Effect of odd viscosity on the stability of thin viscoelastic liquid film flowing along an inclined plate

Abstract

A theory for the stability of a viscoelastic film flowing along an inclined wall which is considered the odd viscosity effect is investigated. Using the lubrication theory, a new liquid-gas interface evolution equation involving odd viscosity effect is derived. Linear stability analysis shows that the larger odd viscosity leads to the higher critical Reynolds number. While the higher viscoelastic parameter makes the critical Reynolds number lower. The weakly nonlinear study reveals that in the limited amplitude range, the supercritical stable region and the explosion region will decrease with the increase of the odd viscosity. Conversely, the unconditional stable region and the subcritical unstable region increase. Interestingly, the threshold of the supercritical stable region decreases with the increase of the odd viscosity. Therefore, by analysing the linear and non-linear stability of the evolution equation, we find that the odd viscosity stabilizes the flow, while the viscoelastic property destabilizes the flow.

Role of odd viscosity in falling viscous fluid

“…Obviously, the capillary number arises in the expression of the complex wave speed in the approximation. Using the neutral stability condition ( as ), we obtain the critical Reynolds number for the surface mode in the limit as It should be noted that the critical Reynolds number for the surface mode increases in the presence of the viscosity ratio of odd viscosity coefficient to even viscosity coefficient, as reported in the studies of Zhao & Jian (2021 a , b ) and Chattopadhyay (2021) which were carried out under the framework of the Benney-type surface evolution equation. Furthermore, (3.17) reveals that the vertical falling viscous fluid can be stable if the odd viscosity coefficient is present because the critical Reynolds number is non-zero when .…”

“…The above governing equations are subjected to the following boundary conditions. At the inclined plane, , fluid velocity components must satisfy no-slip and no-penetration boundary conditions At the fluid surface, , hydrodynamic stresses of the fluid must satisfy tangential stress and normal stress boundary conditions (see, for example Kirkinis & Andreev 2019; Zhao & Jian 2021 a , b ) where is the surface tension, is the direction cosine of the unit tangent vector and is the direction cosine of the unit normal vector , placed on the fluid surface. Using the components of the Cauchy stress tensor, tangential stress and normal stress boundary conditions at the fluid surface, , can be read as where is the ambient pressure.…”

“…Weakly nonlinear analysis was also performed, which revealed that the supercritical stable region is reduced by the odd viscosity coefficient. Zhao & Jian (2021 b ) further performed a similar analysis for a viscoelastic liquid (Walters’ liquid ), where it was assumed that the viscoelastic liquid has a property of broken time-reversal symmetry. They mentioned that viscoelasticity destabilizes the surface instability.…”