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 Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity

2015 ◽  
Vol 145 (4) ◽  
pp. 791-883 ◽  
Author(s):  
M. D. Groves ◽  
E. Wahlén
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Applied Mathematics ◽  
Finite Depth ◽  
Constant Vorticity ◽  
Existence And Stability ◽  
Strong Surface ◽  
Weak Surface ◽  
The Stability ◽  
The Waves

We present an existence and stability theory for gravity–capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy𝓗subject to the constraint𝓘= 2µ, where𝓘is the wave momentum and 0 <µ≪ 1. Since𝓗and𝓘are both conserved quantities, a standard argument asserts the stability of the setDµof minimizers: solutions starting nearDµremain close toDµin a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg–de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation asµ↓ 0.

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.

Developing Laminar Gravity-Driven Thin Liquid Film Flow Down an Inclined Plane

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
H. Lan ◽  
J. L. Wegener ◽  
B. F. Armaly ◽  
J. A. Drallmeier
Keyword(s):  
Surface Tension ◽  
Film Thickness ◽  
Liquid Film ◽  
Flow Rate ◽  
Film Flow ◽  
Thin Liquid Film ◽  
Inclined Plane ◽  
Liquid Film Flow ◽  
Gravity Driven ◽  
Surface Inclination

Three-dimensional (3D)—steady-developing-laminar-isothermal—and gravity-driven thin liquid film flow adjacent to an inclined plane is examined and the effects of film flow rate, surface tension, and surface inclination angle on the film thickness and film width are presented. The film flow was numerically simulated using the volume of fluid model and experimental verification was conducted by measuring film thickness and width using a laser focus displacement instrument. The steady film flow that is considered in this study does not have a leading contact line, however, it has two steady side contact lines with the substrate surface at the outer edge of its width. Results reveal that the film width decreases and the average film thickness increases as the film flows down the inclined plane. The film thickness and width decrease but its streamwise velocity increases as surface inclination angle (as measured from the horizontal plane) increases. A higher film flow rate is associated with a higher film thickness, a higher film width, and a higher average film velocity. Films with higher surface tension are associated with a smaller width and a higher average thickness. A ripple develops near the side contact line, i.e., the spanwise distribution of the film thickness exhibits peaks at the outer edges of the film width and the height of this ripple increases as the surface tension or the film flow rate increases. The width of the film decreases at a faster rate along the streamwise direction if liquid film has higher surface tension. Measurements of the film thickness and the film width compare favorably with the numerically simulated results.

Imaging of Surface-Tension-Driven Convection Using Liquid Crystal Thermography

2010 ◽  
Vol 132 (12) ◽  
Author(s):  
T. W. Dutton ◽  
L. R. Pate ◽  
D. K. Hollingsworth
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Liquid Crystal ◽  
Temperature Distribution ◽  
Liquid Film ◽  
Lower Boundary ◽  
Processing System ◽  
Metal Foil ◽  
Lower Boundary Condition ◽  
Liquid Crystal Thermography

Surface-tension forces can drive fluid motion within thin liquid layers with a free surface. Spatial variations in the temperature of the free surface create surface tractions that drive cellular motions. The cells are most commonly hexagonal in shape and they scale on the thickness of the fluid layer. This investigation documents the formation of cells in the liquid film in the presence of a uniform-heat-flux lower boundary condition. Liquid crystal thermography was used to image the cells and measure the temperature distribution on the lower surface of the liquid layer. A 1.1 mm deep pool of silicone oil was supported on a 50 μm thick electrically heated metal foil. The oil was retained inside an independently heated acrylic ring mounted on the top surface of the foil and a dry-ice cooling plate served as the low-temperature sink above the free surface of the oil. Color images of hexagonal convection cells were captured using liquid crystal thermography and a digital image acquisition and processing system. The temperature distribution inside a typical cell was measured using thermographic image analysis. Experimental issues, such as the use of an independently heated retaining ring to control the height of the liquid film and the utility of a flux-based Marangoni number are discussed.

