Linear analysis of the instability of two-dimensional non-Newtonian liquid sheets

Abstract

The instability of two-dimensional non-Newtonian liquid sheets moving in an inviscid gaseous environment is investigated. A linearized stability analysis shows that non-Newtonian liquid sheets have a higher growth rate than Newtonian liquid sheets for both symmetric and antisymmetric disturbances, indicating that non-Newtonian liquid sheets are more unstable than Newtonian liquid sheets. It is found that the surface tension effects always resist, whereas the aerodynamic effects promote, the occurrence and development of instability of non-Newtonian liquid sheets. Similar results were obtained for inviscid and Newtonian liquid sheets by many other authors before. It is observed that in non-Newtonian liquid sheets the liquid viscosity tends to damp the instability, whereas the liquid elasticity results in an enhancement of instability. It is seen that both the growth rate and the instability range of non-Newtonian liquid sheets increase greatly with the gas Weber number, the Reynolds number, and the ratio of gas to liquid density for symmetric and antisymmetric disturbances, indicating that non-Newtonian liquid sheets destabilize more easily at high values of these three parameters. When the liquid viscosity is increased, the growth rate of symmetric and antisymmetric disturbances on Newtonian and non-Newtonian liquid sheets decreases, but the cutoff wave number remains unchanged. At low liquid viscosity, the growth rate of non-Newtonian liquid sheets is close to that of Newtonian ones. The growth rates of symmetric and antisymmetric disturbances decrease with the ratio of deformation retardation to stress relaxation time but increase with the liquid elasticity, whereas the instability range does not change with these two parameters. Moreover, increasing the gas Weber number, the Reynolds number, and the ratio of gas to liquid density substantially, enhances the maximum growth rate and the dominant wave number of non-Newtonian liquid sheets for symmetric and antisymmetric types of disturbances. However, the effects of the ratio of deformation retardation to stress relaxation time, the Ohnesorge number, and the Elasticity number on both the maximum growth rate and the dominant wave number are relatively weak. It is discovered that the maximum growth rate of antisymmetric disturbances is always larger than that of symmetric disturbances, while the dominant wave number of antisymmetric disturbances is always smaller than that of symmetric disturbances. This indicates that antisymmetric disturbances always prevail over symmetric disturbances for non-Newtonian liquid sheets.