A second-order instability analysis has been performed for sinuous disturbances on two-dimensional planar viscous sheets moving in a stationary gas medium using a perturbation technique. The solutions of second-order interface disturbances have been derived for both temporal instability and spatial instability. It has been found that the second-order interface deformation of the fundamental sinuous wave is varicose or dilational, causing disintegration and resulting in ligaments which are interspaced by half a wavelength. The interface deformation has been presented; the breakup time for temporal instability and breakup length for spatial instability have been calculated. An increase in Weber number and gas-to-liquid density ratio extensively increases both the temporal or spatial growth rate and the second-order initial disturbance amplitude, resulting in a shorter breakup time or length, and a more distorted surface deformation. Under normal conditions, viscosity has a stabilizing effect on the firstorder temporal or spatial growth rate, but it plays a dual role in the second-order disturbance amplitude. The overall effect of viscosity is minor and complicated. In the typical condition, in which the Weber number is 400 and the gas-to-liquid density ratio is 0.001, viscosity has a weak stabilizing effect when the Reynolds number is larger than 150 or smaller than 10; when the Reynolds number is between 150 and 10, viscosity has a weak destabilizing effect.

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A two-dimensional viscoelastic planar liquid sheet subjected to a considerable temperature gradient perpendicular to the surfaces, moving in a gas medium, was investigated in a linear scope. The sheet instability was explored by solving the dispersion relation in the sinuous mode. Results suggested that the viscoelastic liquid sheet could behave with greater stability than its Newtonian counterpart when the temperature difference was sufficiently large. Thermal effects improved sheet instability, while the liquid elasticity had a dual effect when considering the temperature difference. It should be noted that thermal effects could retard the breakup process of viscoelastic planar liquid sheets at a large liquid Weber number. Deformation retardation time was a destabilizing factor when there were great temperature differences, which was polar to the case without thermal effects. However, the effects of liquid viscosity, liquid velocity, gas-to-liquid density ratio, and surface tension were analogous, whether or not there existed a difference in temperature. Finally, the competition between thermocapillar and aerodynamic instabilities on sheet instability was examined.

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