By Arnold Ringstad
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Extra resources for Forest Habitats
Formula (33) shows t h a t one may change the order of the ordinary and the general spatial differentiation of the radius-vector r of the γ^θΰ'+γ^άί Yxy=90%yxydt Jix-W FIG. 16. fluid particle. Using the velocity vector V = -dr r - , equation (33) can be written in the form (34) d(ôr) = ÔVdt. dz (35) We denote by ôrv ôr2, ôr3 (Fig. 16) the infinitesimal line elements of fluid parallel to the coordinate axes and joined at the point M. The components of these displacements are respectively àr^àx, 0, 0), <5ra(0, dy, 0), <5r3(0, 0, ôz).
In the motion of a fluid volume element not only the position of the instantaneous axis and the value of the angular velocity of rotation change b u t also the shape of the moving volume. The vector ω can be taken as the angular velocity of the hypothetical solid body which would be formed on the instantaneous solidification of the fluid volume element which is being considered. Condition (10) for the existence of surfaces orthogonal to the streamlines now takes on a clear mechanical significance: the directions of the axes of the instantaneous rotation of the particles must be perpendicular to the directions of the velocity of their translatory component of motion.
Lines such t h a t at any point, at a given moment, the velocity is tangential to them) are 32 MECHANICS OF LIQUIDS AtfD GASES called streamlines. The following simple experiment gives a simple and obvious representation of streamlines. We scatter on the surface of some water in a channel a light and easily visible powder which is insoluble in water. We suppose t h a t the particles of powder are completely carried along by the water as it moves, so t h a t the movements of the particles of water and of powder are identical.