Broadwell Model in a Thin Channel Peter Smereka Collaborators:Andrew Christlieb James Rossmanith Affiliation:University of Michigan Mathematics Department

Motivation Example: –Gas at Low Density Satellites and Solar Winds Plasma Thrusters Space Planes –High Density Gases Flow in a Nano-Tube –Applications: Chemical Sensors NASA Oxford University's Carbon and Nanotech Group

Starting Point Boltzmann’s Equation: Maxwell’s Boundary condition (v>0): y=0

Limiting Behavior with No Walls Fluid Dynamic Limit: –Large length scales, K n <<1, highly collisional. –Solution of Boltzmann equation can be expressed as Free Molecular Flow: –Small length scales, K n >>1, fluid appears collisionless –In this case, there is no ‘simple’ reduction where  is density, u is velocity and T is temperature which are governed by the Navier-Stokes Equations

Flow In a Thin Channel Mean Free Path Air ~ 70 nm Nano-Tube Diameter ~ 30 nm Knudsen Number, K n ~ O(1) We make the collisionless flow approximation but keep the wall collisions

Knudsen Gas Collisionless Flow Maxwell’s Boundary Condition on walls h

Diffusive Behavior Knudsen Gas has Diffusive Behavior The depth averaged density,,under appropriate scaling, satisfies a diffusion equation h Average and “ wait long enough’’ Maxwell’s Boundary Condition

Diffusive Behavior Diffusion Coefficient: Thin Tube: time scale = 1/h Babovsky (1986) Thin channel : time scale = 1/(h log h) Cercignani (1963), Borgers et.al. (1992), Golse (1998)

Discrete Velocity Models Discrete velocity models are very simplified versions of the Boltzmann equation which preserve some features, namely: H-theorem: Entropy must increase K n small-> Chapman-Enskog -> Fluid equations Reference: T. Platkowski and R. Illner (1988) ‘Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory.’ SIAM REVIEW, 30(2):213.

The Broadwell Model No Long Range Forces 6 velocities with magnitude = 1 6 velocities are source and losses due to collisions

Collisions Result: No Gain or Loss for 1 Gain for 1 from 3-4 collision Loss for 1 from a 1-2 collision

BROADWELL MODEL

Broadwell Model Broadwell (1964): 1D Shock Formation: Kinetic vs. Fluid Gatignol (1975): H- Theorem + Kinetic theory Caflisch (1979): Proved validity of 1D fluid-dynamical to Broadwell model up to formation of shocks Beale (1985): Proved existence of time global solutions to a1D Broadwell model There is large body of work on Broadwell models mainly focusing on the fluid dynamic limit. This is the regime in which inter-particle collisions dominate.

Flow in a Thin Channel

Use Broadwell Model to Understand Flow in a Thin Channel Assumptions: –Channel height, h, is small compared to length, L. –Channel depth is infinite –Dominant collisional effect: WALL Set Up d L h y x z

Broadwell with Boundaries N3N3 N1N1 N2N2 N4N4 N2N2 N4N4 N3N3 N1N1 y=0 y=h To incorporate wall effects we “rotate’’ the Broadwell model by 45 degrees in the x-y plane. The other velocities are parallel to the wall.

Boundary Conditions  : Accommodation Coefficient N4N4 N1N1 Specular N3N3 N1N1 Diffuse N 4 has specular reflections into N 1 : N 1 =(1  N 4 N 4 has diffusive reflection into N 1 : N 1 =(  N 4 )/2 At lower wall: Inward Flux N 2 has diffusive reflection into N 1 : N 1 =(  N 2 )/2

FULL MODEL y=0y=h

Free Molecular Flow y=0 y=h N3N3 N1N1 N2N2 N4N4 N2N2 N4N4 N3N3 N1N1 y=0 y=h

Depth Average Define: Depth Average Equation: y=0 y=h

Depth Average Applying the boundary conditions gives:

Depth Average Define: Adding N 1 through N 4 gives: Adding cN 1 and cN 4 then subtracting cN 2 and cN 3 gives:

Thin Channel Approximation Taylor Series: Combined with: Gives:

Thin Channel Approximation This approximation for We have the system of equations are: Loss of Momentum To Wall approximations for the other boundary terms gives along with similar

Telegraph Equation These maybe combined to give:

Previous Results Solutions to Telegraph Equation Converge to Diffusion Equation on a long time scale. (Zauderer: Partial Differential Equations of Applied Mathematics) So we Expect that Solutions of Broadwell Model Converge to Solutions of Diffusion Equation

Limiting Behavior Rescale so that c=h=1 Domain we consider: Define: Define an inner product: Define: 1=(1,1,1,1) T and 1 +/- =(1,-1,-1,1) T

Theorem 1 - Diffusive Behavior Diffusive scaling: X=x/ and T=t/  Theorem 1: If the initial conditions are N(x,y,0)= M o (x/,y)/  where M o (x,y) is in B(D), then as   (X,T) converges weakly to  (X,T) where Scaled Number Density : M (X,y,T) = N( X,y,  T) Define Scaled Density:

Theorem 2 - Hyperbolic Behavior Hyperbolic scaling: X=x/ T=t/ Scaled Number Density: P (X,y,T) = N( X,y, T,  =2  ) Define Scaled Density: Theorem 2: If the initial conditions are N(x,y,0)= M o (x/,y)/ in B(D), then as ,  (X,T) converges weakly to  (X,T) which is a solution of the telegraph equation: with initial conditions :

Theorem 3 Long-Time Behavior Theorem 3: If N(x, y,0) = N o (x, y) in B(D) and vector-valued eigenfunctions, then the density has the following asymptotic behavior: D=(2  2  and the c’s are determined initial conditions (continued) where are

Theorem 3 Long-Time Behavior Furthermore, if N o =(f(x)/4)1 then  (x,0)=f(x) and it follows from the above expressions that This shows the convergence in Thm 1 cannot be better than weak

Results - Initial Condition = f(x)

Results-Initial condition = f(x,y)

Effects of Collisions

Depth Averaging The boundary terms,, are treated using the thin channel approximation. Need to approximate the terms By Taylor expanding one can show The approximation is O(h) provided

Collisional Thin Channel Defining the averaged variables: After similar algebra as before we arrive at:

Long time behavior When = O(1) and t = O(1/h) then one has where is the diffusion coefficient in the collisionless case

Results

Conclusions We have provided a coarse-grained description for the Broadwell model with and without collisions which is valid over a wide range of time scales. We expect this model to provide insight for the more realistic case when the gas is modeled by the Boltzmann equation.