An operator expansion method for computing nonlinear surface waves on a ferrofluid jet

Abstract

We present a new numerical method to simulate the time evolution of axisym- metric nonlinear waves on the surface of a ferrofluid jet. It is based on the reduction of this problem to a lower-dimensional computation involving surface variables alone. To do so, we describe the associated Dirichlet–Neumann op- erator in terms of a Taylor series expansion where each term can be efficiently computed by a pseudo-spectral scheme using the fast Fourier transform. We show detailed numerical tests on the convergence of this operator and, to illus- trate the performance of our method, we simulate the long-time propagation and pairwise collisions of axisymmetric solitary waves. Both depression and elevation waves are examined by varying the magnetic field. Comparisons with weakly nonlinear predictions are also provided

Travelling wave solutions on an axisymmetric ferrofluid jet

“…The importance of the existence of a minimum in the linear dispersion relation has been demonstrated, and periodic, solitary and generalised solitary waves have been found for both models. The stability of the solutions is as of yet unknown, and would require a time dependent numerical scheme to find out, as done by [19] for pure solitary waves on the one-layer model. As well as time dependent models, it would be interesting to see if symmetry breaking bifurcations can be found with the numerical scheme described in this paper (by removing the imposed symmetry condition), as has been found by [16] for gravity-capillary waves.…”

“…This is in good agreement with Rannacher & Engel's KdV equation, who found B 1 = 3/2 and B 2 = 9 when d = 0. Time dependant computations on solutions of this type are considered by [19]. Furthermore, BP also found branches of depression solitary waves bifurcating from non-zero amplitude for 1 < B < B 1 , and likewise elevation solitary waves bifurcating from non-zero amplitude for B 1 < B  2.…”

HOS Simulations of Nonlinear Water Waves in Complex Media

“…Equations ( 31)-( 32) or ( 33)- (34) provide recursion formulas to evaluate the DNO in its series form (30) given η and ξ. Depending on the physical situation under consideration, each set may be used separately, i.e.…”

“…Considering the periodic boundary conditions in θ, we use a pseudo-spectral method to discretize the DNO and equations of motion ( 21)- (22) in space [5]. This is a natural choice for the computation of G(η) because each term in its Taylor series (30) is evaluated via recursion formulas ( 35)- (36) involving concatenations of Fourier multipliers with powers of η/R. More specifically, both functions η and ξ are expressed as truncated Fourier series…”

“…Their study was strictly mathematical and did not produce any numerical result. Guyenne and Pȃrȃu [30] adopted a similar approach in the axisymmetric cylindrical setting to compute solitary waves on the surface of a ferrofluid jet. This formalism has also been applied to scattering problems in acoustics and electromagnetics with non-Cartesian (e.g.…”