Slow spreading of a sheet of Bingham fluid on an inclined plane

1989 ◽  
Vol 207 ◽  
pp. 505-529 ◽  
Author(s):  
Ko Fei Liu ◽  
Chiang C. Mei
Keyword(s):  
Free Surface ◽  
Fluid Layer ◽  
Thin Sheet ◽  
Nonlinear Partial Differential Equation ◽  
Gravity Current ◽  
Bingham Fluid ◽  
Constant Speed ◽  
Fluid Mud ◽  
Inclined Plane ◽  
First Case

To study the dynamics of fluid mud with a high concentration of cohesive clay particles, we present a theory for a thin sheet of Bingham-plastic fluid flowing slowly on an inclined plane. The physics is discussed on the approximate basis of the lubrication theory. Because of the yield stress, the free surface need not be horizontal when the Bingham fluid is in static equilibrium, nor parallel to the plane bed when in steady flow. We then show that there is a variety of gravity currents that can advance at a constant speed and with the same profile. Experimental confirmation of one type is presented. By solving a nonlinear partial differential equation, transient flows due either to a steady upstream discharge or to the sudden release of a finite fluid mass on another fluid layer are studied. In the first case there is a mud front which ultimately propagates as a constant speed as a steady gravity current. In the second case, when the ambient layer is sufficiently shallow that there is no initial motion, the flow induced by the new fluid can terminate after the disturbance has travelled a finite distance. The extent of the final spread is examined. Disturbances due to an external pressure travelling parallel to the free surface are also examined. It is found in particular that a travelling localized pulse of pressure gradient not only generates a localized mud disturbance which travels along with the forcing pressure, but further leaves behind a permanent footprint.

Spreading and instability of a viscous fluid sheet

1990 ◽  
Vol 211 ◽  
pp. 373-392 ◽  
Author(s):  
L. M. Hocking
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Related Problem ◽  
Numerical Solutions ◽  
Thin Sheet ◽  
Leading Edge ◽  
Small Disturbance ◽  
Inclined Plane ◽  
Moving Contact Line ◽  
Moving Contact

Experiments by Huppert (1982) have demonstrated that a finite volume of fluid placed on an inclined plane will elongate into a thin sheet of fluid as it slides down the plane. If the fluid is initially placed uniformly across the plane, the sheet retains its two-dimensionality for some time, but when it has become sufficiently long and thin, the leading edge develops a spanwise instability. A similarity solution for this motion was derived by Huppert, without taking account of the edge regions where surface tension is important. When these regions are examined, it is found that the conditions at the edges can be satisfied, but only when the singularity associated with the moving contact line is removed. When the sheet is sufficiently elongated, the profile of the free surface shows an upward bulge near the leading edge. It is suggested that the observed instability of the shape of the leading edge is a result of the dynamics of the fluid in this bulge. The related problem of a ridge of fluid sliding down the plane is examined and found to be linearly unstable. The spanwise lengthscale of this instability is, however, dependent on the width of the channel occupied by the fluid, which is at variance with the observed nature of the instability. Preliminary numerical solutions for the nonlinear development of a small disturbance to the position of a straight leading edge show the incipient development of a finger of fluid with a width that does not depend on the channel size, in accordance with the observations.

scholarly journals On a slender dry patch in a liquid film draining under gravity down an inclined plane

2001 ◽  
Vol 12 (3) ◽  
pp. 233-252 ◽  
Author(s):  
S. K. WILSON ◽  
B. R. DUFFY ◽  
S. H. DAVIS
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Liquid Film ◽  
Similarity Solutions ◽  
Inclined Plane ◽  
First Case ◽  
Transverse Profile ◽  
Dry Patch ◽  
Strong Surface ◽  
Weak Surface

In this paper two similarity solutions describing a steady, slender, symmetric dry patch in an infinitely wide liquid film draining under gravity down an inclined plane are obtained. The first solution, which predicts that the dry patch has a parabolic shape and that the transverse profile of the free surface always has a monotonically increasing shape, is appropriate for weak surface-tension effects and far from the apex of the dry patch. The second solution, which predicts that the dry patch has a quartic shape and that the transverse profile of the free surface has a capillary ridge near the contact line and decays in an oscillatory manner far from it, is appropriate for strong surface-tension effects (in particular, when the plane is nearly vertical) and near (but not too close) to the apex of the dry patch. With the average volume flux per unit width (or equivalently with the uniform height of the layer far from the dry patch) prescribed, both solutions contain a free parameter. For each value of this parameter there is a unique solution in the first case and either no solution or a one-parameter family of solutions in the second case. The solutions capture some of the qualitative features observed in experiments.

scholarly journals Experiments on generation of surface waves by an underwater moving bottom

2015 ◽  
Vol 471 (2178) ◽  
pp. 20150069 ◽  
Author(s):  
Timothée Jamin ◽  
Leonardo Gordillo ◽  
Gerardo Ruiz-Chavarría ◽  
Michael Berhanu ◽  
Eric Falcon
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Surface Deformation ◽  
Fluid Layer ◽  
Fluid Velocity ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Spatio Temporal ◽  
Experimental Findings

We report laboratory experiments on surface waves generated in a uniform fluid layer whose bottom undergoes an upward motion. Simultaneous measurements of the free-surface deformation and the fluid velocity field are focused on the role of the bottom kinematics (i.e. its spatio-temporal features) in wave generation. We observe that the fluid layer transfers bottom motion to the free surface as a temporal high-pass filter coupled with a spatial low-pass filter. Both filter effects are often neglected in tsunami warning systems, particularly in real-time forecast. Our results display good agreement with a prevailing linear theory without any parameter fitting. Based on our experimental findings, we provide a simple theoretical approach for modelling the rapid kinematics limit that is applicable even for initially non-flat bottoms: this may be a key step for more realistic varying bathymetry in tsunami scenarios.