scholarly journals The effects of surface tension on the initial development of a free surface adjacent to an accelerated plate

2015 ◽  
Vol 776 ◽  
pp. 37-73 ◽  
Author(s):  
J. Uddin ◽  
D. J. Needham
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Local Structure ◽  
Contact Point ◽  
Initial Development ◽  
Early Stages ◽  
Inviscid Incompressible Fluid ◽  
Stationary Layer ◽  
Weak Surface ◽  
Theoretical Results

When a vertical rigid plate is uniformly accelerated horizontally from rest into an initially stationary layer of inviscid incompressible fluid, the free surface will undergo a deformation in the locality of the contact point. This deformation of the free surface will, in the early stages, cause a jet to rise up the plate. An understanding of the local structure of the free surface in the early stages of motion is vital in many situations, and has been developed in detail by King & Needham (J. Fluid Mech., vol. 268, 1994, pp. 89–101). In this work we consider the effects of introducing weak surface tension, characterized by the inverse Weber number $\mathscr{W}$, into the problem considered by King & Needham. Our approach is based upon matched asymptotic expansions as $\mathscr{W}\rightarrow 0$. It is found that four asymptotic regions are needed to describe the problem. The three largest regions have analytical solutions, whilst a numerical method based on finite differences is used to solve the time-dependent harmonic boundary value problem in the last region. Our results identify the local structure of the jet near the vicinity of the contact point, and we highlight a number of key features, including the height of this jet as well as its thickness and strength. We also present some preliminary experimental results that capture the spatial structure near the contact point, and we then show promising comparisons with the theoretical results obtained within this paper.

Roles of surface tension and Reynolds stresses on the finite amplitude stability of a parallel flow with a free surface

1970 ◽  
Vol 40 (2) ◽  
pp. 307-314 ◽  
Author(s):  
S. P. Lin
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Gravity Waves ◽  
Wave Amplitude ◽  
Finite Amplitude ◽  
Parallel Flow ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Inclined Plane ◽  
The Mean

Subcritically stable motion of long gravity waves of finite amplitude in a liquid layer flowing down an inclined plane is shown to be impossible. However, super-critically stable wave régimes for such flows are found and curves of constant wave amplitude in such régimes are obtained. The mechanism of non-linear stability is investigated by considering the energy transfer between the mean flow and the disturbances. The results obtained show that the mechanism of stability in a parallel flow with a free surface is quite different from that in a parallel flow without a free surface.

Liquid Crystal Imaging of Surface-Tension-Driven Convection

2000 ◽  
Author(s):  
T. W. Dutton ◽  
L. R. Pate ◽  
D. K. Hollingsworth
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Liquid Crystal ◽  
Temperature Distribution ◽  
Liquid Film ◽  
Lower Boundary ◽  
Processing System ◽  
Metal Foil ◽  
Lower Boundary Condition ◽  
Liquid Crystal Thermography

Abstract Surface tension forces can drive fluid motion within thin liquid layers with a free surface. Spatial variations in the temperature of the free surface create surface tractions that drive cellular motions. The cells are most commonly hexagonal in shape and they scale on the thickness of the fluid layer. This investigation documents the formation of cells in the liquid film in the presence of a uniform-heat-flux lower boundary condition. Liquid crystal thermography was used to image the cells and measure the temperature distribution on the lower surface of the liquid layer. A 1.1-mm deep pool of silicone oil was supported on a 50-μm-thick electrically heated metal foil. The oil was retained inside an independently heated acrylic ring mounted on the top surface of the foil, and a dry-ice cooling plate served as the low-temperature sink above the free surface of the oil. Color images of hexagonal convection cells were captured using liquid crystal thermography and a digital image acquisition and processing system. The temperature distribution inside a typical cell was measured using thermographic image analysis. Experimental issues such as the use of an independently heated retaining ring to control the height of the liquid film, and the utility of a flux-based Marangoni number are discussed.

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.

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