Low-Reynolds-number hydrodynamics

it is acceptable to reverse the order of mixed second partial differentiation of an exact differential without affecting the final result. 

The generalized form of Newton s law relates r to linear combinations of velocity gradients with the following restrictions  

Viscous forces should vanish for fluids (a) at rest, (b) in a state of pure translation (i.e., all a, are constant), and (c) in a state of pure rotation (see Landau and Lifshitz, 1959, p. 48). 

To satisfy these conditions, the following linear transport law was conslmcted for isotropic fluids, where the viscosity /r is a scalar instead of a fourth-rank 

Creeping Flow of an Incompressible Newtonian Fluid. It is reasonable to assume that p constant for liquids that are not subjected to large variations in temperature and pressure. This assumption of incompressibility leads to the following form of the equation of continuity (i.e., see 8-35) and Newton s law of viscosity  

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Happel, J. and Brenrmer, H. (1983) Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Martinus Nijhoff Publishers, The Hague, Chapter 6. 

M. Ripoll, K. Mussawisade, R. G. Winkler, and G. Gompper, Low-Reynolds-number hydrodynamics of complex fluids by multi-particle-collision dynamics, Europhys. Lett. 68, 106... 

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1965. 

H3. Happel, J., and Brenner, H., Low Reynolds Number Hydrodynamics, 2nd ed. Noordhoff, Leyden, Netherlands, 1973. 

B. Solutions of the Many-Body Problem in Low Reynolds-Number Hydrodynamics... 

HappelJ, Brenner H (1965) Low Reynolds number hydrodynamics with special application to particulate media, Prentice-Hall, Englewood Cliffs NJ... 

Microfluidics is the manipulation of fluids in channels, with at least two dimensions at the micrometer or submicrometer scale. This is a core technology in a number of miniaturized systems developed for chemical, biological, and medical applications. Both gases and liquids are used in micro-/nanofluidic applications, ° and generally, low-Reynolds-number hydrodynamics is relevant to bioMEMS applications. Typical Reynolds numbers for biofluids flowing in microchannels with linear velocity up to 10 cm/s are less than Therefore, viscous forces dominate the response and the flow remains laminar. 

The metrics (or scale factors) for a large number of orthogonal curvilinear coordinate systems can be found in the appendix by Happel and Brenner (1973) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Noordhoff International, Leyden, The Netherlands, 1973). 

In addition to these general symmetry properties, considerable effort has been made to understand the relationships between symmetries in the geometry of the problem and the forms of the resistance tensors. It is beyond our present scope to discuss these relationships in a comprehensive manner the interested reader can refer to Brenner (1972) or the textbook Low Reynolds Number Hydrodynamics by Happel and Brenner (1973) for a detailed discussion of these questions.8 Here we restrict ourselves to the results for several particularly simple cases. First, if we consider the motion of a body with spherical symmetry in an unbounded fluid, with the origin of coordinates at the geometric center of the body, it can be shown that... 

The preceding sections have been concerned primarily with direct solution techniques for problems in creeping-flow theory. Here, we discuss several general topics that evolve directly from these developments. The first two involve application of the so-called reciprocal theorem of low-Reynolds-number hydrodynamics. 

E. P. Ascoli, D. S. Dandy, and L. G. Leal, Low Reynolds number hydrodynamic interaction of a solid particle with a planar wall, Int. J. Numer. Methods Fluids 9, 651-88 (1989) E. P. Ascoli, D. S. Dandy, and L. G. Leal, Buoyancy-driven motion of a deformable drop toward a planar wall at low Reynolds number, J. Fluid Mech. 213, 287-311 (1990). 

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