**Analytical Study of Particle Motion in Unsteady Stokes Flows**

**Particle Motion in Unsteady Stokes Flows**

The
general solution to the particle momentum equation for unsteady Stokes flows is
obtained

analytically.
The method used to obtain the solution consists on applying a
fractional-differential

operator
to the first-order, integro-differential equation of motion in order to
transform the original

equation
into a second-order, non-homogeneous equation, and then solving this last
equation by the

method
of variation of parameters. The fractional differential operator consists of a
three-time-scale,

linear
operator that stretches the order of the Riemann-Liouville fractional
derivative associated with

the
history term in the equation of motion. In order to illustrate the application
of the general

solution
to particular background flow fields, the particle velocity is calculated for
three specific flow

configurations.
These flow configurations correspond to the gravitationally induced motion of a

particle
through an otherwise quiescent fluid, the motion of a particle caused by a
background

velocity
field that accelerates linearly in time, and the motion of a particle in a
fluid that undergoes

an
impulsive acceleration. The analytical solutions for these three specific cases
are analyzed and

compared
to other solutions found in the literature.

**Motion of Rigid Particles in Harmonic Stokes Flow**

The
particle momentum equation for harmonic Stokes flows is solved exactly in order
to study the

velocity
response of rigid spherical particles. Fluid-to-particle density ratios varying
from 0.001 to

1000 are
studied to identify the proper scaling of the Virtual Mass, Stokes drag and
History drag

forces.
The Fractional Calculus method used to render the governing
integro-differential equation of

motion
analytically tractable is shown to be useful in determining the scaling of the
above referenced

forces.
These forces scale as (a w )^n, where a is the fluid-to-particle density ratio and
w is the

dimensionless
forcing frequency. The power coefficient 'n' is the order of the derivative of
the velocity

potential
that characterizes the force in question in the particle momentum equation.
This coefficient

is
numerically equal to 0 for the Stokes drag, 1/2 for the History drag, and 1 for
the Virtual Mass

force.
The ratio of the History drag and Virtual Mass forces to the Stokes drag is
independent of the

density
of the particles, depending only on the radius of the particle and on the fluid
characteristics

and
forcing frequency. The Virtual Mass force thus dominates the high-frequency
response of light

particles.
The low-frequency response of heavy particles is dominated by the Stokes drag,
and the

intermediate
frequency response is dominated by the Stokes drag for values of aw less than
0.01,

and by
the Virtual Mass force for values of aw larger than 100. The history drag
contribution is

maximum
when the product aw = 2 and when the amplitude of oscillations are smaller than
the

radius
of the particle.

**Publications**

**Journal
Papers**

C.F.M.
Coimbra, and R.H. Rangel (1998). "General Solution of the Particle
Momentum Equation in

Unsteady
Stokes Flows" - Journal of Fluid Mechanics (370) pp. 53-72.

**Conference
Papers**

C.F.M.
Coimbra and R.H. Rangel (1998). "Fractional Calculus Solution of the
Particle Momentum

Equation
in Stokes Flows" - Proceedings of the 7th Brazilian Thermal Science
Meeting (VII Encit) - Rio

de
Janeiro, Brazil.

C.F.M.
Coimbra and R.H. Rangel (1999). "Spherical Particle Motion in Unsteady
Viscous Flows" -

AIAA
paper 99-1031 - presented at the 37th AIAA Aerospace Sciences Meeting and
Exhibit - Reno -

USA. (Best Paper in
Microgravity Sciences Award - AIAA, 1999).

J.D.
Trolinger, R.H. Rangel, C.F.M. Coimbra, W. Whiterow, and J. Rogers (2000).
"SHIVA: A

Spaceflight
Holography Investigation in a Virtual Apparatus" - presented at the 38th
AIAA Aerospace

Sciences
Meeting and Exhibit - Reno - USA.