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1
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33748587051
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H. Brenner, Unsolved problems in fluid mechanics: on the historical misconception of fluid velocity as mass motion, rather than volume motion, in: L.-S. Fan, M. Feinberg, G. Hulse, T.L. Sweeney, J.L. Zakin (Eds.), Unsolved Problems in Chemical Engineering, Proceedings of the Ohio State University, Department of Chemical Engineering Centennial Symposium, April 24-25, 2003, pp. 31-39. This reference is available on-line as 〈http://www.che.eng.ohio-state.edu/centennial/brenner.pdf〉, while the lecture on which the reference is based can be seen on-line as a streaming video presentation at 〈http://www.chbmeng.ohio-state.edu/centennial/〉.
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3
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33748598281
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H. Brenner, Inequality of the tracer and mass velocities of physicochemically-inhomogeneous fluid continua, J. Fluid Mech., 2006, submitted for publication.
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10
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33748593134
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M.N. Kogan, Non-Navier-Stokes gas dynamics and thermal-stress phenomena, in: Rarefied Gas Dynamics, vol. 15, 1986, p. 15.
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11
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33748614623
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M.N. Kogan, Some solved and unsolved problems in kinetic theory, in: A.D. Ketsdever, E.P. Muntz (Eds.), Rarefied Gas Dynamics, 23rd International Symposium, American Institute of Physics, 2003.
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13
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0000911805
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reprinted in: W.D. Niven (Ed.), The Scientific Papers of James Clerk Maxwell, vol. 2, Cambridge University Press, Cambridge, 1890, p. 681
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Maxwell J.C. Phil. Trans. R. Soc. London A 170 (1879) 231 reprinted in: W.D. Niven (Ed.), The Scientific Papers of James Clerk Maxwell, vol. 2, Cambridge University Press, Cambridge, 1890, p. 681
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(1879)
Phil. Trans. R. Soc. London A
, vol.170
, pp. 231
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Maxwell, J.C.1
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19
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33748924841
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H. Brenner, An elementary kinematical model of thermal diffusion in liquids and gases, Phys. Rev. E (2006, in press).
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20
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33748608303
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In earlier publications [2,3] we used the symbol over(m, ^) for the quantity here denoted by v.
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21
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33748582657
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Low Reynolds Number Hydrodynamics, Englewood Cliffs, NJ
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Happel J., Brenner H., and Low Reynolds Number Hydrodynamics. Prentice-Hall (1965), Englewood Cliffs, NJ
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(1965)
Prentice-Hall
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Happel, J.1
Brenner, H.2
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26
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33748631888
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T) - (frac(1, 3)) I : D represents its symmetric traceless counterpart.
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27
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84891553311
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Wiley, Hoboken, New Jersey The apt phrase, "Something is missing," which neatly encapsulates the momentum density problem, is taken verbatim from p. 61 of this reference
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Öttinger H.C. Beyond Equilibrium Thermodynamics (2005), Wiley, Hoboken, New Jersey The apt phrase, "Something is missing," which neatly encapsulates the momentum density problem, is taken verbatim from p. 61 of this reference
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(2005)
Beyond Equilibrium Thermodynamics
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Öttinger, H.C.1
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28
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33748586540
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"If your theory is found to be against the second law of thermodynamics, I give you no hope; there is nothing for it [your theory] but to collapse in the deepest humiliation," in A.S. Eddington, The Nature of the Physical World, Macmillan, New York, 1928, p. 74; "[Thermodynamics] is the only physical theory of universal content which I am convinced that within the framework of applicability of its basic concepts will never be overthrown." (A. Einstein), quoted in M.J. Klein, Science 157 (1967) 509.
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33
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33748603495
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H. Brenner, originally submitted to Phys. Rev. E (March 2003) under the title "A molecular basis for the Euler/Lagrange velocity disparity. The demise of the Navier-Stokes paradigm." Revised version to be submitted to Physica A (2005) under the title: "On Klimontovich's proposed modification of the Boltzmann equation and its consequences for the Navier-Stokes-Fourier equations."
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37
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84945670458
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Mazo, in the historical background to his book on Brownian motion (R.M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications, Clarendon Press, Oxford, 2002) discussing the period between Robert Brown's observations in 1829 and Einstein's 1905 publication, states that (p. 3): "It is striking, however, that the founders and main developers of kinetic theory, Maxwell, Boltzmann and Clausius, never published anything on Brownian motion."
