Surface-tension-driven flow on a moving curved surface
Tóm tắt
The leading-order equations governing the flow of a thin viscous film over a moving curved substrate are derived using lubrication theory. Three possible distinguished limits are identified. In the first, the substrate is nearly flat and its curvature enters the lubrication equation for the film thickness as a body force. In the second, the substrate curvature is constant but an order of magnitude larger; this introduces an extra destabilising term to the equation. In the final regime, the radius of curvature of the substrate is comparable to the lengthscale of the film. The leading-order evolution equation for the thin film is then hyperbolic, and hence can be solved using the method of characteristics. The solution can develop finite-time singularities, which are regularised by surface tension over a short lengthscale. General inner solutions are found for the neighbourhoods of such singularities and matched with the solution of the outer hyperbolic problem. The theory is applied to two special cases: flow over a torus, which is the prototype for flow over a general curved tube, and flow on the inside of a flexible axisymmetric tube, a regime of interest in modelling pulmonary airways.
Tài liệu tham khảo
L. D. Landau and V. G. Levich, Dragging of a liquid by a moving plate. Acta Physicochim. URSS 17 (1942) 42–54.
S. D. R. Wilson, The drag-out problem in film coating theory. J. Engng. Math. 16 (1982) 209–221.
W. S. Overdiep, The levelling of paints. Prog. Org. Coat. 14 (1986) 159–175.
H. Wong, I. Fatt and C. J. Radke, Deposition and thinning of the human tear film. J. Colloid Interface Sci. 184 (1996) 44–51.
D. Halpern and J. B. Grotberg, Fluid-elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244 (1992) 615–632.
A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films. Rev. Modern Phys. 69 (1997) 931–980.
T. G. Myers, Thin films with high surface tension. SIAM Rev. 40 (1998) 441–462.
L. E. Stillwagon and R. G. Larson, Fundamentals of topographic substrate levelling. J. Appl. Phys. 63 (1988) 5251–5258.
L. E. Stillwagon and R. G. Larson, Leveling of thin films over uneven substrates during spin coating. Phys. Fluids A 2 (1990) 1937–1944.
S. Kalliadasis, C. Bielarz and G. M. Homsy, Steady free-surface thin film flows over topography. Phys. Fluids 12 (2000) 1889–1898.
A. Mazouchi and G. M. Homsy, Free surface Stokes flow over topography. Phys. Fluids 13 (2001) 2751–2761.
L. W. Schwartz and D. E. Weidner, Modelling of coating flows on curved surfaces. J. Engng. Math. 29 (1995) 91–103.
M. Hayes, S. B. G. O'Brien and J. H. Lammers, Green's function for steady flow over a small twodimensional topography. Phys. Fluids 12 (2000) 2845–2858.
F. P. Bretherton, The motion of long bubbles in tubes. J. Fluid Mech. 10 (1961) 166–188.
P. S. Hammond, Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular pipe. J. Fluid Mech. 137 (1983) 363–384.
R. W. Atherton and G. M. Homsy, On the derivation of evolution equations for interfacial waves. Chem. Engng. Comm. 2 (1976) 57–77.
M. Cheng and H.-C. Chang, Stability of axisymmetric waves on liquid films flowing down a vertical column to azimuthal and streamwise disturbance. Chem. Engng. Comm. 118 (1992) 327–334.
A. L. Frenkel, On evolution equations for thin films flowing down solid surfaces. Phys. Fluids A 5 (1993) 2342–2347.
B. Reisfeld and S. G. Bankoff, Non-isothermal flow of a liquid film on a horizontal cylinder. J. Fluid Mech. 236 (1992) 167–196.
P. A. Gauglitz and C. J. Radke, The dynamics of liquid film breakup in constricted cylindrical capillaries. J. Colloid Interface Sci. 134 (1990) 14–40.
O. E. Jensen, The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 331 (1997) 373–403.
Lord Rayleigh, On the instability of jets. Proc. London Math. Soc. 10 (1878) 4–13.
R. V. Roy, A. J. Roberts and M. E. Simpson, A lubrication model of coating flows over a curved substrate in space. J. Fluid Mech. 454 (2002) 235–261.
H. E. Huppert, Flow and instability of a viscous current down a slope. Nature 300 (1982) 427–429.
S. M. Troian, E. Herbolzheimer, S. A. Safran and J. F. Joanny, Fingering instabilities of driven spreading films. Euro. Phys. Lett. 10 (1989) 25–30.
J. A. Moriarty, L. W. Schwartz and E. O. Tuck, Unsteady spreading of thin liquid films with small surface tension. Phys. Fluids A 3 (1991) 733–742.
B. W. van de Fliert, P. D. Howell and J. R. Ockendon, Pressure-driven flow of a thin viscous sheet. J. Fluid Mech. 292 (1995) 359–376.
P. D. Howell, S. L. Waters and J. B. Grotberg, The propagation of a liquid bolus along a liquid-lined flexible tube. J. Fluid Mech. 406 (2000) 309–335.
E. Kreyszig, Differential Geometry. New York: Dover (1991) 352pp.
S. D. R. Wilson and A. F. Jones, The entry of a falling film into a pool and the air-entrainment problem. J. Fluid Mech. 128 (1983) 219–230.
A. L. Bertozzi and M. P. Brenner, Linear stability and transient growth in driven contact lines. Phys. Fluids 9 (1997) 530–539.
S. Kalliadasis and G. M. Homsy, Stability of free-surface thin-film flows over topography. J. Fluid Mech. 448 (2001) 387–410.