Mass and momentum balance during particle migration in the pressure-driven flow of frictional non-Brownian suspensions

In this paper, we study numerical methods for solving the Navier-Stokes equations in doubly connected domains. Two methods for solving the problem are considered. The first method is based on constructing a difference problem in variables of the stream function and the vortex of velocity using the uniqueness condition for pressure. The numerical solution of the elliptic equation for stream functions is found as the sum of the solutions of two simple problems of an elliptic type. One problem is with homogeneous boundary conditions, and the other is with a homogeneous equation. An alternative approach to solving the problem is the fictitious domain method with the continuation of the least coefficient. This method does not require satisfying the pressure uniqueness condition, and is simple to implement. An important direction in the development of numerical simulation methods is the study of approximate methods for solving problems of mathematical physics in complex multidimensional areas. To solve many applied problems in irregular areas, the fictitious domain method is widely used, which is characterized by a high degree of automation of programming. The main idea of the fictitious domain method is that the problem is solved not in the original complex domain, but in some other, simpler domain. This allows to create software immediately for a fairly wide class of problems with arbitrary computational domains. The possibilities of applying the fictitious domain method to the problems of hydrodynamics in the variables “stream function, vortex of velocity” are considered in many works. In this paper, we study a numerical method for solving the Navier-Stokes equations in doubly connected domains. A computational finite difference algorithm for solving an auxiliary problem of the fictitious domain method has been developed. The results of numerical modeling of the two-dimensional Navier-Stokes equations by the fictitious domain method with continuation by the lowest coefficient are presented. For this problem, a parallel algorithm was developed using the CUDA architecture, which was tested on various grid dimensions.

Particulate flows arise in a wide class of research areas and industrial processes, for example, fluidized suspensions, materials separation, rate of mixing enhancement, filled polymers, etc. In many of the applications, the fluid phase displays complex non-Newtonian flow behavior. Adding particles in a complex fluid further complicates the flow behavior. In order to study the particle motion in complex fluids such as viscoelastic fluids, a numerical analysis is an essential requirement due to inherent nonlinear behavior of the fluids. An extended finite element method (XFEM) has been developed for the direct numerical simulation of particulate complex flows. The main advantage of the method is that the movement of particles can be simulated on a fixed Eulerian mesh without any need of remeshing. The proposed method has been applied to various particulate flow problems: • Flow of a viscoelastic fluid around a stationary cylinder The method is verified by comparing the solutions with those of simulations using a boundary-fitted mesh and a fictitious domain method. Our method shows a significant improvement of local accuracy around the rigid body compared to the fictitious domain method, obtaining solutions similar to those of boundary-fitted mesh solutions. • Particle migration in circular Couette flow of a viscoelastic fluid The particle migrates to a stabilized radial position near the outer cylinder regardless of its initial position. As the fluid elasticity increases, the particle migrates more rapidly toward the outer cylinder, and the stabilized radial position of the particle shifts toward the outer cylinder. With increasing the particle size, the particle migrates more rapidly toward the outer cylinder. • Dynamics of particles suspended in two-phase flows A model for the dynamics of particles suspended in two-phase flows is presented by coupling the Cahn-Hilliard theory with the extended finite element method. To demonstrate and validate the technique, the dynamics of a single particle at a fluid-fluid interface is studied. In particular, the effects of interfacial thickness, surface tension, particle size and viscosity ratio of two fluids are investigated. As interfacial thickness increases; surface tension increases; particle size decreases; or viscosity ratio decreases, the particle moves rapidly towards its equilibrium position. The movement of a particle passing through multiple layers of fluids is also presented to demonstrate the wide applicability of the method to problems associated with complex morphology of the fluids.

Concentration inhomogeneities occur in many flows of non-Brownian suspensions. Their modeling necessitates the description of the relative motion of the particle phase and of the fluid phase, as well as the accounting for their interaction, which is the object of the suspension balance model (SBM). We systematically investigate the dynamics and the steady state of shear-induced migration in a wide-gap Couette flow for a wide range of particle volume fraction, and we test the ability of the SBM to account for the observations. We use a model suspension for which macroscopic particle stresses are known. Surprisingly, the observed magnitude of migration is much lower than that predicted by the SBM when the particle stress in the SBM is equated to the macroscopic particle stress. Another noteworthy observation is the quasi-absence of migration for semidilute suspensions. From the steady-state volume fraction profiles, we derive the local particle normal stress responsible for shear-induced migration according to the SBM. However, the observed dynamics of migration is much faster than that predicted by the SBM when using this stress in the model. More generally, we show that it is not possible to build a local friction law consistent with both the magnitude and the dynamics of migration within the standard SBM framework. This suggests that there is a missing term in the usual macroscopic constitutive law for the particle normal stress driving migration. The SBM is indeed capable of accurately predicting both the magnitude and the dynamics of migration when a tentative phenomenological term involving a concentration gradient is added to the particle normal stresses determined in macroscopic experiments.

Shear-induced particle migration: Predictions from experimental evaluation of the particle stress tensor

An energy map model for colloid transport

This paper presents a simulation model of concentrated colloidal nanoparticulate flows to investigate self-organization of the nanoparticles and rheology of the colloid. The motion of solid nanoparticles is treated by an off-lattice Newtonian dynamics. The flow of solvent is treated by an on-lattice fluctuating Navier-Stokes equation. A fictitious domain method is employed to couple the motion of nanoparticles with the flow of solvent. The surface of nanoparticles is expressed by discontinuous solid-liquid boundary to calculate accurately contact interaction and Derjaguin-Landau-Verwey-Overbeek interaction between the nanoparticles. At the same time, the surface is expressed by continuous solid-liquid boundary to calculate efficiently hydrodynamic interaction between the nanoparticles and the solvent. Unlike other simulation models that focus on the hydrodynamic interaction, the present model includes all crucial interactions, such as contact force and torque, van der Waals force, electrostatic force, hydrodynamic force, and torque including thermal fluctuation of the solvent that causes translational and rotational Brownian motions of the nanoparticles. Especially the present model contains the frictional force that plays a significant role on nanoparticles in contact with one another. A fascinating novelty of the present model is that computational cost is constant regardless of the concentration of nanoparticles. The capability of the present simulation model is demonstrated by two-dimensional simulations of concentrated colloidal nanoparticles in simple shear flows between flat plates. The self-organization of concentrated colloidal nanoparticles and the viscosity of colloid are investigated in a wide range of Péclet numbers.

