Particles in Flows

Table of Contents

1. Frontmatter

2. Chapter 1. A Maximal Regularity Approach to the Analysis of Some Particulate Flows

   D. Maity, M. Tucsnak

   Abstract

   This work presents some recent advances in the mathematical analysis of particulate flows. The main idea we want to emphasize is that, for a variety of fluid models the corresponding coupled systems have a common structure, at least in the linearized case. Within this framework, several model problems are considered and studied in detail. This includes a simple toy model, motion of a piston in a heat conducting gas, motion of a rigid body in a viscous incompressible fluid and motion of a solid in a compressible fluid.

3. Chapter 2. Time-Periodic Linearized Navier–Stokes Equations: An Approach Based on Fourier Multipliers

   T. Eiter, M. Kyed

   Abstract

   The Stokes and Oseen linearizations of the time-periodic Navier–Stokes equations in the n-dimensional whole space for n ≥ 2 are investigated. An approach based on Fourier multipliers is introduced to establish \(\mathrm{L}^{q}\) estimates and to identify function spaces of maximal regularity for the corresponding operators. Moreover, the representation of a solution in terms of a Fourier multiplier is used to introduce the concept of a time-periodic fundamental solution. The main idea is to replace the time axis by a torus group and to study the system in a setting of functions defined on a locally compact abelian group G. For this purpose, we develop the required formalism. More specifically, we introduce the space \(\mathcal{S}(G)\) of Schwartz-Bruhat functions and investigate the Stokes and Oseen systems in the corresponding space of tempered distributions \(\mathcal{S^{{\prime}}}(G)\). Moreover, we give a detailed proof of the so-called Transference Principle, which enables us to employ Fourier multipliers in a group setting in order to establish \(\mathrm{L}^{q}\) estimates.

4. Chapter 3. Motion of a Particle Immersed in a Two Dimensional Incompressible Perfect Fluid and Point Vortex Dynamics

   F. Sueur

   Abstract

   In these notes, we expose some recent works by the author in collaboration with Olivier Glass, Christophe Lacave and Alexandre Munnier, establishing point vortex dynamics as zero-radius limits of motions of a rigid body immersed in a two dimensional incompressible perfect fluid in several inertia regimes.

5. Chapter 4. Stability of Permanent Rotations and Long-Time Behavior of Inertial Motions of a Rigid Body with an Interior Liquid-Filled Cavity

   G. P. Galdi

   Abstract

   A rigid body, with an interior cavity entirely filled with a Navier-Stokes liquid, moves in absence of external torques relative to the center of mass of the coupled system body-liquid (inertial motions). The only steady-state motions allowed are then those where the system, as a whole rigid body, rotates uniformly around one of the central axes of inertia (permanent rotations). Objective of this article is twofold. On the one hand, we provide sufficient conditions for the asymptotic, exponential stability of permanent rotations, as well as for their instability. On the other hand, we study the asymptotic behavior of the generic motion in the class of weak solutions and show that there exists a time t ₀ after that all such solutions must decay exponentially fast to a permanent rotation. This result provides a full and rigorous explanation of Zhukovsky’s conjecture, and explains, likewise, other interesting phenomena that are observed in both lab and numerical experiments.

6. Chapter 5. Dissipative Particle Dynamics: Foundation, Evolution, Implementation, and Applications

   Z. Li, X. Bian, X. Li, M. Deng, Y.-H. Tang, B. Caswell, G. E. Karniadakis

   Abstract

   Dissipative particle dynamics (DPD) is a particle-based Lagrangian method for simulating dynamic and rheological properties of simple and complex fluids at mesoscopic length and time scales. In this chapter, we present the DPD technique, beginning from its original ad hoc formulation and subsequent theoretical developments. Next, we introduce various extensions of the DPD method that can model non-isothermal processes, diffusion-reaction systems, and ionic fluids. We also present a brief review of programming algorithms for constructing efficient DPD simulation codes as well as existing software packages. Finally, we demonstrate the effectiveness of DPD to solve particle-fluid problems, which may not be tractable by continuum or atomistic approaches.

