel C shows a side view
(x-z plane). Panels D and E present alternate views of the trajectories from panel A. Panel F
demonstrates the non-reciprocity of the Stokeslet flow. The velocity fields (small colored arrows)
are generated by point forces (large colored arrows) in an odd viscous fluid.
where
f 1 (θ) =8R − (5 + 3 cos 2θ)1 − (2 + 6 cos 2θ)2 + O 2
(4.2)
f (θ) = − 8R + (6 + 10 cos 2θ)1 + (4 + 20 cos 2θ)2 + O
3 2
(4.3)
f 5 (θ) = − (1 + 7 cos 2θ)1 − (2 + 14 cos 2θ)2 + O 2
(4.4)
with no modifications to vr and vz at leading order.
We note that the far field of the flow is indeed described by the odd Stokeslet solution in
Eq. 3.1-3.3 with Fz = 6πaU µ. However, unlike standard isotropic fluids, Eq. 4.1 contains
a 1/ρ5 term, which is higher order than the Stokeslet or its second derivative appearing
in standard isotropic fluids. Finally, we note that Eq. 4.1 is obtained via a perturbation
about the solution for a standard isotropic fluid. 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.
Application of the Dirichlet Green’s function of the vector Laplacian yields Eq. 4.1.
5. Sedimentation and non-reciprocal Stokeslet interactions
We now examine the role of odd viscosity on hydrodynamic interactions between
particles at low Reynolds number. Features of the particle geometry, such as chirality
and alignability, have received extensive attention in sedimentation problems (Goldfriend
et al. 2017; Chajwa et al. 2019). Here, by contrast, we consider identical spherical particles
immersed in an odd viscous fluid and study its effect on their trajectories. In the dilute
limit, each particle subject to external forces can be treated as a Stokeslet. In this case,
assuming the odd viscosity is small results in the equation of motion (Happel 1983)7
dxα 1 X
= Fα + G(xα − xβ ) · F β (5.1)
dt 6πµa
β
where G(x) is the inverse Fourier transform of Eq. 2.3, F β is the force applied to the
sphere at position xβ , and a is the radius of the spheres. As an illustration of the effects
of odd viscosity, we consider three particles sedimenting under a constant vertical force
−Fz ẑ. In supplementary movie 2 and figure 3a-e, we show the resulting trajectories of the
particles due to their mutual interactions in the frame co-moving with speed −Fz /6πµa.
The initial arrangement of the particles is an equilateral triangle pointing downward in
the x-z plane. As the particles descend, the odd viscosity gives rise to a chiral spiraling
behavior, forbidden by symmetry in an isotropic fluid. For the trajectories shown, we
consider the regime of perturbatively small odd viscosity with η1o = η2o . In figure 3b-
c, we show the trajectories of the particles when odd viscosity is absent: they remain
entirely planar. However, once odd viscosity is added, the azimuthal flow from the odd
Stokeslet induces the distinctive three-dimensional trajectories shown in figure 3a,d-e and
supplementary movie 2.
In addition to modifying the trajectories, odd viscosity also affects the energetic
relationship between the external forces. For example, Onsager reciprocity implies the
following relationship for the work done by two point forces (Masoud & Stone 2019;
Brenner & Nadim 1996),
dW21 dW12
≡ F 2 · v 1 (x2 ) = F 1 · v 2 (x1 ) ≡ , (5.2)
dt dt
where v 1(2) is the velocity field generated by the force F 1(2) on particle 1(2). The quantity
W21 is the additional work done by F 2 owing to the particle motion induced by F 1 ,
and vice versa for W12 . While W12 and W21 must be equal for a fluid respecting time-
reversal symmetry, external forces acting on an odd viscous fluid can violate Eq. 5.2.
As a simple illustration, figure 3f shows two point forces in the x-y plane for a fluid in
which η1o = −2η2o and µ = 0. Upon evaluating Eq. 5.2, we find dW21 /dt = −dW12 /dt.
Indeed, when µ = 0, the interaction terms between the two Stokeslets must be equal and
opposite since the fluid cannot dissipate energy. Notice that violations of Eq. 5.2 do not in
principle require experimental detailed resolution of the flow field or particle motion, but
only the determination of the work done by the external forces. Beyond sedimentation,
reciprocity underlies the conventional treatment of viscous phenomena such as particulate
mixing, drag on extended bodies, and the motion of air bubbles (Masoud & Stone 2019).
