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.
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.
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.