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
Oxford University's
Carbon and Nanotech
Group
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Starting Point
Boltzmann’s Equation:
f
F
f
 v   x f     v f  
t
m
t

collision
y=0
Maxwell’s Boundary condition (v>0):
f (u,v)  (1  ) f (u,v)  f M (u,v)

v'0
f (u',v') | v'| dv'du'
Limiting Behavior with No Walls
• Fluid Dynamic Limit:

Kn 
– Large length scales, Kn<<1, highly collisional.
L
– Solution of Boltzmann equation can be expressed as
 u  u x,t 2 
  

0

f M  x,t exp
 T x,t  

 
where  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
Flow In a Thin Channel
• Mean Free Path Air ~ 70 nm
• Nano-Tube Diameter ~ 30 nm
• Knudsen Number, Kn ~ O(1)


We make the collisionless flow
approximation

but keep the wall collisions
Knudsen Gas
• Collisionless Flow
• Maxwell’s Boundary Condition on walls
f
 v   x f   0
t

h
Diffusive Behavior
• Knudsen Gas has Diffusive Behavior
• The depth averaged density, ,under appropriate
scaling, satisfies a diffusion equation

f
 v   x f   0
t
h
Average and
“ wait long enough’’
Maxwell’s
Boundary
Condition
2 



 D 2 
t
x 
Diffusive Behavior
Diffusion Coefficient:
D 2
2

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
• Kn 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
3
6 velocities
• No Long Range Forces
2
• 6 velocities with magnitude = 1
N i
N i
 vi
 Si  Li
t
x i
6
1
5
4
i  1,2,3,4,5,6
Si and Li are source and losses due to collisions
Collisions
No Gain or
Loss for 1
Gain for 1 from
3-4 collision
3
3
3
1
Loss for 1 from a
1-2 collision
1
2
1
2
1

3
Result:
4
4
1

1
2
S1  L1  c  N 3 N 4  N 5 N 6  N1 N 2 
3

3
3
BROADWELL MODEL
1

N1
N1
1
2
c
 c  N 3 N 4  N 5 N 6  N1 N 2 
3

t
x
3
3
1

N 2
N 2
1
2
c
 c  N 3 N 4  N 5 N 6  N1 N 2 
3

t
x
3
3
1

N 3
N 3
1
2
c
 c  N1 N 2  N 5 N 6  N 3 N 4 
3

t
y
3
3
1

N 4
N 4
1
2
c
 c  N1 N 2  N 5 N 6  N 3 N 4 
3

t
y
3
3
1

N 5
N 5
1
2
c
 c  N1 N 2  N 3 N 4  N 5 N 6 
3

t
z
3
3
1

N 6
N 6
1
2
c
 c  N1 N 2  N 3 N 4  N 5 N 6 
3

t
z
3
3
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.
• 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
Flow in a Thin Channel
Set Up
• Use Broadwell Model to Understand Flow in a Thin
Channel
y
h
x
z
d
L
• Assumptions:
– Channel height, h, is small compared to length, L.
– Channel depth is infinite
– Dominant collisional effect: WALL
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
N2
N3
N1
N2
N4
N4
N3
N1
y=0
Boundary Conditions
 : Accommodation Coefficient
N4
N2
Inward Flux
N1
N3
Specular
N1
Diffuse
N4 has specular reflections into N1 : N1=(1N4
N4 has diffusive reflection into N1 : N1=(N4)/2
N2 has diffusive reflection into N1 : N1=(N2)/2
At lower wall: N1  1  N4 

