Convection in Fluids
In the present monograph, entirely devoted to “Convection in Fluids”, the purpose is to present a unified rational approach of various convective phenomena in fluids (mainly considered as a thermally perfect gas or an expansible liquid), where the main dr
- PDF / 570,027 Bytes
- 48 Pages / 439.37 x 666.142 pts Page_size
- 7 Downloads / 243 Views
The Bénard (1900, 1901) Convection Problem, Heated from Below
4.1 Introduction In this chapter, we take into account the influence of a deformable free surface and, as a consequence, we revisit the mathematical formulation of the classical problem describing the Bénard instability of a horizontal layer of fluid, heated from below, and bounded by an upper deformable free surface. Because the deformation of the free surface, subject to a temperaturedependent surface tension, is taken into account in the full Bénard convection problem, heated from below, we have not specified this convection as being a ‘thermal convection’. Indeed, in the full starting Bénard problem, heated from below, when we take into account the influence of a deformable free surface, subject to a temperature-dependent surface tension, the fluid being an expansible liquid, it is necessary to take into account, simultaneously, four main effects. Namely: (a) the conduction adverse temperature gradient (Bénard) effect in motionless steady-state conduction temperature, (b) the temperature-dependent surface tension (Marangoni) effect, (c) the heat flux across the upper, free surface (Biot) effect, and (d) the buoyancy (Archimedean–Boussinesq) effect arising from the volume (gravity) forces. Particular attention is also paid to the approximate form of the equation of state for a weakly expansible liquid. In his two pioneering papers [1], Henri Bénard (1900, 1901) considered a very thin layer of fluid, about a millimeter in depth, or less, standing on a levelled metallic plate maintained at a constant temperature. The upper
85
86
The Bénard Convection Problem, Heated from Below
surface was usually free and, being in contact with the air, was at a lower temperature. Bénard experimented with several liquids of differing physical constants. He was particularly interested in the role of viscosity (in fact, liquid with a high viscosity and a low Reynolds number, which is linked with the lubrication approximation). In all cases, Bénard found that when the temperature of the lower plate was gradually increased, at a certain instant the layer became reticulated and revealed its dissection into cells. Concerning the experiments on the onset of (thermal) instability in fluids, see [2, pp. 59–75], and also the more recent book by Koschmieder [3]. More precisely, the Bénard cells are primarily induced (in a very thin layer of expansible liquid) by the temperature-dependent surface tension gradients resulting from temperature variations on the deformable free surface. The corresponding instability phenomenon is usually known as the Bénard– Marangoni (BM) thermocapillary instability. The first scientist whose works enlightened the way to our understanding of surface tension gradient-driven flows was Carlo Marangoni (1865, 1871), in [4], who was known to have lively exchanges with Joseph Plateau (1849, 1873), see [5]. Unfortunately, the Bénard cells phenomenon was confused (over the course of many years) with the well-known Rayleigh–Bénard (RB) buoyancy driven in
Data Loading...
