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.