Three-dimensional temporal instability of non-Newtonian liquid sheets

The three-dimensional temporal instability of a non-Newtonian liquid sheet moving in an inviscid gaseous environment is investigated. The corresponding dispersion relations between the growth rate and the wave number of both symmetric and antisymmetric disturbances are derived. The velocity and pressure distributions are presented. The effects of different parameters on the growth rates are explored. The linear stability analysis shows that a sheet of a viscoelastic non-Newtonian fluid has higher growth rates than Newtonian ones for both symmetric and antisymmetric types of two- and three-dimensional disturbances, indicating that such non-Newtonian liquid sheets are more unstable than Newtonian counterparts. It is found that in non-Newtonian liquid sheets the surface tension effects always resist the occurrence and development of two- and three-dimensional instability, while the aerodynamic effects are the source of the onset and growth of the instability. Similar results have been reported for inviscid and Newtonian liquid sheets by many other authors. It is discovered that two-dimensional disturbances dominate the instability of non-Newtonian liquid sheets for both symmetric and antisymmetric disturbances, and the growth rate decreases as the z-direction wave number increases for three-dimensional disturbances. In viscoelastic non-Newtonian fluids, the liquid viscosity tends to stabilize the sheet, whereas the liquid elasticity results in a destabilization. As the surface tension is decreased, the growth rate and the instability range of non-Newtonian liquid sheets increase significantly for both the symmetric and antisymmetric modes. At low liquid Weber numbers the maximum growth rate of antisymmetric: disturbances is larger, whereas the dominant wave number of antisymmetric disturbances is smaller than that of symmetric disturbances. However at high liquid Weber numbers the values of the maximum growth rate and the dominant wave number of symmetric and antisymmetric disturbances approach each other asymptotically. In addition, when the z-direction wave number is greater than zero, the values of the maximum growth rate and the dominant wave number of symmetric and antisymmetric disturbances are almost identical. Both the disturbance growth rate and the instability range of non-Newtonian liquid sheets increase greatly with the gas-to-liquid density ratio for both the symmetric and antisymmetric modes. The growth rates of symmetric and antisymmetric disturbances decrease with increasing liquid viscosity and with increasing ratio of deformation retardation to stress relaxation time, but increase with the liquid elasticity in a relatively weak manner for two- and three-dimensional instabilities. The instability range does not change with these three parameters.