Anomalous distribution functions in sheared suspensions

We investigate velocity probability distribution functions (PDF) of sheared hardsphere suspensions. As observed in our Stokes flow simulations and explained by our single-particle theory, these PDFs can show pronounced deviations from a Maxwell-Boltzmann distribution. The PDFs are symmetric around zero velocity and show a Gaussian core and exponential tails over more than six orders of magnitude of probability. Following the excellent agreement of our theory and simulation data, we demonstrate that the distribution functions scale with the shear rate, the particle volume concentration, as well as the fluid viscosity. Introduction. – To describe the statistics of complex systems, often probability distribution functions (PDF) are utilized. These distributions have been found to be of non-Gaussian shape in numerous fields of physics, including astrophysics [1], flow in porous media [2], turbulence [3], granular media [4–6], or suspensions [7–9]. However, the underlying processes are often not understood. In this letter, we focus on particularly important systems showing non-Gaussian velocity PDFs, namely sedimenting hard-sphere suspensions confined between sheared walls (see Fig. 1). They appear in river beds, blood examinations, industrial food production, the application of paint, and many more situations. Detailed experiments have been performed for more than a hundred years, but questions about the microstructure or structural relaxations of the sediment are still not well understood. Numerous authors have found that the PDF of particle velocities P (v) is not of similar shape as for an ideal gas, i.e., like a Maxwellian. Instead, P (v) can show a pronounced non-equilibrium shape, where the probability of high velocities is substantially larger [7, 8]. In this letter we present a single-particle theory and simulations to show that such non-equilibrium distributions can be described as a consequence of an irreversible driving process, where particles on average gain energy by one mechanism, but loose energy by another one. Here, energy is gained from the shear or gravitational forces causing particles to collide. Contrarily, energy is dissipated due to viscous damping. This causes P (v) to consist of a Gaussian core and exponential tails. Even though we focus on a well defined system here, the processes described are of general nature and can be applicable to ostensibly different setups. Experimentally, Rouyer et al. [10] studied quasi 2D hard-sphere suspensions and found a stretched exponential P (v) with concentration dependent exponents between 1 and 2 corresponding to exponential distributions for high concentrations and Gaussians for small particle counts. These results contradict theoretical predictions of a transition from exponential to Gaussian with increasing volume concentration [7,8]. However, both experimental and theoretical studies do not present sufficient statistics over more than 2-4 decades. It is important to note that if one does not have enough data points for high quality PDFs, final answers on the nature of the function cannot be given. Indeed, in the process of analyzing our data we found that even stretched exponentials can fit PDFs with purely exponential tails and Gaussian centers if only two to four decades of probability are covered. But as soon as more data is added, the exponential nature of the tails becomes distinct and it is impossible to fit the whole PDF with a single function. Here, we overcome such limitations by presenting PDFs consisting of up to 10 particle displacements each – allowing a statistics superior to any previous work. Our