Nonreciprocal response of a two-dimensional fluid with odd viscosity

We discuss the linear hydrodynamic response of a two-dimensional active chiral compressible fluid with odd viscosity. The viscosity coefficient represents broken time-reversal and parity symmetries in the 2D fluid and characterizes the deviation of the system from a passive fluid. Taking into account the hydrodynamic coupling to the underlying bulk fluid, we obtain the odd viscosity-dependent mobility tensor, which is responsible for the nonreciprocal hydrodynamic response to a point force. Furthermore, we consider a finite-size disk moving laterally in the 2D fluid and demonstrate that the disk experiences a nondissipative lift force in addition to the dissipative drag one.

Figure 1

Schematic sketch of an active chiral fluid characterized by odd viscosity. An infinitely large, flat, and thin 2D fluid layer (green) (e.g., a monolayer formed by amphiphiles) is located at $z=h$ having 2D dilatational, shear, and odd viscosities, ${\eta }_{\mathrm{d}}$, ${\eta }_{\mathrm{s}}$, and ${\eta }_{\mathrm{o}}$, respectively. The fluid interface is in contact with air $(z>h)$ and a 3D fluid underneath $(0<z<h)$ having a 3D shear viscosity $\eta$. The 3D fluid is bounded by an impermeable flat wall located at $z=0$, and its velocity is assumed to vanish at $z=0$. On the 2D fluid layer at $z=h$, both time-reversal and parity symmetries are broken (e.g., due to the self-spinning objects injecting energy into the fluid), giving rise to the possibility of an odd viscosity, ${\eta }_{\mathrm{o}}$, in the 2D layer.

Figure 2

Streamlines of the velocity $\mathbf{v}(x,y)$ rescaled by $F/\left(2\pi {\eta }_{\mathrm{s}}\right)$ and generated by a point force, $\mathbf{F}=F{\widehat{\mathbf{e}}}_x\delta \left(\mathbf{r}\right)$, as a function of $\kappa x$ and $\kappa y$ while keeping ${\eta }_{\mathrm{d}}=3{\eta }_{\mathrm{s}}$. The force along the $x$ axis is applied at the origin (the black horizontal arrow) for (a) $\mu ={\eta }_{\mathrm{o}}/{\eta }_{\mathrm{s}}=0$, (b) $\mu =3$, and (c) $\mu =-3$ [see Eqs. (13) and (19)]. The blue arrows indicate the flow direction.

Figure 3

Streamlines of the velocity $\mathbf{v}(x,y)$ rescaled by $F\kappa \ell /\left(2\pi {\eta }_{\mathrm{s}}\right)$ and generated by a force dipole as a function of $\kappa x$ and $\kappa y$ while keeping ${\eta }_{\mathrm{d}}=3{\eta }_{\mathrm{s}}$. The force dipole along the $x$ axis ($\widehat{\mathbf{d}}={\widehat{\mathbf{e}}}_x$) is centered at the origin (the black double arrow) for (a) $\mu ={\eta }_{\mathrm{o}}/{\eta }_{\mathrm{s}}=0$, (b) $\mu =3$, and (c) $\mu =-3$ [see Eq. (23)]. The blue arrows indicate the flow direction.

Figure 4

Plots of the mobility coefficients rescaled by $2\pi {\eta }_{\mathrm{s}}$ as a function of $\kappa r$ for various values of $\mu$. (a) The longitudinal mobility coefficient $G_{xx}$ for $\mu =0$, 3, and 5 (black, red, and blue lines). (b) The transverse mobility coefficient $G_{yy}$ for $\mu =0$, 3, and 5 (black, red, and blue lines). (c) The antisymmetric mobility coefficient $G_{xy}$ for $\mu =-3$, 0, and 3 (red dashed, black solid, and red solid lines).

Figure 5

Plots of the drag (${\mathrm{\Gamma }}_{\parallel }$) and lift (${\mathrm{\Gamma }}_{\perp }$) coefficients rescaled by $2\pi {\eta }_{\mathrm{s}}$ as a function of the rescaled disk radius $\kappa R$. (a) The drag coefficient ${\mathrm{\Gamma }}_{\parallel }$ for $\mu =0$, 3, and 5 (black, red, and blue solid lines). The dotted line represents the full expression ${\mathrm{\Gamma }}_{\parallel }^0$ for the drag coefficient when $\mu =0$ reported in Ref. [15] (see the text for the specific expression). (b) The lift coefficient ${\mathrm{\Gamma }}_{\perp }$ for $\mu =-5$, $-3$, 0, 3, and 5 (blue dashed, red dashed, black solid, red solid, and blue solid lines).

Figure 6

Plot of the rotational friction coefficient ($\mathrm{\Lambda }$) rescaled by $4\pi {\eta }_{\mathrm{s}}/{\kappa }^2$ as a function of the rescaled disk radius $\kappa R$ for $\mu =0$, 3, and 5 (black, red, and blue solid lines). The dotted line represents the full expression ${\mathrm{\Lambda }}^0$ for the rotational coefficient when $\mu =0$ reported in Ref. [20] (see the text for the specific expression).

Figure 7

Sketch of the circular curve, $C_{\mathrm{d}}$, and the unspecified curve, $C_{\mathrm{u}}$, with the accompanying unknown force distribution, $\mathbf{f}$, while $A$ is the fluid area bounded by both $C_{\mathrm{d}}$ and $C_{\mathrm{u}}$ (white area). The curves, $C_{\mathrm{d}}$ and $C_{\mathrm{u}}$, are parameterized by the vectors $\mathbf{R}$ and ${\mathbf{R}}^{\prime }$, respectively, and the two arrows represent the direction of the line integral. In the sketch, we have $|{\mathbf{R}}^{\prime }|>|\mathbf{R}|$ only for presentation purposes. In actual calculations where the condition $|{\mathbf{R}}^{\prime }|\ll |\mathbf{R}|$ is used, the two curves overlap with each other.