Exact response functions for a compressible thin fluid layer with odd viscosity

Abstract

Fluids composed of chiral active components can exhibit odd viscosity, a property that breaks time-reversal and parity symmetries. We investigate the hydrodynamic response to monopole and dipole singularities in a compressible thin fluid layer with odd viscosity, supported by a conventional lubrication layer. Using the two-dimensional Green's function in Fourier space, we derive exact analytical solutions for the flow and pressure fields. These solutions provide a detailed description of the hydrodynamic interactions governing the motion of colloidal particles and microswimmers in confined chiral fluids, offering insight into the role of odd viscosity in modifying particle dynamics and collective behavior. The derived results are directly applicable to modeling transport, control, and self-organization phenomena in active and chiral microfluidic systems.

Figures (6)

• Figure 1: Schematic illustration of a 2D chiral active fluid layer, which breaks time-reversal and parity symmetries (e.g., due to internally actuated spinning constituents, as depicted by the blue arrows, or the intrinsic handedness of the fluid medium). The layer is modeled as an infinitely thin, compressible 2D fluid with 2D shear viscosity $\eta_\mathrm{S}$, dilatational viscosity $\eta_\mathrm{D}$, and odd viscosity $\eta_\mathrm{O}$. The fluid layer rests on a 3D bulk fluid of thickness $h$ (not to scale) and 3D shear viscosity $\eta_\mathrm{B}$, which is bounded from below by a flat, impermeable rigid substrate. The red arrow indicates a point force $\bm{f}$ applied to the 2D fluid layer.
• Figure 2: Integration contour and the corresponding locations of the residues for $\delta<0$ and $\delta\ge 0$. The contour consists of four segments: two semicircles and two straight lines along the real axis. There are two poles in the upper half-plane, whose positions depend on the sign of $\delta$ defined in Eq. \ref{['eq:delta']}. The integrals are evaluated in the limits $\epsilon\to0$ and $R\to\infty$ using the method of residues.
• Figure 3: Contour plot of the velocity magnitude with superimposed quiver plot of the 2D flow field induced by a point-force singularity in a compressible fluid layer without odd viscosity ($\mu=0$). Results are shown for (a) $\xi=0.5$ and (b) $\xi=1$. The scaled velocity is defined as $\bm{v}^*=(\eta_\mathrm{S}/F)\bm{v}$, where $F$ denotes the magnitude of the applied point force.
• Figure 4: Contour plot of the velocity magnitude with a superimposed quiver representation of the 2D flow field induced by a point-force singularity in a compressible fluid layer with odd viscosity ($\mu=4$). Panel (a) corresponds to $\xi=0.5$ and panel (b) to $\xi=1$. The scaled velocity is defined as $\bm{v}^*=(\eta_\mathrm{S}/F)\bm{v}$.
• Figure 5: Contour plot of the pressure field induced by a point force in a 2D fluid: (a) without odd viscosity ($\mu=0$) and (b) with odd viscosity ($\mu=4$). Results are shown for $\xi=1$. The scaled pressure is defined as $p* = h/\left(\kappa F\right) p$.