A series of parametric computational studies are conducted to assess the frequency/wavelength of disturbances that issue into the atomizing jet as a result of instabilities in the vena contracta formed at the orifice entry. Axisymmetric, laminar incompressible computations are performed for a variety of orifice lengths and inlet rounding levels. The results are summarized and compared against classical linear theories. The computations show that as the inlet of the injector is rounded, the amplitude of the unsteadiness is reduced, and for large enough radii the unsteadiness is absent. Furthermore, as the injector length is increased, local maxima in the amplitude of the unsteadiness are observed. Finally, the results are compared with linear theory and it is hypothesized that the wavelengths associated with unsteadiness could be amplified by the boundary-layer instabilities at the exit of the nozzle and lead to the primary atomization of the liquid jet.