ON CONVECTION CURRENTS IN A HORIZONTAL LAYER OF FLUID, WHEN THE HIGHER TEMPERATURE IS ON THE UNDER SIDE.
[Philosophical Magazine, Vol. xxxn. pp. 529—546, 1916.]
THE present is an attempt to examine how far the interesting results obtained by Benard* in his careful and skilful experiments can be explained theoretically. B6nard worked with very thin layers, only about 1 mm. deep, standing on a levelled metallic plate which was maintained at a uniform temperature. The upper surface was usually free, and being in contact with the air was at a lower temperature. Various liquids were employed—some, indeed, which would be solids under ordinary conditions.
The layer rapidly resolves itself into a number of cells, the motion being an ascension in the middle of a cell and a descension at the common boundary between a cell and its neighbours. Two phases are distinguished, of unequal duration, the first being relatively very short. The limit of the first phase is described as the " semi-regular cellular regime "; in this state all the cells have already acquired surfaces nearly identical, their forms being nearly regular convex polygons of. in general, 4 to 7 sides. The boundaries are vertical, and the circulation in 'each cell approximates to that already indicated. This phase is brief (1 or 2 seconds) for the less viscous liquids (alcohol, benzine, etc.) at ordinary temperatures. Even for paraffin or spermaceti, melted at 100° C., 10 seconds suffice; but in the case of very viscous liquids (oils, etc.), if the flux of heat is small, the deformations are extremely slow and the first phase may last several minutes or more.
The second phase has for its limit a permanent regime of regular hexagons. During this period the cells become equal and regular and align
* Revue generate des Sciences, "Vol. xii. pp. 1261, 1309 (1900); Ann. d. Chimie et de Physique, t. xxiir. p. 62 (1901). M. Be'nard does not appear to be acquainted with James Thomson's paper "On a Changing Tesselated Structure in certain Liquids" (Proc. Glasgow Phil. Soc. 1881—2), where is described a like structure in much thicker layers of soapy water cooling from the surface. x) + i e~sx(l + 3x) + ~e-sx (l + 5a>)+ .......... (10)