Weakly nonlinear instability of planar viscoelastic sheets

Abstract

A second-order weakly nonlinear analysis has been made of the temporal instability for the linear sinuous mode of two-dimensional planar viscoelastic liquid sheets moving in an inviscid gas. The convected Jeffreys models including the corotational Jeffreys model, Oldroyd A model, and the Oldroyd B model are considered as the rheology model of the viscoelastic fluid of the sheet. The solution for the secondorder gas-to-liquid interface displacement has been derived, and the temporal evolution leading to the breakup has been shown. The second-order interface displacement of the linear sinuous mode is varicose, which causes the sheet to fragment into ligaments. First-order constitutive relations of the three rheology models become identical after linearization, so the linear instability results are also the same. For the second-order weakly nonlinear instability, the second-order constitutive relation varies among the corotational Jeffreys model, Oldroyd A model, and the Oldroyd B model, but although they have different disturbance pressures, their disturbance velocities and interface displacements are the same, and therefore, the sheets of the corotational Jeffreys fluid, Oldroyd A fluid, and the Oldroyd B fluid have the same instability behavior characterized by the wave profile and breakup time. The reason for the identical instability behavior is that the effect of different codeformations of the corotational frame, covariant frame, and the contravariant frame is counteracted by the corresponding change in the second-order disturbance pressure, leaving no influence on the second-order velocity. At wavenumbers with maximum instabilities, an increase in the elasticity, or a reduction of the deformation retardation time, leads to a larger linear temporal growth rate, greater second-order disturbance amplitude, and shorter breakup time, thereby enhancing instability. The mechanism of linear instability has been examined using an energy approach, which shows that the main cause of instability is the aerodynamic force. C 2015 AIP Publishing LLC.

Effects of elasticity number and time constant ratio on breakup and droplet formation of viscoelastic planar liquid sheet co-flowing with gases of equal velocities

“…Table 1 shows the effect of elasticity number on breakup time predicted by the present study and it shows excellent agreement with the results of Wang et al. (2015). For the purpose of validation, the wavenumber is kept fixed for all values of El.…”

“…Keeping consistency with Wang et al. (2015), liquid Reynolds number is kept at 63.54. Validation of the present model is done with the temporal instability study of viscoelastic fluid by Wang et al.…”

“…Validation of the present model is done with the temporal instability study of viscoelastic fluid by Wang et al. (2015) and is presented in table 1.…”

“…Nonlinear investigation of a viscoelastic liquid sheet in quiescent gas medium by Wang et al. (2015) reported only a destabilizing effect of elasticity. However, they considered a high Weber number ( We = 600) for their study.…”

“…Wang et al. (2015) performed an in-depth nonlinear analysis of three rheological models, namely corotational Jeffrey's model, Oldroyd A model and Oldroyd B model, exposed to initial sinuous disturbances. The study showed that though different rheological models have different disturbance pressure, the interface displacement and disturbance velocities are the same.…”