Abstract
New fundamental solutions for micropolar fluids are derived in explicit form for two- and three-dimensional steady unbounded Stokes and Oseen flows due to a point force and a point couple, including the two-dimensional micropolar Stokeslet, the two- and three-dimensional micropolar Stokes couplet, the three-dimensional micropolar Oseenlet, and the three-dimensional micropolar Oseen couplet. These fundamental solutions do not exist in Newtonian flow due to the absence of microrotation velocity field. The flow due to these singularities is useful for understanding and studying microscale flows. As an application, the drag coefficients for a solid sphere or a circular cylinder that translates in a low-Reynolds-number micropolar flow are determined and compared with those corresponding to Newtonian flow. The drag coefficients in a micropolar fluid are greater than those in a Newtonian fluid.
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References
Papautsky I, Brazzle J, Ameel T and Frazier AB (1999). Laminar fluid behavior in microchannels using micropolar fluid theory. Sensors Actuators A Physical 73(1–2): 101–108
Shu J-J (2004). Microscale heat transfer in a free jet against a plane surface. Superlattices Microstruct 35(3–6): 645–656
Eringen AC (2001). Microcontinuum field theories II: fluent Media. Springer-Verlag, New York, Inc
Stokes GG (1851). On the effect of the internal friction of fluids on the motion of pendulums. Trans Cambridge Philos Soc 9(II): 8–106
Oseen CW (1910). Über die Stokes’sche formel und über eine verwandte aufgabe in der hydrodynamik. Arkiv för Matematik 6(29): 1–20
Lorentz HA (1986) Eene algemeene stelling omtrent de beweging eener vloeistof met wrijving en eenige daaruit afgeleide gevolgen. Zittingsverslag van de Koninklijke Akademie van Wetenschappen te Amsterdam 5:168–175 (in Dutch). Translated into English by Kuiken, HK (1996) A general theorem on the motion of a fluid with friction and a few results derived from it. J Eng Math 30:19–24
Hancock GJ (1953). The self-propulsion of microscopic organisms through liquids. Proc Roy Soc Lond Ser A Math Phys Sci 217(1128): 96–121
Ladyzhenskaya OA (1961) Mathematical problems of the dynamics for viscous incompressible fluids (Fizmatgiz) (in Russian). English translation, by Gordon and Breach (1963) The mathematical theory of viscous incompressible flow
Kuiken HK (1996). H.A. Lorentz: Sketches of his work on slow viscous flow and some other areas in fluid mechanics and the background against which it arose. J Eng Math 30(1–2): 1–18
Shu J-J and Chwang AT (2001). Generalized fundamental solutions for unsteady viscous flows. Phys Rev E 63(5): 051201
Ramkissoon H and Majumdar SR (1976). Drag on an axially symmetric body in the Stokes’ flow of micropolar fluid. Phys Fluids 19(1): 16–21
Olmstead WE and Majumdar SR (1983). Fundamental Oseen solution for the 2-dimensional flow of a micropolar fluid. Int J Eng Sci 21(5): 423–430
Łukaszewicz G (1999). Micropolar fluids: theory and applications. Birkhäuser, Boston
Zwillinger D (1998) Handbook of differential equations. Academic Press
Pozrikidis C (1992) Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press
Kohr M, Pop I (2004) Viscous incompressible flow for low Reynolds numbers. WIT Press
Lamb H (1911). On the uniform motion of a sphere through a viscous fluid. The London, Edinburgh and Dublin Philos Mag J Sci 21(6): 112–121
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Shu, JJ., Lee, J.S. Fundamental solutions for micropolar fluids. J Eng Math 61, 69–79 (2008). https://doi.org/10.1007/s10665-007-9160-8
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DOI: https://doi.org/10.1007/s10665-007-9160-8