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Calibration of lubrication force measurements by lattice Boltzmann simulations
This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.Many experiments explore the hydrodynamic boundary of a surface by approaching a colloidal sphere and measuring the occurring lubrication force. However, in this case many different parameters like wettability and surface roughness influence the result. In the experiment these cannot be separated easily. For a deeper understanding of such surface effects a tool is required that predicts the influence of different surface properties. Here computer simulations can help. In this paper we present lattice Boltzmann simulations of a sphere submerged in a Newtonian liquid and show that our method is able to reproduce the theoretical predictions. In order to provide high precision simulation results the influence of finite size effects has to be controlled. We study the influence of the required system size and resolution of the sphere and demonstrate that already moderate computing ressources allow to keep the error below 1%.This study is funded by DFG priority program SPP 1164
Inf-sup estimates for the Stokes problem in a periodic channel
We derive estimates of the Babu\u{s}ka-Brezzi inf-sup constant for
two-dimensional incompressible flow in a periodic channel with one flat
boundary and the other given by a periodic, Lipschitz continuous function .
If is a constant function (so the domain is rectangular), we show that
periodicity in one direction but not the other leads to an interesting
connection between and the unitary operator mapping the Fourier sine
coefficients of a function to its Fourier cosine coefficients. We exploit this
connection to determine the dependence of on the aspect ratio of the
rectangle. We then show how to transfer this result to the case that is
or even by a change of variables. We avoid non-constructive
theorems of functional analysis in order to explicitly exhibit the dependence
of on features of the geometry such as the aspect ratio, the maximum
slope, and the minimum gap thickness (if passes near the substrate). We
give an example to show that our estimates are optimal in their dependence on
the minimum gap thickness in the case, and nearly optimal in the
Lipschitz case.Comment: 18 pages, 4 figure
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
Rigorous derivation of the thin film approximation with roughness-induced correctors
50 pagesWe derive the thin film approximation including roughness-induced correctors. This corresponds to the description of a confined Stokes flow whose thickness is of order~\eps (designed to be small)~; but we also take into account the roughness patterns of the boundary that are described at order~\eps^2, leading to a perturbation of the classical Reynolds approximation. The asymptotic expansion leading to the description of the scale effects is rigorously derived, through a sequence of Reynolds-type problems and Stokes-type (boundary layer) problems. Well-posedness of the related problems and estimates in suitable functional spaces are proved, at any order of the expansion. In particular, we show that the micro-/macro-scale coupling effects may be analysed as the consequence of two features: the interaction between the macroscopic scale (order~1) of the flow and the microscopic scale (order~\eps of the thin film) is perturbed by the interaction with a microscopic scale of order~\eps^2 related to the roughness patterns (as expected through the classical Reynolds approximation)~; moreover, the converging-diverging profile of the confined flow, which is typical in lubrication theory (note that the case of a constant cross-section channel has no interest) provides additional micro-macro-scales coupling effects
Accurate lubrication corrections for spherical and non-spherical particles in discretized fluid simulations
Discretized fluid solvers coupled to a Newtonian dynamics method are a
popular tool to study suspension flow. As any simulation technique with finite
resolution, the lattice Boltzmann method, when coupled to discrete particles
using the momentum exchange method, resolves the diverging lubrication
interactions between surfaces near contact only insufficiently. For spheres, it
is common practice to account for surface-normal lubrication forces by means of
an explicit correction term. A method that additionally covers all further
singular interactions for spheres is present in the literature as well as a
link-based approach that allows for more general shapes but does not capture
non-normal interactions correctly. In this paper, lattice-independent
lubrication corrections for aspherical particles are outlined, taking into
account all leading divergent interaction terms. An efficient implementation
for arbitrary spheroids is presented and compared to purely normal and
link-based models. Good consistency with Stokesian dynamics simulations of
spheres is found. The non-normal interactions affect the viscosity of
suspensions of spheres at volume fractions \Phi >= 0.3 but already at \Phi >=
0.2 for spheroids. Regarding shear-induced diffusion of spheres, a distinct
effect is found at 0.1 <= \Phi <= 0.5 and even increasing the resolution of the
radius to 8 lattice units is no substitute for an accurate modeling of
non-normal interactions.Comment: 19 pages, 10 figure
Spectral Ewald Acceleration of Stokesian Dynamics for polydisperse suspensions
In this work we develop the Spectral Ewald Accelerated Stokesian Dynamics
(SEASD), a novel computational method for dynamic simulations of polydisperse
colloidal suspensions with full hydrodynamic interactions. SEASD is based on
the framework of Stokesian Dynamics (SD) with extension to compressible
solvents, and uses the Spectral Ewald (SE) method [Lindbo & Tornberg, J.
