S.Hassan and M.Kawaji: Vibration-Induced Particle Drift in a Fluid Cell under Microgravity __________________________________________________________________________________________________________________________________

© Z-Tec Publishing, Bremen Microgravity sci. technol. XIX-3/4 (2007)

A theoretical investigation into the effect of small vibrations

Samer Hassan and Masahiro Kawaji

Vibration-Induced Particle Drift in a FluidCell under Microgravity

Introduction on the behaviour of a small particle contained in a fluid cell

under microgravity is presented. Diffusion-controlled material processing such as protein crystal growth can be adversely affected by small vibrations called g-jitter, if a relative motion is induced between the particle and surrounding fluid. When a fluid cell containing a small

Many semi-conductor and protein crystal growth experiments conducted in the past aboard the Space Shuttle and Mir Space Station have yielded unexpected results possibly due to small vibrations existing on the space platforms called g-jitter. For example, protein crystals growing in small fluid cells under microgravity have been observed to move despite the absence of buoyancy-induced natural convection [1, 2]. Recent studies by Gamache and Kawaji [3] showed that protein crystals can be induced to move in the protein solution by small vibrations, and thus g-jitter may be able to change the diffusion controlled protein crystal growth (PCG) to convection dominated growth.

particle such as a protein crystal is vibrated parallel to the wall nearest to the particle, the particle oscillates with a certain amplitude and a hydrodynamic force in the direction normal to the wall is induced. Theoretical models based on an inviscid fluid assumption are used to predict the particle amplitude variation and drifting motion. Due to an external vibration such as g-jitter, the oscillating Previous papers by Hassan et al. [4, 5] described the effect

of external vibrations such as g-jitter on a particle contained in an infinite or semi-infinite fluid cell. The particle in a fluid cell would oscillate in response to the external vibration, and an inviscid model was developed to predict the vibration amplitude of the particle [4]. Hassan et al. [5] investigated the effect of a vessel wall on the motion of the particle vibrating normal to the nearest wall in a semi-infinite fluid cell. Hassan et al. [6] further showed that the particle vibration induces a hydrodynamic force which causes the particle to drift towards the nearest wall as it vibrates either in parallel with or normal to the nearest cell wall. All of their model predictions were validated by comparisons with experimental data obtained on the ground with a particle suspended in a fluid cell using a thin wire.

particle is predicted to drift towards the wall and the particle oscillation amplitude to decrease slightly as the distance between the particle and wall is reduced. The reduction in particle ampitude also depends on the particle-to-fluid density ratio. The particle drift towards the nearest wall acclerates due to an increasing attraction force, and the drifting speed increases with both the vibration frequency and particle diameter. Even for small protein crystals with a density close to that of the fluid, the time required to drift from the center of the fluid cell to the wall is predicted to be much shorter than the growth time. _______ Authors Samer Hassan and Masahiro Kawaji Dept. of Chemical Engineering & Applied Chemistry, University of Toronto, Toronto, Ontario M5S 3E5, Canada ______________ Correspondence Masahiro Kawaji Dept. of Chemical Engineering & Applied Chemistry, University of Toronto, Toronto, Ontario M5S 3E5, Canada Paper submitted: 16.07.07 Submission of final revised version: 11.08.07 Paper finally accepted: 23.08.07 Paper was presented on the Second International Topical Team Workshop on TWO-PHASE SYSTEMS FOR GROUND AND SPACE APPLICATIONS October 26-28, 2007, Kyoto, Japan.

Recently, Hassan and Kawaji [7] examined the vibration-induced particle motion in a fluid cell under microgravity by using an inviscid analysis. The inviscid fluid assumption could be justified if the following inequality is valid:

2oR2

fπν

>> (1)

where f is the vibration frequency, ν is the kinematic viscosity, and Ro is the particle radius. For example, for a particle of 1-mm diameter immersed in water, the lower limit of the cell vibration frequency at which Eq. (1) is valid would be given by f = 0.64 Hz. The g-jitter encountered aboard space platforms such as the ISS includes vibration frequencies much higher than 0.64 Hz [8]. At higher frequencies, the particle velocity

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S.Hassan and M.Kawaji: Vibration-Induced Particle Drift in a Fluid Cell under Microgravity __________________________________________________________________________________________________________________________________

Microgravity sci. technol. XIX-3/4 (2007)

increases and viscous effects become smaller. Thus, inviscid models for particle motion are expected to apply to many practical situations encountered in space.

In this paper, the vibration-induced motion of a particle in a fluid cell under microgravity is analyzed using the inviscid models previously developed by Hassan et al. [5, 6] for a semi-infinite fluid cell. An additional analysis of the wall effect on the particle amplitude for the case of parallel particle vibration by Hassan [9] will be used to predict the particle trajectory for different particle sizes.

