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Introduced in 1964 by J.E. Broadwell, this model is the classic example of a discrete velocity gas. A discrete velocity model consists of a collection of gas molecules with velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109101.png" /> belonging to some finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109102.png" /> of discrete velocity vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109103.png" />. The molecules collide with each other, obeying specified sets of transformation rules which must satisfy basic conservation laws. The models generate systems of highly coupled semi-linear partial differential equations which approximate the [[Boltzmann equation|Boltzmann equation]], and are particularly useful for studying problems in rarefied gas dynamics, such as Couette flow, Rayleigh flow and shock structure, especially at high Mach number (cf. also [[Gas dynamics, numerical methods of|Gas dynamics, numerical methods of]]; [[Gas flow theory|Gas flow theory]]; [[Gas dynamics, equations of|Gas dynamics, equations of]]).
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In the Broadwell model, each identical molecule of mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109104.png" /> is allowed to move in space with one of the six unit velocity vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b1109109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091010.png" />. Gas particles are allowed to collide only in pairs, a realistic assumption for gases which are not too dense, and the collision must obey the usual conservation laws for mass, momentum and kinetic energy. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091011.png" /> denote a collision of particles with initial velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091013.png" /> and final velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091015.png" />. Conservation of momentum dictates that the only possible collisions are then
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091016.png" /></td> </tr></table>
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Introduced in 1964 by J.E. Broadwell, this model is the classic example of a discrete velocity gas. A discrete velocity model consists of a collection of gas molecules with velocities  $  \mathbf u _ {i} $
 +
belonging to some finite set  $  S $
 +
of discrete velocity vectors in  $  \mathbf R  ^ {n} $.
 +
The molecules collide with each other, obeying specified sets of transformation rules which must satisfy basic conservation laws. The models generate systems of highly coupled semi-linear partial differential equations which approximate the [[Boltzmann equation|Boltzmann equation]], and are particularly useful for studying problems in rarefied gas dynamics, such as Couette flow, Rayleigh flow and shock structure, especially at high Mach number (cf. also [[Gas dynamics, numerical methods of|Gas dynamics, numerical methods of]]; [[Gas flow theory|Gas flow theory]]; [[Gas dynamics, equations of|Gas dynamics, equations of]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091017.png" /></td> </tr></table>
+
In the Broadwell model, each identical molecule of mass  $  m $
 +
is allowed to move in space with one of the six unit velocity vectors  $  \mathbf u _ {1} = ( 1,0,0 ) $,
 +
$  \mathbf u _ {2} = ( - 1,0,0 ) $,
 +
$  \mathbf u _ {3} = ( 0,1,0 ) $,
 +
$  \mathbf u _ {4} = ( 0, - 1,0 ) $,
 +
$  \mathbf u _ {5} = ( 0,0,1 ) $,
 +
$  \mathbf u _ {6} = ( 0,0, - 1 ) $.  
 +
Gas particles are allowed to collide only in pairs, a realistic assumption for gases which are not too dense, and the collision must obey the usual conservation laws for mass, momentum and kinetic energy. Let  $  ( \mathbf v _ {i} , \mathbf v _ {j} ) \rightarrow ( \mathbf v _ {k} , \mathbf v _ {l} ) $
 +
denote a collision of particles with initial velocities  $  \mathbf v _ {i} $
 +
and  $  \mathbf v _ {j} $
 +
and final velocities  $  \mathbf v _ {k} $
 +
and  $  \mathbf v _ {l} $.  
 +
Conservation of momentum dictates that the only possible collisions are then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091018.png" /></td> </tr></table>
+
$$
 +
( v _ {1} , v _ {2} ) \rightarrow ( v _ {1} , v _ {2} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091019.png" /></td> </tr></table>
+
$$
 +
( v _ {1} , v _ {2} ) \rightarrow ( v _ {3} , v _ {4} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091020.png" /></td> </tr></table>
+
$$
 +
( v _ {1} ,v _ {2} ) \rightarrow ( v _ {5} , v _ {6} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091021.png" /></td> </tr></table>
+
$$
 +
( v _ {3} , v _ {4} ) \rightarrow ( v _ {1} , v _ {2} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091022.png" /></td> </tr></table>
+
$$
 +
( v _ {3} , v _ {4} ) \rightarrow ( v _ {3} , v _ {4} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091023.png" /></td> </tr></table>
+
$$
 +
( v _ {3} , v _ {4} ) \rightarrow ( v _ {5} , v _ {6} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091024.png" /></td> </tr></table>
+
$$
 +
( v _ {5} , v _ {6} ) \rightarrow ( v _ {1} , v _ {2} ) ,
 +
$$
 +
 
