Steady waves on an axisymmetric ferrofluid jet: A weakly nonlinear theory

We examine axisymmetric periodic waves that propagate along the surface of an inviscid ferrofluid jet under the influence of an azimuthal magnetic field, using both theoretical and numerical approaches. Our analysis takes into account the competing forces of the magnetic field and surface tension. By applying the Dirichlet–Neumann operator (DNO) expansion to the free-surface boundary conditions, we derive a fully dispersive model known as the cubic DNO-truncation model. This model demonstrates strong agreement with numerical results obtained from the full Euler equations for small to moderate amplitude traveling waves across a wide range of physical parameters. Using the cubic DNO-truncation model, we first develop perturbation series solutions for periodic waves by modifying the Stokes expansion to suit the axisymmetric configuration. In non-resonant cases, we observe that within a narrow range of the magnetic Bond number, the third-order Stokes wave solutions closely align with numerical results from both the DNO-truncation model and the primitive Euler equations, even for moderate amplitude waves. However, for Wilton ripples, the asymptotic solutions diverge from the numerical results, even at small amplitudes.