This study uses spatio-temporal stability analysis to investigate the convective and absolute instability properties of a steady unconfined planar liquid jet. The approach uses a piecewise linear velocity profile with a finite-thickness shear layer at the edge of the jet. This study investigates how properties such as the thickness of the shear layer and the value of the fluid velocity at the interface within the shear layer affect the stability properties of the jet. It is found that the presence of a finite-thickness shear layer can lead to an absolute instability for a range of density ratios, not seen when a simpler plug flow velocity profile is considered. It is also found that the inclusion of surface tension has a stabilizing effect on the convective instability but a destabilizing effect on the absolute instability. The stability results are used to obtain estimates for the breakup length of a planar liquid jet as the jet velocity varies. It is found that reducing the shear layer thickness within the jet causes the breakup length to decrease, while increasing the fluid velocity at the fluid interface within the shear layer causes the breakup length to increase. Combining these two effects into a profile, which evolves realistically with velocity, gives results in which the breakup length increases for small velocities and decreases for larger velocities. This behaviour agrees qualitatively with existing experiments on the breakup length of axisymmetric jets.
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