Hydrodynamic Singularities in freely cooling granular media Itzhak Fouxon Hebrew University of Jerusalem collaboration: Baruch Meerson, Michael Assaf, Eli Livne. Non-equilibrium Statistical Mechanics and Turbulence Workshop, Warwick, 18 July 2006

Clustering and Organization in granular systems We are surrounded by order and structure which understanding is one of the primary purposes of physics. A basic scenario of the structure formation considered in this talk is where initially disorganized matter forms structures in space characterized by singular density. Granular system gives entropy to inner degrees of freedom and can organize.

Gas of inelastic hard spheres as a basic model of a cooling system coefficient of normal restitution Instantaneous inelastic binary collisions in which momentum is preserved but part of the kinetic energy proportional to 1-r is lost to the inner degrees of freedom Left alone granular gas will freeze. Starting with a uniform gas we could expect just seeing homogeneous freezing of the gas. Simplest model of granular flow

Homogeneous cooling state (HCS) Haff (1983) cooling time q=(1-r)/2 inelasticity of collisions n(r,t) = n 0 = const v(r,t) = 0  particle diameter m particle mass

Goldhirsch and Zanetti (1993), McNamara and Young (1996), … Molecular Dynamic simulations exhibit clustering instability: formation of clusters of particles, generation of vortices Linear Theory of Hydrodynamic Instability: Clustering Instability (CI) of Uniform Cooling Non-equilibrium Statistical Mechanics and Turbulence:

Physics of Linear Instability Density Realized when cooling time is shorter than characteristic time of inhomogeneity smoothening by the viscous processes. Always true for sufficiently long-scale perturbations. more cooling by collisions pressure More particles flow into the perturbed region Density Goldhirsch Zanetti

Hydrodynamic equations for granular gases P: stress tensor Q: heat flux  ~ (1-r 2 ) n 2 T 3/2 : rate of energy loss by inelastic collisions (Haff 1983) Bulk energy losses How to study non-linear regime of instability? At small inelasticity LTE holds – can use hydrodynamics Energy Loss Equivalent to Entropy Loss - Structures

The study of non-linear development of instability in 1d We study the typical unstable situation as initial condition v x Consider hydrodynamical equations in 1d disregarding viscous processes Dimensionless fields are

Hydrodynamics of granular gases in 1d In mass-coordinate frame Explicitly the equations in Eulerian frame are At small may expect that the same singularity holds as for ideal gas hydrodynamics. The same for all ? Performed simulations with two different codes based on different algorithms

Density explodes even at small Lambda What went wrong with the shock consideration?

At small Lambda Infinite density forms after the shock is already there

Is singularity self-similar? No saturation of the power-law exponent Here corresponds to “expected”

Asymptotic Field Properties near the singularity Pressure becomes time-independent

The singularity formation law is universal Corresponds to the special solution with time-independent pressure

The special solution Besides time-independent pressure with correct behavior near the singularity point captures parabolic-like behavior of inverse density near the singularity

Parabolic law of singularity formation The law is not purely parabolic, rather minimal inverse density analytically turns turns to zero at the singularity time with the expansion starting from the quadratic term

Granular particles reflection off the wall Reflected wave stops at finite distance from the origin (Goldshtein, Shapiro)

Clustering instability for large systems For sufficiently long wave-length perturbations local cooling always dominates both over viscous processes and over sound. The temperature decays before anything has time to happen and pressure become negligible in Euler equation. Hopf dynamics becomes an intermediate asymptotic regime. Connection to Zeldovich approximation in astrophysics?

Beyond the singularity Beyond the singularity one can construct non- equilibrium steady state. In this state cold gas particles come from infinity at constant speed and bounce of the wall at the origin. Homogeneous solution holding at infinity is only perturbed within finite region while at the wall finite mass is concentrated with delta function singularity of density. The amplitude grows linearly in time while in the rest of the space solution is steady. Viscosity only changes the law of singularity formation not stopping it.

Summary Hydrodynamic equations of granular gases describe finite-time formation of infinite density out of smooth initial conditions. The law of infinite density formation is close to 1/t 2 but it is not self- similar and there is no power-law plateau. Still the law is universal, locally corresponding to the special solution with time-independent pressure. These singularities develop along with the usual shock singularities of gas dynamics. Example of singularity having no scale- invariance, continuous or discrete

Summary 2 At small cooling, for smooth initial conditions first a shock forms and then infinite density appears gradually At large cooling, Hopf equation appears as intermediate asymptotic regime which shortly before the density singularity of Hopf equation is switched by another regime of infinite density formation Beyond the singularity time, finite-mass, point singularities of density are present in the gas. The density turns infinite in space as a power-law so that local pressure is constant in space Viscosity does not stop this kind of singularities though it changes the law of singularity formation

Future Research Directions Formation of singularities in higher dimensions. Numerical experiments indicate formation of (multi) fractal structures in higher dimensions. What are the properties of the structure? Importance of vorticity. Checking density singularities in other dissipative systems. Applications to astrophysics? Is intermediate Burgers-like regime related to its appearance in astrophysics?

The SDSS 3D Universe Map Credit & Copyright: Sloan Digital Sky Survey Team, NASA, NSF, DOE Is the resemblance coincidental?