Mechanics of Fluids

polygon. Now a polygon can be completely specified by the magnitude and
direction of all the sides except one. The remaining, unspecified, side is
fixed automatically by the condition that it must just close the polygon.
Consequently, if for any particular particle this hypothetical inertia force is
specified, one of the other forces may remain unspecified; it is fixed by the
condition that the force polygon must be completely closed, in other words,
that the addition of the inertia force would give zero resultant force.
For dynamic similarity between two systems, the forces on any fluid
particle in one system must bear the same ratios of magnitude to one another
as the forces on the corresponding particle in the other system. In most cases
several ratios of pairs of forces could be selected for consideration. But,
because the accelerations of particles (and hence the inertia forces) play an
important part in practically every problem of fluid motion, it has become
conventional to select for consideration the ratios between the magnitude of
the inertia force and that of each of the other forces in turn. For example,
in a problem such as that studied in Section 1.9.2, the only relevant forces
are inertia forces, viscous forces and the forces due to differences of pressure.
The ratio chosen for consideration in this instance is that of |Inertia
force| to |Net viscous force| and this ratio must be the same for corresponding
particles in the two systems if dynamic similarity between the systems is
to be realized. (There is no need to consider separately the ratio of |Inertia
force| to |Pressure force| since, once the inertia force and net viscous force are
fixed, the pressure force is determined automatically by the condition that
the resultant of all three must be zero.) In a case where the forces involved
are weight, pressure force and inertia force, the ratio chosen is |Inertia force|
to |Weight|.
We shall now consider the various force ratios in turn.
5.3.1 Dynamic similarity of flow with viscous forces acting
There are many instances of flow that is affected only by viscous, pressure
and inertia forces. If the fluid is in a full, completely closed conduit, gravity
cannot affect the flow pattern; surface tension has no effect since there is no
free surface, and if the velocity is well below the speed of sound in the fluid
the compressibility is of no consequence. These conditions are met also in the
flow of air past a low-speed aircraft and the flow of water past a submarine
deeply enough submerged to produce no waves on the surface.
Now for dynamic similarity between two systems, the magnitude ratio of
any two forces must be the same at corresponding points of the two systems
(and, if the flow is unsteady, at corresponding times also). There are three
possible pairs of forces of different kinds but, by convention, the ratio of
|Inertia force| to |Net viscous force| is chosen to be the same in each case.
The inertia force acting on a particle of fluid is equal in magnitude to
the mass of the particle multiplied by its acceleration. The mass is equal to
the density ϱ times the volume (and the latter may be taken as proportional
to the cube of some length l which is characteristic of the geometry of the
system). The mass, then, is proportional to ϱl 3 . The acceleration of the
particle is the rate at which its velocity in that direction changes with time and
Ratios of forces arising in dynamic similarity 163