scholarly journals Thermocapillary effects on a thin viscous rivulet draining steadily down a uniformly heated or cooled slowly varying substrate

2001 ◽  
Vol 441 ◽  
pp. 195-221 ◽  
Author(s):  
D. HOLLAND ◽  
B. R. DUFFY ◽  
S. K. WILSON
Keyword(s):  
Surface Tension ◽  
Finite Depth ◽  
Transverse Flow ◽  
Finite Width ◽  
Lubrication Approximation ◽  
Volume Flux ◽  
Infinite Depth ◽  
Surrounding Atmosphere ◽  
Helical Vortices ◽  
Horizontal Circular Cylinder

We use the lubrication approximation to investigate the steady flow of a thin rivulet of viscous fluid with prescribed volume flux draining down a planar or slowly varying substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere, when the surface tension of the fluid varies linearly with temperature. Utilizing the (implicit) solution of the governing ordinary differential equation that emerges, we undertake a comprehensive asymptotic and numerical analysis of the flow. In particular it is shown that the variation in surface tension drives a transverse flow that causes the fluid particles to spiral down the rivulet in helical vortices (which are absent in the corresponding isothermal problem). We find that a single continuous rivulet can run from the top to the bottom of a large horizontal circular cylinder provided that the cylinder is either warmer or significantly cooler than the surrounding atmosphere, but if it is only slightly cooler then a continuous rivulet is possible only for a sufficiently small flux (though a rivulet with a discontinuity in the free surface is possible for larger values of the flux). Moreover, near the top of the cylinder the rivulet has finite depth but infinite width, whereas near the bottom of the cylinder it has finite width and infinite depth if the cylinder is heated or slightly cooled, but has infinite width and finite depth if the cylinder is significantly cooled.

EFFECTS OF SURFACE TENSION ON TRAPPED MODES IN A TWO-LAYER FLUID

2015 ◽  
Vol 57 (2) ◽  
pp. 189-203 ◽  
Author(s):  
S. SAHA ◽  
S. N. BORA
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Circular Cylinder ◽  
Boundary Value Problems ◽  
Finite Depth ◽  
Trapped Modes ◽  
Trapped Waves ◽  
Linearized Theory ◽  
Horizontal Circular Cylinder ◽  
Effect Of Surface

We consider a two-layer fluid of finite depth with a free surface and, in particular, the surface tension at the free surface and the interface. The usual assumptions of a linearized theory are considered. The objective of this work is to analyse the effect of surface tension on trapped modes, when a horizontal circular cylinder is submerged in either of the layers of a two-layer fluid. By setting up boundary value problems for both of the layers, we find the frequencies for which trapped waves exist. Then, we numerically analyse the effect of variation of surface tension parameters on the trapped modes, and conclude that realistic changes in surface tension do not have a significant effect on the frequencies of these.

scholarly journals Steady rimming flows with surface tension

2008 ◽  
Vol 597 ◽  
pp. 91-118 ◽  
Author(s):  
E. S. BENILOV ◽  
M. S. BENILOV ◽  
N. KOPTEVA
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Viscous Fluid ◽  
Small Parameter ◽  
Outer Region ◽  
Lubrication Approximation ◽  
Steady Flows ◽  
Matched Asymptotics ◽  
Weak Surface ◽  
Rimming Flows

We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the ‘rising’ side of the cylinder and, for large ones, pools at the cylinder's bottom. The paper examines rimming flows with a pool affected by weak surface tension. Using the lubrication approximation and the method of matched asymptotics, we find a solution describing the pool, the ‘outer’ region, and two transitional regions, one of which includes a variable (depending on the small parameter) number of asymptotic zones.

Reflexion of water waves by a permeable barrier

1979 ◽  
Vol 95 (1) ◽  
pp. 141-157 ◽  
Author(s):  
C. Macaskill
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Finite Depth ◽  
Infinite Depth ◽  
Permeable Barrier ◽  
Impermeable Barrier ◽  
Linearized Problem ◽  
Special Case ◽  
Thin Barrier ◽  
Standard Techniques

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

Hydraulic control by a wide weir in a rotating fluid

1976 ◽  
Vol 73 (3) ◽  
pp. 521-528 ◽  
Author(s):  
E. Sambuco ◽  
J. A. Whitehead
Keyword(s):  
Experimental Investigations ◽  
Length Scale ◽  
Finite Width ◽  
Hydraulic Control ◽  
Volume Flux ◽  
Rotating Fluid ◽  
Coriolis Parameter ◽  
Bernoulli Function ◽  
Scale Measure ◽  
Crest Height

Flow control by a wide, deep weir in a rotating fluid is investigated theoretically and experimentally. A strong (vertical) vorticity constraint due to frame rotation is combined with conservation of the Bernoulli function along streamlines and a standard hydraulic control assumption to show that the volume flux over the barrier is \[ Q = g^{-1}\left[\frac{2}{3}g(H - b_0)-\frac{1}{3}f^2l^2 \right]^{\frac{3}{2}}, \] where H is the depth of the fluid column upstream, bo is the crest height, f is the Coriolis parameter, and l is a length-scale measure of the breadth of the weir. The component of the velocity parallel to the weir crest is computed from conservation of potential vorticity to be v = −fl; perpendicular to the crest, we recover the standard hydraulic relation u = (gh0)½.Experimental investigations of upstream height and streamline deflexion as functions of rotation are described. It is found that agreement with theory is good up to a certain rate of rotation, above which the finite width of the experimental weir becomes important.

