Life as a manifestation of the second law of thermodynamics

A number of apparently disparate problems from engineering, meteorology, geophysics, fluid mechanics and applied mathematics are considered under the unifying heading of natural convection. After a review of the mathematical framework that serves to delineate these problems, the heuristic approach to Benard and Rayleigh convection is discussed with special attention to buoyancy and surface tension. Then consideration is given to some aspects of scaling, and the nondimensionalization of equations to a given problem. The thermohydrodynamic description of a Newtonian fluid is presented, and the Boussinesq--Oberbeck model. This is followed by a treatment of the linear stability problem, and a description of the basic ideas of Landau and Hopf concerning the bifurcation of secondary solutions. Quantitative, though approximate, estimations are given for quantities belonging to the nonlinear steady convective regime: flow velocity and temperature distribution. Higher-order, though steady, bifurcations are discussed, as well as the transition to turbulence, along with such time-dependent phenomena as relaxation oscillations. The paper concludes with an Appendix showing a simple application of the Leray--Schauder topological degree of a mapping.

Convective instability: A physicist's approach

A survey is given of different aspects of fluid motion induced by the Coulomb force exerted by the electric field on an injected space charge. The instability problem in highly symmetrical electrode configurations is considered with positive coupling between velocity and charge perturbations, linear and nonlinear criteria and the phenomena during the transient regime of unipolar injection. Then the features of electroconvection are described for both strong and weak injection and also in the case of coaxial cylinders. The last part deals with the liquid motion induced by injection from very restricted areas, as occurs in configurations of low symmetry with particular attention devoted to charge plumes.

Electrohydrodynamic instability and motion induced by injected space charge in insulating liquids

The Coulomb-driven EHD instabilities, chaos, and turbulence generated by unipolar injection in insulating liquids are reviewed. Only systems of the greatest simplicity, well suited for a fundamental study, are considered. As the paper is intended to be introductory, the emphasis is on the physical mechanisms responsible for EHD phenomena. >

Coulomb-driven convection in electrohydrodynamics

The functioning of an electrostatic precipitator in the light of previous studies on various regimes of electroconvection in both parallel and divergent electric fields is examined. Coupling between velocity and charge density fluctuations for both ions and charged particles is discussed. It is shown that for a certain diameter range of the particles, their nondimensional mobility parameter takes values similar to those characterizing ions in liquids. An experimental simulation using insulating liquids is proposed.

The Electrohydrodynamic Origin of Turbulence in Electrostatic Precipitators

The problem of electric charge convection in a dielectric liquid layer of high ionic purity, when subjected to unipolar injection, is in many ways analogous to that of thermal convection in a horizontal fluid layer heated from below, although no formal analogy can be established. The problem treated is intrinsically more nonlinear than the thermal problem. We consider two asymptotic states of convection: one where the whole motion is dominated by viscosity, and one where inertial effects dominate. In each state, two or three spatial regions are distinguished. From the approximate equations that hold in the different regions, information about the variation of the different quantities with distance from the injector is obtained, and further approximations permit us to establish the dependence of the current density ratio I/I0 (called the electric Nusselt number) on the stability parameter T = M2R = eϕ0/Kρν, and on 1/R = ν/Kϕ0, which is an equivalent Prandtl number (e is the permittivity, ρ the fluid density, K the mobility, ν the kinematic viscosity, and ϕ0 the applied voltage). In the viscous state, the analysis gives I/I0 ∞ T½; in the inertial state the law I/I0 ∞ (T/R)1/4 = M½ is obtained. Since M is independent of the applied voltage, the latter law shows the saturation in the electric Nusselt number observed in earlier experiments. The transition in the states is associated with a transition number (MR)T [gap ] 30, which is an electric Reynolds number, related to an ordinary Reynolds number of about 10.The experimental results, obtained in liquids of very different viscosities and dielectric constants, verify these theoretical predictions; further, they yield more precise numerical coefficients. As for the transition criteria, the experiments confirm that the viscous and inertial effects are of the same order when Re [gap ] 10. It was also possible to determine roughly the limits of the viscous and inertial states. The viscous analysis remains valid up to a Reynolds number of about 1; the inertial state can be considered valid down to a Reynolds number of 60. Schlieren observations show that the motion has the structure of very stable hexagonal cells at applied voltages just above the critical voltage, which are transformed into unstable filaments when the voltage is increased further. At even higher voltages, the motion finally breaks down into turbulence. It may be of interest to point out that, when M < 3, the electric Nusselt number approaches 1, which is equivalent to the situation in thermal convection at low Prandtl numbers.

J. C. Lacroix

Papers

Electro-convection in a dielectric liquid layer subjected to unipolar injection