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The study of SLE in multiply connected domains is very interesting. Their conformal classes are characterised by a set of moduli, which change as the curve grows. Friedrich and Kalkkinen [40] have argued that SLE in such a domain is characterised by diffusion in moduli space as well as diffusion on the boundary.

It is possible to rewrite the differential equations which arise from null state conditions in extended CFTs (for example super-conformal CFTs [41] and WZWN models [42]) in terms of the generators of stochastic conformal mappings which generalise that of Loewner. However, a physical interpretation in terms of the continuum limit of lattice curves appears so far to be missing.

6.3. Other growth models

SLE is in fact just one very special, solvable, example of an approach to growth processes in two dimensions using conformal mappings which has been around for a number of years. For a recent review see [43]. The prototypical problem of this type is diffusion-limited aggregation (DLA). In this model of cluster formation, particles of finite radius diffuse in, one by one, from infinity until they hit the existing cluster, where they stick. The probability of sticking at a given point is proportional to the local electric field, if we imagine the cluster as being charged. The resultant highly branched structures are very similar to those observed in smoke particles, and in viscous fingering experiments where one fluid is forced into another in which it is immiscible. Hastings and Levitov [44] proposed an approach to this problem using conformal mappings. At each time t, the boundary of the cluster is described by the conformal mapping f_t (z) which takes it to the unit disc. The cluster is grown by adding a small semicircular piece to the boundary. The way this changes f_t is well known according to a theorem of Hadamard. The difficulty is that the probability of adding this piece at a given point depends on the local electric field which itself depends on . The equation of motion for f_t is therefore more complicated than in SLE. Moreover, it may be shown that almost all initially smooth boundary curves evolve towards a finite-time singularity: this is thought to be responsible for branching, but just at this point the equations must be regularised to reflect the finite size of the particles (or, in viscous fingering, the effects of finite surface tension.)

It is also possible to generate branching structures by making the driving term a_t in Loewner’s equation discontinuous, for example taking it to be a Levy process. Unfortunately this does not appear to describe a physically interesting model.

Finally, Hastings [45] has proposed two related growth models which each lead, in the continuum limit, to SLE. These are very similar to DLA, except that growth is only allowed at the tip. The first, called the arbitrary Laplacian random walk, takes place on the lattice. The tip moves to one of the neighbouring unoccupied sites r with relative probability E (r)^η, where E (r) is the lattice electric field, that is the potential difference between the tip and r, and η is a parameter. The second growth model takes place in the continuum via iterated conformal mappings, in which pieces of length ℓ₁ are added to the tip, but shifted to the left or right relative to the previous growth direction by a random amount ±ℓ₂. This model depends on the ratio ℓ₂/ℓ₁, and leads, in the continuum limit, to SLE_κ with κ = ℓ₂/4ℓ₁. For the lattice model there is no universal relation between κ and η, except for η = 1, which is the same as the loop-erased random walk (Section 2.2) and converges to SLE₂.

Mathematical modeling of the movement of suspended particles subjected to acoustic and flow fields

N. Aboobaker^a, D. Blackmore^b and J. Meegoda<sup>c, ,

a</sup>Bureau of Research, New Jersey Department of Transportation, Trenton, NJ, USA
^bDepartment of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA
^cDepartment of Civil and Environmental Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA

Received 1 July 2002;  revised 1 August 2004;  accepted 6 September 2004.  Available online 1 December 2004.

Abstract

When particles are subjected to an acoustic field particle trajectories depend on the particle and fluid compressibility and density values. Hence a combination of acoustic and flow fields on particles can be used to deflect and trap, or to segregate and/or fractionate fine particles in fluid suspensions. Using particle physics in an acoustic field, a mathematical model was developed to calculate trajectories of deflected particles due to the application of acoustic standing waves. The resulting second order ordinary differential equation was quite stiff and hence difficult to solve numerically and did not have a closed form solution. The analysis of the above equation showed that the basic problems with numerical solutions could not be ameliorated through the use of standard rescaling techniques. A combination of phase space and asymptotic analysis turns out to be far more useful in obtaining approximate solutions. An approximate solution was derived which enabled the calculation of the particle trajectories and concentration at collection planes in the acoustic field. Analysis of the solution showed that all the particles move toward the pressure node to which the particles are supposed to move. Particles with 2 μm diameter took approximately 20 s to reach that node. Then at the bench scale, the above technology was implemented by building a flow chamber with two transducers at opposite ends to generate an acoustic standing wave. SiC particle trajectories were tracked using captured digital images from a high-resolution microscope. The displacements of SiC particles due to an acoustic force were compared with the mathematical model predictions. For input power levels between 3.0 and 5.0 W, the experimental data were comparable to mathematical model predictions. Hence it was concluded that the proposed approximate solution was both quantitatively and qualitatively closer to experimental results than the simplified form ignoring the second order term reported in the literature.

