Axisymmetric Particles Research Articles

Evaluation of models for numerical simulation of the non-neutral region of sheath plasma

Four different electron models are used to simulate the nonequilibrium plasma flow around a representative cylindrical Faraday probe geometry. Each model is implemented in a two-dimensional axisymmetric hybrid electron fluid and particle in cell method. The geometric shadowing model is derived from kinetic theory on the basis that physical obstruction of part of the velocity distribution leads to many of the expected sheath features. The Boltzmann electron fluid model relates the electron density to the plasma potential through the Boltzmann relation. The non-neutral detailed electron fluid model is derived from the electron conservation equations under the assumption of neutrality, and then modified to include non-neutral effects through the electrostatic Poisson equation. The Poisson-consistent detailed electron fluid model is also derived from the conservation equations and the electrostatic Poisson equation, but uses an alternative method that is inherently non-neutral from the outset. Simulations using the geometric shadowing and non-neutral detailed models do not yield satisfactory sheath structures, indicating that these models are not appropriate for sheath simulations. Simulations using the Boltzmann and Poisson-consistent models produce sheath structures that are in excellent agreement with the planar Bohm sheath solution near the centerline of the probe. The computational time requirement for the Poisson-consistent model is much higher than for the Boltzmann model and becomes prohibitive for larger domains.

Separation of variables method for multilayered nonspherical particles

A separation of variables method based on expansions of the electromagnetic fields in terms of spherical wave functions is expanded at nonspherical (axisymmetric) particles with a rather large number of layers. Commonly used alternative approaches to systems of linear algebraic equations relative to unknown field expansion coefficients for layered particles are considered in some detail. The SVM code developed is compared with the EBCM, GMT and DDA codes designed for multilayered scatterers and some numerical results obtained for nonspherical scatterers with up to 100 layers are presented as illustrations.

Multiscale structure-phenomenological study of magnetorheological suspensions in blood

The rheological equations for dilute suspensions of rigid axisymmetric elongated particles possessing a permanent magnetic moment with blood as a carrier fluid are derived within the frame of a structure-phenomenological approach. The Stokes fluid with couple stresses and the uniaxial dumbbell are used, respectively, as the rheological and hydrodynamic models of blood and suspended particles. The influence of an external magnetic field on a rheological behaviour of the suspensions under study is examined.

Thermophoresis of axisymmetric aerosol particles along their axes of revolution

AbstractThe axisymmetric thermophoretic motion of an aerosol particle of revolution in a uniformly prescribed temperature gradient is studied theoretically. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model. A method of distribution of a set of spherical singularities along the axis of revolution within a prolate particle or on the fundamental plane within an oblate particle is used to find the general solutions for the temperature distribution and fluid velocity field. The jump/slip conditions on the particle surface are satisfied by applying a boundary‐collocation technique to these general solutions. Numerical results for the thermophoretic velocity of the particle are obtained with good convergence behavior for various cases. For the axisymmetric thermophoresis of an aerosol spheroid with no temperature jump and frictional slip at its surface, the agreement between our results and the available analytical solutions is very good. The thermophoretic velocity of a spheroid along its axis of revolution in general increases with an increase in its axial‐to‐radial aspect ratio, but there are exceptions. For most practical cases of a spheroid with a specified aspect ratio, its thermophoretic mobility is not a monotonic function of its relative jump/slip coefficients and thermal conductivity. © 2008 American Institute of Chemical Engineers AIChE J, 2009

Light scattering by multilayered axisymmetric particles: Solution of the problem by the separation of variables method

A new solution to the problem of light scattering by multilayered particles possessing axial symmetry is obtained. Two methods are applied for this purpose. One is the separation of variables method with expansion of fields in terms of spherical wave functions, and the other is a novel approach based on the separation of fields into axisymmetric and nonaxisymmetric parts and on the choice of specific scalar potentials for each of them. A specific feature of the new solution is that the dimension of truncated linear algebraic systems used for determining unknown expansion coefficients of fields does not increase with an increasing number of layers. Using double-and three-layer spheroidal and Chebyshev particles of different shape and size as examples, the domain of applicability of the solution presented is compared with that of the solution previously obtained by the extended boundary conditions method. Except for nearly spherical particles, the solution presented is shown to be more favorable than the previously obtained solution.

