The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow
The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration σ) of a rigid plate, inclined at an angle α ∈ (0, π/2) to the exterior horizontal, as it advances (σ > 0) or retreats (σ < 0) from an initially stationary and horizontal strip of in...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Published: |
Cambridge University Press
2018
|
| Online Access: | https://eprints.nottingham.ac.uk/48619/ |
| _version_ | 1848797808581148672 |
|---|---|
| author | Gallagher, M.T. Needham, D.J. Billingham, John |
| author_facet | Gallagher, M.T. Needham, D.J. Billingham, John |
| author_sort | Gallagher, M.T. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration σ) of a rigid plate, inclined at an angle α ∈ (0, π/2) to the exterior horizontal, as it advances (σ > 0) or retreats (σ < 0) from an initially stationary and horizontal strip of inviscid, incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied in Needham et al. (2008). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters α and μ (where μ = 1+σ tan α), with a bifurcation in structure as μ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (2016) . In particular, when (α, μ) ∈ (0, π/2) × R+, the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when (α, μ) ∈ (0, π/2)×[1,∞), or a collapsing jet when (α, μ) ∈ (0, π/2) × (0, 1). However, when (α, μ) ∈ (0, π/2) × R−, the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles α = α∗n ∈ (0, π/2) (n = 1, 2, . . .). |
| first_indexed | 2025-11-14T20:09:46Z |
| format | Article |
| id | nottingham-48619 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:09:46Z |
| publishDate | 2018 |
| publisher | Cambridge University Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-486192020-05-04T19:33:50Z https://eprints.nottingham.ac.uk/48619/ The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow Gallagher, M.T. Needham, D.J. Billingham, John The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration σ) of a rigid plate, inclined at an angle α ∈ (0, π/2) to the exterior horizontal, as it advances (σ > 0) or retreats (σ < 0) from an initially stationary and horizontal strip of inviscid, incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied in Needham et al. (2008). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters α and μ (where μ = 1+σ tan α), with a bifurcation in structure as μ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (2016) . In particular, when (α, μ) ∈ (0, π/2) × R+, the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when (α, μ) ∈ (0, π/2)×[1,∞), or a collapsing jet when (α, μ) ∈ (0, π/2) × (0, 1). However, when (α, μ) ∈ (0, π/2) × R−, the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles α = α∗n ∈ (0, π/2) (n = 1, 2, . . .). Cambridge University Press 2018-04-25 Article PeerReviewed Gallagher, M.T., Needham, D.J. and Billingham, John (2018) The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow. Journal of Fluid Mechanics, 841 . pp. 109-145. ISSN 1469-7645 https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/initial-development-of-a-jet-caused-by-fluid-body-and-free-surface-interaction-with-a-uniformly-accelerated-advancing-or-retreating-plate-part-1-the-principal-flow/559767DF5FE57A3B doi:10.1017/jfm.2018.4 doi:10.1017/jfm.2018.4 |
| spellingShingle | Gallagher, M.T. Needham, D.J. Billingham, John The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow |
| title | The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow |
| title_full | The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow |
| title_fullStr | The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow |
| title_full_unstemmed | The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow |
| title_short | The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow |
| title_sort | initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. part 1. the principal flow |
| url | https://eprints.nottingham.ac.uk/48619/ https://eprints.nottingham.ac.uk/48619/ https://eprints.nottingham.ac.uk/48619/ |