Stress boundary-conditions in ferrohydrodynamics

Industrial and Engineering Chemistry Research

Volumn 46, Issue 19, 2007, Pages 6113-6117

  Rosensweig, Ronald E ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; MAGNETIZATION; STRESSES; TENSORS;

FLUID MAGNETIC BEHAVIOR; MAGNETIC STRESS TENSOR;

HYDRODYNAMICS;

BOUNDARY CONDITIONS; HYDRODYNAMICS; MAGNETIZATION; STRESSES; TENSORS;

EID: 34748875303     PISSN: 08885885     EISSN: None     Source Type: Journal    
DOI: 10.1021/ie060657e     Document Type: Article

References (19)

1. See, for example: Rosensweig, R. E. Ferrohydrodynamics; Cambridge University Press: New York, 1985. (Reprinted with minor corrections and updates by Dover Publications, Mineola, NY, 1997.)
2. i), with subscript i denoting the orientation of the surface upon which stress component of orientation; acts. Other definitions are stated where introduced in the text.
3. Brenner, H. Antisymmetric stresses induced by the rigid-body rotation of dipolar suspensions. Int. J. Eng. Sci. (Oxford, U.K.) 1984, 22, 645.
4. 2 are the radii of surface curvature in two planes at right angles. A radius of curvature is positive within the medium, and negative outside the medium.
5. (a) In addition to the usual, symmetric, pressure-viscous stress terms, eq 7 incorporates effects that result from particle rotation in the viscous carrier liquid and particle interaction with the magnetic field. The magnetic field terms (last two terms) have the same form as that for equilibrated flow. A recent review by the investigator initially responsible for formulating unequilibrated dynamics of magnetic fluids is given in: Shliomis, M. In Ferrofluids: Magnetically Controllable Fluids and Their Applications; Odenbach, S., Ed.; Lecture Notes in Physics, Vol. 594; Springer: Berlin, 2002, p 85.
6. (b) Proof of the applicability of the magnetic field terms in unequilibrated systems is developed in: Rosensweig, R. E. Continuum equations for magnetic and dielectric fluids with internal rotations. J. Chem. Phys. 2004, 121, 1228,
7. and less completely in: Felderhof, B. U.; Kroh, H. J. Hydrodynamics of magnetic and dielectric fluids in interaction with the electromagnetic field. J. Chem. Phys. 1999, 110, 7403. The vortex viscosity terms are determined to ensure positive rates of entropy production. The third term on the right of eq 7 and the last term are asymmetric, whereas their sum is symmetric under the conditions of eq 20, see ref 6.
8. (c) In preparing the present paper, the author was unaware of the Shliomis formulation of boundary stress relationships in his chapter cited above. Although the Shliomis results are not quite as general as those presented in this work, and apparently are not intended to be, the chapter is highly recommended as an alternative resource, with the formulation developed independently using a somewhat different methodology.
9. The total stress dyadic contains two antisymmetric terms. One term is that which has the vortex viscosity ζ as a coefficient, as seen from eq A.1 in the Appendix. Expanding BH = 1/2(MH - HM) + 1/2(BH + HB) in eq 7 shows that 1/2(MH - HM) is the other. When particle angular acceleration and couple stress are negligible, the antisymmetric terms combine to yield a symmetric total stress dyadic.
10. Feng, S.; Graham, A. L.: Abbot, J. R.; Brenner, H. Antisymmetric stresses in suspensions. Vortex viscosity and energy dissipation. Phys. Fluids 2006, 563, 97-122.
11. Except for the ∇·v term, which comes into play for compressible fluids, this is the well-known Shliomis relaxation equation. Originally proposed on phenomenological grounds, the relationship is derived with the aid of irreversible thermodynamics in the citations to Rosensweig and Felderhof noted in ref 5. A microscopic derivation of a relaxation equation thought to be more accurate at large deviations from equilibrium is discussed in: Shliomis, M. I. Ferrohydrodynamics: Retrospective and Issues. In Ferrofluids; Odenbach. S., Ed.: Springer: Berlin, 2002; pp 85-111.
12. Rosensweig, R. E.; Popplewell, J.: Johnston, R. J. Magnetic fluid motion in rotating field. J. Magn. Magn. Mater. 1990, 55, 171.
13. Krauss, R.; Liu, M.; Reimann, B.; Richter R.; Rehberg, I. Pumping fluid by magnetic surface stress. New J. Phys. 2006, 8, 18.
14. Bacri, J.-C.: Cebers, A. O.; Perzynski, R. Behavior of a magnetic microdrop in a rotating magnetic field. Phys. Rev. Lett. 1994, 72, 2705.
15. Lebedev, A. V.; Enge, A.; Morozov, K. I.; Bauke H. Ferrofluid drops in rotating magnetic fields. New J. Phys. 2003, 5, 57.
16. Rhodes, S.; Perez, J.; Elborai. S.; Lee, S-H.; Zahn, M. Ferrofluid spiral formations and continuous-discrete phase transitions under simultaneously applied DC axial and AC in-plane rotating magnetic fields. J. Magn. Magn. Mater. 2005, 259, 353.
17. As examples: Stress boundary conditions in unequilibrated magnetic fluid are not treated in ref 1. A formula for the magnetic component of tangential antisymmetric stress is given in Section 5.2 (Blums, E.; Cebers. A.; Mairov, M. M. Magnetic Fluids; Walter de Gruyter: Berlin and New York, 1997)
18. and relates to the early work of Cebers (Cebers, A. Interphase pressure in the hydrodynamics of liquid with internal rotation. Magnetohydrodynamics 1975, 11 (1), 63).
19. Reference 9 assumes that the form of the equilibrium stress tensor can be used for the unequilibrated case but writes only one of the tangential stress terms. In an earlier work (Rosenthal, A. D.; Rinaldi, C.; Franklin, T.; Zahn. M. Torque measurements in spin-up flow of ferrofluids. J. Fluids Eng. 2004, 126, 198), the authors applied the stress tensor methodology to find wall torque without explicitly writing the stress difference formula. Other examples could be given.