Abstract
The present review deals with the effects of topography and inertia on gravity-driven film flows. The article is organized like a rope ladder, with the rungs of topography and inertia being scaled one after another. We begin with an introduction, where we specify the literature reviewed in our article and highlight the physical significance of this type of fluid motion. Next, we address the effects of different types of topographies on creeping film flow and films in lubrication approximation, and on inertial flow. Then, findings on inertial flow with sidewalls as bounding topography are reviewed. In all these cases, the impact of topography and inertia on both the free surface and the flow field structure is shown. Subsequently, we briefly highlight inverse problem theory. The following penultimate section focuses on the stability of film flows. After a short review on the stability of films over flat inclines which we give for convenience, the stability of films over topography is considered. A discussion on the stability of films with sidewalls as bounding topography follows. In each case, the interaction between topography, flow field, and free surface is shown with the theoretical and experimental methods being discussed. Finally, the paper closes with some concluding remarks and an outlook from the authors’ perspective—one century after the groundbreaking work of Wilhelm Nusselt.
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References
de Gennes, P.-G., Brochard-Wyart, F., Quere, D.: Capillarity and Wetting Phenomena. Springer, Berlin (2004)
Braun, R.J.: Dynamics of the tear film. Annu. Rev. Fluid Mech. 44, 267–297 (2011)
Luca, I., Hutter, K., Tai, Y.C., Kuo, C.Y.: A hierarchy of avalanche models on arbitrary topography. Acta Mech. 205, 121–149 (2009)
Greve, R., Blatter, H.: Dynamics of Ice Sheets and Glaciers. Springer, Berlin (2009)
Kumar, A., Karig, D., Acharya, R., Neethirajan, S., Mukherjee, P.P., Retterer, S., Doktycz, M.J.: Microscale confinement features can affect biofilm formation. Microfluid. Nanofluid. 14, 895–902 (2013)
Webb, R.L.: Principles of Enhanced Heat Transfer. Wiley, New York (1994)
Kistler, S.F., Schweizer, P.M.: Liquid Film Coating. Springer, Netherlands (1997)
Weinstein, S.J., Ruschak, K.J.: Coating flows. Annu. Rev. Fluid Mech. 36, 29–53 (2004)
Gugler, G., Beer, R., Mauron, M.: Operative limits of curtain coating due to edges. Chem. Eng. Process. Process Intensif. 50, 462–465 (2011)
Oron, A., Davis, S.H., Bankoff, S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980 (1997)
Chang, H.C., Demekhin, E.A.: Complex Wave Dynamics on Thin Films. Elsevier, Amsterdam (2002)
Craster, R.V., Matar, O.K.: Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198 (2009)
Nusselt, W.: Die Oberflächenkondensation des Wasserdampfes. VDI Z 60, 541–546 (1916)
Spurk, J.-H., Aksel, N.: Fluid Mechanics, 2nd edn. Springer, Berlin (2008)
Kalliadasis, S., Bielarz, C., Homsy, G.M.: Steady free-surface thin film flows over topography. Phys. Fluids 12, 1889–1898 (2000)
Kalliadasis, S., Bielarz, C.: Erratum: steady free-surface thin film flows over topography [Phys. Fluids 12, 1889 (2000)]. Phys. Fluids 12, 3305 (2000)
Mazouchi, A., Homsy, G.M.: Free surface Stokes flow over topography. Phys. Fluids 13, 2751–2761 (2001)
Aksel, N.: Influence of the capillarity on a creeping film flow down an inclined plane with an edge. Arch. Appl. Mech. 70, 81–90 (2000)
Heining, C., Sellier, M., Aksel, N.: The inverse problem in creeping film flows. Acta Mech. 223, 841–847 (2012)
Gaskell, P.H., Jimack, P.K., Sellier, M., Thompson, H.M., Wilson, M.C.T.: Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography. J. Fluid Mech. 509, 253–280 (2004)
