Transient Thin-Film Flow on a Moving Boundary of Arbitrary Topography

2009 ◽  
Vol 131 (10) ◽  
Author(s):  
Roger E. Khayat ◽  
Tauqeer Muhammad
Keyword(s):  
Thin Film ◽  
Steady State ◽  
Moving Boundary ◽  
Initial Conditions ◽  
Flow Behavior ◽  
Galerkin Projection ◽  
Thin Film Flow ◽  
Flat Substrate ◽  
Thin Film Equations ◽  
Two Dimensional Flow

The transient two-dimensional flow of a thin Newtonian fluid film over a moving substrate of arbitrary shape is examined in this theoretical study. The interplay among inertia, initial conditions, substrate speed, and shape is examined for a fluid emerging from a channel, wherein Couette–Poiseuille conditions are assumed to prevail. The flow is dictated by the thin-film equations of the “boundary layer” type, which are solved by expanding the flow field in terms of orthonormal modes depthwise and using the Galerkin projection method. Both transient and steady-state flows are investigated. Substrate movement is found to have a significant effect on the flow behavior. Initial conditions, decreasing with distance downstream, give rise to the formation of a wave that propagates with time and results in a shocklike structure (formation of a gradient catastrophe) in the flow. In this study, the substrate movement is found to delay shock formation. It is also found that there exists a critical substrate velocity at which the shock is permanently obliterated. Two substrate geometries are considered. For a continuous sinusoidal substrate, the disturbances induced by its movement prohibit the steady-state conditions from being achieved. However, for the case of a flat substrate with a bump, a steady state exists.

Influence of Upstream Conditions and Gravity on Highly Inertial Thin-Film Flow

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
Roger E. Khayat
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Flow Field ◽  
Film Flow ◽  
Flow Direction ◽  
Vertical Wall ◽  
Galerkin Projection ◽  
Thin Film Flow ◽  
Thin Film Equations ◽  
Horizontal Wall

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.

scholarly journals Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

2021 ◽  
Author(s):  
Konstantinos Dareiotis ◽  
Benjamin Gess ◽  
Manuel V. Gnann ◽  
Günther Grün
Keyword(s):  
Thin Film ◽  
Energy Estimates ◽  
Approximation Procedure ◽  
Thin Film Flow ◽  
Thin Film Equations ◽  
Scalar Conservation Law ◽  
Thin Film Equation ◽  
Degenerate Parabolic ◽  
Scalar Conservation ◽  
Martingale Solutions

AbstractWe prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

Non-Equilibrium Thin-Film Flow in Rotating Disk and Cone

2008 ◽  
Author(s):  
Wallace Woon-Fong Leung
Keyword(s):  
Thin Film ◽  
Mammalian Cells ◽  
Flow Behavior ◽  
Experimental Tests ◽  
Rotating Disk ◽  
Disk Surface ◽  
Thin Film Flow ◽  
Liquid Stream ◽  
Highly Nonlinear ◽  
Rotating Cone

The acceleration of a continuous feed liquid stream in a film “down” the rotating cone and disk is of great interest in centrifuges [1, 2], thin-film reactors and process intensifiers. The mechanism of feed acceleration is determined by an interaction of several different effects. Circumferential viscous forces act to increase the angular momentum. The centrifugal field thus produced establishes a body-force component along the cone/disk surface, thereby driving the flow “down” toward larger radius. The longitudinal flow is however impeded by longitudinal resistance forces. These different effects compete with each other as the flow proceeds, never quite coming to an unchanging equilibrium state. An approximate integral method which was used to explore the “near-equilibrium” flow behavior in earlier work has been extended to investigate the case with large departure from equilibrium. The latter exhibits complicated highly nonlinear effect. Despite this, useful information can be obtained from the theoretical analysis. Experimental results on feed acceleration of liquid streams at various feed rates and rotation speeds in a rotating cone have been used to validate the study. The theoretical study with complementary experimental tests provides insights into how continuous liquid stream in form of a thin film is being accelerated using rotating cones and disks, and the associated shear rates involved. The latter has important bearing in processing shear-sensitive mammalian cells in biopharmaceutical separation with centrifuges and mass transfer in thin-film reactors.

scholarly journals A model for dip-coating of a two liquid mixture

2002 ◽  
Vol 29 (6) ◽  
pp. 313-324 ◽  
Author(s):  
S. B. G. O'Brien ◽  
M. Hayes
Keyword(s):  
Thin Film ◽  
Film Flow ◽  
Liquid Mixture ◽  
Dip Coating ◽  
Industrial Applications ◽  
Thin Film Flow ◽  
Flat Substrate ◽  
Liquid Suspension ◽  
Volatile Liquid ◽  
Miscible Liquids

