skip to main content


Title: Stokes flows in three-dimensional fluids with odd and parity-violating viscosities
The Stokes equation describes the motion of fluids when inertial forces are negligible compared with viscous forces. In this article, we explore the consequence of parity-violating and non-dissipative (i.e. odd) viscosities on Stokes flows in three dimensions. Parity-violating viscosities are coefficients of the viscosity tensor that are not invariant under mirror reflections of space, while odd viscosities are those which do not contribute to dissipation of mechanical energy. These viscosities can occur in systems ranging from synthetic and biological active fluids to magnetized and rotating fluids. We first systematically enumerate all possible parity-violating viscosities compatible with cylindrical symmetry, highlighting their connection to potential microscopic realizations. Then, using a combination of analytical and numerical methods, we analyse the effects of parity-violating viscosities on the Stokeslet solution, on the flow past a sphere or a bubble and on many-particle sedimentation. In all the cases that we analyse, parity-violating viscosities give rise to an azimuthal flow even when the driving force is parallel to the axis of cylindrical symmetry. For a few sedimenting particles, the azimuthal flow bends the trajectories compared with a traditional Stokes flow. For a cloud of particles, the azimuthal flow impedes the transformation of the spherical cloud into a torus and the subsequent breakup into smaller parts that would otherwise occur. The presence of azimuthal flows in cylindrically symmetric systems (sphere, bubble, cloud of particles) can serve as a probe for parity-violating viscosities in experimental systems.  more » « less
Award ID(s):
2011854
NSF-PAR ID:
10325756
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Journal of Fluid Mechanics
Volume:
934
ISSN:
0022-1120
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. null (Ed.)
    We examine the axisymmetric and non-axisymmetric flows of thin fluid films over a spherical glass dome. A thin film is formed by raising a submerged dome through a silicone oil mixture composed of a volatile, low surface tension species (1 cSt, solvent) and a non-volatile species at a higher surface tension (5 cSt, initial solute volume fraction $\phi _0$ ). Evaporation of the 1 cSt silicone oil establishes a concentration gradient and, thus, a surface tension gradient that drives a Marangoni flow that leads to the formation of an initially axisymmetric mound. Experimentally, when $\phi _0 \leqslant 0.3\,\%$ , the mound grows axisymmetrically for long times (Rodríguez-Hakim et al. , Phys. Rev. Fluids , vol. 4, 2019, pp. 1–22), whereas when $\phi _0 \geqslant 0.35\,\%$ , the mound discharges in a preferred direction, thereby breaking symmetry. Using lubrication theory and numerical solutions, we demonstrate that, under the right conditions, external disturbances can cause an imbalance between the Marangoni flow and the capillary flow, leading to symmetry breaking. In both experiments and simulations, we observe that (i) the apparent, most amplified disturbance has an azimuthal wavenumber of unity, and (ii) an enhanced Marangoni driving force (larger $\phi _0$ ) leads to an earlier onset of the instability. The linear stability analysis shows that capillarity and diffusion stabilize the system, while Marangoni driving forces contribute to the growth in the disturbances. 
    more » « less
  2. We investigate the modes of deformation of an initially spherical bubble immersed in a homogeneous and isotropic turbulent background flow. We perform direct numerical simulations of the two-phase incompressible Navier–Stokes equations, considering a low-density bubble in the high-density turbulent flow at various Weber numbers (the ratio of turbulent and surface tension forces) using the air–water density ratio. We discuss a theoretical framework for the bubble deformation in a turbulent flow using a spherical harmonic decomposition. We propose, for each mode of bubble deformation, a forcing term given by the statistics of velocity and pressure fluctuations, evaluated on a sphere of the same radius. This approach formally relates the bubble deformation and the background turbulent velocity fluctuations, in the limit of small deformations. The growth of the total surface deformation and of each individual mode is computed from the direct numerical simulations using an appropriate Voronoi decomposition of the bubble surface. We show that two successive temporal regimes occur: the first regime corresponds to deformations driven only by inertial forces, with the interface deformation growing linearly in time, in agreement with the model predictions, whereas the second regime results from a balance between inertial forces and surface tension. The transition time between the two regimes is given by the period of the first Rayleigh mode of bubble oscillation. We discuss how our approach can be used to relate the bubble lifetime to the turbulence statistics and eventually show that at high Weber numbers, bubble lifetime can be deduced from the statistics of turbulent fluctuations at the bubble scale. 
