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 correspondin g dispersion relations between the growth rate and the wave number of both symmetric and antisymmetric disturbances are derived. The velocity and pres sure distributions are presented. The effects of different parameters on th e growth rates are explored. The linear stability analysis shows that a she et of a viscoelastic non-Newtonian fluid has higher growth rates than Newto nian ones for both symmetric and antisymmetric types of two- and three-dime nsional disturbances, indicating that such non-Newtonian liquid sheets are more unstable than Newtonian counterparts. It is found that in non-Newtonia n liquid sheets the surface tension effects always resist the occurrence an d development of two- and three-dimensional instability, while the aerodyna mic effects are the source of the onset and growth of the instability. Simi lar results have been reported for inviscid and Newtonian liquid sheets by many other authors. It is discovered that two-dimensional disturbances domi nate the instability of non-Newtonian liquid sheets for both symmetric and antisymmetric disturbances, and the growth rate decreases as the z-directio n wave number increases for three-dimensional disturbances. In viscoelastic non-Newtonian fluids, the liquid viscosity tends to stabilize the sheet, w hereas the liquid elasticity results in a destabilization. As the surface t ension is decreased, the growth rate and the instability range of non-Newto nian liquid sheets increase significantly for both the symmetric and antisy mmetric modes. At low liquid Weber numbers the maximum growth rate of antis ymmetric: disturbances is larger, whereas the dominant wave number of antis ymmetric disturbances is smaller than that of symmetric disturbances. Howev er at high liquid Weber numbers the values of the maximum growth rate and t he dominant wave number of symmetric and antisymmetric disturbances approac h each other asymptotically. In addition, when the z-direction wave number is greater than zero, the values of the maximum growth rate and the dominan t wave number of symmetric and antisymmetric disturbances are almost identi cal. Both the disturbance growth rate and the instability range of non-Newt onian liquid sheets increase greatly with the gas-to-liquid density ratio f or both the symmetric and antisymmetric modes. The growth rates of symmetri c and antisymmetric disturbances decrease with increasing liquid viscosity and with increasing ratio of deformation retardation to stress relaxation t ime, but increase with the liquid elasticity in a relatively weak manner fo r two- and three-dimensional instabilities. The instability range does not change with these three parameters.