Steady two-dimensional thin-film flow of a Newtonian fluid is examined in this theoretical study. The influence of exit conditions and gravity is examined in detail. The considered flow is of moderately high inertia. The flow is dictated by the thin-film equations of boundary layer type, which are solved by expanding the flow field in orthonormal modes in the transverse direction and using Galerkin projection method, combined with integration along the flow direction. Three types of exit conditions are investigated, namely, parabolic, semiparabolic, and uniform flow. It is found that the type of exit conditions has a significant effect on the development of the free surface and flow field near the exit. While for the parabolic velocity profile at the exit, the free surface exhibits a local depression, for semiparabolic and uniform velocity profiles, the height of the film increases monotonically with streamwise position. In order to examine the influence of gravity, the flow is studied down a vertical wall as well as over a horizontal wall. The role of gravity is different for the two types of wall orientation. It is found that for the horizontal wall, a hydraulic-jump-like structure is formed and the flow further downstream exhibits a shock. The influence of exit conditions on shock formation is examined in detail.
Issue Section:
Fundamental Issues and Canonical Flows
1.
Pearson
, J. R. A.
, 1985, Mechanics of Polymer Processing
, Elsevier Applied Sciences
, New York
.2.
Middleman
, S.
, 1977, Fundamentals of Polymer Processing
, McGraw-Hill
, New York
.3.
Oron
, A.
, Davis
, S. H.
, and Bankoff
, S. G.
, 1997, “Long-Scale Evolution of Thin Liquid Films
,” Rev. Mod. Phys.
0034-6861, 69
(3
), pp. 931
–980
.4.
Chang
, H.-C.
, and Demekhin
, E. A.
, 2002, Complex Wave Dynamics on Thin Films
, Elsevier Sciences B. V.
, Amsterdam
.5.
Szeri
, A. Z.
, 1987, “Some Extensions of the Lubrication Theory of Osborne Reynolds
,” Trans. ASME, J. Tribol.
0742-4787, 109
(1
), pp. 21
–36
.6.
Quere
, D.
, 1999, “Fluid Coating on a Fiber
,” Annu. Rev. Fluid Mech.
0066-4189, 31
, pp. 347
–384
.7.
Tuck
, E. O.
, and Bentwitch
, M.
, 1983, “Sliding Sheets: Lubrication With Comparable Viscous and Inertia Forces
,” J. Fluid Mech.
0022-1120, 135
, pp. 51
–69
.8.
Watson
, E. J.
, 1964, “The Radial Spread of a Liquid Jet Over a Horizontal Plane
,” J. Fluid Mech.
0022-1120, 20
, pp. 481
–499
.9.
Hamrock
, B. J.
, 1994, Fundamentals of Fluid Film Lubrication
, McGraw-Hill
, New York
.10.
Takeshi
, O.
, 1999, “Surface Equation of Falling Film Flows With Moderate Reynolds Number and Large but Finite Weber Number
,” Phys. Fluids
1070-6631, 11
(11
), pp. 3247
–3269
.11.
Shkadov
, V. Ya.
, 1967, “Wave Condition of Flow in a Thin Viscous Layer Under the Action of Gravitational Forces
,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza
0568-5281, 1
, pp. 43
–51
.12.
Alekseenko
, S. V.
, Nakoryakov
, V. E.
, and Pokusaev
, B. G.
, 1985, “Wave Formation on a Vertical Falling Liquid Film
,” AIChE J.
0001-1541, 31
, pp. 1446
–1460
.13.
Kim
, K.
, and Khayat
, R. E.
, 2002, “Transient Coating Flow of a Thin Non-Newtonian Fluid Film
,” Phys. Fluids
1070-6631, 14
(7
), pp. 2202
–2215
.14.
Khayat
, R. E.
, 2000, “Transient Two-Dimensional Coating Flow of a Viscoelastic Fluid Film on a Substrate of Arbitrary Shape
,” J. Non-Newtonian Fluid Mech.
0377-0257, 95
, pp. 199
–233
.15.
Nakoryakov
, V. E.
, Pokusaev
, B. G.
, and Troyan
, E. N.
, 1978, “Impingement of an Axisymmetric Liquid Jet on a Barrier
,” Int. J. Heat Mass Transfer
0017-9310, 21
, pp. 1175
–1184
.16.
Craik
, A. D. D.
, Latham
, R. C.
, Fawkes
, M. J.
, and Gribbon
, P. W. F.
, 1981, “The Circular Hydraulic Jump
,” J. Fluid Mech.
0022-1120, 112
, pp. 347
–362
.17.
Bohr
, T.
, Ellegaard
, C.
, Hansen
, A. E.
, and Haaning
, A.
, 1996, “Hydraulic Jump, Flow Separation and Wave Breaking: An Experimental Study
,” Physica B
0921-4526, 228
, pp. 1
–10
.18.
Bohr
, T.
, Ellegaard
, C.
, Hansen
, A. E.
, Hansen
, K.
, Haaning
, A.
, Putkaradze
, V.
, and Watanabe
, S.
