Stokes Force Research Articles

On the mechanism of noncoalescence in a droplet cluster

Experimental data are presented, from which it follows that the mechanism of noncoalescence in a cluster of drops is not attributable to the electrostatic charging and the presence of surfactants. The Stokes force acting upon drops of the cluster in a convective plume amounts to a fraction of a percent of the drop weight. New effects are described, which provide evidence for a fast proper rotation of drops in the cluster. Estimates show that this rotation is explained in terms of the hypothesis of a thermocapillary nature of the noncoalescence phenomenon.

Nonlinear differential equations with fractional damping with applications to the 1dof and 2dof pendulum

Existence, uniqueness and dissipativity is established for a class of nonlinear dynamical systems including systems with fractional damping. The problem is reduced to a system of fractional-order differential equations for numerical integration. The method is applied to a non-linear pendulum with fractional damping as well as to a nonlinear pendulum suspended on an extensible string. An example of such a fractional damping is a pendulum with the bob swinging in a viscous fluid and subject to the Stokes force (proportional to the velocity of the bob) and the Basset-Boussinesq force (proportional to the Caputo derivative of order 1/2 of the angular velocity). An existence and uniqueness theorem is proved and dissipativity is studied for a class of discrete mechanical systems subject to fractional-type damping. Some particularities of fractional damping are exhibited, including non-monotonic decay of elastic energy. The 2:1 resonance is compared with nonresonant behavior.

Sliding of fine particles on the slip surface of rising gas bubbles: Resistance of liquid shear flows

In this paper a model was developed to describe the shear flow resistance force and torque acting on a fine particle as it slides on the slip surface of a rising gas bubble. The shear flow close to the bubble surface was predicted using a Taylor series and the numerical data obtained from the Navier–Stokes equations as a function of the polar coordinates at the bubble surface, the bubble Reynolds number, and the gas hold-up. The particle size was considered to be sufficiently small relative to the bubble size that the bubble surface could be locally approximated to a planar interface. The Stokes equation for the disturbance shear flows was solved for the velocity components and pressure using series of bispherical coordinates and the boundary conditions at the no-slip particle surface and the slip bubble surface. The solutions for the disturbance flows were then used to calculate the flow resistance force and torque on the particle as a function of the separation distance between the bubble and particle surfaces. The resistance functions were determined by dividing the actual force and torque by the corresponding (Stokes) force and torque in the bulk phase. Finally, numerical and simplified analytical rational approximate solutions for force correction factors for sliding particles as a function of the (whole range of the) separation distance are presented, which are in good agreement with the exact numerical result and can be readily applied to more general modelling of the bubble–particle interactions.

Long-time dynamics of Rouse–Zimm polymers in dilute solutions with hydrodynamic memory

The dynamics of flexible polymers in dilute solutions is studied taking into account the hydrodynamic memory, as a consequence of fluid inertia. As distinct from the Rouse-Zimm (RZ) theory, the Boussinesq friction force acts on the monomers (beads) instead of the Stokes force, and the motion of the solvent is governed by the nonstationary Navier-Stokes equations. The obtained generalized RZ equation is solved approximately using the preaveraging of the Oseen tensor. It is shown that the time correlation functions describing the polymer motion essentially differ from those in the RZ model. The mean-square displacement (MSD) of the polymer coil is at short times approximately t(2) (instead of approximately t). At long times the MSD contains additional (to the Einstein term) contributions, the leading of which is approximately t. The relaxation of the internal normal modes of the polymer differs from the traditional exponential decay. It is displayed in the long-time tails of their correlation functions, the longest lived being approximately t(-3/2) in the Rouse limit and t(-5/2) in the Zimm case, when the hydrodynamic interaction is strong. It is discussed that the found peculiarities, in particular, an effectively slower diffusion of the polymer coil, should be observable in dynamic scattering experiments.

New principle of electromagnetophoretic adsorption–desorption microchromatography

A new principle for the chromatographic micro-separation of micrometer-sized particles in liquid has been invented by switching the electromagnetophoretic force in a capillary flow system. The principle is the combination of the Stokes force by the bulk flow and the adsorption–desorption force on a capillary inner surface controlled by an electromagnetophoretic buoyancy generated by an alternative current and a homogenous magnetic field. The observed retention profiles of test micro-particles was explained by the “zigzag” migration model mainly depended on particle size and their adsorption force to the capillary wall. By this method, we could succeed to separate polystyrene particles of 10 μm and 20 μm in diameter dispersed in 1 M KCl solution containing 0.01% Triton X-100 using only 1 mm long fused-silica capillary under 10 T.

