Page 24 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 9

The definition of streamlines (tangency condition) implies that the
streamlines should be the integral curves of the differential system

dx dz2 dx3
= = —— (in Cartesian coordinates)
Vy v2 V3

or

dz! dz* dz?
—— = => (tn curvilinear coordinates),
vi v2 v3

where the time t, which appears explicitly in v;(x,,t) or v*(x*, t), has to
be considered as a parameter with a fixed value.
At every fixed moment, the set of the streamlines constitutes the mo-
tion pattern (spectrum). These motion patterns are different at different
times.

When the motion is steady, the motion spectrum (pattern) is fixed
in time and the pathlines and streamlines are the same, the definable
differential system becoming identical. This coincidence could be real-
ized even for an unsteady motion provided that the restrictive condition

v x gy = 0 is fulfilled. This result can be got, for instance, from the
so-called Helmholtz—Zorawski’ criterion which states that a necessary
and sufficient condition for the lines of a vectorial field e(r, t) to become
material curves (i.e., locus of material points) is

de
c x + rot(cx v) +vdive| =0,
Ot

Identifying c = vwe get the necessary and sufficient condition that

the lines of the v field (1.e., the streamlines) become material curves (i.e.,
trajectories), precisely v x ov = 0.
A stream tube is a particular streamsurface made by streamlines drawn
from every point of a simple closed curve. A stream tube of infinitesimal
cross section is called a stream filament.

1.2.3 Vortex Lines and Vortex Surfaces

By curl or vorticity or rotation we understand the vector w= V xv =
rotv. The rationale for such a definition is the fact that, at every point
of the continuum motion, the particles rotate about an instantaneous

axis and the vector w has the direction of this axis, the value of the
; ; 1
rotation being also 5:

+See [33]