, 1998, “Separation and Pattern Formation in Hydraulic Jumps
,” Physica A
0378-4371, 249
, pp. 111
–117
.19.
Bush
, J. W. M.
, and Aristoff
, J. M.
, 2003, “The Influence of Surface Tension on the Circular Hydraulic Jump
,” J. Fluid Mech.
0022-1120, 489
, pp. 229
–238
.20.
Bowles
, R.
, and Smith
, F.
, 1992, “The Standing Hydraulic Jump: Theory, Computations and Comparisons With Experiments
,” J. Fluid Mech.
0022-1120, 242
, pp. 145
–168
.21.
Higuerra
, F.
, 1994, “The Hydraulic Jump in a Viscous Laminar Flow
,” J. Fluid Mech.
0022-1120, 274
, pp. 69
–92
.22.
Watanabe
, S.
, Putkaradze
, V.
, and Bohr
, T.
, 2003, “Integral Methods for Shallow Free Surface Flows With Separation
,” J. Fluid Mech.
0022-1120, 480
, pp. 233
–265
.23.
Chippada
, S.
, Jue
, T. C.
, Joo
, S. W.
, Wheeler
, M. F.
, and Ramaswamy
, B.
, 1996, “Numerical Simulation of Free-Boundary Problems
,” Int. J. Comput. Fluid Dyn.
1061-8562, 7
, pp. 91
–118
.24.
Lee
, J.-J.
, and Mei
, C. C.
, 1996, “Stationary Waves on an Inclined Sheet of Viscous Fluid at High Reynolds and Moderate Weber Numbers
,” J. Fluid Mech.
0022-1120, 307
, pp. 191
–229
.25.
Khayat
, R. E.
, and Welke
, S.
, 2001, “Influence of Inertia, Gravity and Substrate Topography on the Two-Dimensional Transient Coating Flow of a Thing Newtonian Fluid Film
,” Phys. Fluids
1070-6631, 13
(2
), pp. 355
–367
.26.
Siddique
, M. R.
, and Khayat
, R. E.
, 2002, “Influence of Inertia and Topography in Thin-Cavity Flow
,” Phys. Fluids
1070-6631, 14
(5
), pp. 1703
–1719
.27.
Khayat
, R. E.
, 2001, “Influence of Inertia on the Transient Axisymmetric Free-Surface Flow Inside Thin Cavities of Arbitrary Shape
,” Phys. Fluids
1070-6631, 13
(12
), pp. 3636
–3651
.28.
Zienkiewicz
, O. C.
, and Heinrich
, J. C.
, 1979, “A Unified Treatment of Steady-State Shallow Water and Two-Dimensional Navier-Stokes Equations—Finite-Element Penalty Function Approach
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 17/18
, pp. 673
–698
.29.
Sell
, G. R.
, Foias
, C.
, and Temam
, R.
, 1993, Turbulence in Fluid Flows: A Dynamical Systems Approach
, Springer-Verlag
, New York
.30.
Deane
, A. E.
, Kevrekidis
, I. G.
, Karniadakis
, G. E.
, and Orszag
, S. A.
, 1991, “Low-Dimensional Models for Complex Geometry Flows: Application to Grooved Channels and Circular Cylinders
,” Phys. Fluids A
0899-8213, 3
(10
), pp. 2337
–2354
.31.
Khayat
, R. E.
, 1999, “Finite-Amplitude Taylor-Vortex Flow of Viscoelastic Fluids
,” J. Fluid Mech.
0022-1120, 400
, pp. 33
–58
.32.
Ashrafi
, N.
, and Khayat
, R. E.
, 2000, “A Low-Dimensional Approach to Nonlinear Plane-Couette Flow of Viscoelastic Fluids
,” Phys. Fluids
1070-6631, 12
(2
), pp. 345
–365
.33.
Khayat
, R. E.
, and Ashrafi
, N.
, 2002, “A Low-Dimensional Approach to Nonlinear Plane-Poiseuille Flow of Viscoelastic Fluids
,” Phys. Fluids
1070-6631, 14
(5
), pp. 1757
–1767
.34.
Khayat
, R. E.
, 2000, “Transient Free-Surface Flow Inside Thin Cavities of Viscoelastic Fluids
,” J. Non-Newtonian Fluid Mech.
0377-0257, 91
, pp. 15
–29
.35.
Chang
, H.-C.
, 1994, “Wave Evolution on a Falling Film
,” Annu. Rev. Fluid Mech.
0066-4189, 26
, pp. 103
–136
.36.
Muhammad
, T.
, and Khayat
, R. E.
, 2004, “Effect of Substrate Movement on Shock Formation in Pressure-Driven Coating Flow
,” Phys. Fluids
1070-6631, 16
(5
), pp. 1818
–1821
.37.
Kalliadasis
, S.
, Bielarz
, C.
, and Homsy
, G. M.
, 2000, “Steady Free-Surface Thin Film Flows Over Topography
,” Phys. Fluids
1070-6631, 12
(8
), 1889
–1898
.Copyright © 2008
by American Society of Mechanical Engineers
You do not currently have access to this content.
