1916]
HORIZONTAL LAYER OF FLUID
435
except in so far as they modify the action of gravity. Of course, such neglect can be justified only under certain conditions, which Boussinesq has discussed. They are not so restrictive as to exclude the approximate treatment of many problems of interest.
When the fluid is inviscid and the higher temperature is below, all modes of disturbance are instable, even when we include the conduction of heat during the disturbance. But there is one class of disturbances for which the instability is a maximum.
When viscosity is included as well as conduction, the problem is more complicated, and we have to consider boundary conditions. Those have been chosen which are simplest from the mathematical point of view, and they deviate from those obtaining in Benard's experiments, where, indeed, the conditions are different at the two boundaries. It appears, a little unexpectedly, that the equilibrium may be thoroughly stable (with higher temperature below), if the coefficients of conductivity and viscosity are not too small. As the temperature gradient increases, instability enters, and at ' first only for a particular kind of disturbance.
The second phase of Benard, where a tendency reveals itself for a slow transformation into regular hexagons, is not touched. It would seem to demand the inclusion of the squares of quantities here treated as small. But the size of the hexagons (under the boundary conditions postulated) is determinate, at any rate when they assert themselves early enough.
A.n appendix deals with a related analytical problem having various physical interpretations, such as the symmetrical vibration in two dimensions of a layer of air enclosed by a nearly circular wall.
The general Eulerian equations of fluid motion are in the usual notation :—
Du _ Y 1 dp Dv _ v _\dp Dw __ 7 _ 1 dp ^n
Dt~' " ~~
where
p dx' 'Dt ~ pdy' ~Dt-" P dz^ ......
D d .-£ + v-^+i d
Dt~ dt + l dx 'V dy dz'
(2)
and X, Y, Z are the components of extraneous force reckoned per unit of mass. If, neglecting viscosity, we suppose that gravity is the only impressed force,
, Z = 0, F=0, Z = -g, .....................(3)
z being measured upwards. In equations (1) p is variable in consequence of variable temperature and variable pressure. But, as Boussinesq* h'as shown, in the class of problems under consideration the influence of pressure is
* TMorie Analytique de la Chuleur, t. n. p. 172 (1903).
28—2e free vibration of an infinite stretched membrane vibrating with given frequency.