Abstract

The optical force on a non-spherical particle subjected to a loosely focused laser beam was calculated using the dynamic ray tracing method. Ellipsoidal particles with different aspect ratios, inclination angles, and positions were modeled, and the effects of these parameters on the optical force were examined. The vertical component of the optical force parallel to the laser beam axis decreased as the aspect ratio decreased, whereas the ellipsoid with a small aspect ratio and a large inclination angle experienced a large vertical optical force. The ellipsoids were pulled toward or repelled away from the laser beam axis, depending on the inclination angle, and they experienced a torque near the focal point. The behavior of the ellipsoids in a viscous fluid was examined by analyzing a dynamic simulation based on the penalty immersed boundary method. As the ellipsoids levitated along the direction of the laser beam propagation, they moved horizontally with rotation. Except for the ellipsoid with a small aspect ratio and a zero inclination angle near the focal point, the ellipsoids rotated until the major axis aligned with the laser beam axis.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
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2012 (1)

2011 (5)

C. G. Hebert, A. Terray, and S. J. Hart, “Toward label-free optical fractionation of blood-optical force measurements of blood cells,” Anal. Chem.83(14), 5666–5672 (2011).
[CrossRef] [PubMed]

S. H. Simpson and S. Hanna, “Computational study of the optical trapping of ellipsoidal particles,” Phys. Rev. A84(5), 053808 (2011).
[CrossRef]

W.-X. Huang, C. B. Chang, and H. J. Sung, “An improved penalty immersed boundary method for fluid-flexible body interaction,” J. Comput. Phys.230(12), 5061–5079 (2011).
[CrossRef]

M. Kinnunen, A. Kauppila, A. Karmenyan, and R. Myllylä, “Effect of the size and shape of a red blood cell on elastic light scattering properties at the single-cell level,” Biomed. Opt. Express2(7), 1803–1814 (2011).
[CrossRef] [PubMed]

Z. J. Li, Z. S. Wu, and Q. C. Shang, “Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam,” Opt. Express19(17), 16044–16057 (2011).
[CrossRef] [PubMed]

2010 (1)

2009 (1)

S. B. Kim, K. H. Lee, H. J. Sung, and S. S. Kim, “Nonlinear particle behavior during cross-type optical particle separation,” Appl. Phys. Lett.95(26), 264101 (2009).
[CrossRef]

2008 (2)

S. B. Kim, S. Y. Yoon, H. J. Sung, and S. S. Kim, “Cross-type optical particle separation in a microchannel,” Anal. Chem.80(7), 2628–2630 (2008).
[CrossRef] [PubMed]

A. Jonás and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis29(24), 4813–4851 (2008).
[CrossRef] [PubMed]

2007 (3)

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A24(2), 430–443 (2007).
[CrossRef] [PubMed]

S. Yan and B. Yao, “Transverse trapping forces of focused Gaussian beam on ellipsoidal particles,” J. Opt. Soc. Am. B24(7), 1596–1602 (2007).
[CrossRef]

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(2), 026613 (2007).
[CrossRef] [PubMed]

2006 (2)

2005 (2)

S. Mohanty, K. S. Mohanty, and P. K. Gupta, “Dynamics of Interaction of RBC with optical tweezers,” Opt. Express13(12), 4745–4751 (2005).
[CrossRef] [PubMed]

M. Uhlmann, “An immersed boundary method with direct forcing for the simulation of particulate flows,” J. Comput. Phys.209(2), 448–476 (2005).
[CrossRef]

2004 (1)

2003 (1)

S. J. Hart and A. V. Terray, “Refractive-index driven separation of colloidal polymer particles using optical chromatography,” Appl. Phys. Lett.83(25), 5316–5318 (2003).
[CrossRef]

2002 (2)

C. S. Peskin, “The immersed boundary method,” Acta Numer.11, 479–517 (2002).
[CrossRef]

K. Kim, S.-J. Baek, and H. J. Sung, “An implicit velocity decoupling procedure for incompressible Navier-Stokes equations,” Int. J. Numer. Methods Fluids38(2), 125–138 (2002).
[CrossRef]

