Longwave Instabilities and Patterns in Fluids

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction

   Sergey Shklyaev, Alexander Nepomnyashchy

   Abstract

   In the last decades, the spontaneous formation of spatially nonhomogeneous patterns was an object of extensive investigations.

3. Chapter 2. Convection in Cylindrical Cavities

   Sergey Shklyaev, Alexander Nepomnyashchy

   Abstract

   The thermal convection in a viscous fluid heated from below, which provides a classical example of the pattern formation in a nonequilibrium system, is a subject of several monographs [1‐6]. In contrast to the cited books, we focus on the specific features of large-scale convective motions produced by long-wavelength instabilities of the conductive state. There are two main physical origins of convection instabilities: the buoyancy force due to an inhomogeneous density distribution in a gravity field (buoyancy or Rayleigh convection) and the thermocapillary effect caused by the dependence of the surface tension on temperature (surface tension driven or Marangoni convection).

4. Chapter 3. Convection in Liquid Layers

   Sergey Shklyaev, Alexander Nepomnyashchy

   Abstract

   In the previous chapter, we considered convection in cavities with a fixed boundary temperature distribution, which gives an example of instability “without a conservation law” (type III): a spatially homogeneous disturbance with the wavenumber k = 0 has a nonzero growth rate. In the present chapter, we present examples of systems “with a conservation law” (type II instabilities).

5. Chapter 4. Convection in Binary Liquids: Amplitude Equations for Stationary and Oscillatory Patterns

   Sergey Shklyaev, Alexander Nepomnyashchy

   Abstract

   While the Rayleigh–Bénard convection in a pure liquid serves as a paradigmatic example of stationary patterns generated by monotonic instability, convection in a binary liquid provides a basic example of an oscillatory instability generating wave patterns.

6. Chapter 5. Instabilities of Parallel Flows

   Sergey Shklyaev, Alexander Nepomnyashchy

   Abstract

   The stability theory of viscous parallel flows is a traditional part of the fluid mechanics that has a long history (see [1‐4]). As well as the Rayleigh–Benard problem, the abovementioned problem, which has important applications like laminar-turbulent transition in channel flows and boundary layers, is a touchstone for different approaches of the nonlinear science. The nonlinear evolution of disturbances on the background of a parallel flow in a channel was the first physical problem that was considered by means of the complex Ginzburg–Landau equation [5, 6]. The most obvious difference between the problems connected with parallel flows and the convection problems is the anisotropy and (as a rule) lack of the reflection symmetry characteristic for the former problem. The typical instability of a parallel flow, predicted by the linear theory, is the non-degenerated oscillatory instability rather than a stationary instability or twofold degenerated oscillatory instability of convective problems.

7. Chapter 6. Instabilities of Fronts

   Sergey Shklyaev, Alexander Nepomnyashchy

   Abstract

   In the previous chapters, we considered problems where patterns appeared due to an instability of a motionless state or a steady flow. In both cases, the base state of the system was characterized by a set of time-independent fields of variables u(r). In the present chapter, we consider instabilities of moving fronts described by traveling waves u(r −v t).

8. Chapter 7. Longwave Modulations of Shortwave Patterns

   Sergey Shklyaev, Alexander Nepomnyashchy

   Abstract

   The description of longwave instabilities would be incomplete without mentioning longwave instabilities of shortwave spatially periodic patterns. The investigation of that kind of instabilities has a long history [1‐3], and it is described in the literature in detail [4‐9].

9. Chapter 8. Control of Longwave Instabilities

   Sergey Shklyaev, Alexander Nepomnyashchy

   Abstract

   In the previous chapters, we have considered longwave instabilities that appear and develop in “a natural way.” The applications need, however, controlling the instabilities. In some cases, the instabilities should be eliminated, and in other cases, some definite patterns should be selected and controlled.

10. Chapter 9. Outlook

    Sergey Shklyaev, Alexander Nepomnyashchy

    Abstract

    In conclusion, we present an outlook of some important directions of the investigation that are still on the stage of development.

11. Backmatter