As an application of the study of Pellew and Southwell on maintained convective motion in a fluid heated from below, the author makes an attempt to draw the stream lines of the vertical sections of a regular hexagon, which is of great interest in relation to our thermal problem, because it appears from Bénard's experiments that hexagonal cells characterize the permanent regime in a layer of unlimited extent.
If the midpoint of a layer is taken as origin and the directions of the axes of the cartesian coordinates
x,
y,
z are drawn so that the
x-axis is perpendicular to the side of the hexagon, the
y axis coincides with the diagonal and the
z axis is vertical, and if
u,
v w denote the component velocities, the function Ψ which decides the direction of the stream line is not an ordinary stream function, that is, denotes, for example,
w=- ∂Ψ/∂
x,
u=∂Ψ/∂
z, because
u=0 but ∂
v/∂
y_??_0
Therefore, we do not get Ψ from the differential equation
w/
dz=
u/
dx as usual, but put
w and
u in the following form
w=- ∂Ψ/∂
x A(
x,
z),
u=- ∂Ψ/∂
z A(
x,
z) and then from the equation of continuity A(
x,
z) can be decided. As a result A(
x,
z) has a physical meaning of inversely proportional to the width of the stream tube and Ψ denotes an ordinary stream function when
w',
u' are adopted instead of
w,
u (where
w'=
w/
A,
u'=
u/
A).
Finally Ψ
1, Ψ
2 and A(
x,
z) (at
z=0) are drawn in figures, where
Ψ
1 is the stream line on the vertical section along the straight line through the origin and perpendicular to each side of the hexagon.
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