Wang, C.Y.: Liquid film flowing slowly down a wavy incline. AIChE J. 27, 207–212 (1981)
Scholle, M., Wierschem, A., Aksel, N.: Creeping films with vortices over strongly undulated bottoms. Acta Mech. 168, 167–193 (2004)
Scholle, M., Rund, A., Aksel, N.: Drag reduction and improvement of material transport in creeping films. Acta Mech. 75, 93–112 (2006)
Pozrikidis, C.: The flow of a liquid film along a periodic wall. J. Fluid Mech. 188, 275–300 (1988)
Wierschem, A., Scholle, M., Aksel, N.: Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers. Phys. Fluids 15, 426–435 (2003)
Nguyen, P.K., Bontozoglou, V.: Steady solutions of inertial film flow along strongly undulated substrates. Phys. Fluids 23, 052103 (2011)
Moffatt, H.K.: Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18 (1964)
Pozrikidis, C., Thoroddsen, S.T.: The deformation of a liquid film flowing down an inclined plane wall over a small particle arrested on the wall. Phys. Fluids A 3, 2546–2558 (1991)
Hayes, M., O’Brien, S.B.G., Lammers, J.H.: Green’s function for steady flow over a small two-dimensional topography. Phys. Fluids 12, 2845–2858 (2000)
Blyth, M.G., Pozrikidis, C.: Film flow down an inclined plane over a three-dimensional obstacle. Phys. Fluids 18, 052104 (2006)
Baxter, S.J., Power, H., Cliffe, K.A., Hibberd, S.: Three-dimensional thin film flow over and around an obstacle on an inclined plane. Phys. Fluids 21, 032102 (2009)
Lee, Y.C., Thompson, H.M., Gaskell, P.H.: An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localised topographic features. Comput. Fluids 36, 838–855 (2007)
Sellier, M., Lee, Y.C., Thompson, H.M., Gaskell, P.H.: Thin film flow on surfaces containing arbitrary occlusions. Comput. Fluids 38, 171–182 (2009)
Lee, Y.C., Thompson, H.M., Gaskell, P.H.: Three-dimensional thin film and droplet flows over and past surface features with complex physics. Comput. Fluids 46, 306–311 (2011)
Lee, Y.C., Thompson, H.M., Gaskell, P.H.: Dynamics of thin film flow on flexible substrate. Chem. Eng. Process. Process Intensif. 50, 525–530 (2011)
Luo, H., Pozrikidis, C.: Gravity-driven film flow down an inclined wall with three-dimensional corrugations. Acta Mech. 188, 209–225 (2007)
Bontozoglou, V., Serifi, K.: Falling film flow along steep two-dimensional topography: the effect of inertia. Int. J. Multiph. Flow 34, 734–747 (2008)
Wierschem, A., Scholle, M., Aksel, N.: Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel. Exp. Fluids 33, 429–442 (2002)
Wierschem, A., Aksel, N.: Influence of inertia on eddies created in films creeping over strongly undulated substrates. Phys. Fluids 16, 4566–4574 (2004)
Scholle, M., Haas, A., Aksel, N., Wilson, M.C.T., Thompson, H.M., Gaskell, P.H.: Competing geometric and inertial effects on local flow structure in thick gravity-driven fluid films. Phys. Fluids 20, 123101 (2008)
Bontozoglou, V., Kalliadasis, S., Karabelas, A.J.: Inviscid free-surface flow over a periodic wall. J. Fluid Mech. 226, 189–203 (1991)
Bontozoglou, V., Papapolymerou, G.: Laminar film flow down a wavy incline. Int. J. Multiph. Flow 23, 69–79 (1997)
Trifonov, Y.Y.: Viscous liquid film flows over a periodic surface. Int. J. Multiph. Flow 24, 1139–1161 (1998)
Bontozoglou, V.: Laminar film flow along a periodic wall. CMES-Comp. Model Eng. 1, 133–142 (2000)
Wierschem, A., Aksel, N.: Hydraulic jumps and standing waves in gravity-driven flows of viscous liquids in wavy open channels. Phys. Fluids 16, 3868–3877 (2004)
Wierschem, A., Bontozoglou, V., Heining, C., Uecker, H., Aksel, N.: Linear resonance in viscous films on inclined wavy planes. Int. J. Multiph. Flow 34, 580–589 (2008)
Anshus, B.E., Goren, S.L.: A method of getting approximate solutions to the Orr-Sommerfeld equation for flow on a vertical wall. AICHE J. 12, 1004–1008 (1966)
