Thermal collapse of a granular gas under gravity Baruch Meerson Hebrew University of Jerusalem in collaboration with Dmitri Volfson - UCSD Lev Tsimring - UCSD Support: US DOE Israel Science Foundation German-Israeli Foundation for Scientific Research and Development Southern Workshop on Granular Materials, Viña del Mar, 2006 Phys. Rev. E 73, (2006)

Granular gas: instantaneous inelastic binary collisions (constant) coefficient of normal restitution Momentum preserved, part of kinetic energy lost Simplest model of granular flow tangential velocity components: normal velocity components:

Continuum modeling: hydrodynamics of dilute granular gases at q=(1-r)/2 << 1 P: stress tensor Q: heat flux  ~ (1-r 2 ) n 2 T 3/2 : rate of energy loss by inelastic collisions (Haff 1983) Constitutive relations are derivable from the Boltzmann equation (generalized to account for inelastic collisions) under scale separation: mean free path << hydrodynamic length scales Bulk energy losses mass conservation momentum conservation energy balance

Homogeneous Cooling State: a paradigm of kinetic theory and hydrodynamics Haff’s law cooling time in two dimensions n(r,t) = n 0 = const v(r,t) = 0 d: particle diameter particle mass=1 g=0

How does gravity modify the cooling dynamics? Qualitative picture: gravity forces grains to sink to the bottom of the container, where increased density enhances the collision rate and causes “freezing” of the granulate. Surprisingly, no quantitative analysis has ever been performed. We combined MD simulations and numerical and analytical solutions of hydrodynamic equations to develop a detailed quantitative understanding of the cooling process. Main result: in contrast to Haff's law, the cooling gas undergoes thermal collapse: it cools down to zero temperature and condenses on the bottom plate in a finite time exhibiting, close to collapse, a universal scaling behavior. Why should we care? 1. A non-trivial test of hydrodynamics 2. Aesthetic beauty

Event-driven MD simulations Circles: Total kinetic energy normalized to value at t=0 Total kinetic energy drops to zero in a finite time t c. Apparent scaling ~ (t c -t) 2 close to t c. N=5642, L x =10 2, r=0.995, T 0 =10, g=0.01 t=0: barometric density profile Dash-dot: same for a different initial condition units of time: see later

Hydrodynamic theory deals with hydrodynamic fields n(y,t), T(y,t) and v(y,t) mass content between the bottom plate and the (Eulerian) point y One-dimensional time-dependent flow: easier to solve using the Lagrangian coordinate y: vertical coordinate Having solved the problem in the Lagrangian coordinates [that is, having found n(m,t), T(m,t) and v(m,t)], we can return to the Eulerian coordinate y:

gravity length scale at t=0 relative role of heat losses and heat conduction Hydrodynamic equations Λ 2 =(1-r 2 )/(4ε 2 ) Two scaled parameters ε ~1 /( number of granular layers at rest( ε >(λ/g) 1/2 ε = π -1/2 L x /(Nd)<<1 Bromberg, Livne and Meerson (2003) Lagrangian mass coordinate Boundary conditions: zero fluxes of mass, momentum and energy at y=0 and y=∞ (that is, at m=0 and m=1). heat diffusion time

Numerical solution of hydrodynamic equations A variable mesh/variable time step solver (Blom and Zegeling). Short-time behavior is complicated: shock waves emerge and heat the gas at large heights. Circles: MD simulations Black solid line: hydrodynamics N=5642, L x =10 2, r=0.995, T 0 =10, g=0.01 Hydrodynamic parameters ε=0.01 and Λ=6 t=0: barometric density profile

late-time behavior is describable in terms of a quasi-static flow If ε<<min(1,Λ -2 ), then These three eqns. yield a single nonlinear PDE for ω=T 1/2 (m,t): This is the ω-equation [Bromberg, Livne and Meerson (2003)]. nT=1-m hydrostatic balance Λ 2 the only parameter Once ω is found, T, n and v can be calculated, too.

Numerical solution of the ω-equation Same variable mesh/variable time step solver Computations launched at scaled time t=0.04 when the flow is already quasi-static. Temperature profile produced by the full hydrodynamic solver used as initial condition. Circles: MD simulations Black solid line: hydrodynamics Red dashed line: ω-equation Λ=6, T(m,t=0)=1

Numerical solution of the ω-equation Another example: Λ=1, T(m,t=0)=1 Simulations performed at different Λ and different initial conditions. Thermal collapse always observed at a finite time t c which goes down as Λ increases. T(m,t) vanishes, at t=t c, on the whole Lagrangian interval 0<m<1, while the density n=(1-m)/T blows up there. At t=t c this Lagrangian interval corresponds to a single Eulerian point y=0. Therefore, all of the gas condenses at the bottom plate and cools to zero temperature at t=t c.

Analytic theory Remarkably, close to collapse the solution of the initial value problem for the ω- equation becomes separable: Q(m) is determined by the nonlinear ordinary differential equation and the boundary conditions (1-m)Q'(m)=0 at m=0 and m=1. The function Q(m) is uniquely determined by Λ and, for each Λ, can be found numerically, by shooting. The collapse time t c depends on the initial condition. that is,

Λ 2 <<1: perturbation theory Here one can show that Q(m) ~ Λ 2 <<1 Furthermore, as heat conduction dominates over heat losses, the solution must be almost uniform in space, and we arrive at This solution is in excellent agreement with numerical one at Λ 2 <1.

Λ 2 >>1 Here we stretch the coordinate and time in the ω-equation: ξ=Λ(1-m) and τ=Λt. The equation becomes while Λ determines the interval: 0 < ξ ≤ Λ. The separable solution is while the boundary value problem for q(ξ) is the following:

At ξ>>1 (that is, everywhere except the boundary layer at m=1) q(ξ) is exponentially small, and one can drop the term q 2. The resulting linear equation solvable in Bessel functions. Envelope corresponds to limit Λ→∞ Physics: q 2 -term comes from ωω t. At Λ>>1 the energy losses are balanced by heat conduction everywhere, except at very high altitudes. The high altitudes serve as a dynamic bottleneck of the cooling process.

We have also found approximate analytical solutions for the initial stage of the slow cooling, at large and small Λ, and determined the collapse time t c = t c (Λ) For concreteness, isothermal (barometric) density profile at t=0 is assumed. Details: D.Volfson, B. Meerson and L.S. Tsimring, Phys. Rev. E 73, (2006)

A summary of collapse properties The temperature vanishes at t=t c : The density blows up The gas velocity is Though v(m,t) is zero everywhere at t=t c, the gas flux nv blows up. This happens on the whole Lagrangian interval 0<m<1, but this interval corresponds to a single Euleiran point y=0. That is, at t=t c all of the gas condenses on the bottom plate and cools to zero temperature. What is the total kinetic energy of all particles as a function of time? Compare to Haff’s law

Summary: main predictions of theory The gas temperature drops to zero in a finite time t c as (t c -t) 2 All of the gas condenses at the bottom at t=t c The total energy of the gas drops to zero: E(t)~(t c -t) 2. Thank you. Should be amenable to experiment. A freely cooling granular gas under gravity exhibits thermal collapse