Thermal expansion behavior of composites based on axisymmetric ellipsoidal particles

A new model for calculating the coefficients of thermal expansion, CTE, in all three coordinate directions is developed for composites containing aligned, axisymmetric elliptical particles, i.e., characterized by a single aspect ratio, that in the limit approximate the shapes of spheres, fibers and discs. This model is based on Elshelby's method but employs a somewhat different formulation than used in prior papers; a main advantage of the current approach is that it can be readily extended to composites based on ellipsoidal particles with no axes of symmetry, i.e., all three major axes are different, as recently demonstrated for modulus. CTE predictions for the simple case of axisymmetric particles are illustrated by calculations for glass particles in the shape of spheres, fibers and discs in an epoxy resin and are compared to those from the popular Chow theory. For spherical-shaped particles, the CTEs in all directions are the same and decrease modestly as the volume fraction of filler particles increases. As the particle aspect ratio increases from unity, the thermal expansion becomes anisotropic. The coefficient of longitudinal linear thermal expansion always decreases with increasing aspect ratio and filler loading due to the mechanical constraint of the filler. For aligned axisymmetric particles, the coefficients of linear thermal expansion are always the same in two directions. The values in the transverse direction may be higher or lower than that of the matrix depending on the values of aspect ratio and filler loading; these regions are mapped out for this particular set of matrix and filler properties. The two-dimensional constraints on matrix expansion caused by discs versus the one-dimensional effects of fibers cause quantitative differences in behavior for the two shapes.

Particle motion near and inside an interface

The motion of a spherical particle near the interface between two immiscible viscous fluids undergoing simple shear flow is considered in the limit of small Reynolds and capillary numbers where the interface exhibits negligible deformation. Taking advantage of the rotational symmetry of the boundaries of the flow with respect to the axis that is normal to the interface and passes through the particle centre, the problem is formulated as a system of one-dimensional integral equations for the first Fourier coefficients of the unknown components of the traction and velocity along the particle and interface contours. The results document the particle translational and angular velocities, and reveal that the particle slips while rolling over the interface under the influence of a simple shear flow, for any viscosity ratio. In the second part of the investigation, the motion of an axisymmetric particle straddling a planar interface is considered. The results confirm a simple exact solution when a particle with top-down symmetry is immersed half-way in each fluid and translates parallel to the interface, reveal a similar simple solution for a particle that is held stationary in simple shear flow, and document the force and torque exerted on a spherical particle for more general arrangements. The onset of a non-integrable singularity of the traction at the contact line prohibits the computation of the translational and angular velocities of a freely suspended particle convected under the action of a shear flow.

Motion of a rod-like particle between parallel walls with application to suspension rheology

We study the dynamics of elongated axisymmetric particles undergoing shear flow between two parallel planar walls, under creeping-flow conditions. Particles are modeled as linear chains of touching spheres and it is assumed that walls are separated by a distance comparable to particle length. The hydrodynamic interactions of the chains with the walls are evaluated using our Cartesian-representation algorithm [Bhattacharya et al., Physica A 356, 294–340 (2005b)]. We find that when particles are far from both walls in a weakly confined system, their trajectories are qualitatively similar to Jeffery orbits in unbounded space. In particular, the periods of the orbits and the evolution of the azimuthal angle in the flow-gradient plane are nearly independent of the initial orientation of the particle. For stronger confinements, however, (i.e., when the particle is close to one or both walls) a significant dependence of the angular evolution on the initial particle configuration is observed. The phases of partic...