Heining, C., Bontozoglou, V., Aksel, N., Wierschem, A.: Nonlinear resonance in viscous films on inclined wavy planes. Int. J. Multiph. Flow 35, 78–90 (2009)
Duffing, G.: Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz. F. Vieweg und Sohn, Braunschweig (1918)
Malamataris, N.A., Bontozoglou, V.: Computer aided analysis of viscous film flow along an inclined wavy wall. J. Comput. Phys. 154, 372–392 (1999)
Pak, M.I., Hu, G.H.: Numerical investigations on vortical structures of viscous film flows along periodic rectangular corrugations. Int. J. Multiph. Flow 37, 369–379 (2011)
Vlachogiannis, M., Bontozoglou, V.: Experiments on laminar film flow along a periodic wall. J. Fluid Mech. 457, 133–156 (2002)
Argyriadi, K., Vlachogiannis, M., Bontozoglou, V.: Experimental study of inclined film flow along periodic corrugations: the effect of wall steepness. Phys. Fluids 18, 012102 (2006)
Wierschem, A., Pollak, T., Heining, C., Aksel, N.: Suppression of eddies in films over topography. Phys. Fluids 22, 113603 (2010)
Valluri, P., Matar, O.K., Hewitt, G.F., Mendes, M.A.: Thin film flow over structured packings at moderate Reynolds numbers. Chem. Eng. Sci. 60, 1965–1975 (2005)
Varchanis, S., Dimakopoulos, Y., Tsamopoulos, J.: Steady film flow over a substrate with rectangular trenches forming air inclusions. Phys. Rev. Fluids 2, 124001 (2017)
Decré, M.M.J., Baret, J.-C.: Gravity-driven flows of viscous liquids over two-dimensional topographies. J. Fluid Mech. 487, 147–166 (2003)
Veremieiev, S., Thompson, H.M., Gaskell, P.H.: Inertial thin film flow on planar surfaces featuring topography. Comput. Fluids 39, 431–450 (2010)
Veremieiev, S., Thompson, H.M., Gaskell, P.H.: Free-surface film flow over topography: full three-dimensional finite element solutions. Comput. Fluids 122, 66–82 (2015)
Wang, C.Y.: Low Reynolds number film flow down a three-dimensional bumpy surface. J. Fluids Eng. 127, 1122–1127 (2005)
Luo, H., Pozrikidis, C.: Effect of inertia on film flow over oblique and three-dimensional corrugations. Phys. Fluids 18, 078107 (2006)
Luo, H., Pozrikidis, C.: Publisher’s note: effect of inertia on film flow over oblique and three-dimensional corrugations [Phys. Fluids 18, 078107 (2006)]. Phys. Fluids 18, 129901 (2006)
Heining, C., Pollak, T., Aksel, N.: Pattern formation and mixing in three-dimensional film flow. Phys. Fluids 24, 042102 (2012)
Scholle, M., Aksel, N.: An exact solution of visco-capillary flow in an inclined channel. Zeitschrift für Angewandte Mathematik und Physik ZAMP 52, 749–769 (2001)
Scholle, M., Aksel, N.: Thin film limit and film rupture of the visco-capillary gravity-driven channel flow. Zeitschrift für Angewandte Mathematik und Physik ZAMP 54, 517–531 (2003)
Haas, A., Pollak, T., Aksel, N.: Side wall effects in thin gravity-driven film flow: steady and draining flow. Phys. Fluids 23, 062107 (2011)
Sellier, M.: Inverse problems in free surface flows: a review. Acta Mech. 227, 913–935 (2016)
Sellier, M.: Substrate design or reconstruction from free surface data for thin film flows. Phys. Fluids 20, 062106 (2008)
Heining, C., Aksel, N.: Bottom reconstruction in thin-film flow over topography: steady solution and linear stability. Phys. Fluids 21, 083605 (2009)
Heining, C.: Velocity field reconstruction in gravity-driven flow over unknown topography. Phys. Fluids 23, 032101 (2011)
Heining, C., Pollak, T., Sellier, M.: Flow domain identification from free surface velocity in thin inertial films. J. Fluid Mech. 720, 338–356 (2013)
Anjalaiah, Y., Chakraborty, S., Usha, R.: Steady solution of an inverse problem in gravity-driven shear-thinning film flow: reconstruction of an uneven bottom substrate. J. Non-Newton Fluid Mech. 219, 65–77 (2015)
