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 $h$ much smaller than its extension $L$ is described by Navier-Stokes equations:

$$\partial_t {\mbox{\boldmath$u$}} + {\mbox{\boldmath$u$}} \cd... ...o}} + \nu \Delta {\mbox{\boldmath$u$}} + {\mbox{\boldmath$f$}}$$ (2.1)

where $P$ is the pressure, $\rho$ is the density of the fluid, $\nu$ is its kinematic viscosity, and ${\mbox{\boldmath$f$}}$ the external forcing.

In the passage from a three-dimensional to a two-dimensional description the vertical components of velocity $u_z$ are neglected, since their magnitude with respect to the horizontal ones $u_H$ is assumed to be of the same order of the aspect ratio:

$${(u_z)_{rms} \over (u_H)_{rms}} \sim {h \over L}$$ (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 $z$-direction: $u(z) - u(h) \sim (z-h)^2$. 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:

$$\nu (\partial_x \partial_x + \partial_y \partial_y + \partia... ...artial_y) {\mbox{\boldmath$u$}} - \alpha {\mbox{\boldmath$u$}}$$ (2.3)

The resulting friction coefficient $\alpha$ is proportional to the inverse of the square of the total thickness of the layer $h$:

$$\alpha \sim {\nu \over h^2}$$ (2.4)

according to the intuitive idea that the thinner is the layer, the stronger it feels the bottom friction.