Inertial effects on the orientation of nearly spherical particles in simple shear flow

We investigate theoretically the first effects of inertia on the orientation dynamics of a torque-free spheroidal particle in simple shear flow when the deviation from sphericity is small. The inertialess motion of any axisymmetric particle in simple shear represents a degenerate limit, the spheroidal geometry being a special case; as originally found by Jeffery (Proc. R. Soc. Lond. A, vol. 102, 1922, p. 161), the orientation vector moves indefinitely along any one of a single-parameter family of closed orbits centred around the vorticity axis, the distribution across orbits being determined by initial conditions. We consider both the inertia of the particle and that of the suspending fluid, characterized by the Stokes (.

Orientation dynamics of weakly Brownian particles in periodic viscous flows

Evolution equations for the orientation distribution of axisymmetric particles in periodic flows are derived in the regime of small but nonzero Brownian rotations. The equations are based on a multiple time scale approach that allows fast computation of the relaxation processes leading to statistical equilibrium. The approach has been applied to the calculation of the effective viscosity of a thin disk suspension in an oscillating strain flow.

Zoom quadrupole focusing systems producing an image of an object

The main difference between quadrupole and axisymmetric charge particle beam focusing systems is that quadrupole systems do not produce a regular image of an object like axisymmetric systems. Because of the special symmetry in geometry and electric or magnetic fields along the axis, only the Russian quadruplet (RQ) can produce an image, however, for each geometry and given maximum field gradient of the RQ, the number of experimentally attainable focal distances in one system is limited to one or two.A new quadrupole focusing system with the same symmetry as the RQ is now under development at the Louisiana Accelerator Center that produces an image and has the possibility to vary the focal distance over a wide range by changing the excitation of the middle lenses. As a result, the magnification may be changed from 10−3 to 10+3 so that this versatile design can be adapted for several applications of high-energy focusing ion beam (HEFIB) systems, including nuclear nanoprobe, scanning microscope or transmission microscope. The configuration of this lens system consists of six quadrupole lenses and although these lenses can be combined either as magnetic (RMS) or electrostatic (RES) sextuplets, the Louisiana Accelerator Center is presently constructing an RMS system due to operational requirements. The zoom capability of the RMS is numerically investigated and the smallest beam spot size and appropriate optimal parameters of the RMS are presented for different emittances and compared with the corresponding parameters of the RQ.

Inertial effects on fibre motion in simple shear flow

The motion of a torque-free slender axisymmetric fibre in simple shear flow is examined theoretically for small but finite ${\textit{Re}}$, where ${\textit{Re}}$ is the Reynolds number based on the fibre length, and is a measure of the inertial forces in the fluid. In the limit of zero inertia, an axisymmetric particle in simple shear is known to rotate indefinitely in any of an infinite single-parameter family of periodic orbits, originally found by Jeffery (1922) – a degenerate situation wherein the particular choice of orbit is dictated by the initial orientation of the particle. We use a generalization of the well-known reciprocal theorem for Stokes flow to derive the orbit equations, to $O({\textit{Re}})$, for the slender fibre. The structure of the equations bears some resemblance to those previously derived by Leal (1975) for a neutrally buoyant fibre in a viscoelastic (second-order) fluid. It is thereby shown that fluid inertia, for small ${\textit{Re}}$, leads to a slow $O({\textit{Re}})$ drift of the rotating fibre toward the shearing plane, thereby eliminating the aforementioned degeneracy. For Reynolds numbers above a critical value, ${\textit{Re}}_c\,{=}\, ({15}/{4 \pi})(\ln \kappa/\beta \kappa)\sin^{-2}\theta$, the fibre ceases to rotate, however, instead drifting monotonically towards the shearing plane. The limiting stationary orientation in the flow–gradient plane makes an angle $\phi_f$ with the flow direction, where $\phi_f \,{=}\, 4\pi {\textit{Re}}/(15 \ln \kappa) + \{16 \pi {\textit{Re}}^2/(225(\ln \kappa)^2-1/(\beta^2\kappa^2) \}^{1/2}$ is an increasing function of ${\textit{Re}}$. Here, $\kappa$ is the fibre aspect ratio, $\theta$ is the angle made by the fibre with the vorticity axis, and $\beta$ is an $O(1)$ coefficient related to the Jeffery period of the rotating fibre.

