Source-linked AI summary
Linear Rayleigh-Taylor instability for viscous, compressible fluids
Yan Guo, Ian Tice
TL;DR
Viscosity removes the natural variational structure used to construct growing modes for the linearized Rayleigh–Taylor problem. The paper introduces modified variational problems to construct modes and estimates arbitrary solutions using their fastest growth rates. It finds that viscosity and surface tension bound or restrict unstable frequencies, and sufficiently small periodicity eliminates growing modes when surface tension is present.
Problem
With viscosity, λ enters the linearized equations both quadratically and linearly, preventing constrained minimization from directly producing growing-mode solutions.
Source: §2 · p. 10Method
The paper removes the linear dependence on λ through modified viscosities and studies modified variational problems to construct growing modes.
Source: §2 · p. 10Results
With viscosity and surface tension, only a critical interval of spatial frequencies is unstable; with periodicity satisfying (2.13), no growing modes exist.
Source: §2 · p. 12Takeaways & Limitations
For arbitrary solutions, the paper estimates growth in terms of the fastest growing-mode rates in the non-periodic case and in the periodic case when L satisfies (2.12).
Source: §2 · p. 14The stated H^k(Ω) space is piecewise Sobolev, not the usual Sobolev space of order k.
Source: §2 · p. 12
Abstract
from arXiv · showhide
We study the equations obtained from linearizing the compressible Navier-Stokes equations around a steady-state profile with a heavier fluid lying above a lighter fluid along a planar interface, i.e. a Rayleigh-Taylor instability. We consider the equations with or without surface tension, with the viscosity allowed to depend on the density, and in both periodic and non-periodic settings. In the presence of viscosity there is no natural variational framework for constructing growing mode solutions to the linearized problem. We develop a general method of studying a family of modified variational problems in order to produce maximal growing modes. Using these growing modes, we construct smooth (when restricted to each fluid domain) solutions to the linear equations that grow exponentially in time in Sobolev spaces. We then prove an estimate for arbitrary solutions to the linearized equations in terms of the fastest possible growth rate for the growing modes. In the periodic setting, we show that sufficiently small periodicity avoids instability in the presence of surface tension.
1 Formulation of the problem
The paper formulates a two-fluid compressible Navier–Stokes free-boundary problem and recasts it on fixed domains. It constructs a hydrostatic Rayleigh–Taylor state, linearizes about it, and reduces growing-mode solutions to a frequency-dependent system.
1.1 Formulation in Eulerian coordinates
- The model couples two immiscible viscous, compressible fluids across a moving interface through Navier–Stokes evolution, velocity continuity, and a normal-stress jump proportional to curvature.The fluids occupy an infinite slab, with no-slip conditions at its fixed upper and lower boundaries.Source: §1.1 · pp. 1–2
- Viscosities may depend smoothly on density, while shear viscosity is positive, bulk viscosity is nonnegative, and the pressure laws are smooth and strictly increasing.The Rayleigh–Taylor setup requires distinct pressure laws and a steady state denser above the interface.Source: §1.1 · p. 2
1.2 Reformulation in Lagrangian coordinates
- Lagrangian coordinates fix the fluid domains and interface in time, replacing the moving-boundary formulation with equations on reference domains.The mapping must be continuous across the interface and invertible, or invertible on its image in the periodic case.Source: §1.2 · pp. 3–4
1.3 Steady-state solution
- A steady state has zero velocity, identity flow map, and density depending only on height, with the hydrostatic profile chosen to satisfy the pressure jump and heavier-upper-fluid condition.The resulting density is positive, bounded above and below, and smooth within each fluid domain.Source: §1.3 · pp. 5–6
1.4 Linearization around the steady-state
- The equations are linearized around the hydrostatic state, with the linearized boundary conditions and compatible initial data imposed on the fixed domains.The density and viscosity coefficients in the linear system are evaluated at the steady-state profile.Source: §1.4 · pp. 6–7
1.5 Growing mode ansatz
- A normal-mode ansatz combines exponential time growth with horizontal Fourier modes, reducing the linear problem to ODEs in the vertical coordinate for each frequency.In the periodic case, admissible horizontal frequencies lie in L^-1 Z × L^-1 Z.Source: §1.5 · pp. 7–8
- Rotational symmetry permits aligning each horizontal frequency with one coordinate, after which the transverse component vanishes for a growing mode with λ > 0.The remaining equations govern the horizontal and vertical components, subject to boundary and interface jump conditions.Source: §1.5 · p. 9
2 Main results and discussion
The paper restores a variational route to growing modes despite viscosity, then constructs exponentially growing solutions and bounds arbitrary solutions by the fastest mode rates. Surface tension and periodicity restrict instability, with sufficiently small periodicity ruling out growing modes.