2001 (1)

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J.81(2), 767–784 (2001).
[CrossRef] [PubMed]

2000 (3)

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron.6(6), 841–856 (2000).
[CrossRef]

S. C. Grover, R. C. Gauthier, and A. G. Skirtach, “Analysis of the behaviour of erythrocytes in an optical trapping system,” Opt. Express7(13), 533–539 (2000).
[CrossRef] [PubMed]

F. Cirak, M. Ortiz, and P. Schroder, “Subdivision surfaces: a new paradigm for thin-shell finite-element analysis,” Int. J. Numer. Methods Eng.47, 2039–2072 (2000).
[CrossRef]

1997 (2)

R. C. Gauthier, “Trapping model for the low-index ring-shaped micro-object in a focused, lowest-order Gaussian laser-beam profile,” J. Opt. Soc. Am. B14(4), 782–789 (1997).
[CrossRef]

T. Moller and B. Trumbore, “Fast, minimum storage ray-triangle intersection,” J. Graphics, GPU, Games Tools2, 21–28 (1997).

1996 (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun.124(5-6), 529–541 (1996).
[CrossRef]

1995 (2)

R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of the force equations,” J. Opt. Soc. Am. B12(9), 1680–1686 (1995).
[CrossRef]

R. C. Gauthier, “Ray optics model and numerical computations for the radiation pressure micromotor,” Appl. Phys. Lett.67(16), 2269–2271 (1995).
[CrossRef]

1994 (1)

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23(1), 247–285 (1994).
[CrossRef] [PubMed]

1992 (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J.61(2), 569–582 (1992).
[CrossRef] [PubMed]

1991 (1)

S. Sato, M. Ishigure, and H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett.27(20), 1831–1832 (1991).
[CrossRef]

1986 (1)

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett.24(4), 156–159 (1970).
[CrossRef]

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J.81(2), 767–784 (2001).
[CrossRef] [PubMed]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun.124(5-6), 529–541 (1996).
[CrossRef]

Ashkin, A.

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron.6(6), 841–856 (2000).
[CrossRef]

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J.61(2), 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett.11(5), 288–290 (1986).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett.24(4), 156–159 (1970).
[CrossRef]

B Bareil, P.

Baek, S.-J.

K. Kim, S.-J. Baek, and H. J. Sung, “An implicit velocity decoupling procedure for incompressible Navier-Stokes equations,” Int. J. Numer. Methods Fluids38(2), 125–138 (2002).
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23(1), 247–285 (1994).
[CrossRef] [PubMed]

Cai, X. S.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(2), 026613 (2007).
[CrossRef] [PubMed]

Chang, C. B.

W.-X. Huang, C. B. Chang, and H. J. Sung, “An improved penalty immersed boundary method for fluid-flexible body interaction,” J. Comput. Phys.230(12), 5061–5079 (2011).
[CrossRef]

Chiou, A.

Chu, S.

Cirak, F.

F. Cirak, M. Ortiz, and P. Schroder, “Subdivision surfaces: a new paradigm for thin-shell finite-element analysis,” Int. J. Numer. Methods Eng.47, 2039–2072 (2000).
[CrossRef]

Cunningham, C. C.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J.81(2), 767–784 (2001).
[CrossRef] [PubMed]

Dziedzic, J. M.

Eggleton, C. D.

Gao, B. Z.

Gauthier, R. C.

Gouesbet, G.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(2), 026613 (2007).
[CrossRef] [PubMed]

Gréhan, G.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(2), 026613 (2007).
[CrossRef] [PubMed]

Grover, S. C.

Guck, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J.81(2), 767–784 (2001).
[CrossRef] [PubMed]

Gupta, P. K.

Hanna, S.

S. H. Simpson and S. Hanna, “Computational study of the optical trapping of ellipsoidal particles,” Phys. Rev. A84(5), 053808 (2011).
[CrossRef]

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A24(2), 430–443 (2007).
[CrossRef] [PubMed]

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun.124(5-6), 529–541 (1996).
[CrossRef]

Hart, S. J.

