Granular fluid in an arbitrary external potential: spontaneous convection, self-phoresis

Abstract

The hydrodynamic stationary states of a granular fluid are addressed theoretically when subject to energy injection and a time-independent, but otherwise arbitrary external potential force. When the latter is not too symmetrical in a well defined sense, we show that a quiescent stationary state does not exist, rather than simply being unstable and, correspondingly, a steady convective state emerges spontaneously. We also unveil an unexpected connection of this feature with the self-diffusiophoresis of catalytically active particles: if an intruder in the granular fluid is the source of the potential, it will self-propel according to a recently proposed mechanism that lies beyond linear response theory, and that highlights the role of the intrinsic nonequilibrium nature of the state of the granular bath. In both scenarios, a state-dependent characteristic length of the granular fluid is identified which sets the scale at which the induced flow is the largest.

Figures (2)

• Figure 1: Results for the flow induced in a 2D granular fluid by a potential ${\mathbb{W}}({\bf r}) = -\cos (2\pi x/L_x) - \cos(2\pi y/L_y)$ in a rectangular box of side--lengths $L_x$ and $L_y$ with periodic boundary conditions. The strength of the flow is characterized by the velocity scale $u_\mathrm{kin}:= \sqrt{2 K/m N}$, where $K$ is the total kinetic energy of the flow and $N$ is the number of grains. The upper plot shows the growth of $u_\mathrm{kin}$ (in arbitrary units) with the aspect ratio $L_x/L_y$ extracted from Molecular Dynamics, consistently with the perturbative prediction SuppInfo. The lower pannel compares the flow in simulations with the perturbative computation for a particular value of the aspect ratio $L_x/L_y$ (the same conclusions hold for other values, see Ref. SuppInfo). The white arrows are the streamlines, the colored background encodes the modulus of the velocity field normalized by $u_\mathrm{kin}$. The estimated Reynolds and Mach numbers for the simulations are $\sim 10^{-3}$, so that Eqs. (\ref{['eq:divu']},\ref{['eq:flowlin']}) for the flow field are likely a good approximation. The variations in particle number density across the domain amount however to up to $70\%$ of the average density, which casts doubt on the validity of Eqs. (\ref{['eq:plin']},\ref{['eq:heatlin']}) and the form of the forcing term in Eq. (\ref{['eq:flowlin']}); nevertheless, the perturbative theory captures the relevant features of the measured flow.
• Figure 2: Self-phoretic translation velocity $|{\bf V}|$ (in arbitrary units) of a spherical intruder of radius $R$ as a function of the ratio ${\xi}/R$. The intruder exerts a potential on the granular fluid bath of the form ${\mathbb{W}}({\bf r})= \mathrm{e}^{-r/R} \left( 1 + \cos\theta \right)$, shown in the inset as a heat map. The intruder also imposes a no-slip boundary condition on the flow at its surface. The dashed lines are the expected asymptotic behaviors.