Dimensionless Numbers
The Reynolds number (Re) is
a dimensionless number that gives a measure of the balance of forces, between
inertial and viscous forces, that are acting on a fluid. Above a critical
number, usually around 2300, a small disturbance grows and eventually leads to
turbulent flow (Philpotts, 1990). The Reynolds
number of a fluid is defined as:
.
Where r is the radius of the
flow, r = density of the fluid, v is velocity, and m is the viscosity. For large
viscosities encountered in normal, felsic magmas (~106 to 108
Pa s), the Re numbers could be subcritical.
Rayleigh number (Ra) is a
dimensionless number that is used to predict the likelihood of convection in a
system. The convection can be driven by either thermal-density inversions, or
compositional gradients (Sparks et al., 1984). Values for RaT (thermal convection),
Prandtl number (Pr), RaS (compositional convection), and diffusivity
ratio (t) can be calculated as follows:
and
,
and
.
Where, g = gravitational
acceleration, aT =
coefficient of thermal expansion, DT =
temperature difference across the layer, d = the layer thickness, kT =
thermal diffusivity, kS = compositional diffusivity, n = kinematic viscosity, b = coefficient of expansion due to compositional
changes, and bDS = fractional density change across a layer due to
compositional changes (Sparks et al., 1984).
Damköhler Numbers are
dimensionless numbers that characterize reaction rates (Ottino,
1989),
, and,
.
Where tD is the
characteristic diffusion time-scale, tR the characteristic reaction
time-scale, and tF is the characteristic fluid mechanical or
deformation time-scale. These numbers are useful in determining the controls on
reaction rates (i.e. diffusion controlled reaction).
Viscosity can be thought as the resistance to flow. More fundamentally,
viscosity is defined as (Turcotte and
Schubert, 2002):
,
Another type of viscosity used in the previous equations is kinematic viscosity.
This is a diffusivity similar to thermal diffusivity, only it describes how
momentum diffuses instead of heat.
,
Where n = kinematic
viscosity, m = viscosity,
and r = density.