Gallagher, M.T. and Needham, D.J. and Billingham, John
(2017)
The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 2. Wellposedness and stability of the principal flow.
Journal of Fluid Mechanics
.
ISSN 14697645
(In Press)
Abstract
We consider the problem of a rigid plate, inclined at an angle α ∈ (0, π/2) to the horizontal, accelerating uniformly from rest into, or away from, a semiinfinite strip of inviscid, incompressible fluid under gravity. Following on from Gallagher et al. (2016) (henceforth referred to as GNB), it is of interest to analyse the wellposedness and stability of the principal flow with respect to perturbations in the initially horizontal free surface close to the plate contact point. In particular we find that the solution to the principal unperturbed problem, denoted by [IBVP] in GNB, is wellposed and stable with respect to perturbations in initial data in the region of interest, localised close to the contact point of the free surface and the plate, when the plate is accelerated with dimensionless acceleration σ≥ − cot α, while the solution to [IBVP] is illposed with respect to such perturbations in initial data, when the plate is accelerated with dimensionless acceleration σ < − cot α. The physical source of the illposedness of the principal problem [IBVP], when σ < − cot α, is revealed to be due to the leading order problem in the innermost region localised close to the initial contact point being in the form of a local RayleighTaylor problem. As a consequence of this mechanistic interpretation we anticipate that, when the plate is accelerated with σ < − cot α, the inclusion of weak surface tension effects will restore wellposedness of the problem [IBVP] which will, however, remain temporally unstable.
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