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39
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0001455990
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(reproduced in Leonhardi Euleri Opera Omnia, Series II, vol. 12, p. 54 (Füssli, Zürich, 1954)). Additional historical information can be found in the "Editor's Introduction" to the latter volume by C. Truesdell, "Rational fluid mechanics, 1687-1765," pp. VII-CXXV
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Euler L. Mém. Acad. Sci. Berlin 11 (1755) 274 (reproduced in Leonhardi Euleri Opera Omnia, Series II, vol. 12, p. 54 (Füssli, Zürich, 1954)). Additional historical information can be found in the "Editor's Introduction" to the latter volume by C. Truesdell, "Rational fluid mechanics, 1687-1765," pp. VII-CXXV
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(1755)
Mém. Acad. Sci. Berlin
, vol.11
, pp. 274
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Euler, L.1
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41
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33748613827
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A concise history of the conceptual foundations of fluid mechanics, from the time of Newton's Principia in 1687 up to the definitive work of Stokes in 1845 [35], can be found in the following article: C. Truesdell, Amer. Math. Monthly 60 (1953) 445
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43
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33748622849
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Oxford University Press, Oxford
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Darrigol O. Worlds of Flow (2005), Oxford University Press, Oxford
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(2005)
Worlds of Flow
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Darrigol, O.1
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45
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33748584357
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reprinted in: Mathematical and Physical Papers, vol. 1, Cambridge University Press, Cambridge, 1901, p. 75
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46
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33748587332
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An account of pre-1845 work on the Navier-Stokes equations can be found in G.G. Stokes, British Assoc. Advance. Sci. (1846) 1; reprinted in: Mathematical and Physical Papers, vol. 1, p. 157, Cambridge University Press, Cambridge, 1901. This refers to earlier works, as follows: C.L.M.H. Navier, Mém. Acad. R. Sci. Paris 6 (1823) 389; S.D. Poisson, J. École Polytech. Paris 13 (1831) 139; and B. Saint-Venant, C. R. Acad. Sci. Paris 17 (1843) 1240.
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49
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33748626298
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m [39]. This fact in conjunction with the continuity equation assures that no net mass crosses the surface of such a body at any point. However, this is a strictly macroscopic statement in the sense that individual molecules are nevertheless free to cross a material surface in either direction without violating this no-flux condition, provided only that in some time-average sense there are no large-scale variations in the number of molecules contained within the domain. Clearly, the material view has nothing to say about the role of such fluctuations in regard to their possible impact upon Newton's laws of motion, which apply strictly only to permanent collections of molecules.
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53
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33748599602
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However, neither his work nor that of Klimontovich [28-30] touches upon the volume velocity-based no-slip boundary condition issue, which plays a prominent role in our unorthodox interpretation of the thermophoretic motion of macroscopic particles [1-3,5].
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55
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33748626013
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note
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According to the chain rule, for any function f (x, y, z) = 0, {Mathematical expression}
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57
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33748612482
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m.
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58
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33748606403
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However, as discussed in the most recent edition of Ref. [18, cf. Table 19.2-4], such differences have recently been recognized in multi-component fluids.
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61
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33748614114
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1 = 3 for Maxwell molecules.
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62
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31344465939
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A continuum model of thermal transpiration
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Bielenberg J.R., and Brenner H. A continuum model of thermal transpiration. J. Fluid Mech. 246 (2006) 1
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(2006)
J. Fluid Mech.
, vol.246
, pp. 1
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Bielenberg, J.R.1
Brenner, H.2
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63
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33748587874
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q = - k ∇ T in that case, as follows from Eq. (34). Accordingly, the derivation of (54) is consistent with that of (52), at least in the case of gases.
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64
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0006886326
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Fourier's law: a challenge to theorists
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Fokas A., Grigoryan A., Kibble T., and Zegarlinski B. (Eds), Imperial College Press, London
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Bonetto F., Lebowitz J.J., and Rey-Bellet L. Fourier's law: a challenge to theorists. In: Fokas A., Grigoryan A., Kibble T., and Zegarlinski B. (Eds). Mathematical Physics 2000 (2000), Imperial College Press, London 128
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(2000)
Mathematical Physics 2000
, pp. 128
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Bonetto, F.1
Lebowitz, J.J.2
Rey-Bellet, L.3
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65
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33748588957
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′ ≥ 0.