A note on the unsteady motion under gravity of a corner point on a free surface: a generalization of Stokes' theory

2008 ◽  
Vol 465 (2101) ◽  
pp. 165-173 ◽  
Author(s):  
D.J Needham ◽  
J Billingham
Keyword(s):  
Free Surface ◽  
Corner Point ◽  
Constant Pressure ◽  
Unsteady Motion ◽  
Constant Speed ◽  
Inviscid Fluid ◽  
Irrotational Motion

In this paper, we develop a theory based on local asymptotic coordinate expansions for the unsteady propagation of a corner point on the constant-pressure free surface bounding an incompressible inviscid fluid in irrotational motion under the action of gravity. This generalizes the result of Stokes and Michell relating to the horizontal propagation of a corner at constant speed.

Three-dimensional coating flow of nematic liquid crystal on an inclined substrate

2015 ◽  
Vol 26 (5) ◽  
pp. 647-669 ◽  
Author(s):  
M. A. LAM ◽  
L. J. CUMMINGS ◽  
T.-S. LIN ◽  
L. KONDIC
Keyword(s):  
Free Surface ◽  
Liquid Crystal ◽  
Nematic Liquid Crystal ◽  
Nonlinear Partial Differential Equation ◽  
Three Dimensional ◽  
Travelling Wave ◽  
Linear Stability Theory ◽  
Newtonian Flow ◽  
Surface Evolution ◽  
Coating Flow

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

scholarly journals An experimental and theoretical study of the dynamics of grounding lines

2013 ◽  
Vol 728 ◽  
pp. 5-28 ◽  
Author(s):  
Samuel S. Pegler ◽  
M. Grae Worster
Keyword(s):  
Theoretical Study ◽  
Fluid Layer ◽  
Gravity Current ◽  
Three Dimensional ◽  
Vertical Shear ◽  
Inviscid Fluid ◽  
West Antarctica ◽  
Full Contact ◽  
Grounding Line ◽  
Longitudinal Extension

AbstractWe present an experimental and theoretical study of a thin, viscous fluid layer that flows radially under gravity from a point source into a denser inviscid fluid layer of uniform depth above a rigid horizontal surface. Near the source, the viscous layer lies in full contact with the surface, forming a vertical-shear-dominated viscous gravity current. At a certain distance from the source, the layer detaches from the surface to form a floating current whose dynamics are controlled by the viscous stresses due to longitudinal extension. We describe the dynamics of the grounded and floating components using distinct thin-layer theories. Separating the grounded and floating regions is the freely moving line of detachment, or grounding line, whose evolution we model by balancing the horizontal forces between the two regions. Using numerical and asymptotic analysis, we calculate the evolution of the system from a self-similar form at early times towards a steady state at late times. We use our solutions to illustrate how three-dimensional stresses within marine ice sheets, such as that of West Antarctica, can lead to stabilization of the grounding line. To assess the validity of the assumptions underlying our model, we compare its predictions with data from a series of laboratory experiments.

scholarly journals A New Compression Test for Determining Free Surface Roughness Evolution in Thin Sheet Metals

Metals ◽  
2019 ◽  
Vol 9 (4) ◽  
pp. 451 ◽  
Author(s):  
Tsuyoshi Furushima ◽  
Kohei Aoto ◽  
Sergei Alexandrov
Keyword(s):  
Surface Roughness ◽  
Free Surface ◽  
Compression Test ◽  
Thin Sheet ◽  
Principal Strain ◽  
Strain Rates ◽  
Thin Sheets ◽  
Strain Paths ◽  
Effect Of The Support ◽  
Metallic Sheet

In sheet microforming processes, in-surface principal strain rates may be compressive such that the thickness of the sheet increases in the process of deformation. In general, the evolution of free surface roughness depends on the sense of the principal strain normal to the free surface. Therefore, in order to predict the evolution of free surface roughness in processes in which this normal principal strain is positive by means of empirical equations, it is necessary to carry out experiments in which the thickness of the sheet increases. Conventional experiments, such as the Marciniak test, do not provide such strain paths. In general, it is rather difficult to induce a sufficiently uniform state of strain in thin sheets of increasing thickness throughout the process of deformation because instability occurs at the very beginning of the process. The present paper proposes a compression test for thin sheets. Teflon sheets are placed between support jigs and the metallic sheet tested to prevent the occurrence of instability and significantly reduce the effect of the support jigs on the evolution of surface roughness. The test is used to determine the evolution of surface roughness in thin sheets made of C1220-O under three strain paths.

scholarly journals The Flow of a Variable Viscosity Fluid down an Inclined Plane with a Free Surface

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
M. S. Tshehla
Keyword(s):  
Free Surface ◽  
Variable Viscosity ◽  
Convective Heating ◽  
Fluid Viscosity ◽  
Inclined Plane ◽  
Flow Equations ◽  
Variation Parameter ◽  
Viscosity Variation ◽  
Viscosity Fluid ◽  
Variable Viscosity Fluid

The effect of a temperature dependent variable viscosity fluid flow down an inclined plane with a free surface is investigated. The fluid film is thin, so that lubrication approximation may be applied. Convective heating effects are included, and the fluid viscosity decreases exponentially with temperature. In general, the flow equations resulting from the variable viscosity model must be solved numerically. However, when the viscosity variation is small, then an asymptotic approximation is possible. The full solutions for the temperature and velocity profiles are derived using the Runge-Kutta numerical method. The flow controlling parameters such as the nondimensional viscosity variation parameter, the Biot and the Brinkman numbers, are found to have a profound effect on the resulting flow profiles.

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