A line sink in a flowing stream with surface tension effects

2015 ◽  
Vol 27 (2) ◽  
pp. 248-263 ◽  
Author(s):  
R. J. HOLMES ◽  
G. C. HOCKING
Keyword(s):  
Surface Tension ◽  
Stream Flow ◽  
Linear Problem ◽  
Sink Strength ◽  
Curvature Singularity ◽  
Infinite Depth ◽  
Line Sink ◽  
Steady Solutions ◽  
Parameter Values ◽  
Flowing Stream

We examine a problem in which a line sink causes a disturbance to an otherwise uniform flowing stream of infinite depth. We consider the fully non-linear problem with the inclusion of surface tension and find the maximum sink strength at which steady solutions exist for a given stream flow, before examining non-unique solutions. The addition of surface tension allows for a more thorough investigation into the characteristics of the solutions. The breakdown of steady solutions with surface tension appears to be caused by a curvature singularity as the flow rate approaches the maximum. The non-uniqueness in solutions is shown to occur for a range of parameter values in all cases with non-zero surface tension.

scholarly journals On the propagation of a disturbance in a fluid under gravity

1910 ◽  
Vol 83 (563) ◽  
pp. 347-356 ◽  
Keyword(s):  
Incompressible Fluid ◽  
Gravity Wave ◽  
Integral Form ◽  
Finite Depth ◽  
Infinite Depth ◽  
Poisson Problem

In a recent paper Lord Rayleigh has called attention to the fact of the instantaneous propagation of a limited disturbance over the surface of heavy incompressible fluid. This instantaneity occurs in spite of the fact that the velocity of any simple-harmonic gravity wave is finite, the depth being finite and constant. As the points thereby raised are of some delicacy, and are not completely settled in the paper quoted, some further remarks on the phenomenon may not be superfluous. 2. Solution of the Cauchy-Poisson Problem for Finite Depth . In his treatment of this case Lord Rayleigh used the integral form of solution. There are, however, great difficulties raised in this way on account of lack of convergence at the surface. I proceed to obtain a solution in the form of a series similar in type to the known serial solution of the problem for infinite depth.

Reverse Marangoni surfing

2016 ◽  
Vol 811 ◽  
pp. 612-621 ◽  
Author(s):  
Vahid Vandadi ◽  
Saeed Jafari Kang ◽  
Hassan Masoud
Keyword(s):  
Surface Tension ◽  
Liquid Layer ◽  
Scalar Fields ◽  
Finite Depth ◽  
Lower Surface ◽  
Temperature Fields ◽  
Spherical Particles ◽  
Surface Tension Gradient ◽  
Oblate Spheroidal ◽  
Active Particles

We theoretically study the surfing motion of chemically and thermally active particles located at a flat liquid–gas interface that sits above a liquid layer of finite depth. The particles’ activity creates and maintains a surface tension gradient resulting in the auto-surfing. It is intuitively perceived that Marangoni surfers propel towards the direction with a higher surface tension. Remarkably, we find that the surfers may propel in the lower surface tension direction depending on their geometry and proximity to the bottom of the liquid layer. In particular, our analytical calculations for Stokes flow and diffusion-dominated scalar fields (i.e. chemical concentration and temperature fields) indicate that spherical particles undergo reverse Marangoni propulsion under confinement whereas disk-shaped surfers always move in the expected direction. We extend our results by proposing an approximate formula for the propulsion speed of oblate spheroidal particles based on the speeds of spheres and disks.

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.

The instability theory of drumlin formation applied to Newtonian viscous ice of finite depth

2010 ◽  
Vol 466 (2121) ◽  
pp. 2673-2694 ◽  
Author(s):  
A. C. Fowler
Keyword(s):  
Transfer Functions ◽  
Three Dimensional ◽  
Analytic Form ◽  
Finite Depth ◽  
Pressure Gradients ◽  
Interfacial Instabilities ◽  
Infinite Depth ◽  
Ice Surface ◽  
Instability Theory ◽  
Ice Flows

The Hindmarsh instability theory of drumlin formation is applied to the study of interfacial instabilities, which may arise when ice flows viscously over deformable sediments. Here, the analytic form of this theory is extended to the case where the ice is Newtonian viscous and of finite depth, and where the basal till can be both sheared by the ice and squeezed by basal effective pressure gradients: previous authors assumed infinitely deep ice, based on the assumption that the developing waveforms had wavelength much less than ice depth. The previous infinite depth theory only allowed transverse instabilities to occur, and these have been associated with the formation of ribbed moraine; one of the purposes of extending the analysis to finite depth is to see whether three-dimensional instabilities, which might be associated with the formation of drumlins or mega-scale glacial lineations, can occur: we find that they do not. A second purpose is to calculate under what circumstances the infinite depth theory provides accurate prediction of bedform development in ice of finite depth d i . We find that this is the case if the waveforms have a wavelength less than approximately 1.2 d i . Finally, the finite depth theory allows us to compute, for the first time, the response of the ice surface to the developing unstable bedforms. We find that this response is rapid, and we give explicit recipes for the surface perturbation transfer functions in terms of the perturbations to the basal stress and the basal topography.

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