Equation for particle trajectories

Date: 2015-10-07; view: 421.

The equation of motion of fine particles in a fluid with an acoustic sound field is given from Eq. (7). In the literature, the inertia term x″ is neglected because it is small and the singular perturbation solution shown in Eq. (8) is used.

Using Eq. (8) forces the initial velocity to be k/c sin βx₀ (it is usually assumed = 0) and leads to errors in the solution that for very small time t, can be as large as k/c. The solution obtained by using the approximation equation (8) with initial value x(0) = x₀ can easily be obtained:

(29)

By separation of variables:

(29)

(30)

(31)

To obtain a more accurate approximation than the singular perturbation solution, let the assumed solution of Eq. (7) be of the form:

Substituting Eq. (32) in Eq. (7), one obtains:

(33)

Assuming that u is small, it follows from Eq. (30) that the terms in brackets above are small, and it is reasonable to approximate u by solving the following initial value problem:

(34)

Here it is assumed that x′(0) = v(0) = 0 in Eq. (7). The solution of Eq. (34) can be readily found by elementary means to be:

(35)

(36)

where,

(37)

(38)

and r₁ >>>r₂. It is therefore plausible that the following approximate solutions (t) are quite accurate for all t:

(39)

(40)

In fact, it can be proved rigorously that

(41)

and

(42)

Note also then that:

(43)

Therefore, particle trajectories in the acoustic medium can be calculated using Eq. (39) with the given values of constants (c, k, β). In graph Fig. 6, the vertical axis represents the time in seconds for the sound field treatment. The values on the horizontal axis represent the position of particles at corresponding times. The distances are in micrometers and for convenience taken at λ/16 of wavelength intervals from the first pressure anti-node at the face of the transducer. All the particles move toward the pressure node to which the particles are supposed to move. Particles with 2 μm diameter take about 20 s to reach the node. Fig. 5 shows where the sinks (pressure nodes) repeat at every 2π/β distance. Thus the behavior of the particle movement shown in Fig. 6 using the trajectory equation represents the actual behavior of the system.

Fig. 6. Particle trajectories.