 +
$$
 +
( v _ {5} , v _ {6} ) \rightarrow ( v _ {3} , v _ {4} ) ,
 +
$$
 +
 
 +
$$
 +
( v _ {5} , v _ {6} ) \rightarrow ( v _ {5} , v _ {6} ) .
 +
$$
  
 
In each of the above collisions, both mass and kinetic energy are also preserved.
 
In each of the above collisions, both mass and kinetic energy are also preserved.
  
Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091025.png" /> denote the number density of molecules with velocity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091026.png" />, the Boltzmann equation can be written as
+
Letting $  N _ {i} = N _ {i} ( \mathbf x,t ) $
 +
denote the number density of molecules with velocity $  \mathbf v _ {i} $,  
 +
the Boltzmann equation can be written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091027.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{\partial  N _ {i} }{\partial  t }
 +
} + \mathbf u _ {i} \cdot \nabla N _ {i} = G _ {i} - L _ {i} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091029.png" /> are the rates of gain and loss in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091030.png" /> as a result of collisions. Assuming spherical symmetry and collisional cross section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091031.png" />, one has, for example,
+
where $  G _ {i} $
 +
and $  L _ {i} $
 +
are the rates of gain and loss in $  N _ {i} $
 +
as a result of collisions. Assuming spherical symmetry and collisional cross section $  \sigma $,  
 +
one has, for example,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091032.png" /></td> </tr></table>
+
$$
 +
G _ {1} = {
 +
\frac{2}{3}
 +
} \sigma N _ {3} N _ {4} + {
 +
\frac{2}{3}
 +
} \sigma N _ {5} N _ {6}  $$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091033.png" /></td> </tr></table>
+
$$
 +
L _ {1} = {
 +
\frac{4}{3}
 +
} \sigma N _ {1} N _ {2} ,
 +
$$
  
since one-third of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091035.png" /> collisions yield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110910/b11091036.png" /> pairs.
+
since one-third of the $  ( \mathbf v _ {3} , \mathbf v _ {4} ) $
 +
and $  ( \mathbf v _ {5} , \mathbf v _ {6} ) $
 +
collisions yield $  ( \mathbf v _ {1} , \mathbf v _ {2} ) $
 +
pairs.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Broadwell,  "Shock structure in a simple discrete velocity gas"  ''Phys. Fluids'' , '''7'''  (1964)  pp. 1243–1247</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Gatignol,  "Théorie cinétique d'un gaz répartition discrète de vitesses" , Springer  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Monaco,  L. Preziosi,  "Fluid dynamic applications of the discrete Boltzmann equation" , World Sci.  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Platkowski,  R. Illner,  "Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory"  ''SIAM Review'' , '''30 (2)'''  (1988)  pp. 213–255</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Broadwell,  "Shock structure in a simple discrete velocity gas"  ''Phys. Fluids'' , '''7'''  (1964)  pp. 1243–1247</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Gatignol,  "Théorie cinétique d'un gaz répartition discrète de vitesses" , Springer  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Monaco,  L. Preziosi,  "Fluid dynamic applications of the discrete Boltzmann equation" , World Sci.  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Platkowski,  R. Illner,  "Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory"  ''SIAM Review'' , '''30 (2)'''  (1988)  pp. 213–255</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


Introduced in 1964 by J.E. Broadwell, this model is the classic example of a discrete velocity gas. A discrete velocity model consists of a collection of gas molecules with velocities $ \mathbf u _ {i} $ belonging to some finite set $ S $ of discrete velocity vectors in $ \mathbf R ^ {n} $. The molecules collide with each other, obeying specified sets of transformation rules which must satisfy basic conservation laws. The models generate systems of highly coupled semi-linear partial differential equations which approximate the Boltzmann equation, and are particularly useful for studying problems in rarefied gas dynamics, such as Couette flow, Rayleigh flow and shock structure, especially at high Mach number (cf. also Gas dynamics, numerical methods of; Gas flow theory; Gas dynamics, equations of).

