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.
The gas–liquid interface (GLI) over superhydrophobic surfaces (SHSs), where the flow slips, is the key to reduce frictional drag in underwater applications. Many-body dissipative particle dynamics simulations are used to explore the slip behavior of a shear flow over a rectangular grooved SHS, and a flat GLI is obtained by tuning the contact angle of the GLI. Due to the slip, the normal profiles of the local velocity, which are perpendicular to the GLI, are curved and shifted away from the linear form near the GLI. Then, a polynomial function is proposed to fit the velocity profile to extract the local shear rate and calculate the slip length. Based on this fitting method, a hybrid slip boundary condition is derived for both longitudinal and transverse flows. That is, the shear stress and slip length are finite near the groove edge, and the stress is nearly zero and the slip length is infinite in the center region of the GLI. This new hybrid slip boundary condition not only explains the inconsistent slip conditions reported in the literature under different groove length scales, but also unifies the existing exclusive slip assumptions.
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