On the Disturbance of a Thin Layer of Liquid by a Moving Obstruction

1990 ◽  
Vol 57 (4) ◽  
pp. 1066-1072
Author(s):  
Roger F. Gans ◽  
Chung-Hai Wang
Keyword(s):  
Surface Tension ◽  
Reynolds Number ◽  
Free Surface ◽  
Thin Layer ◽  
Viscous Flow ◽  
Length Scale ◽  
Two Dimensional ◽  
Thin Liquid Layer ◽  
Horizontal Length ◽  
Small Ratio

We calculate the free surface shapes upstream and downstream of an obstacle obstructing a thin liquid layer on a moving surface, taking into account gravity and surface tension. We assume low Reynolds number viscous flow, a two-dimensional layer, and small ratio of vertical to horizontal length scale. The upstream and downstream shapes are very different. The upstream liquid piles up against the obstacle to provide an overpressure sufficient to drive the Poiseuille component of the lubrication flow under the obstacle. The downstream liquid is disturbed only by surface tension.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

Linear Spatial Evolution Formulation of Two-Dimensional Waves on Liquid Films Under Evaporating/Isothermal/Condensing Conditions

2002 ◽  
Author(s):  
Xuemin Ye ◽  
Chunxi Li ◽  
Weiping Yan
Keyword(s):  
Boundary Layer ◽  
Surface Tension ◽  
Reynolds Number ◽  
Evolution Equation ◽  
Film Thickness ◽  
Boundary Conditions ◽  
Liquid Films ◽  
Boundary Layer Theory ◽  
Spatial Evolution ◽  
Two Dimensional

The linear spatial evolution formulation of the two-dimensional waves of the evaporating or isothermal or condensing liquid films falling down an inclined wall is established for the film thickness with the collocation method based on the boundary layer theory and complete boundary conditions. The evolution equation indicates that there are two different modes of waves in spatial evolution. And the flow stability is highly dependent on the evaporation or condensation, thermocapillarity, surface tension, inclination angle and Reynolds number.

Alveolar and saccular lung phospholipids of the anaconda, Eunectes murinus

1978 ◽  
Vol 56 (4) ◽  
pp. 1009-1013 ◽  
Author(s):  
Charles F. Phleger ◽  
David G. Smith ◽  
Douglas H. Macintyre ◽  
Brian S. Saunders
Keyword(s):  
Surface Tension ◽  
Thin Layer ◽  
Thin Layer Chromatography ◽  
Lipid Extract ◽  
Two Dimensional ◽  
Surface Tensions ◽  
Methanol Extraction ◽  
Layer Chromatography ◽  
Chloroform Methanol ◽  
Standard Techniques

Phospholipids and their mode of synthesis in lung samples from the alveolar and saccular regions of an anaconda (Eunectes murinus) were investigated by standard techniques of chloroform–methanol extraction and two-dimensional thin-layer chromatography. The alveolar lung has six times as much phosphatidylcholine in its lung wash lipid extract as saccular lung. Phosphatidylcholine and sphingomyelin are the two principal phospholipids of the tissue of both lungs. Alveolar lung incorporates a higher percentage although a smaller total amount of [1-14C]acetate (54%) into phosphatidylcholine (including lysophosphatidylcholine), whereas saccular lung only incorporates 8% [1-14C]acetate into phosphatidylcholine (including lysophosphatidylcholine) and 60% [1-14C]acetate into sphingomyelin. Saccular lung synthesized 31% disphosphatidylglycerol from [1-14C]acetate; alveolar lung did not synthesize any. Surface tension plots of lung wash lipid extracts show slight surpellic activity with minimum surface tensions of 22 dyn/cm (1 dyn = 10 μN) for both alveolar and saccular lung, at 37 °C.

Two-dimensional simulation of unsteady heat transfer from a circular cylinder in crossflow

2007 ◽  
Vol 570 ◽  
pp. 177-215 ◽  
Author(s):  
SALEM BOUHAIRIE ◽  
VINCENT H. CHU
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Circular Cylinder ◽  
Reynolds Numbers ◽  
Boundary Layer Separation ◽  
Length Scale ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Transfer Rates ◽  
Two Dimensional Model

