Sangtae Kim

INTEGRAL EQUATIONS OF THE SECOND KIND FOR STOKES FLOW: DIRECT SOLUTION FOR PHYSICAL VARIABLES AND REMOVAL OF INHERENT ACCURACY LIMITATIONS

Boundary integral methods offer the most attractive combination of generality and computational efficiency for a wide class of particulate Stokes flow problems. Integral equations of the first kind have been numerically applied for more than a decade, whereas those of the second kind are numerically better behaved but involve abstract nonphysical density distributions and have not gained much popularity in applications. We show how the latter may be used for the direct solution of mobility problems, and how the surface tractions corresponding to rigid body motion of a particle may be easily found, thus removing the major disadvantages of the second kind formulations. For the numerical examples we also show how Fourier analysis may be applied to non-axisymmetric problems with axisymmetric boundaries to yield one-dimensional Fredholm integral equations of the second kind. As an application we solve the resistance problem with a numerically efficient quadrature collocation method that avoids the complication...

Parallel Computational Strategies for Hydrodynamic Interactions Between Rigid Particles of Arbitrary Shape in a Viscous Fluid

A fast iterative algorithm is presented for the numerical solution of large linear systems that are encountered in multiparticle Stokes flows. It is applicable to solid particles of arbitrary shape, and finds the translation and rotation velocities when total forces and torques acting on these particles are given (mobility problems). An exact result for the stresslet is also given. The method is based on recently developed boundary integral equations, “the canonical equations for mobility and resistance problems.” These are well‐posed Fredholm equations of the second kind, for mobility problems or problems with arbitrary velocity boundary conditions. They are modified for the direct iterative solution of mobility problems, leading to fast numerical computations. For a single sphere the iteration operator is spectrally decomposed analytically. The convergence rate of the iterations is deduced, and supporting numerical observations are presented. Fast rate of convergence is numerically observed for multisph...

Suspension mechanics for particle contamination control

Abstract This review is intended to bridge the gap between research in particle contamination control for the electronics industry and research in the fluid mechanics of small-particle motions; these two subjects are clearly closely related, but there seems to have been a lack of communication in the past. The technology of contamination control and cleanrooms is described for those readers with no previous knowledge of the field with concentration on those areas for which fluid mechanics is relevant. In particular, this includes the motion of small particles in the presence of extended boundaries and the capture and release of such particles. Research in these areas is reviewed where possible and attention is focused on the modelling of fluid mechanics. The forces associated with small particles are reviewed along with the basic hydrodynamics of dilute suspensions. Some recent developments in low Reynolds number hydrodynamics are emphasized. Finally some contamination control problems for which a fluid mechanics approach seems promising are identified and some directions for future research are suggested.

Computation of rheological properties of suspensions of rigid rods: stress growth after inception of steady shear flow

Abstract The orientation distribution and stress growth for a suspension of rigid rods (or dumbbells) in a Newtonian solvent are calculated for inception of steady shear flow. Galerkins method, with spherical harmonics as trial functions, is used in the spatial coordinates to obtain a system of ordinary differential equations in time which is solved by the spectral method. The method is applicable over a wide range of dimensionless shear rates (Peclet numbers) and has been coded with standard system-solvent and eigensystem packages. For sufficiently large Peclet numbers, the results give the well known rigid-dumbbell prediction of an overshoot in the shear viscosity and normal stress differences. This overshoot is then followed by an undershoot . An explicit analytical approximation for the fluid stresses is presented which is reasonably accurate for Peclet numbers less than unity.

MOBILITY AND STRESSLET FUNCTIONS OF PARTICLES WITH ROUGH SURFACES IN VISCOUS FLUIDS : A NUMERICAL STUDY

A numerical method for the solution of low Reynolds number flow (Stokes flow) past particles of arbitrary shape is described. The method uses a predictor–corrector approach, with a rough initial calculation using the Dabros method followed by an application of the completed double layer boundary integral equation method (CDL‐BIEM). The method is especially well suited for particles with very rough and uneven surfaces. The mobility and stresslet of a sphere with a spiked surface are computed, to illustrate the utility of the new method. For spheres with small protrusions, our results show a small change in the stresslet and particle mobility. For protrusions on the order of the sphere radius, there is a significant increase in the stresslet, leading to Einstein viscosity coefficients on the order of 5, compared to 5/2 for the perfect sphere. We also examine algorithms and the underlying theory behind conjugate gradient and conjugate residual methods that appear to work well with the larger CDL‐BIEM problem...