Comput. Phys. 229 (2010) 8994] for the wave-space mobility computation. To meet
the performance requirement of dynamic simulations, we use Graphic Processing
Units (GPU) to evaluate the suspension mobility, and achieve an order of
magnitude speedup compared to a CPU implementation. For further speedup, we
develop a novel far-field block-diagonal preconditioner to reduce the far-field
evaluations in the iterative solver, and SEASD-nf, a polydisperse extension of
the mean-field Brownian approximation of Banchio & Brady [J. Chem. Phys. 118
(2003) 10323]. We extensively discuss implementation and parameter selection
strategies in SEASD, and demonstrate the spectral accuracy in the mobility
evaluation and the overall computation scaling. We
present three computational examples to further validate SEASD and SEASD-nf in
monodisperse and bidisperse suspensions: the short-time transport properties,
the equilibrium osmotic pressure and viscoelastic moduli, and the steady shear
Brownian rheology. Our validation results show that the agreement between SEASD
and SEASD-nf is satisfactory over a wide range of parameters, and also provide
significant insight into the dynamics of polydisperse colloidal suspensions.Comment: 39 pages, 21 figure
Numerical methods and computers used in elastohydrodynamic lubrication
Some of the methods of obtaining approximate numerical solutions to boundary value problems that arise in elastohydrodynamic lubrication are reviewed. The highlights of four general approaches (direct, inverse, quasi-inverse, and Newton-Raphson) are sketched. Advantages and disadvantages of these approaches are presented along with a flow chart showing some of the details of each. The basic question of numerical stability of the elastohydrodynamic lubrication solutions, especially in the pressure spike region, is considered. Computers used to solve this important class of lubrication problems are briefly described, with emphasis on supercomputers
Inertial Coupling Method for particles in an incompressible fluctuating fluid
We develop an inertial coupling method for modeling the dynamics of
point-like 'blob' particles immersed in an incompressible fluid, generalizing
previous work for compressible fluids. The coupling consistently includes
excess (positive or negative) inertia of the particles relative to the
displaced fluid, and accounts for thermal fluctuations in the fluid momentum
equation. The coupling between the fluid and the blob is based on a no-slip
constraint equating the particle velocity with the local average of the fluid
velocity, and conserves momentum and energy. We demonstrate that the
formulation obeys a fluctuation-dissipation balance, owing to the
non-dissipative nature of the no-slip coupling. We develop a spatio-temporal
discretization that preserves, as best as possible, these properties of the
continuum formulation. In the spatial discretization, the local averaging and
spreading operations are accomplished using compact kernels commonly used in
immersed boundary methods. We find that the special properties of these kernels
make the discrete blob a particle with surprisingly physically-consistent
volume, mass, and hydrodynamic properties. We develop a second-order
semi-implicit temporal integrator that maintains discrete
fluctuation-dissipation balance, and is not limited in stability by viscosity.
Furthermore, the temporal scheme requires only constant-coefficient Poisson and
Helmholtz linear solvers, enabling a very efficient and simple FFT-based
implementation on GPUs. We numerically investigate the performance of the
method on several standard test problems...Comment: Contains a number of corrections and an additional Figure 7 (and
associated discussion) relative to published versio
Characteristic Angles in the Wetting of an Angular Region: Deposit Growth
As was shown in an earlier paper [1], solids dispersed in a drying drop
migrate to the (pinned) contact line. This migration is caused by outward flows
driven by the loss of the solvent due to evaporation and by geometrical
constraint that the drop maintains an equilibrium surface shape with a fixed
boundary. Here, in continuation of our earlier paper [2], we theoretically
investigate the evaporation rate, the flow field and the rate of growth of the
deposit patterns in a drop over an angular sector on a plane substrate.
Asymptotic power laws near the vertex (as distance to the vertex goes to zero)
are obtained. A hydrodynamic model of fluid flow near the singularity of the
vertex is developed and the velocity field is obtained. The rate of the deposit
growth near the contact line is found in two time regimes. The deposited mass
falls off as a weak power Gamma of distance close to the vertex and as a
stronger power Beta of distance further from the vertex. The power Gamma
depends only slightly on the opening angle Alpha and stays between roughly -1/3
and 0. The power Beta varies from -1 to 0 as the opening angle increases from 0
to 180 degrees. At a given distance from the vertex, the deposited mass grows
faster and faster with time, with the greatest increase in the growth rate
occurring at the early stages of the drying process.Comment: v1: 36 pages, 21 figures, LaTeX; submitted to Physical Review E; v2:
minor additions to Abstract and Introductio
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