Foundations of the Navier-Stokes boundary conditions in fluid mechanics

• Autorzy: Badur J. , Karcz M. , Lemański M. , Nastałek L.
• Języki publikacji: EN

Abstrakty

EN

Until recently it was believed that Navier’s boundary condition could be given as a rigorous foundation for slip phenomena. Due to the latest measurements in the mass flow rate of a gas flowing through nano- and microchannels, several discrepancies in the mathematical modelling have been found. Thus, in the literature, the opinion persists for the Navier slip condition to be correct only under certain circumstances, particularly those restricted to the first order boundary conditions. One of many ways to eliminate this discrepancy, which is extensively employed in the contemporary literature, is to develop a variety of the so-called second order boundary conditions. This path, however, seems incorrect since it lacks consistency between the bulk stress tensor and its boundary representation. In the paper we propose to replace the classical Navier slip condition with the new, more general Navier-Stokes slip boundary condition. Instead of the usual method of consideration, the boundary condition is presented as following from the mass and momentum balances within a thin, shell-like moving layer. Owing to this, the problem of consistency between the internal and external friction in a viscous fluid is solved within the framework of new layer balances, and a proper form of constitutive relations for friction and mobility forces. Finally, the common features of the Navier, Stokes, Maxwell and Reynolds concepts of a boundary slip layer are compared and revalorized. The classifications of different mobility mechanisms, important for flows in nano-, microchannels are also discussed.

Słowa kluczowe

EN

enhancement; slip; Navier number; Navier-Stokes; nanoflow

• Wydawca: Wydawnictwo Instytutu Maszyn Przepływowych PAN
• Czasopismo: Transactions of the Institute of Fluid-Flow Machinery
• Rocznik: 2011
• Tom: nr 123
• Strony: 3--55
• Opis fizyczny: Bibliogr. 72 poz., rys., tab.

Twórcy

autor

Badur J.

autor

Karcz M.

autor

Lemański M.

autor

Nastałek L.

• The Szewalski Institute of Fluid-Flow Machinery of the Polish Academy of Sciences, Energy Conversion Department, Fiszera 14, 80-231 Gdańsk, Poland

