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# On the Place of Energy Methods in a Global Theory of Hydrodynamic Stability

• D. D. Joseph
Conference paper
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

## Abstract

The point of departure for the global theory to be described is the system of the nonlinear Boussinesq equations (1, 2) governing the disturbance of some given motion. For simplicity, let (U, T, T) be a basic steady velocity, temperature and concentration (say, salt) field which satisfies the Boussinesq equations in a bounded domain (or in a period cell) and such that these fields take on prescribed values on ∂V (or periodic values on appropriate parts of ∂V ). Let (u,0 6, y) be disturbances of the given motion which are induced at time zero as initial conditions. Subsequently [30]
$$\left( {u,\,\theta ,\,\gamma } \right)/\partial V = 0,\,\,\,\,\,div\,u = 0/V,$$
(1a,b)
$$\frac{{du}}{{dt}} + R\left( {u\cdot\nabla } \right)U + \left( {u\cdot\nabla } \right)u = - \nabla P - \left( {R\theta - \ell \gamma } \right)n + \Delta u,$$
(2a)
$${P_T}\frac{{d\theta }}{{dt}} + {P_T}u \cdot \nabla \theta + Ru \cdot {\eta _T} = \Delta \theta ,$$
(2b)
and
$${P_T}\frac{{d\gamma }}{{dt}} + {P_\Gamma }u \cdot \nabla \gamma + \ell u \cdot {\eta _T} = \Delta \gamma ,$$
(2c)
where
$$\frac{d}{{dt}} = \frac{\partial }{{\partial t}} + RU \cdot \nabla .$$

## Keywords

Global Stability Couette Flow Energy Method Fluid Layer Poiseuille Flow
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## Copyright information

© Springer-Verlag, Berlin/Heidelberg 1971

## Authors and Affiliations

• D. D. Joseph
• 1
1. 1.Imperial College of Science and TechnologyLondonUK