Parameter identification for axi-symmetric density-dependent groundwater flow based on drawdown and concentration data

A new tool for the simultaneous interpretation of a pumping and upconing test of saltwater is developed. The forward model is verified and a synthetic example checks the functioning of the inverse model. The new model interprets the movement of the transition zone between saltwater and freshwater towards the pumping well and the evolution of the drawdowns at different depths and different distances from the pumping well simultaneously. The drawdown and concentration observations are considered in their logarithmic space during the definition of their residuals. These two different types of data are interpreted by minimising a composite object function through a Gauss–Newton algorithm with sensitivities calculated by the perturbation method. The method is fast and relatively free of numerical dispersion because it uses an axi-symmetric grid and particle tracking for the solute transport.

Slow motion of axisymmetric slip particles along their axes of revolution

This paper presents a combined analytical–numerical study for the Stokes flow caused by a rigid particle of revolution translating axisymmetrically in a viscous fluid. The fluid, which may be a slightly rarefied gas, is allowed to slip at the surface of the particle. A singularity method based on the principle of distribution of a set of Sampson spherical singularities along the axis of revolution within a prolate particle or on the fundamental plane within an oblate particle is used to find the general solution for the fluid velocity field that satisfies the boundary condition at infinity. The slip condition on the surface of the translating particle is then satisfied by applying a boundary-collocation technique to this general solution to determine the unknown coefficients. The drag force exerted on the particle by the fluid is evaluated with good convergence behavior for various cases. For the motion of a slip sphere, our drag results agree very well with the exact solution. For the translation of a no-slip spheroid, prolate or oblate, along its axis of symmetry, the agreement between our results and the analytical solutions obtained by using spheroidal coordinates is also quite good. It is found that the normalized drag force on the translating spheroid increases monotonically with an increase in the axial-to-radial aspect ratio of the spheroid for a no-slip or slightly-slip spheroid, and decreases monotonically as the ratio increases for a well-slip spheroid. The drag force on a spheroid with intermediate values of its slip coefficient is not a monotonic function of its aspect ratio. For a spheroid with a given aspect ratio, its drag force is a monotonically decreasing function of the slip coefficient of the particle.

The effect of particle shape on scattering—A study with a collection of axisymmetric particles and sphere clusters

We study the effect of particle shape on the efficiency of light scattering. We show that the projected area of the particle has a key role in scattering, but the effect is different for different sized particles and changes systematically. We derive some practical results for industrial applications of pigments, but the results are also valuable for astronomy and atmospherical studies.

Formation of composite materials in gradient flows of suspensions

The rheological constitutive equations for dilute suspension of rigid axisymmetric elongated particles immersed in the Ericksen anisotropic fluid are obtained within the framework of a structure–phenomenological approach. As a hydrodynamic model of suspended particles possessing zero buoyancy, a symmetric triaxial dumbbell is used. The effect of anisotropy of the suspension carrier fluid is investigated on the dynamics of suspended particles and on the rheological properties of the suspension in simple shear flow. A stationary orientation of the elongated suspended particles under the action of hydrodynamic forces is found. These results offer the possibility of formation of composite materials by friction forces arising in gradient flows of a dilute suspension of axisymmetric rigid particles with anisotropic carrier fluid.