In the Broadwell model, each identical molecule of mass $ m $ is allowed to move in space with one of the six unit velocity vectors $ \mathbf u _ {1} = ( 1,0,0 ) $, $ \mathbf u _ {2} = ( - 1,0,0 ) $, $ \mathbf u _ {3} = ( 0,1,0 ) $, $ \mathbf u _ {4} = ( 0, - 1,0 ) $, $ \mathbf u _ {5} = ( 0,0,1 ) $, $ \mathbf u _ {6} = ( 0,0, - 1 ) $. Gas particles are allowed to collide only in pairs, a realistic assumption for gases which are not too dense, and the collision must obey the usual conservation laws for mass, momentum and kinetic energy. Let $ ( \mathbf v _ {i} , \mathbf v _ {j} ) \rightarrow ( \mathbf v _ {k} , \mathbf v _ {l} ) $ denote a collision of particles with initial velocities $ \mathbf v _ {i} $ and $ \mathbf v _ {j} $ and final velocities $ \mathbf v _ {k} $ and $ \mathbf v _ {l} $. Conservation of momentum dictates that the only possible collisions are then

$$ ( v _ {1} , v _ {2} ) \rightarrow ( v _ {1} , v _ {2} ) , $$

$$ ( v _ {1} , v _ {2} ) \rightarrow ( v _ {3} , v _ {4} ) , $$

$$ ( v _ {1} ,v _ {2} ) \rightarrow ( v _ {5} , v _ {6} ) , $$

$$ ( v _ {3} , v _ {4} ) \rightarrow ( v _ {1} , v _ {2} ) , $$

$$ ( v _ {3} , v _ {4} ) \rightarrow ( v _ {3} , v _ {4} ) , $$

$$ ( v _ {3} , v _ {4} ) \rightarrow ( v _ {5} , v _ {6} ) , $$

$$ ( v _ {5} , v _ {6} ) \rightarrow ( v _ {1} , v _ {2} ) , $$

$$ ( v _ {5} , v _ {6} ) \rightarrow ( v _ {3} , v _ {4} ) , $$

$$ ( v _ {5} , v _ {6} ) \rightarrow ( v _ {5} , v _ {6} ) . $$

In each of the above collisions, both mass and kinetic energy are also preserved.

Letting $ N _ {i} = N _ {i} ( \mathbf x,t ) $ denote the number density of molecules with velocity $ \mathbf v _ {i} $, the Boltzmann equation can be written as

$$ { \frac{\partial N _ {i} }{\partial t } } + \mathbf u _ {i} \cdot \nabla N _ {i} = G _ {i} - L _ {i} , $$

where $ G _ {i} $ and $ L _ {i} $ are the rates of gain and loss in $ N _ {i} $ as a result of collisions. Assuming spherical symmetry and collisional cross section $ \sigma $, one has, for example,

$$ G _ {1} = { \frac{2}{3} } \sigma N _ {3} N _ {4} + { \frac{2}{3} } \sigma N _ {5} N _ {6} $$

and

$$ L _ {1} = { \frac{4}{3} } \sigma N _ {1} N _ {2} , $$

since one-third of the $ ( \mathbf v _ {3} , \mathbf v _ {4} ) $ and $ ( \mathbf v _ {5} , \mathbf v _ {6} ) $ collisions yield $ ( \mathbf v _ {1} , \mathbf v _ {2} ) $ pairs.

References

[a1] J.E. Broadwell, "Shock structure in a simple discrete velocity gas" Phys. Fluids , 7 (1964) pp. 1243–1247
[a2] R. Gatignol, "Théorie cinétique d'un gaz répartition discrète de vitesses" , Springer (1975)
[a3] R. Monaco, L. Preziosi, "Fluid dynamic applications of the discrete Boltzmann equation" , World Sci. (1991)
[a4] T. Platkowski, R. Illner, "Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory" SIAM Review , 30 (2) (1988) pp. 213–255
How to Cite This Entry:
Broadwell model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Broadwell_model&oldid=14034
This article was adapted from an original article by M. Ikle (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article