Hydrodynamic resistance of particles at small Reynolds numbers

  • An attempt has been made in this article to critically survey the field of low Reynolds number flows, with particular regard to the hydrodynamic resistance of particles in this regime. A remarkable burgeoning of interest in such problems has occurred wlthin the past decade. Significant advances have been recorded on both the theoretical and experimental sides, with the former gains far outdlstancing the latter m scope. Problems which would have been impossible to solve rigorously before the advent of singular perturbation techniques are now being regularly solved, though hardly in a routine fashion; insight, intuition, inspiration, and ingenuity are still the order of the day. For those interested in direct engineering applications of the material covered by this review, the perspective from which many of the more general results set forth here should be viewed is, perhaps, best illustrated by an example: The resistance of any solid particle to translational and rotational motions in Stokes flow may be completely calculated from knowledge of a set of 21 scalar coefficients (Section II,C,l). While it seems highly improbable to expect that all these coefficients could be experimentally measured in practice, except perhaps in the trivial case of highly symmetrical bodies for which many of the coefficients vanish identically, this does not detract from the conceptual advantages of knowing exactly how much one does not know. Having an ideal goal against which the extent of present knowledge can be gaged permits a rational decision as to how to optimize one's investment of time, effort, and money in the pursuit of additional data. Furthermore, with the development of high-speed digital computers it may soon be possible to calculate all these coefficients for any given body (O 1 b). The general theory provides a rigorous framework into which such knowledge may be embedded. Use of symbolic" drag coefficients" (Section II,C,2) and symbolic heat- and mass-transfer" coefficients" (Section IV,A) furnishes a unique method for describing the intrinsic, interphase transport properties of particles for a wide variety of boundary conditions. Here, the particle resistance is characterized by a partial differential operator that represents its intrinsic resistance to vector or scalar transfer, independently of the physical properties of the fluid, the state of motion of the particle, or of the unperturbed velocity or temperature fields at infinity. Though restricted as yet in applicability, the general ideas underlying the existence of these operators appear capable of extension in a variety of ways. A recurrent theme arising throughout the analysis pertains to the screwlike properties of particles and of their intrinsic right- and left-handedness (Sections II,C, 1; II,C,2; III,C and IV,B). Such properties reflect an inseparable coupling between the translational and rotational motions of the particle. Helicoidally isotropic particles furnish the simplest examples of bodies manifesting screw-like behavior. These particles are isotropic, in that their properties are the same in all directions. Yet they possess a sense, and spin as they settle in a fluid. These id eas are likely to be of interest to microbiologists, biophysicists, geneticists, and others in the life sciences for whom handedness and life are intimately intertwined. The microscopic dimensions of the objects of interest to them insures ipso Jacto that the motion takes place at very small Reynolds numbers. Readers interested in an elementary but broad survey of sense in the physical and biological sciences are referred to Gardner's delightful book "The Ambidextrous Universe" (01). First-order corrections to the Stokes force on a particle, arising from wallor inertial-effects, can be directly expressed in terms of the Stokes force on the body in the absence of such effects. Thus, with regard to wall-effects in the Stokes regime, Eq. (135) expresses the force experienced by a particle falling in, say, a circular cylinder, in terms of the comparable force experienced by the particle when falling with the same velocity and orientation in the unbounded fluid. Equation (139) expresses a similar relationship for the torque on a rotating particle in a circular cylinder, as does Eq. (166) for the first-order interaction between two particles in an unbounded fluid in terms of the properties of the individual particles. Analogously, Eq. (234) expresses the inertial correction to the Stokes drag force in terms of the Stokes force itself. A comparable relationship exists (Section IV, A) between the heat-transfer coefficient at small, nonzero Peelet numbers and the heat-transfer coefficient at zero Peelet number-that is, the coefficient for conduction heat transfer. Finally, Eqs. (78)-(79) (or their symbolic operator counterparts) permit direct calculation of the Stokes force and torque experienced by a particle in an arbitrary field of flow solely from knowledge of the elementary solutions of Stokes equations for translation and rotation of the particle in a fluid at rest at infinity. The utility of already available knowledge is thus greatly extended by the existence of such relations. It permits one whose interests lie entirely in the macroscopic manifestation of the motion, e.g., the force and torque on the body, to bypass the oftentimes difficult problem of obtaining a detailed solution of the equations of motion, and to proceed directly to the computation of the force and torque on the body from the prescribed boundary conditions alone. The calculation is thereby reduced to a quadrature. The contents of this review may be read simultaneously from two different points of view. First and foremost it may be regarded as a compendium of recent advances in low Reynolds number flows. Secondly, from a pedagogic viewpoint it may be profitably used to illustrate the direct application of invariant techniques, that is, vector-polyadic and tensor methods, to a class of physical problems. Because of the relative simplicity and rich variety of physical problems associated with low Reynolds number motions, intuitive arguments may be employed to gain insight into the nature of polyadics and tensors; the role played by the concept of direction as a primitive entity is brought out here to a degree not usually found in standard works on tensor analysis.

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Metadaten
Author:Howard Brenner
URN:urn:nbn:de:hebis:30-1143878
ISSN:0065-2377
Parent Title (English):Advances in chemical engineering
Document Type:Article
Language:English
Date of Publication (online):2010/01/19
Year of first Publication:1966
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2010/01/19
Volume:6
Page Number:77
First Page:287
Last Page:438
Note:
Signatur: ZsN 528
HeBIS-PPN:35949241X
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 54 Chemie / 540 Chemie und zugeordnete Wissenschaften
Sammlungen:Sonstige
Licence (German):License LogoArchivex. zur Lesesaalplatznutzung § 52b UrhG