In this paper, the lateral migration of a neutrally buoyant spherical particle in the pressure-driven rectangular channel flow of an Oldroyd-B fluid is numerically investigated with a fictitious domain method. The aspect ratio of the channel cross-section considered is 1 and 2, respectively. The particle lateral motion trajectories are shown for the bulk Reynolds number ranging from 1 to 100, the ratio of the solvent viscosity to the total viscosity being 0.5, and a Weissenberg number up to 1.5. Our results indicate that the lateral equilibrium positions located on the cross-section midline, diagonal line, corner and channel centreline occur successively as the fluid elasticity is increased, for particle migration in square channel flow with finite fluid inertia. The transition of the equilibrium position depends strongly on the elasticity number (the ratio of the Weissenberg number to the Reynolds number) and weakly on the Reynolds number. The diagonal-line equilibrium position occurs at an elasticity number ranging from roughly 0.001 to 0.02, and can coexist with the midline and corner equilibrium positions. When the fluid inertia is negligibly small, particles migrate towards the channel centreline, or the closest corner, depending on their initial positions and the Weissenberg number, and the corner attractive area first increases and then decreases as the Weissenberg number increases. For particle migration in a rectangular channel with an aspect ratio of 2, the transition of the equilibrium position from the midline, ‘diagonal line’ (the line where two lateral shear rates are equal to each other), off-centre long midline and channel centreline takes place as the Weissenberg number increases at moderate Reynolds numbers. An off-centre equilibrium position on the long midline is observed for a large blockage ratio of 0.3 (i.e. the ratio of the particle diameter to the channel height is 0.3) at a low Reynolds number. This off-centre migration is driven by shear forces, unlike the elasticity-induced rapid inward migration, which is driven by the normal force (pressure or first normal stress difference).

Numerical simulations of particle migration in rectangular channel flow of Giesekus viscoelastic fluids

Two-dimensional computational fluid dynamical investigation of particle migration in rotating eccentric cylinders using suspension balance model

Macroscopic characteristics of heavy particle resuspension obtained from direct numerical simulations of pressure driven channel flow

ABSTRACTThe interaction between a particle and the viscous fluid and then the particle-wall collision in the flow field plays an important role in the study of particulate flow. In this paper, we examine the velocity characteristics of a spheroidal particle sediment in the fluid and its rebound dynamics by applying the Computational Fluid Dynamics-Discrete Element Method (CFD-DEM). The Fictitious Domain method and Monte Carlo method are combined to improve the accuracy of the hydrodynamic force acting on the particle. A soft-sphere scheme of DEM is used to model the collision of particles, and the hydrodynamic force on the particle is fully solved directly from the CFD-DEM. The numerical results are verified by comparing the previous numerical and experimental results. The results are in good agreement with the corresponding published data. The simulation results show that the critical factor that affects the particle rebound is Stokes number (St). No rebound occurs when Stokes number is equal to 3.74. Initially, the results show that the ellipsoid particle shows large “wiggles” down the square tube at 45° angle with respect to the horizontal axis. These large “wiggles” gradually reduce after a time, and the ellipsoid finally settles into a stable horizontal state in the center of the square tube due to the effect of fluid viscosity dissipation.

An extended finite element method for the simulation of particulate viscoelastic flows

This paper is intended to overview on analytical and numerical methods in shape optimization. We compute and analyse the shape Hessian in order to distinguish well‐posed and ill‐posed shape optimization problems. We introduce different discretization techniques of the shape and present existence and convergence results of approximate solutions in case of well posedness. Finally, we survey on the efficient numerical solution of the state equation, including finite and boundary element methods as well as fictitious domain methods. Copyright © 2008 John Wiley & Sons, Ltd.

A two-fluid model for numerical simulation of shear-dominated suspension flows

We consider in this paper the wave scattering problem by an object with Neumann boundary conditions in an anisotropic elastic body. To obtain an efficient numerical method (permitting the use of regular grids) we follow a fictitious domain approach coupled with a first order velocity stress formulation for elastodynamics. We first observe that the method does not always converge when the $Q_1^{\rm div}-Q_0$ finite element is used. In particular, the method converges for some scattering object geometries but not for others. Note that the convergence of the $Q_1^{\rm div}-Q_0$ finite element method was shown in [E. Becache, P. Joly, and C. Tsogka, SIAM J. Numer. Anal., 39 (2002), pp. 2109-2132] for the elastodynamic problem in the absence of a scattering object (i.e., without the coupling of the mixed finite elements with the fictitious domain method). Therefore we propose here a modification of the $Q_1^{\rm div}-Q_0$ element following the approach in [E. Becache, J. Rodriguez, and C. Tsogka, On the convergence of the fictitious domain method for wave equation problems, Technical report 5802, INRIA, 2006], where the simpler acoustic case was considered. To study the numerical properties of the new element we carry out a dispersion analysis. Several numerical simulations as well as a numerical convergence analysis show that the proposed method provides a good approximate solution.