C. G. Hebert, A. Terray, and S. J. Hart, “Toward label-free optical fractionation of blood-optical force measurements of blood cells,” Anal. Chem.83(14), 5666–5672 (2011).
[CrossRef] [PubMed]

S. J. Hart and A. V. Terray, “Refractive-index driven separation of colloidal polymer particles using optical chromatography,” Appl. Phys. Lett.83(25), 5316–5318 (2003).
[CrossRef]

Hebert, C. G.

C. G. Hebert, A. Terray, and S. J. Hart, “Toward label-free optical fractionation of blood-optical force measurements of blood cells,” Anal. Chem.83(14), 5666–5672 (2011).
[CrossRef] [PubMed]

Huang, W.-X.

W.-X. Huang, C. B. Chang, and H. J. Sung, “An improved penalty immersed boundary method for fluid-flexible body interaction,” J. Comput. Phys.230(12), 5061–5079 (2011).
[CrossRef]

Inaba, H.

S. Sato, M. Ishigure, and H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett.27(20), 1831–1832 (1991).
[CrossRef]

Ishigure, M.

S. Sato, M. Ishigure, and H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett.27(20), 1831–1832 (1991).
[CrossRef]

Jonás, A.

A. Jonás and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis29(24), 4813–4851 (2008).
[CrossRef] [PubMed]

Karmenyan, A.

Käs, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J.81(2), 767–784 (2001).
[CrossRef] [PubMed]

Kauppila, A.

Kim, K.

K. Kim, S.-J. Baek, and H. J. Sung, “An implicit velocity decoupling procedure for incompressible Navier-Stokes equations,” Int. J. Numer. Methods Fluids38(2), 125–138 (2002).
[CrossRef]

Kim, S. B.

S. B. Kim, K. H. Lee, H. J. Sung, and S. S. Kim, “Nonlinear particle behavior during cross-type optical particle separation,” Appl. Phys. Lett.95(26), 264101 (2009).
[CrossRef]

S. B. Kim, S. Y. Yoon, H. J. Sung, and S. S. Kim, “Cross-type optical particle separation in a microchannel,” Anal. Chem.80(7), 2628–2630 (2008).
[CrossRef] [PubMed]

S. B. Kim and S. S. Kim, “Radiation forces on spheres in loosely focused Gaussian beam: ray-optics regime,” J. Opt. Soc. Am. B23(5), 897–903 (2006).
[CrossRef]

Kim, S. S.

S. B. Kim, K. H. Lee, H. J. Sung, and S. S. Kim, “Nonlinear particle behavior during cross-type optical particle separation,” Appl. Phys. Lett.95(26), 264101 (2009).
[CrossRef]

S. B. Kim, S. Y. Yoon, H. J. Sung, and S. S. Kim, “Cross-type optical particle separation in a microchannel,” Anal. Chem.80(7), 2628–2630 (2008).
[CrossRef] [PubMed]

S. B. Kim and S. S. Kim, “Radiation forces on spheres in loosely focused Gaussian beam: ray-optics regime,” J. Opt. Soc. Am. B23(5), 897–903 (2006).
[CrossRef]

Kinnunen, M.

Lee, K. H.

S. B. Kim, K. H. Lee, H. J. Sung, and S. S. Kim, “Nonlinear particle behavior during cross-type optical particle separation,” Appl. Phys. Lett.95(26), 264101 (2009).
[CrossRef]

Li, Y. M.

Li, Z. J.

Mahmood, H.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J.81(2), 767–784 (2001).
[CrossRef] [PubMed]

Marr, D. W. M.

Mohanty, K. S.

Mohanty, S.

Moller, T.

T. Moller and B. Trumbore, “Fast, minimum storage ray-triangle intersection,” J. Graphics, GPU, Games Tools2, 21–28 (1997).