Bibliografia

• [1] Badur J., Karcz. M.: On evolution of the notion of a capillarity tensor, (W. Pietraszkiewicz, I. Kreja, eds.) In: Shell Structures: Theory and Applications, Vol. 2, 159–162. CRC Press Taylor & Francis Group, London 2010.
• [2] Badur J., Karcz M., Lemański M.: On the mass and momentum transport in the Navier-Stokes slip layer. Microfluid Nanofluid 11 (2011), 439–449.
• [3] Beskok A., Karniadakis G.E.: A model for flows in channels, pipes and ducts at micro and nanoscales . Nanoscale and Microscale Thermophys. Eng. 3 (1999), 1, 43–77.
• [4] Bilicki Z., Badur J.: A thermodynamically consistent relaxation model for turbulent binary mixture undergoing phase transition. J. Non-Equilibrium Thermodynamics 28, (2003), 311–340.
• [5] Boussinesq M.J.: Sur l’existence d’une viscosité superficielle, dans le mince couche de transition separant un liquide d’une autre fluide contigu. Ann. Chim. Phys. 29 (1913), 349–357.
• [6] Brenner H.: Navier-Stokes revisited. Physica A349 (2005) 60–132.
• [7] Butcher J.G.: On viscous fluid in motion, Proc. London Math. Soc. 8 (1876) 103–135.
• [8] Cermelli P., Fried E., Gurtin M.E.: Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interfaces. J. Fluid Mech. 544 (2005) 339–351.
• [9] Coulomb C.A.: Expériences destinées ŕ déterminer la cohčrence des fluids et les lois de leur résistance dans les mouvements trés lents. Memoires de sciences mathematiques et physiques 3 (1800) 246–305.
• [10] de Gennes P.G., Brochard-Wyart F., Quere D.: Capillarity and Wetting Phenomena. Springer, New York 2004.
• [11] Deissler R.G.: An analysis of second-order slip flow and temperature jump boundary conditions for rarefied gases. Int. J. Heat Mass Transfer 7(1964), 6, 681–694.
• [12] dell’Isola F., KosińskiW.: Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers. Arch. Mech. 45 (1993) 333–359
• [13] dell’Isola F., Madeo A., Seppecher P.: Boundary conditions at fluidpermeable interfaces in porous media: A variational approach. Int. J. Solid Structures 46 (2009) 3150–3164.
• [14] Duhem P.: Recherches sur l’hydrodynamique. Ann. Toulouse 3 (1901) 315–377; 379–431.
• [15] Duhem P.: Recherches sur l’hydrodynamique. Ann. Toulouse 4, (1902), 101–169.
• [16] Duhem P.: Recherches sur l’hydrodynamique. Ann. Toulouse 5 (1903), 5–61; 197–255; 355–404.
• [17] Euler L.: Sur le frottement des corps solides. Memoires de l’academie des sciences de Berlin 4 (1748), 122–132.
• [18] Fried E., Gurtin M.E.: Tractions, balances and boundary conditions for nonsimple materials with application to liquid flow at small-length scales. Arch. Ration. Mech. Anal. 182 (2005), 3, 513–554.
• [19] Fried E., Gurtin M.E.: Thermodynamics of the interface between a body and its environment. Continuum Mech. Thermodyn. 19 (2007), 5, 253–271.
• [20] Gad-El-Hak M.: The fluid mechanics of microdevices. Trans.ASME J. Fluid Eng. 121 (1999), 7–33.
• [21] Gad-El-Hak M: Handbook of MEMS. CRC Press, Taylor & Francis Group, 2001.
• [22] Gaede W.: Die äussere Reibing der Gase. Ann. Phys. 41 (1913) 289–336.
• [23] Gardeniers H., Van den Berg A.: Micro- and nanofluidic devices for environmental and biomedical applications. Int. J. Environ. Anal. Chem. 84 (2004) 809–819.
• [24] Goodrich F.C.: Sirface viscosity as a capillary excess transport property. In: The Modern Theory of Capillarity, F.C. Goodrich, A.I. Rusanov, eds. Akademie-Verlag, 1981, 19–34.
• [25] Graham T.: On the motion of gases. Phil. Trans. Royal Soc. London 131 (1846), 573–632.
• [26] Graham T.: On the motion of gases. Phil. Trans. Royal Soc. London 137 (1849), 349–362.
• [27] Hadamard J.S.: Mouvement permenent lent d’une sphére liquide et visqueuse dans un liquide visqueux. Comp. Rend. Acad. Sci. 152 (1911), 1735–1738.
• [28] Hadjiconstantinou N.G.: The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics. Phys. Fluid 18 (2006), 111–301.
• [29] Helmholtz H., von Piotrowski G.: Über Reibung tropfbarer Flüssigkeiten. In: Wissenschaftliche Abhandlungen, H. Helmholtz, ed., Sitzungsberichte, Wien 1860.
• [30] Ho C.M., Tai Y.C.: Micro-electro-mechanical-systems (MEMS) and fluid flow. Ann. Rev. Fluid Mech. 30 (1998), 579–612.
• [31] Jebauer S., Czerwińska J.: Implementation of velocity slip and temperature jump boundary conditions for microfluidic devices. IFTR Reports, bf 59 2007.
• [32] Kaczmarek M., Kazimierska-Drobny K.: Simulation of reactive materials in column and reservoir test. Sensitivity analysis of a linear coupled model. Computers and Geotechnics 34 (2007), 4, 247- 253.
• [33] Karniadakis G.E., Beskok A., Aluru N.: Microflows and Nanoflows: Fundamentals and Simulation. Springer, 2005.
• [34] Knudsen M.: Die Gesetze der Molekularströmung und der innere Reibungsströmung der Gase durch Röhren. Ann. Phys. 28(1909), 4, 75–130.
• [35] Knudsen M.: Eine Revision der Gleichgewichtsbedingung der Gase. Thermische Molekularströmung. Ann. Phys. 31 (1910), 205–229.
