Measurement of the dispersion relation of a convectively unstable capillary jet under confinement

The dispersion relation of a confined capillary jet with negligible inertia is measured by Fourier analysis of the jet radius. The real part of the dispersion relation, obtained through the spatiotemporal Fourier spectrum, demonstrates that the phase velocity is independent of the perturbation frequency, at least in the accessible range of frequencies. The imaginary part of the dispersion relation, i.e., the spatial growth rate, is frequency dependent. To measure this dependence, an external forcing is used to amplify modes with different frequencies. The spatial growth rate is then obtained from the exponential growth of the corresponding Fourier mode. The phase velocity and the spatial growth rate are found to increase and decrease, respectively, with the capillary number and to depend only weakly on the degree of confinement. These observations are consistent with a spatial instability that convects the perturbations downstream with velocity proportional to the capillary number. The frequency of the dominant mode in the absence of forcing, on the contrary, depends mainly on the degree of confinement, i.e., on the jet diameter, and only weakly on the capillary number, which is consistent with the mode selection of the Rayleigh-Plateau instability. Experimental measurements are compared to predicted dispersion relations reported in the literature, and good qualitative agreement is observed.