Generalized Taylor dispersion in suspensions of gyrotactic swimming micro-organisms

We calculate the average swimming velocity and dispersion rate characterizing the transport of swimming gyrotactic micro-organisms suspended in homogeneous (simple) shear. These are requisite effective phenomenological coefficients for the macroscale continuum modelling of bioconvection and related collective-dynamics phenomena. The swimming cells are modelled as rigid axisymmetric dipolar particles subject to stochastic Brownian rotations. Calculations are effected via application of the generalized Taylor dispersion scheme. Attention is focused on finite (as opposed to weak) shear. Results indicate that the largest transverse average swimming velocities (essential to gyrotactic focusing) appear shortly after transition from the ‘tumbling’ mode of motion to cells swimming in the equilibrium direction. At sufficiently large shear rates, dispersivity is not monotonically decreasing with external-field intensity. Exceptional dispersion rates which are unique to non-spherical cells appear in the ‘intermediate domain’ of external fields. These are rationalized in terms of the corresponding deterministic problem (i.e. in the absence of diffusion) when cell rotary motion is governed by the simultaneous coexistence of multiple stable attractors.

Phase behavior and structure of Janus fluids.

The equilibrium phase behavior of Janus fluids is examined based on a model potential for the interaction between their constituents. Janus fluids consist of axisymmetric particles possessing two different "faces," e.g., one hydrophobic and one hydrophilic surface, and the interaction depends on the relative orientation. Starting from a short range, isotropic potential we make an ansatz for an anisotropic model interaction potential. Two types of symmetries of the particles are considered. One leads to a polar phase. The Helmholtz free energy and the pressure are calculated by the help of an augmented van der Waals approximation. A qualitative phase diagram is obtained. The appearance of a polar phase and the corresponding transition temperature are examined adapting a Landau-de Gennes expansion of the orientational part of the free energy. Monte Carlo simulations are performed and the results are compared with the ones obtained by the analytical description.

Thermocapillary Coagulations of a Fluid Sphere and a Gas Bubble

This paper presents an analytical study of the thermocapillary motion of an adiabatic gas bubble and a liquid drop with constant temperature by using the method of reflections. The particles are allowed to differ in radius, and the droplet viscosity is arbitrary, which can be limited to a solid sphere. The Peclet and Reynolds numbers are assumed small, so that the temperature and flow fields are governed, respectively, by the Laplace and Stokes equations. The method of reflections is based on an analysis of the thermal and hydrodynamic disturbances produced by an insulated gas bubble and by a single drop with constant temperature, placed in an arbitrarily Laplacian temperature field. The results for two-sphere interactions are correct to O(r12-7), where r12 is the distance between the particle centers. The thermocapillary interactions between a gas bubble and a thermally uniform droplet are discussed in the situations of the prescribed temperature gradient parallel and/or normal to the particles' center line. In addition, the axisymmetric particle interactions in a medium with uniform applied temperature are formulated. The effect of interactions on thermocapillary migrations of an insulated gas bubble and a thermally uniform droplet is generally stronger than that on the motions influenced by the interaction of two gas bubbles or droplets. Meanwhile, the particle behaviors are quite different from those we know.

NUMERICAL STUDY OF SOUND RADIATION BY AXISYMMETRIC VORTEX RINGS

The sound production by vortex rings is investigated by means of an axisymmetric vortex particle method. The predictions are first calibrated by analyzing the noise generated by steady vortex rings that are described by the analytical solutions of Fraenkel and Norbury. The noise produced by isolated vortex rings for both nominally steady and unsteady cores is then analyzed. For nominally steady cores, computed results indicate that the efficiency of sound radiation decreases as the slenderness parameter is reduced, and the acoustic signals reveal a dominant period that is approximately half the eddy turnover time. For unsteady cores, the amplitude of the radiated sound is substantially higher than that of similar steady rings. When the initial core vorticity distribution is nonuniform, complex internal motion may also occur within the core which is also reflected in the corresponding far-field acoustic signal. Finally, the effect of vortex stretching is analyzed based on computations of two coaxial corotating vortex rings.