• [36] Koplik J., Banavar J.R.: Continuum deductions from molecular hydrodynamics. Ann. Rev. Fluid Mech. 27 (1995), 257–292.
• [37] Kundt A., von Warburg E.: Über Reibung und Wärmeleitung verdünnter Gase. Ann.Phys. 155 (1875), 7, 337–365.
• [38] Kundt A., von Warburg E.: Über Reibung und Wärmeleitung verdünnter Gase. Ann.Phys. 156 (1876), 10, 177–211.
• [39] Larronde F.E., Housiadas C., Drossinos Y.: Slip-flow heat transfer in circular tubes. Int. J Heat Mass Transfer 43 (2000), 2669–2680.
• [40] Lauga E., Stone H.A.: Effective slip in pressure-driven stokes flow, J. Fluid Mech., 489, 2003, 55 77.
• [41] Lee Y.-B., KwakH.-D., Kim C.-H., Lee N.-S.: Numerical prediction of slip flow effect on gas lubricated journal bearings for MEMS/MST based microrotating machinery. Tribol. Int. 38 (2005), 1, 89–96.
• [42] Marques W. Jr., Kremer G.M., Sharipov F.M.: Couette flow with slip and jump boundary conditions. Continuum Mech. Thermodyn. 12 (2000), 6, 379– 386.
• [43] Matthews M.T., M.J. Hill: On three simple experiments to determine slip length. Microfluid Nanofluid 6 (2009), 5, 611–619.
• [44] Maxwell J.C.: On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. Royal Soc. London, 170 (1879), 231–256.
• [45] Natanson L.: On the laws of viscosity. Phil. Mag 2 (1901), 342–356.
• [46] Navier C.-L.-M.-H.: Mémoire sur les lois du mouvement des fluides (submitted 1822). Memoires l’Acad. Royale des Sciences de l’Institut de France 2 (1827), 375–393.
• [47] Newton I.: Philosophiae maturalis principia mathematica, 3rd edn., 1726. Transl. A. Motte: Sir Isaac Newtons Mathematical Principles of Natural Philosophy and his System of theWorld, London [revised ed. Univ. California Press, 1962].
• [48] Nie X.B., Chen S.Y., W.N. E, Robbins M.O.: A continuum and molecular hybrid method for micro- and nanofluid flow. J. Fluid Mech. 500 (2004), 55–64.
• [49] Petryk H., Mroz Z.: Time derivatives of integrals and functionals defined on varying volume and surface domain, Arch. Mech. 38 (1986), 694–724.
• [50] Priezjev N.V.: Rate dependent slip boundary conditions for simple fluids. Phys. Rev. E. 75 (2007).
• [51] Prosperetti A.: Boundary conditions at a liquid-vapour interface. Meccanica 14 (1979), 1, 34–47.
• [52] Reynolds O.: On certain dimensional properties of matter in the gaseous state. Phil. Trans. Royal Soc. London 170 (1879), 727–845.
• [53] Reynolds O.: On the equations of motion and the boundary conditions for viscous fluid. British Association Sec. A, Papers II (46), 1883.
• [54] Schnell E.: Slippage of water over nonwettable surfaces. J. Appl. Phys. 27 (1956), 10, 1149–1152.
• [55] Scriven L.E.: On the dynamics of phase growth. Chem. Eng. Sci. 10 (1959), 1-2, 1–13.
• [56] Scriven L.E. Dynamics of fluid interfaces, Chem. Eng. Sci. 12 (1960), 2, 98–108.
• [57] Sinha Ray S., Chando P., Yarin A.L.: Enhanced release of liquid from carbon nanotubes due to entrainmeny by an air layer. Nanotechnology 20 (2009), 095711, 6.
• [58] Slattery J.C., Sagis L., Oh E.-S.: Interfacial Transport Phenomena, 2nd edn. Springer-Verlag, 2007.
• [59] Stokes G.G.: On the theories of the internal friction of fluids in motion, and on the equilibrium and motion of elastic solids. Trans. Cambridge Phil. Soc. 8 (1845), 287–319.
• [60] Stumpf H., Badur J.: On objective surface rate. Quart. Appl. Math. 51 (1993), 161–181.
• [61] Thompson P.A., Trojan S.M.: A general boundary condition for liquid flow at solid surface. Nature 389 (1997), 360–362.
• [62] To Q.D., Bercegeay C., Lauriat G., Leonard C., Bonnet G.: A slip model for micro/nano gas flows induced by body forces. Microfluid Nanofluid 8 (2010), 3, 417–422.
• [63] von Rybczyński W.: Über die fortschreitende Bewegung einer flüssigen Kugel in einem zähen meidum. Bull. Acad. Sci. Cracovie A (1911), 40–46.
• [64] von Smoluchowski M.: Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilichen. Phys. Z. 17 (1916), 557–571.
• [65] Weatherburn C.E.: Differential Geometry of Surfaces. Cambridge Univ. Press, 1930.
• [66] Weidemann G.: Über die Bewegung von Flüssigkeiten im Kriese der geschlossen galvanischen Säule. Ann. Phys. 163 (1852), 11, 321–352.
• [67] Whetham W.C.D.: On the alleged slipping at the boundary of a liquid in motion. Phil. Trans. Royal Soc. London 48 (1890), 225–230.
• [68] Ych L.-H., Hsu J.-P.: Electroporesis of a finite rod along the axis of a long cylinder microchannel filled with carreau fluids. Microfluid Nanofluid 7 (2009), 3, 383–392.
• [69] You X.-Y., Zheng J.-R., Jing Q.: Effects of boundary slip and apparent viscosity on the stability of microchannel flow. Forschung im Ingenieurwesen 71 (2007), 2.
• [70] Zhang Y., Yarin A.L.: Thermo-responsive copolymer coatings for flow regulation on demand in glass microcapillaries. Eur. Phys. J. E 33 (2010), 211–218.
• [71] Zhu Y., Garnick S.: Rate-dependent slip of newtonian liquid at smooth surfaces. Phys. Rev. Lett. 87 (2001), 9, 96–105.
• [72] Zmitrowicz A.: A thermodynamical model of contact, friction and wear. Wear 114 (1987), 135–168; 169–197; 199–221.