2
N2  N4 
FULL MODEL
1

N1
N1
N1
1 2 2
c
c
 2c  N 3 N 4  N 5  N1N 2 
3

t
x
y
3
3
1

N 2
N 2
N 2
1 2 2
c
c
 2c  N 3 N 4  N 5  N1N 2 
3

t
x
y
3
3
1

N 3
N 3
N 3
1 2 2
c
c
 2c  N1N 2  N 5  N 3 N 4 
3

t
x
y
3
3
1

N 4
N 4
N 4
1 2 2
c
c
 2c  N1N 2  N 5  N 3 N 4 
3

t
x
y
3
3
1
N 5
1
2 
 2c  N1N 2  N 3 N 4  N 52 
3
t
3
3 


N

1

N



N 2  N 4 
4
 1
2



N

1

N



N 2  N 4 
2
 3
2
y=0


N

1

N



N1  N 3 
3
 2
2

N  1  N   N  N 
1
1
3
 4
2
y=h
Free Molecular Flow
y=h
N3
N1
N2
N4
N2
N1
N
N
c 1 c 1 0
t
x
y
N 2
N
N
c 2 c 2 0
t
x
y
N 3
N 3
N 3
c
c
0
t
x
y

N 4
N 4
N 4
c
c
0
t
x
y
N4
N3
N1
y=0


N

1

N

 4
N 2  N 4 
 1 
2

N  1  N   N  N 
2
2
4
 3
2
y=0


N

1

N

 3 N1  N 3 
 2 
2

N  1  N   N  N 
1
1
3
 4
2
y=h
Depth Average
Define:
1
N i (t, x) 
h
h
 N (t, x, y)dy
i
0
Depth Average Equation:
N
N1 c
1
c
 N1(t, x,0)  N1 (t, x,h)
t
x h
N 2
N 2 c
c
 N 2 (t, x,h)  N 2 (t, x,0)
t
x
h
N 3
N 3 c
c
 N 3 (t, x,0)  N 3 (t,x,h)
t
x
h
N 4
N 4 c
c
 N 4 (t, x,h)  N 4 (t, x,0)
t
x
h


N1  1  N 4  2 N 2  N 4 

N  1  N   N  N 
2
2
4
 3
2
y=0


N 2  1  N 3  2 N1  N 3 

N  1  N   N  N 
1
1
3
 4
2
y=h
Depth Average
Applying the boundary conditions gives:

N1
N1 c 

c
 (1  )N 4 0  N 2 0  N 4 0  N1 h 

t
x h 
2

N 2
N 2 c 

c
 (1  )N 3 h  N1 h  N 3 h  N 2 0 

t
x
h 
2

N 3
N 3 c 

c
 (1  )N 2 0  N 2 0  N 4 0  N 3 h 

t
x
h 
2

N 4
N 4 c 

c
 (1  )N1 h  N1 h  N 3 h  N 4 0 

t
x
h 
2
Depth Average
Define: (t, x)  N1(t, x)  N 2 (t, x)  N 3 (t, x)  N 4 (t, x)
m(t, x)  c N1(t, x)  N 2 (t, x)  N 3 (t, x)  N 4 (t, x)
Adding N1 through N4 gives:
 m


0
t x
Adding cN1 and cN4 then subtracting cN2 and cN3 gives:
2
m


c
c 2

N1 h  N 2 0  N 3 h  N 4 0 

t
x
h
Thin Channel Approximation
Taylor Series:
h N1 (t, x,h)
N1 (t, x,h)  N1 (t, x) 
 O(h 2 )
2
y
Combined with:
N1
N1
N1
c
c
0
t
x
y
h  N1
 N1 
Gives: N1 (t, x,h)  N1(t, x)    c

2c  t
x 

Thin Channel Approximation
This approximation for N1(t, x,h) along with similar
approximations for the other boundary terms gives
m 2   m 2   c
c
   c
m

t x 2h t
x  h
We have the system of equations are:
 m

0
t x
Loss of
 2 c 
m
2 
c
 
 m Momentum
t
x 2   h 
To Wall
Telegraph Equation
These maybe combined to give:
 2  h  2 
 2

    2  hc 2
2   t c t
x

Previous Results
• Solutions to Telegraph Equation Converge to
Diffusion Equation on a long time scale.
(Zauderer: Partial Differential Equations of Applied Mathematics)
 2  h  2 
2    2 
 2

 hc

    2  hc 2
 2
2   t c t
 2 x
x
t
• 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: D  (x, y) | (,) [0,1]
Define: N(x, y,t)  (N1,N 2,N 3,N 4 )T
1
 product: u,v  u H v dy
Define an inner

