## Mean force on a finite-sized rigid particle, droplet, or bubble in a viscous compressible medium

## Abstract

In this paper, a force formulation to compute the axial acoustic mean second-order force on finite-sized compressible and rigid particles is presented. The flow inside and outside the spherical inclusion is considered viscous and compressible. Other than for volumetric pulsations of the bubble/droplet, the sphericity of the inclusion is maintained (taken to be unity). A far-field derivation approach has been used to compute the force due to standing and traveling waves; and the force is expressed as a multipole expansion (infinite series). In case of a bubble and a rigid particle, there exist three length scales that govern the mean second-order force: mean radius of the spherical inclusion (R _{0}), wavelength of the incoming acoustics (λ), and the momentum diffusion thickness of the ambient fluid (δ ^{o}). While R _{0} and λ are arbitrary, we assume the viscous length scale is negligibly small compared to the acoustic wavelength. In case of a droplet, however, the following additional parameters (inside to outside fluid ratios) also play a role: density ratio ($$\sim\atop{p}$$), viscosity ratio ($$\sim\atop{μ}$$, and speed of sound ratio ($$\sim\atop{c}$$). The force expression yields the correct behavior in several limiting cases considered: (i) inviscid bubble and droplets with R _{0}/λ « 1, (ii) inviscid bubbles with finite R _{0}/λ, and (iii) finite size rigid immovable particles. In general, while the monopole alone is sufficient to capture the force for small bubbles, higher-order terms are found to be important when R _{0}/λ ≥ 0.02. In addition to reporting similar behavior for droplets, we study the effect arising from $$\sim\atop{p}$$, $$\sim\atop{μ}$$, $$\sim\atop{c}$$, and δ ^{o}/R _{0} on the mean second-order force.

- Authors:

- Univ. of Florida, Gainesville, FL (United States). Department of Mechanical and Aerospace Engineering

- Publication Date:

- Research Org.:
- Univ. of Florida, Gainesville, FL (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1469559

- Alternate Identifier(s):
- OSTI ID: 1224221

- Grant/Contract Number:
- NA0002378

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Fluids

- Additional Journal Information:
- Journal Volume: 27; Journal Issue: 10; Journal ID: ISSN 1070-6631

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 42 ENGINEERING

### Citation Formats

```
Annamalai, Subramanian, and Balachandar, S. Mean force on a finite-sized rigid particle, droplet, or bubble in a viscous compressible medium. United States: N. p., 2015.
Web. doi:10.1063/1.4933051.
```

```
Annamalai, Subramanian, & Balachandar, S. Mean force on a finite-sized rigid particle, droplet, or bubble in a viscous compressible medium. United States. doi:10.1063/1.4933051.
```

```
Annamalai, Subramanian, and Balachandar, S. Fri .
"Mean force on a finite-sized rigid particle, droplet, or bubble in a viscous compressible medium". United States. doi:10.1063/1.4933051. https://www.osti.gov/servlets/purl/1469559.
```

```
@article{osti_1469559,
```

title = {Mean force on a finite-sized rigid particle, droplet, or bubble in a viscous compressible medium},

author = {Annamalai, Subramanian and Balachandar, S.},

abstractNote = {In this paper, a force formulation to compute the axial acoustic mean second-order force on finite-sized compressible and rigid particles is presented. The flow inside and outside the spherical inclusion is considered viscous and compressible. Other than for volumetric pulsations of the bubble/droplet, the sphericity of the inclusion is maintained (taken to be unity). A far-field derivation approach has been used to compute the force due to standing and traveling waves; and the force is expressed as a multipole expansion (infinite series). In case of a bubble and a rigid particle, there exist three length scales that govern the mean second-order force: mean radius of the spherical inclusion (R0), wavelength of the incoming acoustics (λ), and the momentum diffusion thickness of the ambient fluid (δo). While R0 and λ are arbitrary, we assume the viscous length scale is negligibly small compared to the acoustic wavelength. In case of a droplet, however, the following additional parameters (inside to outside fluid ratios) also play a role: density ratio ($\sim\atop{p}$), viscosity ratio ($\sim\atop{μ}$, and speed of sound ratio ($\sim\atop{c}$). The force expression yields the correct behavior in several limiting cases considered: (i) inviscid bubble and droplets with R0/λ « 1, (ii) inviscid bubbles with finite R0/λ, and (iii) finite size rigid immovable particles. In general, while the monopole alone is sufficient to capture the force for small bubbles, higher-order terms are found to be important when R0/λ ≥ 0.02. In addition to reporting similar behavior for droplets, we study the effect arising from $\sim\atop{p}$, $\sim\atop{μ}$, $\sim\atop{c}$, and δo/R0 on the mean second-order force.},

doi = {10.1063/1.4933051},

journal = {Physics of Fluids},

number = 10,

volume = 27,

place = {United States},

year = {2015},

month = {10}

}

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Works referenced in this record:

##
Standing‐wave acoustic trap for nonintrusive positioning of microparticles

journal, October 1995

- Hertz, H. M.
- Journal of Applied Physics, Vol. 78, Issue 8, p. 4845-4849

##
Theory of long wavelength acoustic radiation pressure

journal, October 1991

- Löfstedt, Ritva; Putterman, Seth
- The Journal of the Acoustical Society of America, Vol. 90, Issue 4, p. 2027-2033

##
Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity

journal, November 1985

- Trinh, E. H.
- Review of Scientific Instruments, Vol. 56, Issue 11, p. 2059-2065

##
Mean force on a small sphere in a sound field in a viscous fluid

journal, January 2000

- Danilov, S. D.; Mironov, M. A.
- The Journal of the Acoustical Society of America, Vol. 107, Issue 1, p. 143-153

##
Acoustic radiation force on a spherical particle in a viscous heat-conducting fluid. III. Force on a liquid drop

journal, February 1997

- Doinikov, Alexander A.
- The Journal of the Acoustical Society of America, Vol. 101, Issue 2, p. 731-740

##
Acoustic Radiation Pressure on a Rigid Sphere in a Viscous Fluid

journal, December 1994

- Doinikov, A. A.
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 447, Issue 1931, p. 447-466