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Broadwell Model in a Thin Channel

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

  2. 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 Motivation

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

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

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

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

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

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

  9. Discrete velocity models are very simplified versions of the Boltzmann equation which preserve some features, namely: H-theorem: Entropy must increase Kn small-> Chapman-Enskog -> Fluid equations Discrete Velocity Models 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.

  10. No Long Range Forces 6 velocities with magnitude = 1 6 velocities 3 6 2 1 5 4 The Broadwell Model are source and losses due to collisions

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


  13. 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 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.

  14. Flow in a Thin Channel

  15. 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 h d L Set Up y x z

  16. N2 N4 N3 N1 N2 N4 Broadwell with Boundaries 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. y=h N3 N1 y=0

  17. N4 N1 N3 N1 Specular Diffuse At lower wall: Boundary Conditions a : Accommodation Coefficient Inward Flux N4 has specular reflections into N1 : N1=(1-a) N4 N4 has diffusive reflection into N1 : N1=(a N4)/2 N2 has diffusive reflection into N1 : N1=(a N2)/2

  18. y=0 y=h FULL MODEL

  19. N2 N4 N3 N1 N2 N4 y=0 y=h Free Molecular Flow y=h N3 N1 y=0

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

  21. Depth Average Applying the boundary conditions gives:

  22. Define: Adding N1through N4 gives: Depth Average Adding cN1and cN4 then subtracting cN2and cN3 gives:

  23. Thin Channel Approximation Taylor Series: Combined with: Gives:

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

  25. Telegraph Equation These maybe combined to give:

  26. 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 Previous Results

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

  28. Define Scaled Density: Theorem 1 - Diffusive Behavior Diffusive scaling: X=x/l and T=t/l2 Scaled Number Density : Ml(X,y,T) = lN(lX,y, l2 T) Theorem 1: If the initial conditions areN(x,y,0)= Mo(x/l ,y)/l, whereMo(x ,y) is inB(D), then asl-> yl(X,T) converges weakly to y(X,T) where

  29. Theorem 2 - Hyperbolic Behavior Hyperbolic scaling: X=x/lT=t/l Scaled Number Density: Pl(X,y,T) = lN(lX,y, lT,a=2G/l) Define Scaled Density: Theorem 2: If the initial conditions areN(x,y,0)= Mo(x/l ,y)/linB(D), then asl-> , fl(X,T) converges weakly to f(X,T) which is a solution of the telegraph equation: with initial conditions :

  30. Theorem 3Long-Time Behavior Theorem 3: If N(x, y,0) = No(x, y) in B(D) and are where vector-valued eigenfunctions, then the density has the following asymptotic behavior: D=(2-a)/2a and the c’s are determined initial conditions (continued)

  31. Theorem 3Long-Time Behavior Furthermore, if No=(f(x)/4)1 then r(x,0)=f(x) and it follows from the above expressions that This shows the convergence in Thm 1 cannot be better than weak

  32. Results - Initial Condition = f(x)

  33. Results-Initial condition = f(x,y)

  34. Effects of Collisions

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

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

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

  38. Results

  39. 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. Conclusions