189 research outputs found
Hamiltonian model for coupled surface and internal waves in the presence of currents
We examine a two dimensional fluid system consisting of a lower medium
bounded underneath by a flatbed and an upper medium with a free surface. The
two media are separated by a free common interface. The gravity driven surface
and internal water waves (at the common interface between the media) in the
presence of a depth-dependent current are studied under certain physical
assumptions. Both media are considered incompressible and with prescribed
vorticities. Using the Hamiltonian approach the Hamiltonian of the system is
constructed in terms of 'wave' variables and the equations of motion are
calculated. The resultant equations of motion are then analysed to show that
wave-current interaction is influenced only by the current profile in the
'strips' adjacent to the surface and the interface. Small amplitude and
long-wave approximations are also presented.Comment: 33 pages, 1 figur
Extended Camassa-Holm Hierarchy and Conserved Quantities
An extension of the Camassa-Holm hierarchy is constructed in this letter. The
conserved quantities of the hierarchy are studied and a recurrent formula for
the integrals of motion is derived.Comment: 13 page
Hamiltonian formulation and integrability of a complex symmetric nonlinear system
The integrability of a complex generalisation of the 'elegant' system,
proposed by D. Fairlie and its relation to the Nahm equation and the Manakov
top is discussed.Comment: 8 pages, Physics Letters A (accepted
One-dimensional weakly nonlinear model equations for Rossby waves
In this study we explore several possibilities for modelling weakly nonlinear
Rossby waves in fluid of constant depth, which propagate predominantly in one
direction. The model equations obtained include the BBM equation, as well as
the integrable KdV and Degasperis-Procesi equations.Comment: 15 page
The Dynamics of Flat Surface Internal Geophysical Waves with Currents
A two-dimensional water wave system is examined consisting of two discrete
incompressible fluid domains separated by a free common interface. In a
geophysical context this is a model of an internal wave, formed at a pycnocline
or thermocline in the ocean. The system is considered as being bounded at the
bottom and top by a flatbed and wave-free surface respectively. A current
profile with depth-dependent currents in each domain is considered. The
Hamiltonian of the system is determined and expressed in terms of canonical
wave-related variables. Limiting behaviour is examined and compared to that of
other known models. The linearised equations as well as long-wave
approximations are presented.Comment: LaTeX, 21 pages, 1 figure, available online in J. Math. Fluid Mech.
(2016
Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrability
Many engineering and physiological applications deal with situations when a
fluid is moving in flexible tubes with elastic walls. In the real-life
applications like blood flow, there is often an additional complexity of
vorticity being present in the fluid. We present a theory for the dynamics of
interaction of fluids and structures. The equations are derived using the
variational principle, with the incompressibility constraint of the fluid
giving rise to a pressure-like term. In order to connect this work with the
previous literature, we consider the case of inextensible and unshearable tube
with a straight centerline. In the absence of vorticity, our model reduces to
previous models considered in the literature, yielding the equations of
conservation of fluid momentum, wall momentum and the fluid volume. We show
that even when the vorticity is present, but is kept at a constant value, the
case of an inextensible, unshearable and straight tube with elastics walls
carrying a fluid allows an alternative formulation, reducing to a single
compact equation for the back-to-labels map instead of three conservation
equations. That single equation shows interesting instability in solutions when
the vorticity exceeds a certain threshold. Furthermore, the equation in stable
regime can be reduced to Boussinesq-type, KdV and Monge-Amp\`ere equations
equations in several appropriate limits, namely, the first two in the limit of
long time and length scales and the third one in the additional limit of the
small cross-sectional area. For the unstable regime, we numerical solutions
demonstrate the spontaneous appearance of large oscillations in the
cross-sectional area.Comment: 57 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1805.1102
Hamiltonian Approach to Internal Wave-Current Interactions in a Two-Media Fluid with a Rigid Lid
We examine a two-media 2-dimensional fluid system consisting of a lower
medium bounded underneath by a flatbed and an upper medium with a free surface
with wind generated surface waves but considered bounded above by a lid by an
assumption that surface waves have negligible amplitude. An internal wave
driven by gravity which propagates in the positive -direction acts as a free
common interface between the media. The current is such that it is zero at the
flatbed but a negative constant, due to an assumption that surface winds blow
in the negative -direction, at the lid. We are concerned with the layers
adjacent to the internal wave in which there exists a depth dependent current
for which there is a greater underlying than overlying current. Both media are
considered incompressible and having non-zero constant vorticities. The
governing equations are written in canonical Hamiltonian form in terms of the
variables, associated to the wave (in a presence of a constant current). The
resultant equations of motion show that wave-current interaction is influenced
only by the current profile in the 'strip' adjacent to the internal wave.Comment: 13 pages, 1 figur
Empirical balanced truncation of nonlinear systems
Novel constructions of empirical controllability and observability gramians
for nonlinear systems for subsequent use in a balanced truncation style of
model reduction are proposed. The new gramians are based on a generalisation of
the fundamental solution for a Linear Time-Varying system. Relationships
between the given gramians for nonlinear systems and the standard gramians for
both Linear Time-Invariant and Linear Time-Varying systems are established as
well as relationships to prior constructions proposed for empirical gramians.
Application of the new gramians is illustrated through a sample test-system.Comment: LaTeX, 11 pages, 2 figure
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