Keywords: Standing wave; Segregation; Fractionation; Acoustic field; Mathematical model; Particle trajectories; Approximate solution; Silicon carbide

Article Outline

1. Introduction

1.1. Acoustic force field

1.2. Primary axial acoustic force

1.3. Primary and secondary acoustic force

2. Application of Newton’s second law

3. Mathematical model

3.1. Preliminary analysis

4. Equation for particle trajectories

5. Concentration equation

6. Experimental procedure and results

6.1. SiC particle trajectories in an acoustic field

7. Comparison between experimental results and mathematical model

8. Summary and conclusions

Acknowledgements

References

1. Introduction

It is difficult to fractionate ultra-fine particles and particles with neutral buoyancy, or uniform electro-magnetically charged surfaces. Existing methods are extremely slow or require prohibitive pressure drops, or extremely high electric or magnetic fields. Hence this exploratory research (cf. [1] and [2]) using acoustic and flow fields was conducted to evaluate the feasibility of fractionating ultra fine suspended particles and at the same time segregating them. In this technology the particle movements due to density and compressibility differences between fluid and particles rather than the particle size are used to fractionate ultra-fine suspended particles. In the following section, the basic derivation for particles in equilibrium in an acoustic field is given, which will be used for the derivation of particle trajectory and concentration equations.

1.1. Acoustic force field

King and Macdonald [3], proposed a mathematical model for the force acting on a spherical particle suspended in a standing acoustic wave field. This analysis was restricted to a rigid particle with a radius much smaller than the wavelength of sound and for a standing wave field created by oppositely travelling, single frequency, and sinusoidal waves. Fig. 1 shows the forces acting on a particle in an acoustic field. These forces are:

1. Primary axial acoustic radiation force (F_PARF)

2. Primary transverse acoustic radiation force (F_PTRF), and

3. Secondary acoustic radiation force (F_S)

Fig. 1. Acoustic forces on a particle.

1.2. Primary axial acoustic force

Almost immediately following the application of an acoustic field, particles experience a time-averaged primary axial acoustic force, F_ac (F_PARF), which is generated by the interaction between particles and the primary wave field. The primary axial radiation force drives dispersed particles toward the velocity antinodes of the resonance field. The magnitude of the force depends on the difference in compressibility and density between the particle and the medium. The primary acoustic force was derived by Yosioka and Kawasima [4] and given below.

The acoustic energy density, E_ac, is a measure of the energy residing in a wave field. V₀ is the volume of one particle, k = 2π/λ is the wave number of the acoustic radiation, λ is the sound wavelength, x is the axial distance from a pressure node, and k is the unit vector in the axial direction. The response of any solid suspended in a fluid, to a resonant acoustic field, depends on the acoustic contrast factor, G. For any solid particle of size r   λ suspended in a fluid, the acoustic contrast factor is given by:

where, β_f and β_p are the compressibility of the fluid and particle respectively, and ρ_f and ρ_p are the densities of the fluid and particle respectively.

1.3. Primary and secondary acoustic force

The primary transverse force acts normal to the wave propagation. Of the three forces, the primary axial radiation force generally has the greatest magnitude.

2. Application of Newton’s second law

The proposed technology can be implemented in a rectangular narrow channel of upward fluid flow. Two walls of the channel can be made with a piezoelectric transducer and rigid reflecting surface, as shown in Fig. 2. When the transducer is energized at the proper frequency to maintain a resonant acoustic field, there will be a pressure node located on the mid-plane of the chamber, and pressure antinodes located on the chamber walls. The acoustic force on a suspended particle results from the particle–fluid interaction that arises when the particle and suspending fluid have different acoustic properties. When the particle is at the pressure node at quarter wavelength, mid-plane of the chamber width, the magnitude of this force is the maximum. For a dilute suspension, the secondary radiation forces, the body forces and the hydrodynamic interactions are neglected. The rate of change of particle momentum is equal to:

where v is the particle velocity and g is the gravitational acceleration. The mass in the momentum term is the “virtual mass” of the particle, m. Since the particle’s velocity is v, the drag force is given by Stokes’ law (F_D = −6πμrv), where μ is the viscosity of the fluid and r is the radius of the particle in suspension. The summation of the forces in the direction of the acoustic wave propagation gives:

Thus the acoustic force, F_ac, on a particle in an acoustic field is due to the primary axial radiation force and can be used to calculate the particle trajectories.