It is well known that most gas fluidized beds of particles bubble,
while most liquid
fluidized beds do not. It was shown by Anderson, Sundaresan & Jackson
(1995),
through direct numerical integration of the volume-averaged equations of
motion for
the fluid and particles, that this distinction is indeed accounted for
by
these equations, coupled with simple, physically credible closure relations
for
the stresses and interphase drag. The aim of the present study is to investigate
how the model equations afford this
distinction and deduce an approximate criterion for separating bubbling
and
non-bubbling systems. To this end, we have computed, making use of numerical
continuation techniques as well as bifurcation theory, the one- and
two-dimensional travelling wave solutions of the volume-averaged equations
for a
wide range of parameter values, and examined the evolution of these travelling
wave
solutions through direct numerical integration. It is demonstrated that
whether
bubbles form or not is dictated by the value of Ω =
(ρsv3t/Ag)
1/2, where ρs is the density of
particles, vt is the terminal settling velocity of an isolated
particle, g is acceleration due to gravity
and A is a measure of the particle phase viscosity. When
Ω is large (> ∼ 30), bubbles
develop easily. It is then suggested that a natural scale for A
is
ρsvtdp so that Ω2 is
simply a Froude number.