0

Define: 1=(1,1,1,1)T and 1+/-=(1,-1,-1,1)T
(x,t)  1,N(x,,t)



and m(x,t)  1 /,N(x,,t)

N  B(D) if max  N  N  N x dx   N  L1  H1 ()
0y1

2
2
Theorem 1 - Diffusive Behavior
Diffusive scaling: X=x/ and T=t/2
Scaled Number Density : M(X,y,T) =  N(X,y, 2 T)
Define Scaled Density: y (X,T)  1,M (X,,T)
Theorem 1: If the initial conditions are N(x,y,0)=
Mo(x/ ,y)/,where
 Mo(x ,y) is in B(D), then as > 
y(X,T) converges weakly to y(X,T) where
2 
yT 
yXX , with y(X,0)  1, Mo (X,) .
2
Theorem 2 - Hyperbolic Behavior
Hyperbolic scaling: X=x/ T=t/
Scaled Number Density: P(X,y,T) =  N(X,y, T,=2G/)
Define Scaled Density:
 (X,T)  1,P (X,,T)
Theorem 2: If the initial conditions are N(x,y,0)=

Mo(x/ ,y)/ in B(D), then as > , (X,T) converges


weakly to (X,T) which is a solution of the telegraph
T  TT   XX
2G

equation:
with initial conditions :
 (X,0)  1, M o (X,) , m(X,0)  1 , M o (X,)

where: T  mX  0
Theorem 3
Long-Time Behavior
Theorem 3: If N(x, y,0) = No(x, y) in B(D) and
Nˆ o (k, y) 

4
  gˆ
n j1
j,n
(k) j,n (k, y) where  are
j,n
vector-valued
 eigenfunctions, then the density
has the following asymptotic behavior:


2

x 

im(xt )
im(x t )
(x,t) 
exp
e
c m e
 o1
Dt
4Dt m

D=(2/2 and the c’s are determined initial conditions
(continued)
Theorem 3
Long-Time Behavior
Furthermore, if No=(f(x)/4)1 then (x,0)=f(x)
and it follows from the above expressions that

x 2  
i

m(xt
)
i

m(x
t
)
ˆ
(x,t) 
exp
f
e
 o1

  (m )e

Dt
4Dt m


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
N1
N1 c
2c
c
 N1 h  N1 0 
t
x h
3h
h
2
N
N

N
  3 4 5  2N1N 2 dy
0
N 2
N 2 c
2c
c
 N 2 h  N 2 0 
t
x h
3h
2
N
N

N
  3 4 5  2N1N 2 dy
N 3
N 3 c
2c
c
 N 3 h  N 3 0 
t
x h
3h
h
N 4
N 4 c
2c
c
 N 4 h  N 4 0 
t
x h
3h
N 5
t
2c

3h
h
0
2
N
N

N
  1 2 5  2N 3N 4 dy
0
h
2
N
N

N
  1 2 5  2N 3N 4 dy
0
h
2
N
N

N
N

2N
 1 2 3 4
5 dy
0
Depth Averaging
c
The boundary terms, h Ni h  Ni 0 , are treated using
the thin channel approximation.
Need to approximate the terms

h

 NiN j dy i  {1,3,5} and j  {2,4,5}
h
0
By Taylor expanding one can show


h
N N dy  N N


h
i
j
i
2

O(

h
)
j
0
The approximation is O(h) provided
  O(h 1 )


Collisional Thin Channel
Defining the averaged variables:
(t, x)  N1 (t, x) N 2 (t, x) N 3 (t, x) N 4 (t, x) 2N 5 (t, x)
m(t, x)  cN1 (t, x)  N 2 (t, x)  N 3 (t, x) N 4 (t, x)
z(t, x)  N5 (t, x)
After similar algebra as before we arrive at:
 m

0
t x
2 c 
m 2 
c
 m
  2z  
2   h 
t
x
z  
m 2 
   2z  6z  2 
t 24 
c 
Long time behavior
When

= O(1) and t = O(1/h) then one has
 2  
 D 2
t 3 x
2
where

2 
D  ch
2
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.