Modeling of marangoni-induced droplet motion and melt convection during solidification of hypermonotectic alloys

A two-phase volume averaging approach to model Marangoni-induced droplet motion of the minority liquid phase and the convection in the parent melt during solidification of the hypermonotectic alloys is presented. The minority liquid phase decomposed from the parent melt as droplets in the miscibility gap was treated as the second-phase L2. The parent melt including the solidified monotectic matrix was treated as the first phase L1. Both phases were considered as different and spatially interpenetrating continua. The conservation equations of mass, momentum, solute, and enthalpy for both phases, and an additional transport equation for the droplet density, were solved. Nucleation of the L2 droplets, diffusion-controlled growth, interphase interactions such as Marangoni force at the L1-L2 interface, Stokes force, solute partitioning, and heat release of decomposition were taken into account by corresponding source and exchange terms in the conservation equations. The monotectic reaction was modeled by adding the latent heat on the L1 phase during monotectic reaction, and applying an enlarged viscosity to the solidified monotectic matrix. A two-dimensional (2-D) square casting with hypermonotectic composition (Al-10 wt pct Bi) was simulated. This paper focused on Marangoni motion, hence gravity was not included. Results with nucleation, droplet evolution, Marangoni-induced droplet motion, solute transport, and macrosegregation formation were obtained and discussed.

Correction factor of the Stokes force undergone by a sphere in the axis of a cylinder in uniform and Poiseuille flows

To contribute to the existing knowledge of the hydrodynamic force exerted on a spherical particle placed in the axis of a cylinder, at small Reynolds numbers, the influence of the uniform and Poiseuille flows on the wall correction factor are numerically and asymptotically investigated. The Stokes and continuity equations are expressed in the stream function and vorticity formulation and are rewritten in an orthogonal system of curvilinear coordinates. These equations are solved using a finite differences method. The generation of the grid was carried out by the singularities method. The accuracy of the numerical code is tested through comparison with theoretical and experimental results. In both cases we numerically calculated the separate contributions of the pressure and viscosity forces. In concentrated regime these numerical calculations are in very good agreement with those obtained by asymptotic expansions. This analysis allowed us to show the prevalence of the pressure term over the viscosity one in the lubrication regime contrary to what happened for the dilute regime. All our numerical and asymptotical results compared with those of Bungay et al. (Int. J. Multiphase Flow 1 , 25–56 (1973)) seem to give a response to this problem argued for a long time.

Modelling the solidification of hypermonotectic alloys

A two-phase model is developed to simulate the decomposition and spatial phase separation (microstructure evolution) during solidification of hypermonotectic alloys. The minority liquid phase, decomposing in morphology of droplets from the parent melt, is treated as the second phase, L2, while the parent melt, including the solidified monotectic matrix, is the first phase, L1. The conservation equations of mass, momentum, solute and enthalpy for both phases, and an additional transport equation for the droplets are solved. Nucleation of the droplets, diffusion-controlled growth (coarsening) and dissolution of the droplets, interphase interactions such as Marangoni (thermocapillary) force, Stokes force, solute partitioning and heat release of decomposition are modelled by the corresponding source and exchange terms in the conservation equations. The monotectic reaction is modelled by adding the latent heat on the L1 phase and applying a suitable large viscosity to the solidified monotectic matrix. The simulation results of a two-dimensional square casting with hypermonotectic composition (Al–10 wt.% Bi) under normal terrestrial and weightless conditions are presented and discussed.

The influence of the Stokes and Archimedes forces on the stability of a two-phase flow

The problem of the stability of a two-phase flow is considered. The solution of the stability equations is performed by the spectral method using polynomials of Chebyshev. A decrease in the stability region gas flow with the addition of particles of the solid phase. The analysis influence on the stability characteristic of Stokes and Archimedes forces.

Numerical calculation of wall effect and ‘backflow’ on the Stokes force

We give numerical results on the modification of the drag force F x exerted on a sphere positioned eccentrically and moving at very low Reynolds number, at constant velocity within and along a cylindrical tube. The numerical results computed by Lattice-Boltzmann method or by finite volume formulation are in good agreement with the experimental results obtained by Ambari et al. (J. Fluid Mech. 149 (1984) 235–253). In particular, they confirm the existence of a minimum of the force F x away from the axis of the cylinder and a sharp increase when the sphere approaches the sidewall. To cite this article: T. Godin et al., C. R. Mecanique 330 (2002) 837–842.