Thus, the simultaneous breaking of reciprocity and chiral symmetry in three-dimensional
Stokes flow suggests a host of experimental effects due to odd viscosity that do not rely
on detailed measurements of pressure and velocity fields.
Appendix A. Viscosity tensor in three dimensions
In three-dimensions, the viscosity tensor is a rank-4 tensor, ηijkl , with 81 possible
elements. Following the notation in Scheibner et al. (2020b,a), we use a basis of 3 × 3
A
matrices denoted by τij , with A = 1 . . . 9 to decompose the stress tensor and velocity
gradient tensor, and thereby express ηijkl as a 9 × 9 matrix. The basis consists of three
k
anti-symmetric matrices (torques and rotations, Rij = kij ), a diagonal matrix (pressure
q
2
and dilation, Cij = 3 δij ), and five traceless symmetric matrices (shear stresses and8
shear strains, 3 × 3 Pauli matrices, Sij ),
−1
√ 0 0
1 0 0 0 1 0 0 0 0 0 0 1
03 −1
S 1 = 0 −1 0 S 2 = 1 0 3
0 S = √ 0 S 4 = 0 0 1 S 5 = 0 0 0
3
0 0 0 0 0 0 0 0 √2 0 1 0 1 0 0
3
(A 1)
A B
Note the normalization τij τij = 2δ AB . Defining v A ≡ ∂i vj τij
A
and σ A ≡ σij τij
A
, we find
σ A = η AB v B where η AB = 21 τijA B
ηijkl τkl . Subject to cylindrical symmetry about the ẑ
AB
axis, the most general form of η is given by
(A 2)
Writing the viscosity tensor in this way elucidates the physical meaning of each element.
In an isotropic fluid, the only remaining terms are the familiar bulk viscosity ζ, which
couples isotropic dilation to isotropic pressure; shear viscosity µ = µ1 = µ2 = µ3 , which
couples shear strain to shear stress; and rotational viscosity Γ = Γ1 = Γ3 , which couples
vorticities and internal torques. In the main text, we investigate the role of the odd
counterparts to these terms, focusing on the shear (η1o , η2o ) and rotational (η R ) viscosities.
In gases of polyatomic molecules, coefficients η1o and η2o have been measured and they
are sometimes referred to as η4 and η5 , respectively (Groot 1962).
It is instructive to note the relationship between Eq. A 2 and chirality. We say that a
tensor is achiral if and only if every parity operation (i.e. a negation of an odd number of
spatial components) is equivalent to a rotation. Working in three dimensions, every parity
operation on a tensor may be implemented by acting on each index of the tensor with
−Rik , where Rik is a rotation matrix. Since ηijkl is a rank-4 tensor, ηijkl is manifestly
achiral. However, the three-dimensional viscosity tensor can nonetheless possess a distinct
property known as planar chirality. A three-dimensional object is planar chiral if it has
two-dimensional chirality about a given axis. More formally, a tensor is planar achiral
about an axis p̂i if and only if every parity operation preserving p̂i is equivalent to a
rotation preserving p̂i . For the matrix in Eq. A 2, all the components introduced by odd
viscosity are planar chiral about ẑi , except ηso , which respects planar chiral symmetry.
e e e
Moreover, of the even terms, ηA , ηQ,3 , and ηQ,1 are planar chiral, while the remainder
are not.
Appendix B. Stokeslet in two dimensions
Here we provide the Stokeslet solution for isotropic odd viscosity in two dimensions.
In this case, the incompressible Stokes equation is given by (Avron et al. 1995)
Fi δ 2 (r) =∂i p(r) + µ∇2 vi (r) + η o ∇2 ij vj (r) (B 1)
2
=∂i p(r) + η∇ Rij vj (r) (B 2)9
p
where η = µ2 + (η o )2 and Rij is a rotation matrix through angle θ = arctan(η o /µ).
The Green’s function for Eq. B 1 is given by:
−1 −1
" #
−1 −2 Rij r̂j r̂l Rlk
Rik Rik
vi (r) = + log(r) + Fk (B 3)
η µ µ
The logarithmic divergence in Eq. B 3 is a well known feature of two-dimensional Stokes
flows. We note that the coefficient η o disappears from the bulk equations of motion for
isotropic odd viscosity by defining an effective pressure (Ganeshan & Abanov 2017).