Usha, R.: Anjalaiah: Steady solution and spatial stability of gravity-driven thin-film flow: reconstruction of an uneven slippery bottom substrate. Acta Mech. 227, 1685–1709 (2016)
Heining, C., Sellier, M.: Flow domain identification in three-dimensional creeping flows. Phys. Fluids. 29, 012107 (2017)
Schörner, M., Reck, D., Aksel, N.: Does the topography’s specific shape matter in general for the stability of film flows? Phys. Fluids 27, 042103 (2015)
Kapitza, P.L.: Wavy flow of thin layers of a viscous fluid. Zh. Eksp. Teor. Fiz. 18, 3–28 (1948)
Kapitza, P.L., Kapitza, S.P.: Wavy flow of thin layers of a viscous fluid. Zh. Eksp. Teor. Fiz. 19, 105–120 (1949)
Benjamin, T.B.: Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574 (1957)
Yih, C.S.: Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334 (1963)
Orr, W.: The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: a perfect liquid. Proc. R. Ir. Acad. A Math. Phys. Sci. 27, 9–68 (1907)
Orr, W.M.F.: The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: a viscous liquid. Proc. R. Ir. Acad. A Math. Phys. Sci. 27, 69–138 (1907)
Sommerfeld, A.: Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitsbewegungen. In: Proceedings of the 4th International Congress of Mathematicians, vol. 3, pp. 116-124 (1908)
Lin, S.P.: Finite-amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113–126 (1969)
Gjevik, B.: Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1918–1925 (1970)
Benney, D.J.: Long waves on liquid films. J. Math. Phys. 45, 150 (1966)
Liu, J., Paul, J.D., Gollub, J.P.: Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101 (1993)
Liu, J., Gollub, J.P.: Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712 (1994)
Liu, J., Gollub, J.P.: Onset of spatially chaotic waves on flowing films. Phys. Rev. Lett. 70, 2289–2292 (1993)
Liu, J., Schneider, J.B., Gollub, J.P.: Three-dimensional instabilities of film flows. Phys. Fluids 7, 55–67 (1995)
Trifonov, Y.Y.: Stability of the wavy film falling down a vertical plate: the DNS computations and Floquet theory. Int. J. Multiph. Flow 61, 73–82 (2014)
Kalliadasis, S., Homsy, G.M.: Stability of free-surface thin-film flows over topography. J. Fluid Mech. 448, 387–410 (2001)
Bielarz, C., Kalliadasis, S.: Time-dependent free-surface thin film flows over topography. Phys. Fluids 15, 2512–2524 (2003)
Dávalos-Orozco, L.A.: Instabilities of thin films flowing down flat and smoothly deformed walls. Microgravity Sci. Technol. 20, 225–229 (2008)
Dávalos-Orozco, L.A.: Nonlinear instability of a thin film flowing down a smoothly deformed surface. Phys. Fluids 19, 074103 (2007)
Wierschem, A., Aksel, N.: Instability of a liquid film flowing down an inclined wavy plane. Physica D 186, 221–237 (2003)
Wierschem, A., Lepski, C., Aksel, N.: Effect of long undulated bottoms on thin gravity-driven films. Acta Mech. 179, 41–66 (2005)
Trifonov, Y.Y.: Stability and nonlinear wavy regimes in downward film flows on a corrugated surface. J. App. Mech. Tech. Phys. 48, 91–100 (2007)
Trifonov, Y.Y.: Stability of a viscous liquid film flowing down a periodic surface. Int. J. Multiph. Flow 33, 1186–1204 (2007)
Heining, C., Aksel, N.: Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline. Int. J. Multiph. Flow 36, 847–857 (2010)
D’Alessio, S.J.D., Pascal, J.P., Jasmine, H.A.: Instability in gravity-driven flow over uneven surfaces. Phys. Fluids 21, 062105 (2009)
Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations, 2nd edn. Oxford University Press, Oxford (1987)
Ruyer-Quil, C., Manneville, P.: Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357 (2000)
Balmforth, N.J., Mandre, S.: Dynamics of roll waves. J. Fluid Mech. 514, 1–33 (2004)
Kármán, Th. v.: Über laminare und turbulente Reibung. ZAMM 1, 233-252 (1921)
