The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 2. Well-posedness and stability of the principal flow

We consider the problem of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the horizontal, accelerating uniformly from rest into, or away from, a semi-infinite strip of inviscid, incompressible fluid under gravity. Following on from Gallagher et al. (J....

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Bibliographic Details
Published in:Journal of fluid mechanics 2018-04, Vol.841, p.146-166
Main Authors: Gallagher, M. T., Needham, D. J., Billingham, J.
Format: Article
Language:English
Subjects:
Acceleration
Computational fluid dynamics
Flow stability
Fluid flow
Fluid mechanics
Free surfaces
Gravity
Ill posed problems
Incompressible flow
Incompressible fluids
JFM Papers
Mathematical problems
Perturbations
Stability analysis
Surface tension
Well posed problems
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Summary:We consider the problem of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the horizontal, accelerating uniformly from rest into, or away from, a semi-infinite strip of inviscid, incompressible fluid under gravity. Following on from Gallagher et al. (J. Fluid Mech., vol. 841, 2018, pp. 109–145) (henceforth referred to as GNB), it is of interest to analyse the well-posedness 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 well-posed 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 $\unicode[STIX]{x1D70E}\geqslant -\cot \,\unicode[STIX]{x1D6FC}$ , while the solution to [IBVP] is ill-posed with respect to such perturbations in the initial data, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2018.5