On effects of elasticity and magnetic fields in the linear Rayleigh–Taylor instability of stratified fluids

Abstract

It is well known that there exists a threshold such that the linearized stratified viscoelastic Rayleigh–Taylor problem is unstable for the elasticity coefficient κ satisfying . In this paper, we further prove that if , then there exists an unstable solution to the linearized stratified viscoelastic Rayleigh–Taylor problem with a largest growth rate. Moreover, the largest growth rate decreases from a positive constant to 0 as κ increases from 0 to . In addition, we further extend the obtained results in the linearized stratified viscoelastic Rayleigh–Taylor problem to the linearized stratified magnetic Rayleigh–Taylor problem.

On classical solutions of Rayleigh–Taylor instability in inhomogeneous viscoelastic fluids

“…for some escape time T δ := 1 Λ ln 2 m 0 δ ∈ (0, T), where T denotes an existence time of the solution (η, u). Moreover, the initial data (η 0 , u 0 ) ∈ H 4 0 × H 3 0 satisfies the keeping volume condition "det(∇η 0 + I) = 1 in Ω" and necessary compatibility conditions:…”

“…By virtue of O(ξ ) dy = 0 and the existence theory of Stokes problem, we can derive that there exists a solution (η, ) ∈ H 4 0 × H 3 of (4.8). In addition, it holds that…”

“…To mathematically prove the viscoelastic RT instability, we choose a density profileρ := ρ(x 3 ) satisfyinḡ ρ ∈ C 4…”

On effect of surface tension in the Rayleigh–Taylor problem of stratified viscoelastic fluids

“…In the last decades, it has been also widely investigated how the RT instability evolves under the effects of other physical factors, such as elasticity [4,11,27,29,42,46], rotation [3,6,45], internal surface tension [9,16,24,48], magnetic fields [5, 17, 19-23, 25, 26], and so on. We also refer to the other related mathematical problems [1, 12, 13, 30, 32-34, 36, 38, 41].…”

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.…”