A falling film down a slippery inclined plane

2011 ◽  
Vol 684 ◽  
pp. 353-383 ◽  
Author(s):  
A. Samanta ◽  
C. Ruyer-Quil ◽  
B. Goyeau
Keyword(s):  
Travelling Waves ◽  
Falling Film ◽  
Inclined Plane ◽  
Local Flow ◽  
Navier Slip ◽  
Averaged Equations ◽  
Gravity Driven ◽  
The Waves ◽  
Navier Slip Condition ◽  
Good Agreement

AbstractA gravity-driven film flow on a slippery inclined plane is considered within the framework of long-wave and boundary layer approximations. Two coupled depth-averaged equations are derived in terms of the local flow rate $q(x, t)$ and the film thickness $h(x, t)$. Linear stability analysis of the averaged equations shows good agreement with the Orr–Sommerfeld analysis. The effect of a slip at the wall on the primary instability has been found to be non-trivial. Close to the instability onset, the effect is destabilising whereas it becomes stabilising at larger values of the Reynolds number. Nonlinear travelling waves are amplified by the presence of the slip. Comparisons to direct numerical simulations show a remarkable agreement for all tested values of parameters. The averaged equations capture satisfactorily the speed, shape and velocity distribution in the waves. The Navier slip condition is observed to significantly enhance the backflow phenomenon in the capillary region of the solitary waves with a possible effect on heat and mass transfer.

An Accurate Low Dimension Model for the Waves on Thin Layer Fluid Flowing Down an Inclined Plane

2009 ◽  
Vol 76 (6) ◽  
Author(s):  
H. Ait Abderrahmane ◽  
G. H. Vatistas
Keyword(s):  
Film Flow ◽  
Equation Model ◽  
Surface Velocity ◽  
Inclined Plane ◽  
Linear Pulse ◽  
Pulse Structure ◽  
The Waves ◽  
Theoretical Results ◽  
Primary Instability ◽  
Good Agreement

This technical brief deals with the surface instability mode of a liquid film flowing down an inclined plane. A four-equation model that describes the development of the film depth, the flow rate, the free-surface velocity, and the wall shear stress is proposed. The obtained results were found to be in very good agreement with experimental and theoretical results of Liu et al. (1993, “Measurements of the Primary Instability of Film Flow,” J. Fluid Mech., 250, pp. 69–101) and Brevdo et al. (1999, “Linear Pulse Structure and Signalling in Film Flow on an Inclined Plane,” J. Fluid Mech., 396, pp. 37–71).

Modelling film flows down a fibre

2008 ◽  
Vol 603 ◽  
pp. 431-462 ◽  
Author(s):  
C. RUYER-QUIL ◽  
P. TREVELEYAN ◽  
F. GIORGIUTTI-DAUPHINÉ ◽  
C. DUPRAT ◽  
S. KALLIADASIS
Keyword(s):  
Evolution Equations ◽  
Selection Process ◽  
Thin Liquid Film ◽  
Local Flow ◽  
Dispersive Effect ◽  
Linear Waves ◽  
Gravity Driven ◽  
Gravity Driven Flow ◽  
Film Flows ◽  
Good Agreement

Consider the gravity-driven flow of a thin liquid film down a vertical fibre. A model of two coupled evolution equations for the local film thickness h and the local flow rate q is formulated within the framework of the long-wave and boundary-layer approximations. The model accounts for inertia and streamwise viscous diffusion. Evolution equations obtained by previous authors are recovered in the appropriate limit. Comparisons to experimental results show good agreement in both linear and nonlinear regimes. Viscous diffusion effects are found to have a stabilizing dispersive effect on the linear waves. Time-dependent computations of the spatial evolution of the film reveal a strong influence of streamwise viscous diffusion on the dynamics of the flow and the wave selection process.

A falling film on a porous medium

2013 ◽  
Vol 716 ◽  
pp. 414-444 ◽  
Author(s):  
A. Samanta ◽  
B. Goyeau ◽  
C. Ruyer-Quil
Keyword(s):  
Porous Medium ◽  
Composite Layer ◽  
Lower Boundary ◽  
Equation Model ◽  
Falling Film ◽  
Local Flow ◽  
Porous Layers ◽  
Porous Interface ◽  
Gravity Driven ◽  
The Stability

AbstractA gravity-driven falling film on a saturated porous inclined plane is studied via a continuum approach, where the liquid and porous layers are considered as a single composite layer. Using a weighted residual technique, a two-equation model is derived in terms of the local flow rate $q(x, t)$ and the entire layer thickness $H(x, t)$. Its linear stability analysis has been satisfactorily compared to the results of the Orr–Sommerfeld problem. The principal effect of the porous substrate on the film flow is to displace the liquid–porous interface to an effective liquid–solid interface located at the lower boundary of the upper momentum boundary layer in the porous medium. The stability and dynamics of the film is thus only weakly affected by the presence of a permeable substrate. In both the linear and the nonlinear regimes, the spatial response of a falling film on a porous medium is not very different from that observed on an impermeable inclined wall. However, the wavy motion of the film triggers a significant exchange of mass at the liquid–porous interface.

scholarly journals The spin-up of a linearly stratified fluid in a sliced, circular cylinder

2016 ◽  
Vol 806 ◽  
pp. 254-303
Author(s):  
R. J. Munro ◽  
M. R. Foster
Keyword(s):  
Circular Cylinder ◽  
Rossby Waves ◽  
Rotation Axis ◽  
A Priori ◽  
Propagation Speed ◽  
Stratified Fluid ◽  
Decay Rates ◽  
The Waves ◽  
Western Half ◽  
Good Agreement