7. Chapter 6. Numerical Methods for Dispersed Multiphase Flows

   M. Sommerfeld

   Abstract

   This article gives an overview of numerical methods for the calculation of dispersed multi-phase flows. At the beginning, a brief introduction is given on the different flow regimes observed for multi-phase flows in general. Then a characterisation and classification of dispersed multi-phase flows is introduced based on inter-particle spacing and volume fraction. As an introduction to the subject, the numerical methods used for single-phase flows are briefly described based on the turbulent scales being resolved by the numerical grid. Since even dispersed multi-phase flows are extremely complex, the hierarchy of the different numerical methods is highlighted ranging from macro-scale numerical simulations for an entire industrial process down to micro-scale simulations required for analysing particle scale phenomena. Due to constraints in computational power and storage availability, macro-scale simulations can only be done with a limited grid resolution and the assumption of particles being treated as point-masses. Consequently, all transport phenomena occurring on scales smaller than the grid cell and on the scale of the particle have to be considered through additional closures and models. Therefore, essential elements in this multi-scale problem are direct numerical simulations that fully resolve the particles and the flow around them. The different methods for such resolved simulations are briefly described. The major part of this article is focused on the modelling of dispersed multi-phase flows relying on the point-particle assumption. The multi-fluid method or Euler/Euler model is briefly described in order to demark its applicability and limitations. The hybrid Euler/Lagrange approach based on tracking a large number of point particles and its different variants are introduced in more detail, emphasising the two-way coupling approaches for unsteady flows. The importance of accurately modelling particle-scale phenomena is highlighted and an estimate for the significance of particle-wall and inter-particle collisions is given. Finally, three application examples are introduced, emphasising the potential of Euler/Lagrange simulations. For a particle-laden swirling flow the semi-unsteady approach is used for analysing unsteady particle roping phenomena. The simulations of particle suspension in a stirred vessel highlight the importance of inter-particle collisions even at relatively low volume fractions up to 5%. Finally, it is demonstrated that the Euler/Lagrange approach may also be used to study an industrial filtration process where it allows the prediction of particle deposits and filter cake formation. In this respect extensions are possible which provide more information on the internal filter cake structure.

8. Chapter 7. Path Instabilities of Axisymmetric Bodies Falling or Rising Under the Action of Gravity and Hydrodynamic Forces in a Newtonian Fluid

   J. Dušek

   Abstract

   The chapter deals with the effect of instabilities on the paths of sedimenting or rising bodies in Newtonian fluids. To separate the effect of shape, we focus on axisymmetric objects. Spheres, discs, oblate spheroids and flat cylinders are all expected to follow vertical trajectories with axisymmetry axis aligned with the trajectory. This is, indeed, the case, however, this regime remains stable only if viscous effects are sufficiently strong. The loss of stability of the vertical regime leads to a large variety of trajectories depending on the details of shape, namely their flatness (expressed by aspect ratio for cylinders or spheroids), their inertia and viscous effects (expressed by some equivalent of Reynolds number). Defined in this manner, the problem is basically that of axisymmetry breaking.

   Since a significant part of dynamics is expected to arise in the wake, we first focus on axisymmetry breaking of wakes of fixed axisymmetric bodies, the sphere being considered as a prototypical case. We show that the scenario is dominated by two bifurcations following systematically, with increasing Reynolds number, in the order of the regular one as primary and a Hopf one as secondary. The weakly non-linear analysis points out the relevance of Fourier azimuthal decomposition serving as an optimal numerical tool for all presented simulations. Next, the free body degrees of freedom are accounted for. The interplay of the regular and Hopf bifurcations still dominates, however, the scenario is significantly different for spheres and flat objects. The presented parametric study shows that trajectories of spheres become very rapidly chaotic. As an example of flat object, nominally infinitely thin disc is investigated. In this case the Hopf bifurcation is the primary one. The scenario is remarkable by strong subcritical effects due to the significant role of inertia of the combined motion of the solid and of the surrounding fluid.

9. Chapter 8. Microbubbles: Properties, Mechanisms of Their Generation

   V. Tesař

   Abstract

   This chapter discusses microbubbles—small gas bubbles in liquid medium of diameter less than 1 mm. Although they were known to offer a number of advantages, until recently they could be generated only by methods energetically inefficient. New horizons became open by the discovery of generation by aerators provided with an oscillator in their gas supply. Chapter provides in particular an information about no-moving-part fluidic oscillators, recently already almost forgotten but now demonstrated to offer benefits like low manufacturing cost, reliability, long life and absence of maintenance. The empirical fact that small bubbles cannot be obtained simply by making small passages in the aerator is here explained by conjunction of several microbubbles. Because the velocity of bubble motion decreases with decreasing size, small microbubbles tend to dwell near the aerator exits. They then coalesce there into a much larger single bubble (the effect promoted by the latter possessing lower surface energy). The fact that the oscillator prevents this conjunction and thus keeps the microbubbles small has been explained by high-speed camera images which show the effect of oscillatory motions.