The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number

The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a^2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t^-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t^-2

The force on a sphere in a uniform flow with small-amplitude oscillations at finite Reynolds number

The unsteady force acting on a sphere that is held fixed in a steady uniform flow with small-amplitude oscillations is evaluated to O(Re) for small Reynolds number Re. Good agreement is shown with the numerical results of Mei, Lawrence & Adrian (1991) up to Re [approximate] 0.5. The analytical result is transformed by Fourier inversion to allow for an arbitrary time-dependent motion which is small relative to the steady uniform flow. This yields a history-dependent force which has an integration kernel that decays exponentially for large time

The force on a bubble, drop, or particle in arbitrary time-dependent motion at small Reynolds number

The hydrodynamic force on a body that undergoes translational acceleration in an unbounded fluid at low Reynolds number is considered. The results extend the prior analysis of Lovalenti and Brady [to appear in J. Fluid Mech. (1993)] for rigid particles to drops and bubbles. Similar behavior is shown in that, with the inclusion of convective inertia, the long-time temporal decay of the force (or the approach to steady state) at finite Reynolds number is faster than the t-1/2 predicted by the unsteady Stokes equations

Three-dimensional finite element simulation of viscoelastic fluid flow using the EVSS-G

An implementation of the EVSS-G method in a finite element framework is presented for modeling three-dimensional viscoelastic flows. The extension from two to three dimensions is discussed, along with the use of the Picard and Newton-Raphson iterative techniques. A selection of benchmark problems, with known analytical solutions are presented as validation of the implementation as well as results for secondary flows in rectangular ducts. 7 refs

Inertial effects on particle dynamics

While the theory of suspension flows and particle dynamics is well understood under Stokes flow conditions when viscous forces dominate, little is known at finite Reynolds number when the inertial forces of the suspending fluid are important. In the present study, expressions are derived that allow for dynamic calculations of particle, drop, and bubble motion at finite Reynolds number. The results show a significant change in the temporal behavior of the force/velocity relationship from that derived from the unsteady Stokes equations, particularly as a body approaches its steady state. At finite Reynolds number, when the convective inertial effects are included, the hydrodynamic force on a body has much weaker history dependence on the past motion of the body and it reaches its steady state faster than what would be predicted if only the unsteady inertial effects are accounted for. When compared with numerical solutions of the Navier-Stokes equations, the analytical force expressions perform well up to a Reynolds number of 0.5. A common theme to the derivations is the use of the reciprocal theorem which provides for an efficient and elegant means for computing inertial effects in suspension mechanics. Connections with past approaches are made in light of these new applications of the reciprocal theorem