Theoretical analysis of the particle motion

The analysis of the particle motion in an inviscid fluid of density, ρL, in a direction parallel to the nearest fluid cell wall using the method of images was performed by Hassan [9] and will be reported in detail in a separate paper. This method assumes a spherical particle of radius and density, ρoR S, initially located at a distance H between the particle centroid and the nearest cell wall such that the actual gap between the particle edge and the wall is H - . The other cell walls are considered to be sufficiently far away from the particle; thus, the particle motion analyzed would be that in a semi-infinite cell. The boundary conditions to be satisfied are those for the so-called “mirror image” flow fields.

oR

The semi-infinite cell is assumed to be vibrated at a frequency, f (Hz) or ω (=2πf) in radians/s, with an amplitude, a, in the direction parallel with the nearest wall. Using the potential flow theory, the velocity potential for the oscillating fluid surrounding the oscillating particle is first obtained. The pressure around the particle is calculated from the Navier-Stokes equations and integrated over the particle surface to obtain the effective force on the particle. By applying Newton’ second law to the particle of mass, m, Hassan [5] derived the amplitude of the particle vibrating in the direction parallel to the nearest cell wall as given by,

,

W1

R1

K Gρρ

)aρ(ρA

33o

LS

LSp

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛++−

−= (2)

where W = 2H, , and 3/22o

23o )R(W RG −=

3o

2/32o

2 R )R(W 2K −−= .

The above amplitude equation can be written in dimensionless form by introducing the dimensionless parameters for the fluid density, particle-to-wall distance, and particle and cell amplitudes as follows:

S

L

ρρρ~ = ,

oRHH~ = ,

o

pp R

AA~ = ,

oRaa~ = (3)

Using these dimensionless parameters, Equation (1) can be written as follows.

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛ +

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ −

+−

−=

3

3

2/3

2

2/3

2

p

H~811

H~161

H~411

H~411

2ρ~1

a~ )ρ~(1A~ (4)

The dimensionless particle amplitude would depend on the density ratio ρ~ between the particle and the surrounding fluid, the cell amplitude a~ , and the ratio of the particle-wall distance to the particle radius, . H~

Next, the attraction force for a particle vibrating in a direction parallel with the nearest wall was given by Hassan et al. [4] as follows.

4

6o

2

LS

LSL

22attraction H

Rρρ 2ρρ

ρω πa83F ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−= (5)

This force will cause the particle to drift towards the wall as per Newton’s second law, and the particle position, x, normal to the nearest wall is described by the following differential equation:

4

3o

2

LS

LS22

S

L2

2

HR

ρρ 2ρρ

ω a ρρ

329

dtxd

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−⎟⎟⎠

⎞⎜⎜⎝

⎛= (6)

The particle acceleration, d2x/dt2, is inversely proportional to the fourth power of the instantaneous distance, H, between the particle centroid and the nearest cell wall. Equation 6 can be numerically integrated from the starting time (t = 0) over small increments of time, Δt = 0.05 seconds.

Results and Discussion

Using Equations (2) or (4), (5) and (6), the particle vibration amplitude and drifting motion are calculated for a steel particle (ρs= 7,830 kg/m3) of radius Ro= 6.35 mm in water (ρf = 998 kg/m3) as shown in Fig. 1. The cell amplitude is assumed to be constant at a = 1.0 mm and vibration frequencies are set at 10, 15, or 20 Hz. The initial distance of the particle centroid to the nearest cell wall is Ho =11.0 mm. In Fig. 1, one can see clearly that the particle drift from the initial position increases in speed with time and the cell vibration frequency. For the highest vibration frequency of 20 Hz shown in Fig. 1, the particle displacement from the initial position in 1.4 seconds is 4.3 mm which is still less than the initial distance between the particle and the nearest cell wall of Ho - (= 4.65 mm). At a slightly later time just exceeding 1.4 seconds, the particle would hit the nearest cell wall.

oR

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S.Hassan and M.Kawaji: Vibration-Induced Particle Drift in a Fluid Cell under Microgravity __________________________________________________________________________________________________________________________________

Microgravity sci. technol. XIX-3/4 (2007)

Time (s) Fig. 1 Particle drift from the initial position (Ho =11 mm) for a cell amplitude of 1.0 mm and different vibration frequencies.

Figure 2 shows the variations of the particle vibration

amplitude with the distance between the particle and the nearest wall for a cell vibration amplitude of 1.0 mm and any vibration frequency. The results for two particle-fluid density ratios, = 7.85 and 1.11, are shown.

ρ~

Particle-to-wall distance, H (mm)

Particle-to-wall distance, H (mm) Fig. 2 Variation of particle amplitude with the particle-to- wall distance for ρ~ = 7.85 and 1.11

For both density ratios, the particle amplitude decreases slightly (1.1 % for = 7.85 and 5.7% for ρ~ = 1.11) as the particle drifts towards the wall. This is due to an increase in the added mass coefficient above 0.5 for a spherical particle as evident from Eq. 4 when the particle approaches the wall [5].

ρ~

0

0.5

1

5

2

5

3

5

4

4.5

5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

20 Hz15 Hz10 Hz

Parti

cle

drift

(mm

)

1.

2.

3.