Moon, T. J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J.81(2), 767–784 (2001).
[CrossRef] [PubMed]

Myllylä, R.

Nahmias, Y. K.

Odde, D. J.

Ortiz, M.

F. Cirak, M. Ortiz, and P. Schroder, “Subdivision surfaces: a new paradigm for thin-shell finite-element analysis,” Int. J. Numer. Methods Eng.47, 2039–2072 (2000).
[CrossRef]

Peskin, C. S.

C. S. Peskin, “The immersed boundary method,” Acta Numer.11, 479–517 (2002).
[CrossRef]

Ren, K. F.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(2), 026613 (2007).
[CrossRef] [PubMed]

Sato, S.

S. Sato, M. Ishigure, and H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett.27(20), 1831–1832 (1991).
[CrossRef]

Schroder, P.

F. Cirak, M. Ortiz, and P. Schroder, “Subdivision surfaces: a new paradigm for thin-shell finite-element analysis,” Int. J. Numer. Methods Eng.47, 2039–2072 (2000).
[CrossRef]

Shang, Q. C.

Sheng, Y.

Simpson, S. H.

S. H. Simpson and S. Hanna, “Computational study of the optical trapping of ellipsoidal particles,” Phys. Rev. A84(5), 053808 (2011).
[CrossRef]

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A24(2), 430–443 (2007).
[CrossRef] [PubMed]

Skirtach, A. G.

Sraj, I.

Sung, H. J.

W.-X. Huang, C. B. Chang, and H. J. Sung, “An improved penalty immersed boundary method for fluid-flexible body interaction,” J. Comput. Phys.230(12), 5061–5079 (2011).
[CrossRef]

S. B. Kim, K. H. Lee, H. J. Sung, and S. S. Kim, “Nonlinear particle behavior during cross-type optical particle separation,” Appl. Phys. Lett.95(26), 264101 (2009).
[CrossRef]

S. B. Kim, S. Y. Yoon, H. J. Sung, and S. S. Kim, “Cross-type optical particle separation in a microchannel,” Anal. Chem.80(7), 2628–2630 (2008).
[CrossRef] [PubMed]

K. Kim, S.-J. Baek, and H. J. Sung, “An implicit velocity decoupling procedure for incompressible Navier-Stokes equations,” Int. J. Numer. Methods Fluids38(2), 125–138 (2002).
[CrossRef]

Svoboda, K.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23(1), 247–285 (1994).
[CrossRef] [PubMed]

Szatmary, A. C.

Terray, A.

C. G. Hebert, A. Terray, and S. J. Hart, “Toward label-free optical fractionation of blood-optical force measurements of blood cells,” Anal. Chem.83(14), 5666–5672 (2011).
[CrossRef] [PubMed]

Terray, A. V.

S. J. Hart and A. V. Terray, “Refractive-index driven separation of colloidal polymer particles using optical chromatography,” Appl. Phys. Lett.83(25), 5316–5318 (2003).
[CrossRef]

Trumbore, B.

T. Moller and B. Trumbore, “Fast, minimum storage ray-triangle intersection,” J. Graphics, GPU, Games Tools2, 21–28 (1997).

Uhlmann, M.

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Figures (13)

Fig. 1
Fig. 1

Paths of a ray from the Gaussian laser beam passing through a non-spherical object.

Fig. 2
Fig. 2

(a) The oblate spheroid with major and minor semi-axes of a and b, and (b) the inclination angle defined on the x-y plane.

Fig. 3
Fig. 3

(a) The y-component of the dimensionless momentum transfer, and (b) the matched polar angles on the upper and lower surfaces.

Fig. 4
Fig. 4

Distribution of the optical force on rigid particles of different shapes: (a) sphere; (b) ellipsoid with b/a = 0.9; (c) ellipsoid with b/a = 0.5; (d) ellipsoid with b/a = 0.3.

Fig. 5
Fig. 5

The optical forces on the particles with different inclination angles as a function of the x- and y-directional positions: (a) the y-component optical force on a sphere; (b) the x-component optical force on a sphere; (c) the y-component optical force on an ellipsoid with b/a = 0.9; (d) the x- component optical force on an ellipsoid with b/a = 0.9.