The motion of ellipsoids in a second order fluid

Abstract The rigid body motion of an ellipsoid in a second order fluid (SOF) under the action of specified (time independent) external forces and torques has been obtained to first order in the Weissenberg number by inverting the resistance relations for the force and torque under specified rigid body motions. The reciprocal theorem of Lorentz was used to bypass the calculation of the O(W) velocity field. The results agree with known analytic solutions for a SOF with the secondary to primary normal stress ratio of − 1 2 . The solution procedure was also tested by computing the torque on a translating prolate spheroid with aspect ratios ranging from slender bodies to near-spheres. One result is that for a SOF with zero secondary normal stress (Weissenberg fluid), previous asymptotic results for near-spheres were found to be accurate even at fairly large aspect ratios (e.g. 2). New results for non-degenerate ellipsoids suggest that the orientation (as monitored by Euler angles) and trajectory of sedimenting, non-axisymmetric particles such as ellipsoids provide useful information on the rheology of the suspending fluid.

Similarity solutions for the orientation distribution function and rheological properties of suspensions of axisymmetric particles with external couples

Abstract We consider the effect of an external field on dilute suspensions of dipolar axisymmetric Brownian particles in a Newtonian solvent. A family of similarity solutions is derived for the orientation distribution of particles after inception of steady two-dimensional flow in the plane normal to the field. It is assumed that the particles are initially aligned by the field. The solution is uniformly valid for small times, but if the field is strong enough to overcome diffusion, the solution remains valid at all time, correctly predicting the steady state distribution. The rheological properties are obtained in closed form from the similarity solution and the role of the external field is demonstrated. First and second normal stress differences are obtained. The solutions are presented for particles with fixed dipoles, although they apply equally to particles with dipoles induced by the external field.

Towards ab Initio Simulations of Concentrated Suspensions

Determination of many-body interactions between particles of arbitrary shape in a viscous fluid is a key problem in the simulation of concentrated suspensions. Three-dimensional flows involving such complex fluid-solid boundaries are beyond the scope of spatial methods, even on supercomputers. Boundary integral methods convert the three-dimensional PDE to a two-dimensional integral equation. Unfortunately, conventional boundary methods yield Fredholm integral equations of the first kind, and dense linear systems which are too large for accurate solution. We have pursued a different boundary integral formulation, which yields Fredholm integral equations of the second kind; these arc amenable to iterative solution. The velocity representation involves a compact operator, so a discrete spectrum results. Wielandt deflations give dramatic reductions in the spectral radius and accurate solutions are obtained after only a few iterations (typically less than 10). An analytic construction of the spectrum for sphere sphere interactions confirms these numerical results. The mathematics is similar to that encountered in the mixing ofd-atomic orbitals to form bonding/antibonding molecular orbitals in transition metals. The memory-saving version of our code can be implemented directly on a dedicated MicroVAX to solve problems involving clusters of less than a dozen particles. For a fixed number of processors, the algorithm grows essentially asN2, whereN is the system size, so computational times are readily estimated on more powerful super-minicomputers and supercomputers using standard “dot-product” benchmarks. The algorithm is especially ideal for gigaflop and teraflop parallel array processors under construction in a number of computer companies; an analysis of the spectrum reveals that asynchronous iterative methods will converge, leading the way to a rigorous formulation of screening concepts for suspended particles of arbitrary shape.

TRACTION SINGULARITIES ON SHARP CORNERS AND EDGES IN STOKES FLOWS

An integral identity developed by Brenner (1964) is used in the “reversed” context to derive a fixed point iterative scheme for the surface tractions on a rigid particle of arbitrary shape submerged in Stokes flows. The iterative approach facilitates the solution of very large systems of equations and thus the employment of high resolution discretization schemes. The utility of the approach is demonstrated by illustrative computations of the tractions on regular polyhedra, with special emphasis on the results near the edges and corners where (integrable) singular behavior is expected to provide a stringent test for the numerical method. Excellent agreement between our numerical estimates for the exponent of these singularities with the analytical result (local 2-D analysis) were obtained. The emphasis in this work is on validation of the approach, but references to applications, such as fluidic self-assembly of semiconductor microstructures, are provided.