Hydrodynamics of Suspensions of Passive and Active Rigid Particles: A Rigid Multiblob Approach

We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally-resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. For unbounded suspensions and suspensions sedimented against a single no-slip boundary, we rely on existing analytical expressions for the Rotne-Prager tensor combined with a fast multipole method or a direct summation on a Graphical Processing Unit to obtain an simple yet efficient and scalable implementation. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently-developed rigid-body immersed boundary method to suspensions of freely-moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid body equations converges in a bounded number of iterations regardless of the system size. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.Comment: Under revision in CAMCOS, Nov 201

The Singular Hydrodynamic Interactions Between Two Spheres In Stokes Flow

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centres, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs) which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by use of a bipolar coordinate system, we derive the stream function for the flow as $Re\to 0$ and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behaviour of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix $\boldsymbol{R}$ in a number of situations in which positivity is destroyed for multipole/perturbative methods.Comment: 28 pages, 12 Figure

Modelling interfacial tribochemistry in the mixed lubrication regime

The need to reduce the cost of components is driving more and more machine elements to operate under mixed lubrication conditions. With higher operating pressures, the lubricant film is becoming thinner and eventually reaches nanometre scales, comparable to the surface roughness. Thus, understanding the mixed lubrication phenomenon is becoming increasingly important. However, the mixed lubrication phenomenon is difficult to capture experimentally and the lubricant additive ZDDP (Zinc Dialkyl Dithio Phosphate) shows its full antiwear character in the mixed lubrication conditions. This research stems from the need for models that can simulate contact mechanics, lubrication and tribochemistry in a single framework. This is the key to understanding and optimizing the lubrication systems to meet future needs. To this end, a numerically efficient procedure based upon the tridiagonal solution of the Reynolds equation is developed and is implemented, in Fortran to solve the equations line by line to incorporate more information from the current iteration step. The asperity contacts are handled by the unified solution algorithm. A new strategy to simulate plastic deformation in a lubricated contact is developed. Under practical loading conditions, the pressures inside the contact can reach values far above the material yielding limit. Thus, an efficient numerical scheme is devised to include the elastic perfectly plastic behaviour in the EHL solution procedure to simulate realistic contact conditions with minimal increase in computational cost. The Boussinesq deformation integrals result in a convolution of pressure and the deformation which is solved using Fast Fourier Transforms (FFTs) by modifying the solution domain to create a cyclic convolution. Code is developed to allow exploration of the highly optimized C-based library (www.fftw.org). The use of FFTs speeds up the solution process many times and makes the use of denser grids and larger time scales accessible. A mesh size of 129 x 129 is found to give reasonable results. The simulation results from the current study agree very well with the previously published results. The evolution of contact area ratio and the central film thickness exhibit a Stribeck type behaviour and the transition speeds at which the contact transits from EHL to mixed and from mixed to complete boundary lubrication can be precisely identified. Existing tribofilm growth models developed for boundary lubrication are reviewed and a model based on the interface thermodynamics is adapted and integrated with the mixed lubrication model to simulate tribochemistry. The problems with existing EHL concepts such as lambda ratio and central film thickness are identified and new definitions are proposed to allow understanding of the mixed lubrication mechanics. The mutual interaction between the tribofilm growth and lubricant film formation is studied. Finally the wear of the tribological system is studied and the wear track profiles are predicted. The new model is then applied to study a ball-on-disc system to explore wear, tribochemistry and roughness evolution. The ZDDP tribofilm growth is studied and the it is found that the final ZDDP tribofilm thickness is very weakly affected by increasing SRR but the rate of formation and removal are strongly affected by the SRR value. The tribofilm growth results are validated against published numerical and experimental results. It is found that the antiwear action of the ZDDP tribofilm is not only due to its chemical action but the ZDDP tribofilm helps to entrain more lubricant and improves contact performance. The presence of tribofilm roughens the contact and the contact area and load ratio both increase due to tribofilm growth. It was also found that the use of conventional EHL parameters to analyse the behaviour of tribosystem is misleading. The flattening of the roughness inside the contact and the proper identification of the central film thickness are crucial to the interpretation of the mixed lubrication results. The roughness of the ball generally decreases due to wear but the presence of tribofilm limits this reduction. Contrary to this, the surface roughness of the ball generally increases due to wear but the presence of tribofilm results in increased roughness of the ball