The stability of linearized partial differential equations (PDEs) governing disturbance propagation in a one-dimensional medium, such as a perturbed fluid-fluid interface, is traditionally deduced by examining the growth of exponential modes of the form hk = Akei(kx−!t), where hk is a mode associated with the height of the perturbed interface, Ak is its amplitude, k is a real wave number, ! is a complex frequency, x is the spatial direction, and t is time. It is assumed that these modes may be superimposed to construct the solution, h, to any response, and thus, if any of these modes grow in time, the operator admits growing solutions, i.e. unstable. In previous work, King et al. (Stability of algebraically unstable dispersive flows, Phys. Rev. Fluids, 1(073604), 2016) report the breakdown in this stability classification when Im[!] = 0 for all k values. While this result classically indicates that solution responses neither grow nor decay in time, King et al. find, instead, that responses grow algebraically as h ts where s is a positive fractional power. This result is deduced through an asymptotic analysis of the Fourier integral that invokes the continuous superposition of an infinite number of modes. In this work, we explore — through a sequence of three related problems — the response behavior of linear operators that exhibit a similar breakdown in stability classification. First, we study the rectilinear, Newtonian, thin-film flow of a liquid down an inclined plane. In contrast to the structure of King et al, at neutral stability (based on traditional mode characterization discussed above) the operator yields a single mode having Im[!] = 0 for k = 0, and Im[!] < 0 for k > 0. As in King et al., we utilize a Fourier-Laplace approach and conclude that the response amplitude is not constant at neutral stability, and in fact decays in time as t−1/2 — this is yet another breakdown in the traditional classical stability conclusion based on exponential modes. Next, two operators that demonstrate algebraic growth as t1/2 in one dimension (1D) studied by King et al. (Phys. Rev. Fluids, 1, 2016, 073604:1-19) and Huber et al. (IMA J. Appl. Math., 85, 2020, 309-340) are extended to two dimensions (2D) and propagation characteristics are examined. We find that the Spatiotemporal stability increase in dimension leads to a reduction of 1D response magnitude by a factor of t1/2 compared with the 1D cases. Thus, peaks which grew in 1D have a constant height in 2D, and regions which decay (algebraically or exponentially) do so faster in 2D. We also find that both operators admit long-time solutions that are functions of similarity variables, which contract space and time. Lastly, we examine the effect of a local oscillatory forcing on the previously examined operator studied by King et al. — this is referred to as the signaling problem. Whereas signaling has been studied with operators that admit exponential growth and decay, we examine, for the first time, the morphology of the solution response for an operator that admits algebraic growth. As in exponentially growing systems, we find that critical velocities can be determined that observably delineate the response into distinct regions. However, the response is distinctly different from exponentially growing cases, owing to the neutrally stable character of its component modes. The superposition of such modes via Fourier integral analysis reveals a rich structure having broad regions with distinct bounding amplitudes. Overall, the three problems examined provide new insights into the nature of neutral stability and the solution of PDEs under such conditions.
Library of Congress Subject Headings
Differential equations, Partial--Numerical solutions; Stability
Mathematical Modeling (Ph.D)
Department, Program, or Center
School of Mathematical Sciences (COS)
Huber, Colin, "Analysis of growth and decay in neutrally stable flows" (2022). Thesis. Rochester Institute of Technology. Accessed from
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