We consider a thin film flow where a flat substrate is coated with a mixture of two miscible liquids, of equal viscosity, and develop a model to predict the evolving coating thicknesses. The developed model can, under certain circumstances, be used as an approximation for the dip-coating of a liquid suspension of a viscous volatile liquid and solid solute as occurs in many industrial applications.

scholarly journals Patterns formed in a thin film with spatially homogeneous and non-homogeneous Derjaguin disjoining pressure

2021 ◽  
pp. 1-25
Author(s):  
ABDULWAHED S. ALSHAIKHI ◽  
MICHAEL GRINFELD ◽  
STEPHEN K. WILSON
Keyword(s):  
Thin Film ◽  
Steady State ◽  
Bifurcation Theory ◽  
Disjoining Pressure ◽  
Thin Film Equations ◽  
Planar Substrate ◽  
Spatially Homogeneous ◽  
The One ◽  
Steady State Solutions ◽  
Average Film Thickness

We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honisch et al. [Langmuir 31: 10618–10631, 2015] and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady-state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honisch et al., and study the dependence of the steady-state solutions on the wettability contrast in that case.

Influence of initial conditions on transient two-dimensional thin-film flow

2002 ◽  
Vol 14 (12) ◽  
pp. 4448-4451 ◽  
Author(s):  
Roger E. Khayat ◽  
Kyu-Tae Kim
Keyword(s):  
Thin Film ◽  
Film Flow ◽  
Initial Conditions ◽  
Two Dimensional ◽  
Thin Film Flow

Thin-Film Flow of a Power-Law Fluid Down an Inclined Plane

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
A. Ganguly ◽  
M. Reza ◽  
A. S. Gupta
Keyword(s):  
Thin Film ◽  
Power Law ◽  
Film Flow ◽  
Equations Of Motion ◽  
Dimensionless Time ◽  
Two Dimensional ◽  
Thin Film Flow ◽  
Power Law Fluid ◽  
Inclined Plane ◽  
Two Dimensional Flow

An analysis is presented for two-dimensional flow of a thin layer of power-law fluid down an inclined plane. Integration of the equations of motion using lubrication approximations shows that for both pseudoplastic and dilatant fluids, the rate of advance of a blob of fluid of given volume decreases with increasing time. The analysis further reveals that for dimensionless time less than about 0.50, a blob of the fluid (of fixed volume) with given exponent n moves faster than a fluid of same volume with larger n. However, thereafter, a blob of the latter fluid moves faster than the former fluid.

Formulation and Spectral Reduction of the Dynamical Model of a Circulating Fluidized Bed Combustor

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Katarzyna Bizon ◽  
Gaetano Continillo
Keyword(s):  
Steady State ◽  
Fluidized Bed ◽  
Circulating Fluidized Bed ◽  
Initial Conditions ◽  
Galerkin Projection ◽  
State Variables ◽  
Order Reconstruction ◽  
Spectral Reduction ◽  
Isothermal Operation ◽  
Fluidized Bed Combustor

The distributed dynamic model of a circulating fluidized bed combustor in isothermal operation is developed and proposed, consisting of four 1–D evolutionary partial differential equations and relevant boundary and initial conditions. The continuum model is then approximated by a finite–difference method to provide a "reference" solution. Then, proper orthogonal decomposition with Galerkin projection is introduced to derive a reduced order model. The POD modes are then tested in the low-order reconstruction of space profiles of the state variables at different times, during the transient and at steady-state. POD–based models prove to be effective, being able to reproduce steady–state with four basis functions. Quantitative accuracy even during the early transient stage is achieved with as little as 24 modes.

Studies on Water in the Hydration Chamber of a Modified Electron Microscope

1970 ◽  
Vol 28 ◽  
pp. 544-545
Author(s):  
R. C. Moretz ◽  
G. G. Hausner ◽  
D. F. Parsons
Keyword(s):  
Thin Film ◽  
Thin Films ◽  
Electron Microscope ◽  
Steady State ◽  
Room Temperature ◽  
Small Mass ◽  
Equilibrium Pressure ◽  
Biological Objects ◽  
Mechanical Pump ◽  
Differential Pumping

Use of the electron microscope to examine wet objects is possible due to the small mass thickness of the equilibrium pressure of water vapor at room temperature. Previous attempts to examine hydrated biological objects and water itself used a chamber consisting of two small apertures sealed by two thin films. Extensive work in our laboratory showed that such films have an 80% failure rate when wet. Using the principle of differential pumping of the microscope column, we can use open apertures in place of thin film windows.Fig. 1 shows the modified Siemens la specimen chamber with the connections to the water supply and the auxiliary pumping station. A mechanical pump is connected to the vapor supply via a 100μ aperture to maintain steady-state conditions.

Sign in / Sign up

Export Citation Format

Share Document