On the exchange of stability for the subcritical laminar flow

TL;DR

This work investigates exchange of stability for subcritical laminar flows with constant vorticity in a two-dimensional channel. By combining a fixed-domain reformulation of the water-wave problem with a dispersion relation and formal asymptotics, it constructs a branch of Stokes waves emanating from a laminar state and analyzes the sign of the second Fréchet-derivative eigenvalue to determine stability exchange. The authors derive explicit expressions and asymptotics for the second eigenvalue, identify depth regions (including a counter-current domain) where exchange occurs, and formulate a formal stability condition via the Dirichlet–Neumann operator, supported by numerical illustrations. These results link depth, vorticity, and period growth along bifurcation curves, clarifying when subharmonic bifurcations and formal stability hold in rotational flows.

Abstract

We consider steady water waves in two-dimensional channel bounded below by a flat, rigid bottom and above by a free surface. The surface tension is neglected and the water motion is rotational with a constant vorticity $a$. We consider an analytic branch of Stokes waves started from a subcritical laminar flow, where the period is considered as the bifurcation parameter. The first eigenvalue of the Fréchet derivative on this branch is always negative. The main object of our study is the second eigenvalue of the Fréchet derivative at this branch in a neighborhood of the laminar flow. This is a small eigenvalue and the positive sign corresponds to the confirmation of the principle of exchange of stability and the negative sign to its violation. We consider the dependence of the sign on the Bernoulli constant $R$ (or the depth $d$ of the laminar flow) and the value of the constant vorticity $a$. We show that for each $a$ there exist a depth $d_0(a)$ such that the sign of the second eigenvalue is positive for $d<d_0(a)$ and negative for $d>d_0(a)$. If $d_s(a)$ is the depth where a stagnation point appears on the corresponding laminar flow then $d_0(a)<d_s(a)$ for positive $a$ and $d_0(a)>d_s(a)$ for negative $a<-1.01803$. The sign of the second eigenvalue is important in study of subharmonic bifurcations. Another observed property consists of the following: if the sign of the second eigenvalue is positive then the period is increasing along the bifurcation curve in a neighborhood of the bifurcation point, if the sign is negative then the period is decreasing there. We also verify the property of formal stability by a description of the domain in $(a,d)$ variables, where this property holds. Numerical illustrations of these properties are presented in the paper.

Figures (6)

• Figure 1: Positivity of $\mu_2(a,d)$ when the laminar flow $U$ is not unidirectional. Here $d_{\textrm{c}}(a)$ is the critical value of $d$, $d_{\textrm{s}}(a)=\sqrt{2/|a|}$ the depth when the laminar flow has a stagnation point on the surface or at the bottom, and $d_0(a)$ is the root of $\mu_2(a,d_0)=0$. The curves $d_{\textrm{s}}(a)$ and $d_0(a)$ intersect at $a=a_0\approx-1.01803$.
• Figure 2: Stagnation points of the laminar flow $U$: stagnation points are absent in the domain $\Theta$ and they are present in both $\Upsilon_-$ and $\Upsilon_+$.
• Figure 3: Dependence of $\mu_2$ on $d$ in a semilogarithmic scale for fixed $a=-10$, $-3$, $a_0$, $-0.3$, $-0.1$, and $0$ (plots $1,\dots,6$).
• Figure 4: Dependence of $\mu_2$ on $d$ in a semilogarithmic scale for fixed $a=5$, $1.5$, $0.5$, $0.25$, and $0.15$ (plots $1,\dots,5$).
• Figure 5: Dependence of $Y_*(a,d_0(a))$ on $a<a_0\approx-1.01803$. The dotted line shows the limit level $Y_{*,\max}\approx 0.314507...$

Theorems & Definitions (6)

• Lemma 2.1
• proof
• Corollary 2.2
• Remark 3.1
• Proposition 3.2
• proof