Modified variational method
- Modified viscosities remove λ's linear appearance from the viscous problem, restoring a variational structure for constrained minimization.A solution of the modified problem with λ=s also solves the original problem; the method produces λ(|ξ|,s)>0 for sufficiently small s under the stated condition.Source: §2 · p. 10
Growing modes and frequency limits
- For 0<|ξ|<|ξ|c, growing modes exist; without surface tension every nonzero frequency is allowed, while surface tension excludes |ξ|≥|ξ|c.In the periodic case the available growing frequencies are finite, and the method constructs none when L satisfies (2.13).Source: §2 · p. 11
Growth-rate behavior
- The largest rate is attained at a frequency in both settings; the periodic maximum ΛL is generally below the non-periodic maximum Λ.The paper establishes continuity of λ over the unstable frequency interval and defines the maxima over the respective frequency sets.Source: §2 · p. 11
- Viscosity bounds λ and makes it decay at high frequencies without surface tension; with surface tension, only a critical frequency interval is unstable and λ remains bounded.With viscosity and surface tension, periodicity satisfying (2.13) eliminates growing modes.Source: §2 · p. 12
Growing solutions
- The constructed modes yield smooth-on-each-fluid-domain solutions with exact e^(tΛL) Sobolev growth in the periodic case.The stated Hk space is piecewise Sobolev, not the usual Sobolev space.Source: §2 · p. 12
- In the non-periodic case, Fourier synthesis produces L2 solutions whose growth can approach, but need not equal, e^(tΛ).A single horizontal exponential mode is not in L2(Ω).Source: §2 · p. 12
Bounds for arbitrary solutions
- For arbitrary solutions, the paper proves growth estimates in terms of Λ or ΛL, the rates of the fastest growing modes.The periodic estimate uses ΛL and integrates over the periodic horizontal domain.Source: §2 · p. 14
Stability regime
- When surface tension is present and periodicity satisfies (2.13), exponential growth is impossible and a stability estimate is available.The paper describes this setting as one in which its method constructs no growing modes.Source: §2 · p. 14
3 A family of modified variational problems
This section develops modified variational problems whose constrained minima yield smooth solutions and determine a unique viscosity parameter for the original problem. The resulting growth rates vary continuously with horizontal frequency, vanish at endpoint frequencies, and provide modes used to construct growing solutions in periodic and non-periodic settings.
3.1 Solutions to (1.44)–(1.45) via constrained minimization
- The energy attains its infimum on the constraint set, and its minimizer is smooth on each fluid interval and satisfies the modified equations with jump and boundary conditions.The Euler–Lagrange equations for the constrained minimizer yield the modified problem’s equations.Source: §3.1 · pp. 16–17
- When 0 < |ξ|² < g Jρ₀K/σ, the constrained energy minimum μ(s) is negative for all sufficiently small s.The threshold and smallness bound depend on the listed fluid, physical, and domain parameters.Source: §3.1 · p. 18
- The minimum μ(s) is continuous and strictly increasing in s, enabling the unique parameter choice that matches s to the positive growth rate.On the negative-energy range, the growth rate is λ = √−μ.Source: §3.1 · pp. 20, 22
- For each 0 < |ξ|² < g Jρ₀K/σ, a unique parameter choice s = λ(|ξ|, s) yields a positive-growth solution to the original problem.The corresponding modified-problem solutions are smooth on each fluid interval and have ψ(0) ≠ 0.Source: §3.1 · pp. 22–23
3.2 Behavior of the solutions with respect to ξ
- The growth rate λ is bounded and continuous as a function of |ξ| on (0, |ξ|c).Here |ξ|c = √(g Jρ₀K/σ).Source: §3.2 · p. 23
- The growth rate λ(|ξ|) tends to zero as |ξ| approaches 0 and |ξ|c; at zero, the bound 0 ≤ λ²(|ξ|) ≤ g|ξ| establishes this limit.Chaining the cited inequalities establishes the limit at |ξ|c.Source: §3.2 · p. 24
- The supremum of λ is attained in both settings, denoted Λ in the non-periodic case and ΛL in the periodic case.Source: §3.2 · p. 24
- For |ξ| in any compact interval [a,b] inside (0, |ξ|c), the constructed modes have H^k bounds for every k ≥ 0.The bounds depend on a, b, and the fluid parameters.Source: §3.2 · pp. 24–25
3.3 Solutions to (1.35)–(1.36)
- Periodic normal modes give real-valued solutions in H^k(Ω) initially, with the stated exponential growth in time.The construction applies in the periodic case when L satisfies (2.12).Source: §3.3 · p. 26
- In the non-periodic case, Fourier synthesis yields real-valued solutions smooth on each fluid domain, with growth rates arbitrarily close to e^Λt.The synthesis uses frequencies supported compactly inside (0, |ξ|c).Source: §3.3 · p. 26
4 Growth of solutions to the linearized problem
The section develops energy estimates for solutions of the second-order formulation, then uses them to bound arbitrary linearized solutions in terms of the largest growing-mode rates. In the periodic case, the estimates distinguish the general setting from sufficiently small periodicity when surface tension is present.
4.2 Proof of Theorems 2.5 and 2.6
- Bounds for arbitrary solutions are expressed in terms of the largest growing-mode rates Λ in the non-periodic case and ΛL in the periodic case.The estimates apply to real-valued solutions satisfying the jump and boundary conditions; complex solutions can be treated by their real and imaginary parts.Source: §4.1 · p. 27
4.1 Preliminary estimates
- The second-order formulation yields an energy evolution equation for solutions, with the same identity in the periodic setting after changing the integration domain.The periodic identity replaces the integral over R2 with one over (2πLT)2.Source: §4.1 · p. 27
- An energy inequality relates the energy to Λ, and its periodic counterpart holds when σ/(g Jρ0K) < L.Under that condition, the inequality uses ΛL and an integral over (2πLT)2.Source: §4.1 · pp. 28–29
- When σ > 0 and L is sufficiently small, a better result is available in the periodic case.When L satisfies (2.13), Lemma 4.3 may be used to improve the above result in the periodic case.Source: §4.1 · p. 30, §4.2 · p. 32