Hydrodynamic Interactions between Two Rigid or Fluid Particles

A contact of a falling spherical particle with another fixed one in an unbounded viscous fluid is theoretically investigated based on a model of adding the contact interaction to the gravitational and hydrodynamic forces. The hydrodynamic interaction between the two particles is dealt with using an extended successive reflection method, with which the complete solution to the exterior velocity field around the two-particle system is constructed on the basis of the general expression given by Lamb, and then the hydrodynamic forces and torques on the two particles are obtained by integrating the fluid stress over each particle surface. The mechanical contact force is characterized by the standard friction theory with a criterion responsible for the transition from pure rolling to rolling with slip. Resorting to the dynamical equations of motion including the gravitational, hydrodynamic, and contact forces, the settling motion of a spherical particle in the vicinity of another fixed one is depicted using the fourth-order Runge-Kutta-Fehlberg method. Compared with the experimental results available in the literature, the theoretical prediction confirms two moving patterns at contact: pure rolling and rolling with slip, analyzes the dependence of the transition from one to another on the static friction coefficient and the contact separation distance between the particle surfaces, and accounts for a limitation of the quasisteady description of two interacting noncolloidal particles.

Subject of this work is the numerical simulation of the motion of rigid particles in an incompressible viscous or viscoelastic fluid. What makes this task especially challenging is the fact that two different physical models are coupled: The fluid motion is described by a system of partial differential equations (Navier-Stokes equations), while the dynamics of the rigid particles is determined as the solution of a system of ordinary differential equations (Newton's laws of motion). In this work, numerical methods for some prototypical situations are developed, analyzed and validated; all of them are based on the finite element method (FEM). In this context, special emphasis is given to local grid adaptation. The goal is to obtain a high accuracy and, at the same time, to keep the computational effort as low as possible.

The motion of a solid body in a viscous incompressible fluid under the action of external and hydrodynamic forces and momenta

The effects of electrostatic forces on the thermophoretic suppression of particle diffusional deposition onto hot surfaces

The effect of hydrodynamic interactions (HI) on the long-time self-diffusion in quasi-two-dimensional fluids of paramagnetic colloidal particles is investigated using a combination of experiments and Brownian dynamics (BD) simulations. In the BD simulations, the direct interactions (DI) between the particles consist of a short-ranged repulsive part and a long-ranged part that is proportional to 1/r^{3}, with r the interparticle distance. By studying the equation of state, the simulations allow for the identification of the regime where the properties of the fluid are fully controlled by the long-ranged interactions, and the thermodynamic state solely depends on the dimensionless interaction strength Γ. In this regime, the radial distribution functions from the simulations are in quantitative agreement with those from the experiments for different fluid area fractions. This agreement confirms that the DI in the experiments and simulations are identical, which thus allows us to isolate the role of HI, as these are not taken into account in the BD simulations. Experiment and simulation fall onto a master curve with respect to the Γ dependence of D_{L}^{★}=D_{L}/(D_{0}Γ^{1/2}), with D_{0} the self-diffusion coefficient at infinite dilution and D_{L} the long-time self-diffusion coefficient. Our results thus show that, although HI affect the short-time self-diffusion, for a quasi-two-dimensional system with 1/r^{3} long-ranged DI, the reduced quantity D_{L}^{★} is effectively not affected by HI. Interestingly, this is in agreement with prior work on quasi-two-dimensional colloidal hard spheres.

Hydrodynamic forces in annular seals of centrifugal pumps create a significant effect on the vibrational activity of the rotor as a whole. Theoretical and experimental studies of various authors made it possible to establish the structure of hydrodynamic forces and determine the magnitude of the coefficients of radial forces. The authors of this work obtained quantitative load characteristics of the rotor in an annular seal at specially created laboratory experimental stands at various pressure drops and significant angles of misalignment of the rotor axis relative to the seal axis. Static experiments were also conducted with annular seals of various lengths with a constant differential pressure of the liquid and a fixed angle of misalignment of the rotor axis relative to the axis of the annular seal. The given misalignment angle was obtained as a result of the action of various external forces on the rotor. In addition to static experiments, dynamic rotor tests were carried out. Experimentally obtained amplitude - frequency characteristics of the forced radial - angular oscillations of the rotor, which is dynamically mounted in two symmetrical annular seals. From the experimental characteristics, the critical frequency of the rotor angular oscillations in the seals was determined. According to theoretical formulas, taking into account the value of the coefficient of angular stiffness of the annular seal, the frequency of the natural angular oscillations of the rotor in the annular seals was calculated. A comparison of theoretical and experimental values indicates their good quantitative agreement.

Microfiltration is an in-process recycling method that shows great potential to extend fluid life and reduce bacterial concentrations in synthetic and semi-synthetic metalworking fluids (MWFs). The primary problem facing this use of microfiltration is membrane fouling, which is the blocking of membrane pores causing reduced flux. In this paper a fluid dynamic model of partial and complete blocking in sintered alumina membranes is developed that includes hydrodynamic, electrostatic, and Brownian forces. Model simulations are employed to study the impact of electrostatic and Brownian motion forces on the progression of partial blocking. The simulations also examine the effects of fluid velocity, particle size, and particle surface potential. The inclusion of electrostatic and Brownian forces is shown to significantly impact the progression of the partial blocking mechanism. The addition of a strong inter-particle electrostatic force is shown to eliminate the partial blocking build-up of small particles due to the presence of the repulsive forces between the particles. As a result, the time to complete blocking of the test pore was lengthened, suggesting that flux decline is reduced in the presence of electrostatic forces. Brownian motion is shown to have a large impact at low fluid velocities. The most effective parameter set is a low fluid velocity, small particle sizes, high microemulsion surface potential, and high membrane surface potential.