    more » « less
  3. Inward radial Rayleigh-Be'nard-Poiseuille flow can exhibit a buoyancy-driven instability when the Rayleigh number exceeds a critical value. Furthermore, similar to plane Rayleigh-Be'nard-Poiseuille flow, a viscous Tollmien-Schlichting instability can occur when the Reynolds number is high enough. Direct numerical simulations were carried out with a compressible Navier-Stokes code in cylindrical coordinates to investigate the spatial stability of the inward radial flow inside the collector of a hypothetical solar chimney power plant. The convective terms were discretized with fifth-order-accurate upwind-biased compact finite-differences and the viscous terms were discretized with fourth-order-accurate compact finite differences. For cases with buoyancy-driven instability, steady three-dimensional waves are strongly amplified. The spatial growth rates vary significantly in the radial direction and lower azimuthal mode numbers are amplified closer to the outflow. Traveling oblique modes are amplified as well. The growth rates of the oblique modes decrease with increasing frequency. In addition to the purely radial flow, a spiral flow with swept inflow was examined. Overall lower growth rates are observed for the spiral flow compared to the radial flow. Different from the radial flow, the relative wave angles and growth rates of the left and right traveling oblique modes are not identical. A plane RBP case with viscosity-driven instability by Chung et al. was considered as well. The reported growth rate and phase speed were matched with good accuracy. 
    more » « less
  4. Since the generation of green and clean renewable energy is a major concern in the modern era, solar energy conversion technologies such as the solar chimney power plant are gaining more attention. In order to accurately predict the performance of these power plants, hydrodynamic instabilities that can lead to large-scale coherent structures which affect the mean flow, have to be identified and their onset has to be predicted accurately. The thermal stratification of the collector flow (resulting from the temperature difference between the heated bottom surface and the cooled top surface) together with the opposing gravity can lead to buoyancy-driven instability. As the flow accelerates inside the collector, the Reynolds number can get large enough for viscous (Tollmien-Schlichting) instability to occur. A new highly accurate compact finite difference Navier-Stokes code in cylindrical coordinates has been developed for the spatial stability analysis of such radial flows. The new code was validated for a square channel flow. Stability results for different stable and unstable Reynolds/Rayleigh number combinations were in good agreement with temporal stability simulations as well as linear stability theory. To investigate the radial flow effect, spatial stability simulations were then carried out for a computational domain with constant streamwise extent and different outflow radii. The Reynolds and Rayleigh number were chosen such that buoyancy-driven instability occured. For cases with significant radial flow effect, the spatial growth rates of the azimuthal modes were found to vary considerably in the streamwise direction. 
    more » « less
  5. null (Ed.)
    Rayleigh–Bénard convection (RBC) and Taylor–Couette flow (TCF) are two paradigmatic fluid dynamical systems frequently discussed together because of their many similarities despite their different geometries and forcing. Often these analogies require approximations, but in the limit of large radii where TCF becomes rotating plane Couette flow (RPC) exact relations can be established. When the flows are restricted to two spatial independent variables, there is an exact specification that maps the three velocity components in RPC to the two velocity components and one temperature field in RBC. Using this, we deduce several relations between both flows: (i) heat and angular momentum transport differ by $(1-R_{\Omega })$ , explaining why angular momentum transport is not symmetric around $R_{\Omega }=1/2$ even though the relation between $Ra$ , the Rayleigh number, and $R_{\Omega }$ , a non-dimensional measure of the rotation, has this symmetry. This relationship leads to a predicted value of $R_{\Omega }$ that maximizes the angular momentum transport that agrees remarkably well with existing numerical simulations of the full three-dimensional system. (ii) One variable in both flows satisfies a maximum principle, i.e. the fields’ extrema occur at the walls. Accordingly, backflow events in shear flow cannot occur in this quasi two-dimensional setting. (iii) For free-slip boundary conditions on the axial and radial velocity components, previous rigorous analysis for RBC implies that the azimuthal momentum transport in RPC is bounded from above by $Re_S^{5/6}$ , where $Re_S$ is the shear Reynolds number, with a scaling exponent smaller than the anticipated $Re_S^1$ . 
    more » « less