Fig. 6
Fig. 6

The optical forces on particles with different inclination angles, as a function of the x- and y-directional positions: (a) the y-component optical force on an ellipsoid with b/a = 0.5; (b) the x-component optical force on an ellipsoid with b/a = 0.5; (c) the y-component optical force on an ellipsoid with b/a = 0.3; (d) the x-component optical force on an ellipsoid with b/a = 0.3.

Fig. 7
Fig. 7

The z-component torque on particles with different inclination angles, as a function of the x-directional position: (a) sphere; (b) ellipsoid with b/a = 0.9; (c) ellipsoid with b/a = 0.5; (d) ellipsoid with b/a = 0.3.

Fig. 8
Fig. 8

Ellipsoids with an inclination angle of π/4, released at the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5 and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Fig. 13
Fig. 13

Ellipsoids with b/a = 0.3 and different inclination angles, released at a horizontal distance of 1.0 from the focal point: the instantaneous shapes and trajectories of the ellipsoid with (a) an inclination angle of zero, (b) an inclination angle of π/4, and (c) an inclination angle of -π/4; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Fig. 9
Fig. 9

Ellipsoids with inclination angles of π/4, released at a horizontal distance of 0.5 from the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5, and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Fig. 10
Fig. 10

Ellipsoids with an inclination angle of -π/4, released at a horizontal distance of 0.5 from the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5, and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Fig. 11
Fig. 11

Ellipsoids with an inclination angle of π/2, released at a horizontal distance of 0.5 from the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5, and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Fig. 12
Fig. 12

Ellipsoids with an inclination angle of zero, released at a horizontal distance of 0.5 from the focal point: the instantaneous shapes and trajectories of the ellipsoids with (a) b/a = 0.9, (b) b/a = 0.5, and (c) b/a = 0.3; the time histories of (d) the inclination angle, (e) the total x-component optical force, and (f) the optically-induced torque.

Equations (22)

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F o = Δp Δt .
I= 2P πω (y) 2 exp[ 2 d f 2 ω (y) 2 ],
ω(y)= ω 0 [ 1+ ( λ 0 y π ω 0 2 ) 2 ] 1/2 ,
R r = R i +(2cosα)N,
R t = n i n t R i +(cosβ+ n i n t cosα)N,
Q l = R i R(α) R r n(1R(α)) R t ,
R( α )= 1 2 [ sin 2 (αβ) sin 2 (α+β) + tan 2 (αβ) tan 2 (α+β) ],T( α )=1R( α ).
Q u =(1R(α))[ n i n m R i n i n m R( α ) R r n t n m (1R( α )) R t ],
F o,i = n m c P g,i Δ A i Q i ,
ρ 0 ( u t +uu )=p+μ 2 u+f,
u=0,
ρ V P d U c dt = Ω S ( F o F )d s 1 d s 2 ,
ρ I P d ω c dt = Ω S r×( F o F )d s 1 d s 2 ,
U= U c + ω c ×r.
u * t * + u * u * = p * + 1 Re 2 u * + f * ,
F( s 1 , s 2 ,t )=κ[ ( X ib X )+Δt( U ib U ) ],
U ib ( s 1 , s 2 ,t )= Ω f u( x,t )δ( X( s 1 , s 2 ,t )x ) dx,
X= X 0 + 0 t Udt , X ib = X ib 0 + 0 t U ib dt ,
f( x,t )= Ω S F( s 1 , s 2 ,t )δ( xX( s 1 , s 2 ,t ) )d s 1 d s 2 ,
F o,i = F o,x,i 2 + F o,y,i 2 + F o,z,i 2 Δ A i .
F o,x = i=1 NE F o,x,i Δ A i , F o,y = i=1 NE F o,y,i Δ A i .
τ z = i=1 NE ( r x,i F o,y,i r y,i F o,x,i )Δ A i ,

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