Asymptotic and numerical solution of the linear stability problem of boundary layer of relaxing gas on a plate

AbstractConvection in a porous medium at high Rayleigh number $\mathit{Ra}$ exhibits a striking quasisteady columnar structure with a well-defined and $\mathit{Ra}$-dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ‘heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber $k$ and amplitude $\widehat{A}$ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter $A= \widehat{A}\mathit{Ra}/ k$. A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of $A$. The growth rate of the most unstable mode scales with ${A}^{4/ 9} $ for $A\gg 1$, and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh–Bénard (Rayleigh–Darcy) cell at high $\mathit{Ra}$, and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers $k$ greater than $k\sim {\mathit{Ra}}^{5/ 14} $ as $\mathit{Ra}\rightarrow \infty $. This stability criterion is consistent with hitherto unexplained numerical measurements of $k$ in a Rayleigh–Darcy cell.

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Non-linear stability problem of an open conical sandwich shell under external pressure and compression

International Journal of Non-Linear Mechanics ◽

10.1016/0020-7462(84)90009-x ◽

1984 ◽

Vol 19 (3) ◽

pp. 217-233 ◽

Cited By ~ 5

Author(s):

Roman Struk

Keyword(s):

Linear Stability ◽

Stability Problem ◽

External Pressure ◽

Sandwich Shell ◽

Non Linear ◽

Linear Stability Problem

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Polydisperse streaming instability – II. Methods for solving the linear stability problem

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stab111 ◽

2021 ◽

Vol 502 (2) ◽

pp. 1579-1595 ◽

Cited By ~ 1

Author(s):

Sijme-Jan Paardekooper ◽

Colin P McNally ◽

Francesco Lovascio

Keyword(s):

Linear Stability ◽

Stability Problem ◽

Computational Cost ◽

Complex Polynomial ◽

System Of Integral Equations ◽

Root Finding ◽

Streaming Instability ◽

A New Technique ◽

Linear Stability Problem ◽

Python Package

ABSTRACT Occurring in protoplanetary discs composed of dust and gas, streaming instabilities are a favoured mechanism to drive the formation of planetesimals. The polydispserse streaming instability is a generalization of the streaming instability to a continuum of dust sizes. This second paper in the series provides a more in-depth derivation of the governing equations and presents novel numerical methods for solving the associated linear stability problem. In addition to the direct discretization of the eigenproblem at second order introduced in the previous paper, a new technique based on numerically reducing the system of integral equations to a complex polynomial combined with root finding is found to yield accurate results at much lower computational cost. A related method for counting roots of the dispersion relation inside a contour without locating those roots is also demonstrated. Applications of these methods show they can reproduce and exceed the accuracy of previous results in the literature, and new benchmark results are provided. Implementations of the methods described are made available in an accompanying python package psitools.

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Exact solution of the linear-stability problem for the onset of convection in binary fluid mixtures

Physical Review A ◽

10.1103/physreva.35.4349 ◽

1987 ◽

Vol 35 (10) ◽

pp. 4349-4353 ◽

Cited By ~ 34

Author(s):

Barbara J. A. Zielinska ◽

Helmut R. Brand

Keyword(s):

Exact Solution ◽

Linear Stability ◽

Stability Problem ◽

Onset Of Convection ◽

Binary Fluid ◽

Fluid Mixtures ◽

Binary Fluid Mixtures ◽

Linear Stability Problem

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Linear Stability Analysis

Introduction to Modeling Convection in Planets and Stars ◽

10.23943/princeton/9780691141725.003.0003 ◽

2013 ◽

Author(s):

Gary A. Glatzmaier

Keyword(s):

Stability Analysis ◽

Rayleigh Number ◽

Linear Stability ◽

Linear Stability Analysis ◽

Stability Problem ◽

Linear Codes ◽

Linear Equations ◽

Critical Rayleigh Number ◽

Linear Solution ◽

Linear Stability Problem

This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For this linear analysis, each Fourier mode n can be considered a separate and independent problem. The question that needs to be addressed now is under what conditions—that is, what values of Ra, Prandtl number Pr, and aspect ratio a—will the amplitude of the linear solution grow with time for a given mode n. This is a linear stability problem. The chapter first introduces the linear equations before discussing the linear code and explaining how to find the critical Rayleigh number; in other words, the value of Ra for a and Pr that gives a solution that neither grows nor decays with time. It also shows how the linear stability problem can be solved using an analytic approach.

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