The unsteady flow field of an incompressible viscous fluid around
an impulsively started cylinder with slow motion is studied in detail. Integral expressions are derived from the nonlinear vorticity equation, and are solved by the method of
matched
asymptotic expansions. To complete the matching process five regions are
necessary
and their regions are essentially governed by the following relations:
(i)
the initial flow is unsteady Stokes flow (I), (ii) the early transient flow near the cylinder is steady Stokes flow (II), but the far-field flow is unsteady Stokes flow (III), so that Stokes&–Oseen-like matching is necessary, and (iii) as time increases the inertia terms become significant far downstream; thus the far flow is unsteady Oseen flow (IV), but the flow near the cylinder is steady Stokes flow (V), so that the matching of the Stokes–Oseen equations is necessary. The asymptotic analytical solutions are given for five flow fields around a circular cylinder. Also
presented are the drag coefficient, the vorticity, and the streamline.
The
drag coefficient is verified quantitatively by
comparing with earlier theories of the initial flow and the steady flow.
The
streamline patterns calculated show the generation of a circulating zone
close
to the circular cylinder just as for the transient flow around a sphere,
and
the difference between two-dimensional and three-dimensional flows
is discussed.