The stability of axially symmetric cone-and-plate flow of an
Oldroyd-B fluid at
non-zero Reynolds number is analysed. We show that stability is controlled
by two
parameters: [Escr ]1≡DeWe and
[Escr ]2≡Re/We,
where De, We, and Re are the Deborah,
Weissenberg and Reynolds numbers respectively. The linear stability problem
is solved
by a perturbation method for [Escr ]2 small and by a Galerkin
method when
[Escr ]2=O(1).
Our results show that for all values of the retardation parameter β
and for all values of [Escr ]2
considered the base viscometric flow is stable if
[Escr ]1 is sufficiently small. As [Escr ]1
increases past a critical value the flow becomes unstable as a pair of
complex-conjugate eigenvalues crosses the imaginary axis into the right
half-plane. The critical value of [Escr ]1 decreases as
[Escr ]2 increases indicating that increasing inertia destabilizes
the
flow. For the range of values considered the critical wavenumber
kc first decreases
and then increases as [Escr ]2
increases. The wave speed on the other hand decreases
monotonically with [Escr ]2.
The critical mode at the onset of instability corresponds to
a travelling wave propagating inward towards the apex of the cone with
infinitely
many logarithmically spaced toroidal roll cells.