Mechanics of Fluids

A well-known example of kinematic similarity is found in a planetarium.
Here the heavens are reproduced in accordance with a certain length scale
factor, and in copying the motions of the planets a fixed ratio of time intervals
(and hence velocities and accelerations) is used.
When fluid motions are kinematically similar the patterns formed by
streamlines are geometrically similar (at corresponding times). Since the
boundaries consist of streamlines, kinematically similar flows are possible
only past geometrically similar boundaries. This condition, however, is not
sufficient to ensure geometric similarity of the stream-line patterns at a distance
from the boundaries. Geometrically similar boundaries therefore do
not necessarily imply kinematically similar flows.
5.2.3 Dynamic similarity
Dynamic similarity is similarity of forces. If two systems are dynamically
similar then the magnitudes of forces at similarly located points in each system
are in a fixed ratio. Consequently the magnitude ratio of any two forces
in one system must be the same as the magnitude ratio of the corresponding
forces in the other system. In a system involving fluids, forces may be due
to many causes: viscosity, gravitational attraction, differences of pressure,
surface tension, elasticity and so on. For perfect dynamic similarity, therefore,
there are many requirements to be met, and it is usually impossible to
satisfy all of them simultaneously. Fortunately, in many instances, some of
the forces do not apply at all, or have negligible effect, so it becomes possible
to concentrate on the similarity of the most important forces.
The justification for comparing results from one flow system with those for
another is that the behaviour of the fluid is similar in the two systems. As we
have seen, one necessary condition is that the boundaries be geometrically
similar. In addition, however, similarity of forces is necessary because the
direction taken by any fluid particle is determined by the resultant force acting
on it. Consequently, complete similarity of two flows can be achieved
only when corresponding fluid particles in the two flows are acted on by
resultant forces that have the same direction and are in a fixed ratio of
magnitude. Moreover, the same conditions apply to the components of
these resultant forces. The directions of component forces are determined
either by external circumstances (as for gravity forces, for example) or by
the flow pattern itself (as for viscous forces). In dynamically similar flows,
therefore, the force polygons for corresponding individual particles are geometrically
similar, and so the component forces too have the same ratio
of magnitude between the two flows. Dynamic similarity, then, produces
geometric similarity of the flow patterns. It should be noted, however, that
the existence of geometric similarity does not, in general, imply dynamic
similarity.
Before examining dynamic similarity in more detail we may note in passing
that we have not exhausted the list of types of similarity, even of fluids.
For example, some investigations may call for thermal similarity in which
differences of temperature are in fixed ratio between model and prototype.
Types of physical similarity 161