Abstract
In the preceding chapter, we examined the stability of a wide range of flows, most of them steady solutions to their governing fluid equations. In every case, our approach was to perturb the steady flow in order to define a new problem for the evolution of the perturbations themselves, or in other words the robustness of the steady base flow. We then drew conclusions about the stability properties of the original flow based on the evolution of the perturbations. To make these problems tractable, we assumed that the perturbations were small, neglecting all nonlinear terms, that is, terms involving products of the small perturbation quantities. As a linear instability grows, it will eventually become large enough in magnitude that nonlinear terms can no longer be neglected. In this case, nonlinearity plays several roles, but their main role, and of greatest interest to us, is the ability to saturate the exponential growth.
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Notes
- 1.
The Landau equation first appeared as a model for first-order phase transitions for which u represents an order parameter.
- 2.
In an initial value problem, the amplitudes \(A_{{j}}\) would be determined through projection of the initial condition V′(x, 0) onto the set of eigenvectors \(\hat{\mathbf{V}}_{j}\).
- 3.
It is true that \(\mathbb{L}_{{1}}\psi _{{2}} = 0\) implies that there is a solution of arbitrary amplitude, say, B(X, T), which is of the same form as expressed in (8.30). We are not concerned with these solutions, because they do not influence the evolution equation for A that is determined later. It is common practice to absorb this solution implicitly into the one appearing at the previous order.
- 4.
Because we are dealing here with differential operators, the definition of \(\mathbb{L}^{\dag }\) has to include a choice of a Hilbert space with a well-defined inner product. In the current calculation, the relevant operators are self-adjoint; the reader is referred to references in the Bibliographical Notes for more detailed discussion of adjoint operators.
- 5.
A typical inner product for square integrable functions f, g defined on the interval \([x_{L},x_{R}]\) is \(\langle f,g\rangle \equiv \int _{x_{L}}^{x_{R}}f^{{\ast}}gdx\).
- 6.
A similar equation would have been found had one adopted, instead, rigid boundary conditions throughout.
- 7.
In the literature these equations often appear in the following form: \(\dot{x} =\sigma (x - y)\), Â Â \(\dot{y} =\rho x - y - xz,\) Â Â \(\dot{z} = -\beta z + xy.\)
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Regev, O., Umurhan, O.M., Yecko, P.A. (2016). Weakly Nonlinear Instability. In: Modern Fluid Dynamics for Physics and Astrophysics. Graduate Texts in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3164-4_8
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DOI: https://doi.org/10.1007/978-1-4939-3164-4_8
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