Benard convection, heating from

The formation of Benard convection cells takes place as follows if water is heated from below in a vessel, macroscopic convection currents occur under certain conditions seen from above, these have the structure of uniform, honeycombshaped cells. 

BENARD CONVECTION CELLS. When a layer of liquid is heated from below, the onset of convection is marked by the appearance of a regular array of hexagonal cells, the liquid rising in the center and falling near the wall of each cell. The criterion for the appearance of the cells is that the Rayleigh number should exceed 1700 (for rigid boundaries). 

F. NATURAL CONVECTION IN A HORIZONTAL FLUID LAYER HEATED FROM BELOW-THE RAYLEIGH-BENARD PROBLEM... 

In this section, we consider the classic problem of a fluid layer of depth d, with an upper surface that is an interface with air that is maintained at an ambient temperature 7o. The fluid layer is heated from below, and we shall assume that the lower fluid boundary is isothermal with temperature T (> To). This problem sounds exactly like the Rayleigh-Benard problem with a free upper surface. However, we consider the fluid layer to be very thin (i.e., d small) so that the Rayleigh number, which depends on d3, is less than the critical value for this configuration. Nevertheless, as previously suggested, the fluid layer may still undergo a convective motion that is due to Marangoni instability. 

At the turn of the century, Henri Benard, a young French physicist, published the first truly systematic study of natural convection in a horizontal fluid layer (B4, B5, B6). In a horizontal liquid layer heated from below B6nard, sought to measure and to define the most stable steady-state convection currents prevailing under given conditions. He utilized liquid layers only a few millimeters in thickness, initially in an apparatus giving a free upper surface, and of considerable horizontal extent (about 20 cm) so that edge effects could not influence the form of the convection pattern. For these studies. 

The most well-known example of pattern formation is Rayleigh-Benard convection which appears when a fluid layer is uniformly heated from below... 

Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid. When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value A 7. The flow has a stationary cellular character with a spatial periodicity of about 2d. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T in temperature creates warmer and cooler regions, and due lO buoyancy effects the former tends to move upwards and the latter downwards. When AT < AT, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T and the velocity (normal to the layer) vary as exp (i j,y) with x ji/rf, the threshold is given by the dimensionless Rayleigh number... 

Benard cell A structure associated with a layer of liquid that is confined by two horizontal parallel plates, in which the lateral dimensions are much larger than the width of the layer. Before heating the liquid is homogeneous. However, if after heating from below the temperatures of the plates are Tj and T2, at a critical value of the temperature gradient AT=Ti T2 the liquid abruptly starts to convect. The liquid spontaneously... 

Methods from heat transfer. The modeling of Benard convection cells in hydrodynamic stability is relevant to our purposes, since it leads to a modal equation identical to Equation 12-9 (Yih, 1969). It turns out interestingly that Equation 12-9 itself can be solved by separation of variables once more. For example, the choice... 

Benard in the year 1900 observed that hexagonal convection cells are formed within thin film of molten spermaceti of about 0.5-1 mm depth that were heated from below, with the cell spacing somewhat more than three times the liquid depth. These cells are now referred as Benard cells (Figure 5.31). Bdnard initially assumed that surface tension at the free surface of... 

Secondly, heat is transferred towards the surface, within the 0.4 mm mixed con-duction/convection layer, via a very large temperature gradient of the order of 5000-10,000 K/m, by a relatively weak thermal process. With a high thermal impedance, the process consists of a static thermal conductance enhanced about 1.5-2.5 times by penetration of some of the intermittent convection from the Rayleigh/Benard convection below. 

There is another type of bifurcation called Turing bifurcation, which results in a spatial pattern rather than oscillation. A typical example where a new spatial structure emerges from a spatially unique situation is Benard s convection cells. These have been well examined and are formed with increasing heat conduction.53 Prigogine called this type of structure a dissipative structure.54-56... 

One of the best-known physical ordering phenomena is the Benard cells, which occur during the heating a fluid held between two parallel horizontal plates separated by a small distance. The lower plate is heated, and the temperature is controlled. The upper plate is kept at a constant temperature. When the temperature difference between the two plates reaches a certain critical value, the elevating effect of expansion predominates, and the fluid starts to move in a structured way the fluid is divided into horizontal cylindrical convection cells, in which the fluid rotates in a vertical plane. At the lower hot plate, the hot fluid rises later, it is cooled at the upper plate, and its density increases again this induces a movement downward, as seen in Figure 13.2. The Benard cells are one of the best-known physical examples of spontaneous structurization as a result of sufficient distance from equilibrium, which is the large temperature difference between the plates. The critical temperature difference ( A 7 )c can be determined from the... 

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