However, odd viscosity can still be detected by the application of point forces described
by Eq. B 3, because the Stokeslet acts as a stress boundary condition.
Appendix C. Stokeslet Reciprocity
Here, we comment on the equivalence between Eq. 5.2 and the Onsager reciprocity
relation ηijkl = ηklij . Suppose force distributions F 1 and F 2 are associated with velocity
distributions v 1 and v 2 over some volume V . Let us define Wαβ = V F α (x) · v β (x)dV .
R
Using the incompressiblity of the flow, we obtain
Z
dW21 dW12
− = (ηijkl − ηklij ) ∂i vj2 ∂k vl1 dV
dt dt V
Z
+ [ηijkl vj ∂k vl2 − ηijkl vj2 ∂k vl1 − p2 vi1 + p1 vi2 ]n̂i dA
1
(C 1)
∂V
where n̂i is the outward normal. Hence, when the flow and pressure vanish at the
boundary of the fluid, dW21 /dt − dW12 /dt is determined by the major symmetry of
the viscosity tensor. The far-field boundary term, however, is not always negligible. As
an illustration, consider the two Stokeslets obeying the solution in Eqs. 2.5-2.7 at γ = ∞.
In this case, ηijkl = −ηklij , and yet one obtains dW21 /dt 6= −dW12 /dt. This apparent
discrepancy arises since the flow along θ = 0 does not vanish at ρ → ∞ and hence the
boundary term of Eq. C 1 must be considered.
REFERENCES
Abanov, Alexander G., Can, Tankut & Ganeshan, Sriram 2018 Odd surface waves in
two-dimensional incompressible fluids. SciPost Phys. 5, 10.
Abanov, Alexander G. & Monteiro, Gustavo M. 2019 Free-surface variational principle
for an incompressible fluid with odd viscosity. Phys. Rev. Lett. 122, 154501.
Avron, J. E. 1998 Odd viscosity. Journal of statistical physics 92 (3-4), 543–557.
Avron, J. E., Seiler, R. & Zograf, P. G. 1995 Viscosity of quantum hall fluids. Phys. Rev.
Lett. 75, 697–700.
Baardink, Guido, Cassella, Gino, Neville, Luke, Milewski, Paul A. & Souslov,
Anton 2020 Complete absorption of topologically protected waves, arXiv: 2010.07342.
Banerjee, D., Souslov, A., Abanov, A. G. & Vitelli, V. 2017 Odd viscosity in chiral active
fluids. Nature communications 8 (1), 1–12.
Banerjee, Debarghya, Souslov, Anton & Vitelli, Vincenzo 2020 Hydrodynamic
correlation functions of chiral active fluids, arXiv: 2005.00621.
Berdyugin, A. I., Xu, S. G., Pellegrino, F. M. D., Krishna Kumar, R., Principi, A.,
Torre, I., Ben Shalom, M., Taniguchi, T., Watanabe, K., Grigorieva, I. V.,
Polini, M., Geim, A. K. & Bandurin, D. A. 2019 Measuring hall viscosity of graphene’s
electron fluid. Science 364 (6436), 162–165.
Bogatskiy, A 2019 Vortex flows on closed surfaces. Journal of Physics A: Mathematical and
Theoretical 52 (47), 475501.
Brenner, Howard & Nadim, Ali 1996 The Lorentz reciprocal theorem for micropolar fluids,
pp. 169–176. Dordrecht: Springer Netherlands.10
Chajwa, Rahul, Menon, Narayanan & Ramaswamy, Sriram 2019 Kepler orbits in pairs
of disks settling in a viscous fluid. Phys. Rev. Lett. 122, 224501.
Chapman, Sydney 1939 The mathematical theory of non-uniform gases; an account of the
kinetic theory of viscosity, thermal conduction, and diffusion in gases. Cambridge Eng.:
University Press.
Elfring, Gwynn J. & Lauga, Eric 2020 Active particles in linear viscoelastic fluids and the
scallop theorem, arXiv: 2009.08514.