Numerical Simulation of Scavenging of Small Particles by Charged Droplets

The trajectories of fine aerosol particles in the vicinity of a free falling collector droplet and their deposition on it were investigated numerically by solving the equations of motion of the particle and the droplet in quiescent air. The droplet was assumed to be charged to one half of the Rayleigh limit. The Coulomb, image, Stokes, inertial, and gravitational forces acting upon the particle near the droplet were taken into consideration in the equations of motion. The equations of the droplet motion were also incorporated into the set of equations including the Coulomb and image forces on the droplet due to the particle charge. The flow field in the vicinity of the droplet was determined by numerical solution of the Navier-Stokes equations. The equations of particle motion were solved in threedimensional (3-D) space by the Runge-Kutta method of the fourth order. The collection efficiency of the particles on the droplet was determined by searching the limiting trajectory within the entire space. The results for particles charged to 10 elementary charges of the same and opposite polarity as the droplet, as well as the electrically neutral ones, were compared. The assumption on the charge of the particle was rather arbitrary. It was assumed that particles are not intentionally charged but only possess a charge generated by tribocharging due to random contacts and were independent of the particle size. Charging the collector causes the Coulomb forces between these 2 species to improve particle deposition on the droplet and in this way the aerosol is removed from the gas. For the aerosol particles charged to the same polarity as the collector, the collection efficiency is still higher than for uncharged particles due to the action of the image forces. In this case, the collection efficiency increases for smaller droplets and for particles with increasing diameter.

Modelling and asymptotic analysis of particle-interface interaction

An asymptotic study of the interaction between a foreign particle and a solidifying interface is undertaken. The analysis focuses primarily on the influence of the disjoining pressure on particle pushing or engulfment. The analysis considers only thermal effects in a pure substance in which a neutrally buoyant spherical particle, whose thermal conductivity is equal to that of the melt, is positioned near the solid-liquid interface. The interface equilibrium temperature includes the undercooling effects due to both the front curvature (the Gibbs-Thomson effect) and the long-range intermolecular forces (nonretarded van der Waals interactions) in the thin film behind the particle. A uniformly valid asymptotic representation for the front shape is derived and used to infer the gap width separating the particle from the solid front. The latter is utilized to calculate the growth rate that results from a balance of the van der Waals and Stokes forces. An attempt is also made at deriving an expression for the critical velocity for continuous pushing from the balance of forces.

Single-Microparticle Measurements: Laser Trapping-Absorption Microspectroscopy under Solution-Flow Conditions

A laser trapping-microspectroscopy system combined with a fluid manifold was developed to manipulate and analyze "single" microparticles. A sample solution containing microparticles was introduced to a flow cell set on a microscope stage, and a single particle was trapped by a 1064-nm laser beam. With the particle being trapped, the other particles were pumped out by flowing water to hold the unique microparticle in the flow cell. Under solution-flow conditions, a single microparticle was laser trapped in balance with the gradient (F(g)) and Stokes forces (F(s)) experienced by the particle, and thus, the trapped position was shifted to the downstream side of the 1064-nm laser beam focus. Flow rate and particle size dependencies of this particular positional displacement of the particle were discussed in terms of F(g) and F(s). On the basis of these studies, optical requirements to conduct absorption microspectroscopy of a laser-trapped particle were optimized, and the technique was applied to study a time course of dye adsorption processes in single microparticles. The adsorption rate of Rhodamine B was determined for individual microparticles for the first time.

Theory of local electrification of cavitation bubbles: new approaches

The theory of local electrification of cavitation bubbles has been generalised. The major cases for a local electrification of bubbles in a cavitation field were considered; i.e., fragmentation and deformation of cavitation bubbles. The splitting of cavitation bubbles was considered taking into account surface tension, bubble perturbation, Stokes force and electrostatic forces between like charges on the wall of the collapsing neck of the fragment bubble. The problem of the uncompensated charge on the surface of the deformed cavitation bubble is solved. For this purpose radial deformations are considered in terms of the paraboloid of rotation and axial deformation approximated by one cavity hyperboloid of rotation. The maximum electric strength is accounted for. An explanation for some physical and physico-chemical effects in cavitation fields is proposed in terms of the electrical theory of the local electrification of cavitation bubbles.

