Non-Euclidean Adventures on the Lenart Sphere: Activities by Istvan Lenart

Show description

Div v = 0; otherwise div v does not, in general, vanish. If the motion is irrotational curl v = 0. The above analysis is the justification for the names divergence and curl (or, as used by some authors, the rotation) of a vector field, and indicates how the distinction between translation and rotation of a rigid body appears in fluid motion as a stretching and a rotation. 3. Flow, circulation and vorticity If we consider a uniform flow of fluid and draw lines in that fluid, we can say that there is no "flow" of fluid along any line which is perpendicular to the velocity, and that the maximum flow occurs along a line parallel to the velocity.

1/V2. 2. Multiply connected regions If liquid moving irrotationally is enclosed in the volume be­ tween two coaxial cylinders closed at the ends by planes perpen­ dicular to the axis, Fig. 15, then a motion in which the streamlines are circles concentric with the axis, the magnitude of the velocity 56 ELEMENTARY CLASSICAL HYDRODYNAMICS on a circle of radius r being A/r, is irrotational and satisfies the boundary conditions on all surfaces/br any value of A. Hence there exists an infinity of possible irrotational motions.

There are three directions, those of the principal axes of the quadric, in which the divergence of the fluid is in a direct line through A, and if we choose the coordinate axes in these directions (or rotate them to coincide with these directions) then etj = / a i 0 \0 0 0\ a2 0 01. a3/ The velocities of points situated at distances ll9 /2, / 3 from A on these principal directions are ai/i, a2/2, zh respectively. Consider now a cuboidal element of fluid having ll9 /2, / 3 as its edges and a volume x — /i /2 /3.