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66
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0004080557
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Princeton University Press, Princeton, NJ
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Debenedetti P.G. Metastable liquids (1996), Princeton University Press, Princeton, NJ
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(1996)
Metastable liquids
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Debenedetti, P.G.1
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68
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0000472452
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translated in: R. Fürth, (Ed.), A. Einstein, Investigations on the Theory of the Brownian Movement, Dover, New York, 1956 (reprint)
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Einstein A. Ann. Phys. 17 (1905) 549 translated in: R. Fürth, (Ed.), A. Einstein, Investigations on the Theory of the Brownian Movement, Dover, New York, 1956 (reprint)
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(1905)
Ann. Phys.
, vol.17
, pp. 549
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Einstein, A.1
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71
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33748583519
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In this connection, see also R.F. Streater, Statistical Dynamics, Imperial College Press, London, 1995
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84
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0003430259
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Princeton University Press, Princeton, NJ see also the revised online second edition (2001) of this book at 〈http://www.math.princeton.edu/~nelson/books/bmotion.pdf〉
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Nelson E. Dynamical Theories of Brownian Motion (1967), Princeton University Press, Princeton, NJ. http://www.math.princeton.edu/~nelson/books/bmotion.pdf see also the revised online second edition (2001) of this book at 〈http://www.math.princeton.edu/~nelson/books/bmotion.pdf〉
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(1967)
Dynamical Theories of Brownian Motion
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Nelson, E.1
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85
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0004266655
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Princeton University Press Princeton, NJ see also 〈http://www.math.princeton.edu/~nelson/books/Quantum_Fluctuations_1 985.pdf〉
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Nelson E. Quantum Fluctuations (1985), Princeton University Press Princeton, NJ. http://www.math.princeton.edu/~nelson/books/Quantum_Fluctuations_1985.pd f see also 〈http://www.math.princeton.edu/~nelson/books/Quantum_Fluctuations_1 985.pdf〉
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(1985)
Quantum Fluctuations
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Nelson, E.1
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87
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33748632696
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v in inhomogeneous multicomponent fluid mixtures, is well known in the irreversible thermodynamics literature [19,20].
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90
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0006098842
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see also 〈http://telemaco.uab.es/eit/home/principal.php〉
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Jou D., Casas-Vázquez J., and Lebon G. Rep. Prog. Phys. 62 (1999) 1035 see also 〈http://telemaco.uab.es/eit/home/principal.php〉
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Jou, D.1
Casas-Vázquez, J.2
Lebon, G.3
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33748605084
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H.C. Öttinger, "Hydrodynamics from Boltzmann's kinetic equation after proper coarse-graining," Phys. Rev. Lett., 2005, submitted for publication.
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14644410451
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Boettinger W.J., McFadden G.B., Coriell S.R., Sekerka R.F., and Warren J.A. Acta Materialia 53 (2005) 1995
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(2005)
Acta Materialia
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Boettinger, W.J.1
McFadden, G.B.2
Coriell, S.R.3
Sekerka, R.F.4
Warren, J.A.5
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105
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33748626574
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J. Philibert, Atomic Movements, Diffusion and Mass Transport in Solids. Translated from the French by S.J. Rothman, Les Editions de Physique, Les Ulis, Paris, 1991
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107
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24144495037
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Foss J., Tropea C., and Yarin A. (Eds), Springer, New York (Chapter 15)
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Lauga E., Brenner M.P., and Stone H.A. In: Foss J., Tropea C., and Yarin A. (Eds). Handbook of Experimental Fluid Dynamics (2005), Springer, New York (Chapter 15)
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(2005)
Handbook of Experimental Fluid Dynamics
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Lauga, E.1
Brenner, M.P.2
Stone, H.A.3
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Neto C., Evans D.R., Bonaccurso E., Butt H.-J., and Craig V.S.J. Rep. Prog. Phys. 68 (2005) 2859
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(2005)
Rep. Prog. Phys.
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Neto, C.1
Evans, D.R.2
Bonaccurso, E.3
Butt, H.-J.4
Craig, V.S.J.5
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