Ganeshan, Sriram & Abanov, Alexander G. 2017 Odd viscosity in two-dimensional
incompressible fluids. Phys. Rev. Fluids 2, 094101.
Goldfriend, Tomer, Diamant, Haim & Witten, Thomas A. 2017 Screening,
hyperuniformity, and instability in the sedimentation of irregular objects. Phys. Rev. Lett.
118, 158005.
Groot, S. R. de (Sybren Ruurds), 1916 1962 Non-equilibrium thermodynamics,. Amsterdam,
New York,: North-Holland Pub. Co.; Interscience Publishers.
Han, M., Fruchart, M., Scheibner, C., Vaikuntanathan, S., Irvine, W. T. M.,
de Pablo, J. & Vitelli, V. 2020 Statistical mechanics of a chiral active fluid. arXiv
preprint arXiv:2002.07679 .
Happel, John 1983 Low Reynolds number hydrodynamics with special applications to particulate
media, 1st edn. The Hague; Boston: Hingham, MA, USA: M. Nijhoff; Distributed by
Kluwer Boston.
Hoyos, Carlos & Son, Dam Thanh 2012 Hall viscosity and electromagnetic response. Physical
review letters 108 (6), 066805.
Huang, H.-W., Uslu, F. E., Katsamba, P., Lauga, E., Sakar, M. S. & Nelson, B. J.
2019 Adaptive locomotion of artificial microswimmers. Science Advances 5 (1).
Hulsman, H., Van Waasdijk, E.J., Burgmans, A.L.J., Knaap, H.F.P. & Beenakker,
J.J.M. 1970 Transverse momentum transport in polyatomic gases under the influence of
a magnetic field. Physica 50 (1), 53 – 76.
Korving, J., Hulsman, H., Scoles, G., Knaap, H.F.P. & Beenakker, J.J.M. 1967 The
influence of a magnetic field on the transport properties of gases of polyatomic molecules;:
Part i, viscosity. Physica 36 (2), 177 – 197.
Lapa, Matthew F. & Hughes, Taylor L. 2014 Swimming at low reynolds number in fluids
with odd, or hall, viscosity. Phys. Rev. E 89, 043019.
Markovich, T. & Lubensky, T.C. 2020 Odd viscosity in active matter: microscopic origin
and 3d effects. arXiv preprint arXiv:2006.05662 .
Masoud, Hassan & Stone, Howard A. 2019 The reciprocal theorem in fluid dynamics and
transport phenomena. Journal of Fluid Mechanics 879, P1.
Monteiro, G. M. & Ganeshan, S. 2020 Non-linear shallow water dynamics with odd viscosity.
arXiv preprint arXiv:2009.00118 .
Purcell, E. M. 1977 Life at low reynolds number. American Journal of Physics 45 (1), 3–11.
Scheibner, Colin, Irvine, William T. M. & Vitelli, Vincenzo 2020a Non-hermitian band
topology and skin modes in active elastic media. Phys. Rev. Lett. 125, 118001.
Scheibner, C., Souslov, A., Banerjee, D., Surowka, P., Irvine, W. T. M. & Vitelli,
V. 2020b Odd elasticity. Nature Physics 16 (4), 475–480.
Soni, V., Bililign, E. S., Magkiriadou, S., Sacanna, S., Bartolo, D., Shelley, M. J.
& Irvine, W. T. M. 2019 The odd free surface flows of a colloidal chiral fluid. Nature
Physics 15 (11), 1188–1194.
Souslov, A., Dasbiswas, K., Fruchart, M., Vaikuntanathan, S. & Vitelli, V. 2019
Topological waves in fluids with odd viscosity. Physical review letters 122 (12), 128001.
Souslov, Anton, Gromov, Andrey & Vitelli, Vincenzo 2020 Anisotropic odd viscosity
via a time-modulated drive. Phys. Rev. E 101, 052606.
Tauber, Clément, Delplace, Pierre & Venaille, Antoine 2020 Anomalous bulk-edge
correspondence in continuous media. Physical Review Research 2 (1), 013147.
Vollhardt, Dieter 1990 The superfluid phases of helium 3 . London ; New York: Taylor &
Francis.
Wiegmann, P. & Abanov, A. G. 2014 Anomalous hydrodynamics of two-dimensional vortex
fluids. Physical review letters 113 (3), 034501.You can also read
