Re. Khayat, NONLINEAR OVERSTABILITY IN THE THERMAL-CONVECTION OF VISCOELASTIC FLUIDS, Journal of non-Newtonian fluid mechanics, 58(2-3), 1995, pp. 331-356
The existence and stability of elastic overstability in the presence o
f non-negligible inertia for an Oldroyd-B fluid are examined for the R
ayleigh-Bernard thermal convection problem. The study is based on the
four-dimensional non-linear dynamical system presented by Khayat (J. N
on-Newtonian Fluid Mech., 53 (1994) 227) which constitutes a generaliz
ation of the classical Lorenz system for a Newtonian fluid. It is show
n that elastic overstability can only set in once the Deborah number e
xceeds a critical value which depends on the Prandtl number and fluid
retardation. Fluid elasticity is found to precipitate the onset of ove
rstability while retardation tends to delay it. The conditions of exis
tence of the corresponding Hopf bifurcation are examined as functions
of fluid elasticity, retardation and thermal conductivity. The stabili
ty of the periodic orbit (in phase space) is investigated using center
manifold theory. It is found that the orbit is asymptotically stable
to perturbations about the conductive state, with the initial period o
f oscillation decreasing with Deborah number, reaching a minimum, and
increasing asymptotically toward a constant value.