Abstracts - KTH Mechanics

Three-dimensional gravity-capillary interfacial solitary waves
and related problems
E. I. Părău ∗ , J.-M. Vanden-Broeck ∗ and M.J. Cooker ∗
Fully localised three-dimensional gravity-capillary solitary interfacial waves, which
travel on the interface between two superposed fluids, with a lighter fluid lying above a
heavier one, are calculated. The fluids are assumed to be inviscid and incompressible
and the flow is assumed to be irrotational. The fluid layers can be semi-infinite or of
finite thickness. The three-dimensional problem is formulated as a nonlinear integrodifferential
equation by using the Green’s identity in each layer and the dynamic
boundary condition, which includes capillarity. These waves have damped oscillations
in the direction of propagation, as in the two-dimensional case 1 , and also decay in the
transverse direction. When the density ratio is zero, the three-dimensional solitary
interfacial waves reduce to free-surface solitary water waves which were studied in
recent papers. 2 3 4 The stability of the solutions is discussed. A typical solution
is presented in figure 1 for central-elevation and central-depression interfacial solitary
waves.
The algorithm is modified to compute three-dimensional flows due to an immersed
disturbance that propagates at a constant velocity U along the interface between the
two fluids, and the effect of density ratio on the wave patterns is studied. Possible
generalisations of the algorithm include the computation of waves when both on
interface and an upper free-surface are present.
∗ School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK.
1 Dias and Iooss, Eur.J. Mech./B Fluids, 15, 367(1996).
2 Părău et al., J. Fluid Mech. 536, 99(2005).
3 Kim and Akylas J. Fluid Mech. 540, 337(2005).
4 Părău et al., Phys. Fluids 17, 122101 (2005).
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Figure 1: (a) Central elevation wave (b) Central depression wave. Only half of the
each symmetric solution is shown. The waves propagate in the x direction.
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