Interactions between hard spheres sedimenting at low Reynolds number

The experimental investigation of the wall effect on the cluster particle settling in the Newtonian and non-Newtonian fluid medium is carried out in three different diameter flow channels. The cluster usually forms during the sedimentation of particles in a fluid medium, so it becomes necessary to study the behavior of the cluster particles. Different cluster particles are considered depending on the number of spheres (N) attached to the cluster and the cluster's shape. The present experiment covers the following range of conditions: 0.05 ≤ deq/D ≤ 0.24, 2 ≤ N ≤ 7, 0.64 ≤ n ≤ 1, and 0.14 ≤ K ≤ 1.81. The results reveal that the terminal velocity varies with the blockage ratio (deq/D), N, and cluster shape. For a particular deq/D ratio and the same N, the terminal velocity of polyhedron particles is high compared to the planar and chain shape particles. The blockage ratio and the Reynolds number affect the wall factor of the cluster particles. However, from the experiment, it is observed that the wall effect also depends on the orientation of the particles. The impact of the wall on the cluster particle is high in Newtonian fluid when compared to cluster particles in non-Newtonian fluid. The present work additionally investigates the influence of drag on cluster particles in the presence and absence of the wall effect. The numerical relationships are developed to predict experimental results in Newtonian and non-Newtonian fluid mediums.

Retarded or frequency-dependent hydrodynamic interactions are relevant for velocity relaxation of colloidal particles immersed in a fluid, sufficiently close that their flow patterns interfere. The interactions are also important for periodic motions, such as occur in swimming. Analytic expressions are derived for the set of scalar mobility functions of a pair of spheres. Mutual hydrodynamic interactions are evaluated in one-propagator approximation, characterized by a single Green function acting between the two spheres. Self-mobility functions are evaluated in a two-propagator approximation, characterized by a single reflection between the two spheres. The approximations should yield accurate results for intermediate and long distances between the spheres. Both translations and rotations are considered. For motions perpendicular to the line of centers, there is a translation-rotation coupling. Extensive use is made of Faxén theorems, which yield the hydrodynamic force and torque acting on a sphere in an incident oscillating flow. The derived results are important for the study of velocity relaxation of two interacting spheres immersed in a fluid and for the study of swimming of assemblies of spheres.

Analytical study on accelerating falling of non-spherical particle in viscous fluid

A theoretical and numerical study of yield-stress fluid creeping flow about a particle is presented. Yield-stress fluids can hold rigid particles statically buoyant if the yield stress is large enough. In addressing sedimentation of rigid particles in viscoplastic fluids, we should know this critical ‘yield number’ beyond which there is no motion. As we get close to this limit, the role of viscosity becomes negligible in comparison to the plastic contribution in the leading order, since we are approaching the zero-shear-rate limit. Admissible stress fields in this limit can be found by using the characteristics of the governing equations of perfect plasticity (i.e. the sliplines). This approach yields a lower bound of the critical plastic drag force or equivalently the critical yield number. Admissible velocity fields also can be postulated to calculate the upper bound. This analysis methodology is examined for three families of particle shapes (ellipse, rectangle and diamond) over a wide range of aspect ratios. Numerical experiments of either resistance or mobility problems in a viscoplastic fluid validate the predictions of slipline theory and reveal interesting aspects of the flow in the yield limit (e.g. viscoplastic boundary layers). We also investigate in detail the cases of high and low aspect ratio of the particles.

In the extension of the electrostatic approach developed in the previous paper on the basis of the available experimental and ab initio calculations data, the geometric structure of acyclic organic molecules containing one isolated chemical bond C=C, C=O, or C=N was analyzed. In agreement with conventional σ/π-representation for this type of bonds, the electrostatic model was suggested, comprising two kinds of local electron densities (LEDs) differing in topology and polarizability, which are in the close spin–spin interaction and show some asymmetry in behavior under the action of external perturbing electrostatic forces (ESFs). Based on the previously introduced concepts of LEDs, point positive charges (PPCs), directions of their most effective interaction (MEI) and using the procedure of fixed molecule (FM), the capabilities of the model were demonstrated on the examples of typical alkenes, carbonyl compounds, azomethines and oximes.

A study is made of the statistical characteristics of isotropic velocity and scalar fields in supply of the kinetic energy of turbulence from the energy of the average flow. Two models based on the distributions of the kinetic turbulence energy and the intensity of scalar-field pulsations by wave numbers and length scales are used for calculation of the statistical characteristics of turbulent velocity and scalar fields. The calculation results are compared to the data of the direct numerical modeling performed under the same initial conditions.

We can basically distinguish two types of problems in the study of forces acting on a mechanism. On one side, we can determine the type of motion produced by a known system of forces (direct problem). On the other side, and provided that we know the variables defining the motion of the links, we can find the forces producing it (inverse problem). In this chapter we will focus on the second problem. Forces acting on a mechanism can be due to several reasons, such as the weight of the links (gravity forces), external forces, friction forces, acceleration of the links (inertial forces) and so on. However, the weight of the links is usually a negligible force compared to the other forces mentioned. Furthermore, if the mechanism is well lubricated, it is possible to neglect friction forces and obtain sufficiently accurate results, simplifying the problem significantly. For this reason we will only consider external and inertial forces in this book. We will first study the action of external forces applied to a mechanism in static equilibrium. We will begin by developing a method for the static analysis of linkages which, in addition to facilitating the comprehension of force transmission in mechanisms, will be useful to explain methods that will subsequently be applied when considering inertial forces of the links, that is, when performing a dynamic analysis. We will only study the action of forces in articulated mechanisms with planar motion and assume that all forces occur on the same plane, not on parallel ones. When the neglected moments are considerable, it will be necessary to make a second analysis on a plane perpendicular to the one on which forces are acting.