DSMC simulations of Rb gas flow A. Petrenko, G. Plyushchev AWAKE PEB CERN Direct Simulation Monte Carlo method (developed by G. A. Bird in the 60’s): Slightly modified Matlab scripts from A. L. Garcia’s book “Numerical Methods for Physics” were used. More details are given in this article: F. J. Alexander and A. L. Garcia. “The Direct Simulation Monte Carlo Method”. Computers in Physics, Vol. 11. Num. 6, 1997.Matlab scriptsA. L. Garcia’s book “Numerical Methods for Physics”“The Direct Simulation Monte Carlo Method”. Computers in Physics, Vol. 11. Num. 6, )Move particles as in vacuum. 2)Apply boundary conditions. 3)Calculate collisions in each cell depending only on local gas density and particle relative velocities. 1)Move particles as in vacuum. 2)Apply boundary conditions. 3)Calculate collisions in each cell depending only on local gas density and particle relative velocities. First divide the gas sample into cells smaller than the mean free path between collisions. Select time step smaller than typical time between collisions. Repeat this cycle over and over again:

n 0 = 7· /cm 3 : t = 0 t = 10 ms (m) n 0 = 7· /cm 3 : t = 0 t = 10 ms (m) 1D DSMC simulation of Rb density profile after 10 ms for the baseline density and for the free molecular flow: The upper limit on the gas outflow, i. e. the worst possible case. Basic parameters: T = 200 °C, Rb atom diameter = 0.6 nm, The most probable thermal velocity = 300 m/s, Mean free path between collisions = 0.9 mm (for n 0 = 7· /cm 3 ). 3 m = 300 m/s * 10 ms 300 m/s,

1D DSMC simulation of Rb density profile after 10 ms for the free molecular flow: The upper limit on the gas outflow, i. e. the worst possible case. n 0 = 7· /cm 3 : t = 0 t = 10 ms (m) Red line is the analytical calculation for the free molecular flow regime:

1D DSMC simulation of Rb density profile after 10 ms for the baseline density: Red line is the analytical calculation: n 0 = 7· /cm 3 : t = 0 t = 10 ms (m)

2D DSMC simulation up to 0.6 ms (Rb flow in a 4 cm wide slit)

2D DSMC simulation up to 3 ms: 300 m/s * 3 ms = 90 cm Pretty close to the simple estimate:

On-axis injection requirements From our recent article: K.V. Lotov, A.P. Sosedkin, A.V. Petrenko, L.D. Amorim, J. Vieira, R.A. Fonseca, L.O. Silva, E. Gschwendtner, P. Muggli. Electron trapping and acceleration by the plasma wakefield of a self-modulating proton beam. Electron trapping and acceleration by the plasma wakefield of a self-modulating proton beam z Plasma density 0 L0L0 There is simply no way to make plasma density ramp length L 0 to be less than 10 cm with a valve opening time of several ms. (3 ms * 300 m/s = 90 cm!)

Conclusions 8 DSMC method is easy to apply in our case (from free molecular flow to density of a few times /cm 3 ) using simple Matlab or C++ programs. 1D model confirmed by analytical calculations shows that density drop travels with the speed of sound (i.e. 3 m in °C). 2D model (Rb flow in a 4 cm wide slit) shows no significant difference from the 1D model so far (this is preliminary). Next steps to implement in simulations: – Moving valve (easy) – Irises (easy) – 3D model (easy but computationally expensive) – Equilibrium density profile with hot/cold spots in the plasma pipe. The only straightforward solution to the density ramp problem seems to be an additional lightweight and super-fast shutter (as already suggested by Mikhail Martyanov). On-axis injection needs 1 cm wide shutter must open in less than 0.1 m / 300 m/s = 0.3 ms => the speed of shutter plate will be 30 m/s => maybe something feasible with lightweight conducting/magnetic materials in a pulsed magnetic field? Spinning wheel with a hole? There is no need for this additional shutter to be leak tight!