The heat transfer from the surface of a circular cylinder into a crossflow has been computed using a two-dimensional model, for a range of Reynolds numbers from Re=200 to 15550. The boundary-layer separation, the local and overall heat-transfer rates, the eddy- and flare-detachment frequencies and the width of the flares were determined from the numerical simulations. In this range of Reynolds numbers, the heat-transfer process is unsteady and is characterized by a viscous length scale that is inversely proportional to the square root of the Reynolds number. To ensure uniform numerical accuracy for all Reynolds numbers, the dimensions of the computational mesh were selected in proportion to this viscous length scale. The small scales were resolved by at least three nodes within the boundary layers. The frequency of the heat flares increases, and the width of each flare decreases, with the Reynolds number, in proportion to the viscous time and length scales. Despite the presence of three-dimensional structures for the range of Reynolds numbers considered, the two-dimensional model captures the unsteady processes and produced results that were consistent with the available experimental data. It correctly simulated the overall, the front-stagnation and the back-to-total heat-transfer rates.

scholarly journals Numerical Analysis of Two-Dimensional Compressible Viscous Flow in Turbomachinery Cascades Using an Improved k-ε Turbulence Model

1997 ◽  
Author(s):  
Debasish Biswas ◽  
Hideo Iwasaki ◽  
Masaru Ishizuka
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Viscous Flow ◽  
Gas Turbine ◽  
Turbine Blade ◽  
Turbulence Model ◽  
Higher Order ◽  
Tvd Scheme ◽  
Two Dimensional ◽  
Gas Turbine Blade

In the present work two-dimensional viscous flows through compressor and gas turbine blade cascades at low subsonic and transonic speed are analyzed by solving compressible N-S equations in the generalized co-ordinate system, so that sufficient number of grid points could be distributed in the boundary layer and wake regions. An efficient Implicit Approximate Factorization (IAF) finite difference scheme, originally developed by Beam-Warming, is used together with a higher order Total Variation Diminishing (TVD) scheme based on the MUSCL-type approach with the Roe’s approximate Rieman solver for shock capturing. In order to predict the boundary layer turbulence characteristics, shock boundary layer interaction, transition from laminar to turbulent flow, etc. with sufficient accuracy, an improved low Reynolds number k-ε turbulence model developed by the authors is used. In this k-ε model, the low Reynolds number damping factors are defined as a function of turbulence Reynolds number which is only a rather general indicator of the degree of turbulence activity at any location in the flow rather than a specific function of the location itself. Computations are carried out for different flow conditions of compressor and gas turbine blade cascades for which detailed and reliable information about shock location, shock losses, viscous losses, blade surface pressure distribution and overall performance are available. Comparison of computed results with the experimental data showed a very good agreement. The results demonstrated that the Navier-Stokes approach using the present k-ε turbulence model and higher order TVD scheme would lead to improved prediction of viscous flow phenomena in turbomachinery cascades.

Hydraulic jumps in a shallow flow down a slightly inclined substrate

2015 ◽  
Vol 782 ◽  
pp. 5-24 ◽  
Author(s):  
E. S. Benilov
Keyword(s):  
Reynolds Number ◽  
Free Surface ◽  
Free Surface Flows ◽  
Hydraulic Jumps ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Surface Flows ◽  
Asymptotic Equations ◽  
Two Dimensional Flow ◽  
Existence Criterion

This work examines free-surface flows down an inclined substrate. The slope of the free surface and that of the substrate are both assumed small, whereas the Reynolds number $Re$ remains unrestricted. A set of asymptotic equations is derived, which includes the lubrication and shallow-water approximations as limiting cases (as $Re\rightarrow 0$ and $Re\rightarrow \infty$, respectively). The set is used to examine hydraulic jumps (bores) in a two-dimensional flow down an inclined substrate. An existence criterion for steadily propagating bores is obtained for the $({\it\eta},s)$ parameter space, where ${\it\eta}$ is the bore’s downstream-to-upstream depth ratio, and $s$ is a non-dimensional parameter characterising the substrate’s slope. The criterion reflects two different mechanisms restricting bores. If $s$ is sufficiently large, a ‘corner’ develops at the foot of the bore’s front – which, physically, causes overturning. If, in turn, ${\it\eta}$ is sufficiently small (i.e. the bore’s relative amplitude is sufficiently large), the non-existence of bores is caused by a stagnation point emerging in the flow.