For practical situations in microgravity such as protein crystal growth, a small particle-to-fluid density ratio close to unity is usually involved. Thus, calculations were performed for a density ratio of ρs/ρf =1.1, with ρs= 1,100 kg/m3 and ρf = 998 kg/m3, and particle diameters between 1.0 and 6.35mm. The drifting motion for different particle diameters for a vibration frequency of 20 Hz is shown in Fig. 3. It is observed to take less time for larger particles to drift and reach the wall compared to smaller particles due to a greater attraction force, Eq. 5.

0

5

1

5

2

5

3

5

4

5

5

0 20 40 60 80 100 120

Fig. 3. Particle drift with time for different particle diameters and a cell vibration amplitude of a=1.0 mm.

For smaller particle densities and lower vibration

frequencies, the attraction force is reduced and the time needed to reach the wall becomes greater as summarized in Table 1. However, even for a 1.0 mm diameter particle at a vibration frequency of 5 Hz, it is predicted to take less than 8 minutes to drift over a distance of 10.5 mm and reach the wall under sustained vibration conditions. Thus, in protein crystal growth in a confined PCG cell, g-jitter could cause even small protein crystals to move in a preferential direction towards the nearest cell wall over crystallization periods of many hours or days.

0.

1.

2.

3.

4.

Parti

cle

drift

(mm

)

808

810

812

814

816

818

820

(a) ρ~ = 7.85

Parti

cle

ampl

itude

(μm

)

Dia: 1 mmDia: 2 mmDia: 4 mmDia: 6.35 mm

Time (s) 6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6

58

58.5

59

59.5

60

60.5

61

61.5

62

6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6

(b) ρ~ = 1.11

Parti

cle

ampl

itude

(μm

)

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S.Hassan and M.Kawaji: Vibration-Induced Particle Drift in a Fluid Cell under Microgravity __________________________________________________________________________________________________________________________________

Microgravity sci. technol. XIX-3/4 (2007)

Table 1. Collision time in seconds of the particle with the wall for different particle diameters and vibration frequencies. Particle density: 1,100 kg /m3, Fluid density: 998 kg /m3; Ho=11.0 mm.

Conclusion A study of vibration-induced particle motion in a fluid cell under microgravity has been conducted theoretically. Inviscid models of the particle motion and attraction force were used to predict the trajectory of a particle oscillating in parallel with the nearest cell wall. The particle was predicted to accelerate its drifting motion towards the wall, and the particle vibration amplitude was predicted to slightly decrease as it drifted towards the wall. The effects of the particle diameter, cell vibration frequency and particle and fluid densities have been analyzed theoretically to predict the time needed for the particle to reach the nearest cell wall.

Acknowledgements

This work was financially supported by a contract from the Canadian Space Agency. One of the authors (S. Hassan) also received a graduate fellowship from the Government of Ontario.

References: [1] Chayen, N., Snell, E. H., Helliwell, J. R., and Zagalsky, P. F.,

1997, “CCD Video Observation of Microgravity Crystallization”. J. Crystal Growth, 171(1), 219–225.

[2] Lorber B., Ng J. D., Lautenschlager P., and Giege R., 2000, “Growth Kinetics and Motion of Thaumatin Crystals during USML-2 and LMS Microgravity Missions and Comparison with Earth Controls”. J. Crystal Growth, 208, 665 – 677.

Particle diameter (mm)

[3] Gamache, O. and Kawaji, M., 2005,“Experimental Investigation of Marangoni Convection and Vibration-Induced Crystal Motion during Protein Crystal Growth”. Microgravity Science and Technology, XVI-I, 342-347.

[4] Hassan, S., Kawaji, M., Lyubimova, T.P. and Lyubimov, D.V., 2005,“Motion of a Sphere Suspended in a Vibrating Liquid-Filled Container”. ASME Journal of Applied Mechanics, 73(1), 72-78.

[5] Hassan, S., Kawaji, M., Lyubimova, T.P. and Lyubimov, D.V., 2005,“The Effects of Vibrations near a Wall in a Semi-infinite Cell”. ASME Journal of Applied Mechanics, 73(4), 610-621.

[6] Hassan, S., Kawaji, M., Lyubimova, T.P. and Lyubimov, D.V., 2006,“ Effects of Vibrations on Particle Motion near a Wall: Existence of Attraction Force”. Int. J. Multiphase Flow, 32, 1037-1054.

[7] Hassan, S. And Kawaji, M., 2007, “Wall Effects on Vibration-Induced Particle Motion in a Fluid Cell in Space”. AIAA journal, 45(8), 2090-2092.

[8] Jules, K., McPherson, K., Hrovat, K., Kelly, E., and Reckart, T., 2004, “A Status Report on the Characterization of the Microgravity Environment of the International Space Station,” Acta Astronautica, 55(3–9), 335–364.

[9] Hassan, S. 2005, “Experimental and Theoretical study of Vibration-Induced Particle Motion in Normal Gravity and under Microgravity”. Ph.D. Thesis, Department of Chemical Engineering and Applied Chemistry, University of Toronto.

Vibration

1.0 2.0 4.0 6.35

Frequency (Hz)

5 440 156 55 27.5 10 220 77.8 27.5 13.8 20 110 38.9 13.8 6.9

112