On the Stream Lines of the Two Vertical Sections when the Thermal Convective Pattern is a Re_??_ular and Symmetrical Hexa_??_on

Abstract

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 Ψ₁, Ψ₂ and A(x, z) (at z=0) are drawn in figures, where
Ψ₁ is the stream line on the vertical section along the straight line through the origin and perpendicular to each side of the hexagon.