Abstract
In this article, we investigate the effect of surface tension in the Rayleigh–Taylor (RT) problem of stratified incompressible viscoelastic fluids. We prove that there exists an unstable solution to the linearized stratified RT problem with a largest growth rate Λ under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold vartheta _{c}). Moreover, for this instability condition, the largest growth rate varLambda _{vartheta } decreases from a positive constant to 0, when ϑ increases from 0 to vartheta _{c}, which mathematically verifies that the internal surface tension can constrain the growth of the RT instability during the linear stage.
Highlights
It is well known that the equilibrium state of the heavier fluid on top of the lighter one under the gravity is unstable to sustain a small disturbance
In 1953, Bellman–Pennington [2] first analyzed the inhibition of RT instability by surface tension, where the study was based on a linearized two-dimensional (2D) motion equation of stratified incompressible inviscid fluids defined on the domain T1 × (–h, h+) (i.e., μ = 0 in the corresponding 2D case of (1.13))
3 Main results In this paper, we investigate the effect of surface tension on the linear RT instability by the linearized motion (1.13)
Summary
It is well known that the equilibrium state of the heavier fluid on top of the lighter one under the gravity is unstable to sustain a small disturbance. In 1953, Bellman–Pennington [2] first analyzed the inhibition of RT instability by surface tension, where the study was based on a linearized two-dimensional (2D) motion equation of stratified incompressible inviscid fluids defined on the domain T1 × (–h–, h+) (i.e., μ = 0 in the corresponding 2D case of (1.13)). They precisely proved that there exists a threshold gtρuL21 of surface tension coefficient for linear stability and instability of RT problem. As documented in [14, 18], the results of nonlinear RT instability in inhomogeneous fluid (without interface), by the classical bootstrap instability method, were obtained for inviscid and viscous cases
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