Deposition of aerosol particles on a charged spherical collector

A numerical model for determining the trajectories of a spherical aerosol particle in the vicinity of a spherical collector is presented. The Coulomb, image, Stokes, inertial and gravitational forces acting upon the particle and the collector are taken into consideration in the equations of motion. The system of equations is solved in 3-D space. The flow field in the vicinity of the collector is determined by numerical solution of the Navier-Stokes equations. The examples of aerosol particles' trajectories are presented for a few values of the Coulomb and Reynolds numbers.

Viscosity reduction of polymeric liquid by dissolved carbon dioxide

The viscosities of polydimethylsiloxane (PDMS)/CO2 solutions were measured over the range of pressures of 1–3 MPa and at the ambient temperature. The viscosities were measured by using a specially designed falling ball viscometer (FBV). The Stokes equation was used to determine the viscosities and the Stokes force expressing the viscous drag of the sphere was corrected for the effect of the lateral cylindrical wall. The Kelley-Bueche (KB) free-volume treatment of the viscosities of polymeric solutions was modified to account for the gas solvent and applied to interpret our data on PDMS/CO2 systems. It was shown that the theoretical equation, based on the assumption of the additivity of free volumes of the components, was capable of predicting with remarkable accuracy the concentration and pressure dependence of the viscosities of the investigated polymeric solutions. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 63: 459–466, 1997

A Numerical Study of Heterogeneous Nucleation and Mist Development in Turbulent Flows.

In order to predict fog in an air-induced tunnel, heterogeneous nucleation and water-mist development in turbulent duct flows with cold bottom/top surfaces are numerically investigated. The distribution function of fine aerosol, considered as seeds of water-mist, and the Stokes force of rather large particles are taken into consideration regarding the particle diffusivity and the k-e turbulent model as well as gravity precipitation. For a 2-dimensional duct of 2 m×360 m with air flow rate of 1.0 m/s, mist formation occurs mainly near the upper wall and water-mist develops up to 10μm in radius. In the mist regions of the tunnel deposition flux at the bottom surface is much larger than the vapor flux because of the contribution of precipitation. The effects of the distribution function of seeds and the Reynolds number are also discussed in detail.

Dispersed phase motion in laminar boundary layer flow over a wedge

The motion of a dispersed phase in the laminar boundary layer on a wedge is considered with allowance for the effect of not only the Stokes force, which coincides in direction with the flow velocity, but also the transverse force (Saffman force) resulting from the transverse nonuniforrnity of the flow over the individual particle [1–3].

Stokes resistance, added mass, and Basset force for arbitrarily oriented, finite-length cylinders

The Stokes resistance, added mass, and Basset force are numerically computed for finite-length, circular cross-section cylinders using a boundary integral formulation. Numerical results are presented for cylinders translating parallel and perpendicular to their symmetry axes with length-to-width ratios in the range: 0.01≤a/b≤100. There is good agreement with the limited experimental and numerical results of previous studies and slender-body theory. Simple, analytical formulas for the Stokes force, added mass, and Basset force of spheroids are contrasted with the numerical results for cylinders of the same aspect ratio. For some of these parameters, significant differences persist for disk- and rod-shaped particles.

Force on single actin filaments in a motility assay measured with an optical trap.

We have used an optical trap to measure or exert a force on single actin filaments via the attachment of polystyrene beads which were coated with NEM-modified HMM. In the simplest experiment, beads were attached to rhodamine phalloidin labelled actin filaments and observed to move on an HMM coated surface in the presence of ATP. Moving beads were steered into the vicinity of the trap using a PZT operated microscope stage. The minimum force needed to stop a moving bead was measured by lowering the trap strength until the bead resumed movement. By aligning the optical trap with the centre of a quadrant detector placed in an image plane of the microscope, it was possible to measure the force exerted on a filament by measuring the displacement of the bead position from the centre of the trap. In each of these experiments, the trap was calibrated by applying a Stokes force to a bead in free solution. The characteristics of the trap were studied, and the displacement of the bead from the centre of the trap was shown to be directly proportional to the applied force over a large part of the total range of the trap. The compliance of the trap could be substantially reduced by the use of feedback control to deflect the laser beam via an acousto-optic modulator. The advantages and limitations of this technique will be discussed.