Next: Steepening of the energy
Up: Effects of friction in
Previous: Effects of friction in
  Contents
Origin of the friction term
A linear friction term naturally arises in a wide
range of different physical situations, and its origin should
be considered within each specific context.
Here we will briefly consider the case of thin stratified
layers of fluids electromagnetically forced [31,32],
in which the origin of the friction term can be easily understood
starting from the classical three-dimensional Navier-Stokes equations.
The dynamics of a shallow layer of incompressible fluid,
with a thickness
much smaller than its extension
is described by Navier-Stokes equations:
 |
(2.1) |
where
is the pressure,
is the density of the fluid,
is its kinematic viscosity, and
the external forcing.
In the passage from a three-dimensional to a
two-dimensional description the vertical
components of velocity
are neglected, since their magnitude with
respect to the horizontal ones
is assumed to be of the
same order of the aspect ratio:
 |
(2.2) |
Then we need to parameterize the vertical dependence
of horizontal velocities.
Experimental results [31] suggest that the flow structure
within the layer is close to a Poiseuille flow, so
we can assume a laminar viscous profile of velocities in the
-direction:
. With this assumption the
three dimensional viscous term in eq.(2.1)
gives origin to a two dimensional viscous term plus
an additional linear damping term, which represent the effects
of the bottom friction of the fluid:
 |
(2.3) |
The resulting friction coefficient
is proportional
to the inverse of the square of the total thickness of the layer
:
 |
(2.4) |
according to the intuitive idea that the thinner is the layer,
the stronger it feels the bottom friction.
Next: Steepening of the energy
Up: Effects of friction in
Previous: Effects of friction in
  Contents
Stefano Musacchio
2004-01-09