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8. Hydrodynamics
How do you describe a flow?
Some basic truths:
General equation of motion for the flow field
in a
compressible viscous fluid: NavierStokes equation

(8.1) 
Here, is the viscosity, and

(8.2) 
defines the NavierStokes tensor. (In 2 dimensions, write in place
of ).
Character of NavierStokes PDE:
 Contains advective (hyperbolic) and diffusive
(parabolic) terms
 Small viscosity: advective terms dominate
hyperbolic
 High viscosity: diffusive terms important
parabolic
 Stationary case (
):
elliptic
The NS equation results from the conservation of momentum.
In addition, we have conservation of mass,

(8.3) 
and conservation of energy,

(8.4) 
where is the energy density
( ... internal energy per unit mass of the fluid).
To close the set of equations some equation of state
is assumed.
The following approaches will be discussed:
 Conventional methods of solving the NavierStokes PDE
 Discretized ``Lattice gas''or ``Lattice Boltzmann''dynamics
 Direct Simulation Monte Carlo
Subsections
Next: 8.1 Compressible Flow without
Up: III. SELECTED APPLICATIONS
Previous: 7.4 Density Functional Molecular
Franz J. Vesely Oct 2005
See also: "Computational Physics  An Introduction," KluwerPlenum 2001