On exponential stability of gravity driven viscoelastic flows

Abstract

We investigate stability of an equilibrium state to a nonhomogeneous incompressible viscoelastic fluid driven by gravity in a bounded domain Ω ⊂ R 3 of class C 3. First, we establish a critical number κ C , which depends on the equilibrium density and the gravitational constant, and is a threshold of the elasticity coefficient κ for instability and stability of the linearized perturbation problem around the equilibrium state. Then we prove that the equilibrium sate is exponential stability provided that κ > κ C and the initial disturbance quantities around the equilibrium state satisfy some relations. In particular, if the equilibrium densityρ is a Rayleigh-Taylor (RT) type andρ ′ is a constant, our result strictly shows that the sufficiently large elasticity coefficient can prevent the RT instability from occurrence.

On the stability of Rayleigh–Taylor problem for stratified rotating viscoelastic fluids

“…However, we need some more complicated mathematical techniques for the estimate of η 3 . Thus our derivation of the estimate of η 3 is very different with the one in [27]. In fact, since w := u + (κρ/μ)η may not be continuous on , we shall apply the regularity theory of one-phase steady Stokes problem to the momentum equations of upper and lower viscoelastic fluids.…”

“…Recently, Jiang et al [27] have proved that the elasticity can inhibit the RT instability based on the energy method. To obtain (2.8), we naturally use the energy method in [27] to prove Theorem 2.1. However, we need some more complicated mathematical techniques for the estimate of η 3 .…”

“…Recently, Jiang et al [16,27] have found that the elasticity can inhibit the RT instability based on the following idea, i.e., Oldroyd-B model of an incompressible viscoelastic fluid in the presence of a uniform gravitational field in a three-dimensional domain :…”

“…Recently, Baldwin et al investigated the effect of rotation on the RT instability by an experiment. Motivated by the experiment of Baldwin et al and the mathematical result of Jiang et al [16,27], we further mathematically investigate the effect of rotation on the RT instability in the stratified viscoelastic fluids in this article. Before stating our result, we shall introduce the mathematical model.…”

On effects of viscosity and magnetic fields on the largest growth rate of linear Rayleigh–Taylor instability

“…In the last decades, this phenomenon has been extensively investigated from mathematical, physical and numerical aspects, see [2,8,29] for instance. It has been also widely investigated how the RT instability evolves under the effects of other physical factors, such as elasticity [3,11,21,23,30], rotation [2,4], (internal) surface tension [9,14,34], magnetic fields [16-20, 22, 31, 32] and so on. In this article, we are interested in the effect of surface tension on the linear RT instability in stratified incompressible viscous fluids.…”