Stability of algebraically unstable dispersive flows

Abstract

A largely unexplored type of hydrodynamic instability is examined: long-time algebraic growth. Such growth is possible when the dispersion relation extracted from classical stability analysis indicates neutral stability. A physically motivated class of partial differential equations that describes the response of a system to disturbances is examined. Specifically, the propagation characteristics of the response are examined in the context of spatiotemporal stability theory. Morphological differences are identified between system responses that exhibit algebraic growth and the more typical case of exponential growth. One key attribute of predicted algebraically growing solutions is the prevalence of transient growth in almost all of the response, with the long-time growth occurring asymptotically at precisely one wave speed.

Instability of a thin viscous film flowing under an inclined substrate: steady patterns

“…We can make an analogy with a spatio-temporal analysis, where the front is defined by a particular ray x/t = v for which the perturbation is marginally stable, as t → ∞ (Huerre & Monkewitz 1990;Van Saarloos 2003;King et al 2016). In this approach, the front is the velocity, i.e.…”

“…the front velocity c g0 , we look for the velocity on rays where the spatiotemporal growth rate is equal to zero, i.e. (Van Saarloos 2003;King et al 2016) This calculation was done in Duprat et al (2007) and Brun et al (2015) and was applied in Limat et al (1992) to compute the front velocity of the perturbation in the horizontal case θ = π/2.…”

On the two-dimensional extension of one-dimensional algebraically growing waves at neutral stability

“…However, at the boundary between the two regimes -Eq. (3b)-, the method fails to accurately predict the long-time linear stability of the system, as shown in the previous work by King et al [12] and Huber et al [13]. Specifically, King et al examined a one-dimensional (1D) operator (henceforth referred to as 1D-KRK) 1 that governs the response to varicose perturbations in a curtain flow, and Huber et al studied a 1D operator (henceforth referred to as 1D-CMH) 1 that enabled the neutral stability threshold in Eq.…”

“…In Eq. (4a), c is the underlying convective fluid flow vector with components c x and c z and B is a real valued parameter that is related to the Weber number as described in [12]. Note that the forcing function in Eq.…”

“…Section 3 similarly examines the 2D-CMH operator. A comparison between the 2D responses in Sections 2 and 3 and their 1D counterparts in [12] and [13] is provided in Section 4, and concluding remarks are provided in Section 5.…”