Three-sphere low-Reynolds-number swimmer near a wall

ZARGAR, NAJAFI, AND MIRIOur work can be extended in many directions. Our resultscan be examined by a simple extension of the recent experimentof Leoni et al. 23. Swimming in a complex geometry,e.g., a microchannel of rectangular or circular cross section isof immediate interest. Inspired by the colonies of swimmingbacteria near biological membranes, we are investigatingmany propellers adjacent to a wall.APPENDIX: HYDRODYNAMIC MOBILITY TENSORSThe hydrodynamic mobility tensors appearing in Eq. 6areM xx ij = 11 R 38 1 r − 1 R + x i − x j 2 1 r 3 − 1 R 3 +2z j2− 3x i − x j 2R 5 −2z j z i + z j R 3− 3x i − x j 2 z i + z j R 5 ,M xz ij =8 1 x i − x j z i − z j r 3 − x i − x j z i + z j R 3+6z j2 x i − x j z i + z j R 5+2z j x i − x j R 3 − 3x i − x j z i + z j 2R 5 ,M zx ij =8 1 x i − x j z i − z j r 3 − x i − x j z i + z j R 3M zz ij = 1−6z j2 x i − x j z i + z j R 5+2z j x i − x j R 3 + 3x i − x j z i + z j 2R 5 ,r 3 − z i + z j 2R 3 −2z2 1 jR 38 1 r − 1 R + z i − z j 2 +2z j z i + z j − 3z i + z j 2R 5R 3 − 3z i + z j 3R 5 ,where r 2 =x i −x j 2 +z i −z j 2 and R 2 =x i −x j 2 +z i +z j 2 .We definePHYSICAL REVIEW E 80, 026308 2009S 1 = M 12 + M 21 − M 11 − M 22 ,S 2 = M 13 + M 21 − M 11 − M 23 ,S 3 = M 12 + M 31 − M 11 − M 32 ,S 4 = M 13 + M 31 − M 11 − M 33 .Now the matrix elements appearing in Eq. 11 can be introducedasa 11 = − cos S 1 xx − sin S 1 zx ,a 12 = − cos S 1 xz − sin S 1 zz ,a 13 = − cos S 2 xx − sin S 2 zx ,a 14 = − cos S 2 xz − sin S 2 zz ,a 21 = cos S 3 xx + sin S 3 zx ,a 22 = cos S 3 xz + sin S 3 zz ,a 23 = cos S 4 xx + sin S 4 zx ,a 24 = cos S 4 xz + sin S 4 zz ,a 31 = cos hS 1 zx + gS 3 zx − sin hS 1 xx + gS 3 xx ,a 32 = cos hS 1 zz + gS 3 zz − sin hS 1 xz + gS 3 xz ,a 33 = cos hS 2 zx + gS 4 zx − sin hS 2 xx + gS 4 xx ,a 34 = cos hS 2 zz + gS 4 zz − sin hS 2 xz + gS 4 xz ,a 41 = g sin ,a 42 =−g cos ,a 43 =−h sin ,a 44 = h cos .1 E. R. Kay, D. A. Leigh, and F. Zerbetto, Angew. Chem., Int.Ed. 46,722007; K. Kinbara and T. Aida, Chem. Rev. Washington,D.C. 105, 1377 2005.2 J. P. Sauvage and V. Amendola, Molecular Machines and MotorsSpringer-Verlag, Berlin, 2001.3 V. Balzani et al., Angew. Chem., Int. Ed. 39, 3348 2000.4 S. H. Kim, J. H. Jeong, S. H. Lee, S. W. Kim, and T. G. Park,J. Controlled Release 129, 107 2008.5 P. S. Dittrich and A. Manz, Nat. Rev. Drug Discovery 5, 2102006.6 H. Bruus, Theoretical Microfluidics Oxford University Press,Oxford, 2008.7 H. C. Berg, E. coli in Motion Springer, New York, 2004.8 G. Taylor, Proc. R. Soc. London, Ser. A 209, 447 1951.9 E. M. Purcell, Am. J. Phys. 45, 31977.10 L. E. Becker, S. A. Koehler, and H. A. Stone, J. Fluid Mech.490, 152003.11 D. Tam and A. E. Hosoi, Phys. Rev. Lett. 98, 068105 2007.12 J. E. Avron, O. Gat, and O. Kenneth, Phys. Rev. Lett. 93,186001 2004; J. E. Avron, O. Kenneth, and D. H. Oaknin,026308-6