A linearly stratified fluid contained in a circular cylinder with a linearly sloped base, whose axis is aligned with the rotation axis, is spun-up from a rotation rate $\unicode[STIX]{x1D6FA}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}$ to $\unicode[STIX]{x1D6FA}$ (with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}\ll \unicode[STIX]{x1D6FA}$) by Rossby waves propagating across the container. Experimental results presented here, however, show that if the Burger number $S$ is not small, then that spin-up looks quite different from that reported by Pedlosky & Greenspan (J. Fluid Mech., vol. 27, 1967, pp. 291–304) for $S=0$. That is particularly so if the Burger number is large, since the Rossby waves are then confined to a region of height $S^{-1/2}$ above the sloped base. Axial vortices, ubiquitous features even at tiny Rossby numbers of spin-up in containers with vertical corners (see van Heijst et al.Phys. Fluids A, vol. 2, 1990, pp. 150–159 and Munro & Foster Phys. Fluids, vol. 26, 2014, 026603, for example), are less prominent here, forming at locations that are not obvious a priori, but in the ‘western half’ of the container only, and confined to the bottom $S^{-1/2}$ region. Both decay rates from friction at top and bottom walls and the propagation speed of the waves are found to increase with $S$ as well. An asymptotic theory for Rossby numbers that are not too large shows good agreement with many features seen in the experiments. The full frequency spectrum and decay rates for these waves are discussed, again for large $S$, and vertical vortices are found to occur only for Rossby numbers comparable to $E^{1/2}$, where $E$ is the Ekman number. Symmetry anomalies in the observations are determined by analysis to be due to second-order corrections to the lower-wall boundary condition.

Modeling gravity-driven flows on an inclined plane

1996 ◽  
Vol 101 (B5) ◽  
pp. 11565-11577 ◽  
Author(s):  
Barbara C. Bruno ◽  
Stephen M. Baloga ◽  
G. Jeffrey Taylor
Keyword(s):  
Inclined Plane ◽  
Gravity Driven

On the Effect of Hull Girder Flexibility on the Vertical Wave Bending Moment for Ultra Large Container Vessels

2012 ◽  
Author(s):  
Ingrid Marie Vincent Andersen ◽  
Jørgen Juncher Jensen
Keyword(s):  
Bending Moment ◽  
Main Concern ◽  
Hull Girder ◽  
Numerical Predictions ◽  
Strip Theory ◽  
Reliability Method ◽  
Non Linear ◽  
Large Container ◽  
The Waves ◽  
Good Agreement

Currently, a number of very large container ships are being built and more are on order, and some concerns have been expressed about the importance of the reduced hull girder stiffness to the wave-induced loads. The main concern is related to the fatigue life, but also a possible increase in the global hull girder loads as consequence of the increased hull flexibility must be considered. This is especially so as the rules of the classification societies do not explicitly account for the effect of hull flexibility on the global loads. In the present paper an analysis has been carried out for the 9,400 TEU container ship used as case-ship in the EU project TULCS (Tools for Ultra Large Container Ships). A non-linear time-domain strip theory is used for the hydrodynamic analysis of the vertical bending moment amidships in sagging and hogging conditions for a flexible and a rigid modelling of the ship. The theory takes into account non-linear radiation forces (memory effects) through the use of a set of higher order differential equations. The non-linear hydrostatic restoring forces and non-linear Froude-Krylov forces are determined accurately at the instantaneous position of the ship in the waves. Slamming forces are determined by a standard momentum formulation. The hull flexibility is modelled as a non-prismatic Timoshenko beam. Generally, good agreement with experimental results and more accurate numerical predictions has previously been obtained in a number of studies. The statistical analysis is done using the First Order Reliability Method (FORM) supplemented with Monte Carlo simulations. Furthermore, strip-theory calculations are compared to model tests in regular waves of different wave lengths using a segmented, flexible model of the case-ship and good agreement is obtained for the longest of the waves. For the shorter waves the agreement is less good. The discrepancy in the amplitudes of the bending moment can most probably be explained by an underestimation on the effect of momentum slamming in the strip-theory applied.

Contact line instability of gravity driven thin films flowing down an inclined plane with wall slippage

2020 ◽  
Vol 214 ◽  
pp. 115418 ◽  
Author(s):  
Chicheng Ma ◽  
Jianlin Liu ◽  
Shilin Xie ◽  
Yongqi Liu
Keyword(s):  
Thin Films ◽  
Contact Line ◽  
Inclined Plane ◽  
Gravity Driven ◽  
Wall Slippage

A Lagrangian Model for Irregular Waves and Wave Kinematics

2003 ◽  
Vol 125 (2) ◽  
pp. 94-102 ◽  
Author(s):  
Svein Helge Gjøsund
Keyword(s):  
Iterative Method ◽  
Irregular Waves ◽  
Lagrangian Model ◽  
Surface Zone ◽  
Wave Interactions ◽  
Wave Flume ◽  
Wave Kinematics ◽  
Eulerian Formulation ◽  
The Waves ◽  
Good Agreement

It has proven difficult to describe the kinematics in irregular waves satisfactorily, in particular for the surface zone in broad-banded waves. A Lagrangian approach offers distinct advantages in this respect, eliminating the need for extrapolation of solutions or “stretching” of coordinates. This paper presents a model of irregular waves based on superposition of linear Lagrangian wave components, using an iterative method to obtain the Eulerian solution. This approach yields theoretically consistent results everywhere in the waves, and comparisons with wave flume measurements show good agreement. Also, the linear Lagrangian model includes wave interactions that would be nonlinear in an Eulerian formulation.

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