Pohlhausen, K.: Zur näherungsweisen Integration der Differentialgleichung der laminaren Reibungsschicht. ZAMM 1, 252–268 (1921)
Pollak, T., Aksel, N.: Crucial flow stabilization and multiple instability branches of gravity-driven films over topography. Phys. Fluids 25, 024103 (2013)
Trifonov, Y.Y.: Stability of a film flowing down an inclined corrugated plate: the direct Navier–Stokes computations and Floquet theory. Phys. Fluids 26, 114101 (2014)
Cao, Z., Vlachogiannis, M., Bontozoglou, V.: Experimental evidence for a short-wave global mode in film flow along periodic corrugations. J. Fluid Mech. 718, 304–320 (2013)
Schörner, M., Reck, D., Aksel, N.: Stability phenomena far beyond the Nusselt flow: revealed by experimental asymptotics. Phys. Fluids 28, 022102 (2016)
Trifonov, Y.Y.: Viscous liquid film flow down an inclined corrugated surface. Calculation of the flow stability to arbitrary perturbations using an integral method. J. Appl. Mech. Tech. Phys. 57, 195–201 (2016)
Trifonov, Y.Y.: Nonlinear waves on a liquid film falling down an inclined corrugated surface. Phys. Fluids 29, 054104 (2017)
Schörner, M., Reck, D., Aksel, N., Trifonov, Y.Y.: Switching between different types of stability isles in films over topographies. Acta Mech. 229, 423–436 (2018)
Schörner, M., Aksel, N.: The stability cycle: a universal pathway for the stability of films over topography. Phys. Fluids 30, 012105 (2018)
Dauth, M., Schörner, M., Aksel, N.: What makes the free surface waves over topographies convex or concave? Phys. Fluids 29, 092108 (2017)
Reck, D., Aksel, N.: Experimental study on the evolution of traveling waves over an undulated incline. Phys. Fluids 25, 102101 (2013)
Vlachogiannis, M., Samandas, A., Leontidis, V., Bontozoglou, V.: Effect of channel width on the primary instability of inclined film flow. Phys. Fluids 22, 012106 (2010)
Leontidis, V., Vatteville, J., Vlachogiannis, M., Andritsos, N., Bontozoglou, V.: Nominally two-dimensional waves in inclined film flow in channels of finite width. Phys. Fluids 22, 112106 (2010)
Georgantaki, A., Vatteville, J., Vlachogiannis, M., Bontozoglou, V.: Measurements of liquid film flow as a function of fluid properties and channel width: evidence for surface-tension-induced long-range transverse coherence. Phys. Rev. E 84, 026325 (2011)
Pollak, T., Haas, A., Aksel, N.: Side wall effects on the instability of thin gravity-driven films: from long-wave to short-wave instability. Phys. Fluids 23, 094110 (2011)
Guzanov, V.V., Bobylev, A.V., Heinz, O.M., Kvon, A.Z., Markovich, D.M.: Characterization of 3-D wave flow regimes on falling liquid films. Int. J. Multiph. Flow 99, 474–484 (2018)
Thompson, A.B., Gomes, S.N., Pavoliotis, G.A., Papageorgiou, D.T.: Stabilising falling liquid film flows using feedback control. Phys. Fluids 28, 012107 (2016)
Gomes, S.N., Kalliadasis, S., Papageorgiou, D.T., Pavoliotis, G.A.: Controlling roughening processes in the stochastic Kuramoto–Sivashinsky equation. Physica D 348, 33–43 (2017)
Usha, R.: Effects of velocity slip on the inertialess instability of a contaminated two-layer film flow. Acta Mech. 226, 3111–3132 (2015)
Ghosh, S., Usha, R.: Stability of viscosity stratified flows down an incline: role of miscibility and wall slip. Phys. Fluids 28, 104101 (2016)
Tseluiko, D., Blyth, M.G., Papageorgiou, D.T.: Stability of film flow over inclined topography based on a long-wave nonlinear model. J. Fluid Mech. 729, 638–671 (2013)
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Aksel, N., Schörner, M. Films over topography: from creeping flow to linear stability, theory, and experiments, a review. Acta Mech 229, 1453–1482 (2018). https://doi.org/10.1007/s00707-018-2146-y
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DOI: https://doi.org/10.1007/s00707-018-2146-y