scholarly journals On periodic Stokesian Hele-Shaw flows with surface tension

2008 ◽  
Vol 19 (6) ◽  
pp. 717-734 ◽  
Author(s):  
J. ESCHER ◽  
B.-V. MATIOC
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Incompressible Fluid ◽  
Classical Solution ◽  
Small Perturbation ◽  
Darcy’S Law ◽  
Viscous Incompressible Fluid ◽  
Darcy's Law ◽  
Two Dimensional ◽  
Unique Classical Solution

In this paper we consider a 2π-periodic and two-dimensional Hele-Shaw flow describing the motion of a viscous, incompressible fluid. The free surface is moving under the influence of surface tension and gravity. The motion of the fluid is modelled using a modified version of Darcy's law for Stokesian fluids. The bottom of the cell is assumed to be impermeable. We prove the existence of a unique classical solution for a domain which is a small perturbation of a cylinder. Moreover, we identify the equilibria of the flow and study their stability.

scholarly journals Hamiltonian long–wave expansions for water waves over a rough bottom

2005 ◽  
Vol 461 (2055) ◽  
pp. 839-873 ◽  
Author(s):  
Walter Craig ◽  
Philippe Guyenne ◽  
David P. Nicholls ◽  
Catherine Sulem
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Kdv Equation ◽  
Bottom Topography ◽  
Multiple Scale ◽  
Three Dimensions ◽  
Length Scale ◽  
Two Dimensional ◽  
Long Wave ◽  
Starting Point

This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long–wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou (1983 Stud. Appl. Math. 68 , 89–102) on periodic bottoms for two–dimensional flows.We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being Zakharov's Hamiltonian (V. E. Zakharov 1968 J. Appl. Mech. Tech. Phys. 9, 1990–1994) for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length–scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length–scale as, or longer than, the order of the wavelength of the surface waves. We do not take up the question of random bottom variations, a topic which is considered in Rosales & Papanicolaou (1983). In the two–dimensional case of waves in a channel, we give an alternate derivation of the effective Korteweg–de Vries (KdV) equation that is obtained in Rosales & Papanicolaou (1983). In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long–scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three–dimensional long–wave equations in a Boussinesq scaling regime, and again in certain cases an effective Kadomtsev–Petviashvili (KP) system in the appropriate unidirectional limit. The computations for these results are performed in the framework of an asymptotic analysis of multiple–scale operators. In the present case this involves the Dirichlet–Neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.

scholarly journals Anisotropy in shrinkage during sintering

2006 ◽  
Vol 38 (1) ◽  
pp. 13-25 ◽  
Author(s):  
A. Zavaliangos ◽  
J.M. Missiaen ◽  
D. Bouvard
Keyword(s):  
Surface Tension ◽  
Viscous Flow ◽  
Two Dimensional ◽  
Significant Progress ◽  
Dimensional Array ◽  
Recent Developments

While significant progress in modeling of sintering has been accomplished since the original paper by Frenkel "Viscous flow of crystalline bodies under action of surface tension", there are still several issues that remain open. One of them is anisotropy during sintering. In this paper we present some recent developments that improve our understanding of sintering anisotropy based on simulations of a two- dimensional array of particles. A number of possible sources of anisotropy are examined and evaluated. .

scholarly journals Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

2019 ◽  
Vol 874 ◽  
pp. 891-925 ◽  
Author(s):  
A. I. Dyachenko ◽  
S. A. Dyachenko ◽  
P. M. Lushnikov ◽  
V. E. Zakharov
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Half Plane ◽  
Second Order ◽  
Zero Gravity ◽  
Two Dimensional ◽  
Branch Points ◽  
Integrals Of Motion ◽  
Complex Half Plane ◽  
Constants Of Motion

We address the problem of the potential motion of an ideal incompressible fluid with a free surface and infinite depth in a two-dimensional geometry. We admit the presence of gravity forces and surface tension. A time-dependent conformal mapping $z(w,t)$ of the lower complex half-plane of the variable $w$ into the area filled with fluid is performed with the real line of $w$ mapped into the free fluid’s surface. We study the dynamics of singularities of both $z(w,t)$ and the complex fluid potential $\unicode[STIX]{x1D6F1}(w,t)$ in the upper complex half-plane of $w$. We show the existence of solutions with an arbitrary finite number $N$ of complex poles in $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ which are the derivatives of $z(w,t)$ and $\unicode[STIX]{x1D6F1}(w,t)$ over $w$. We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for non-zero surface tension. We find that the residues of $z_{w}(w,t)$ at these $N$ points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations, arXiv:1206.2046) for the preliminary results. All these constants of motion commute with each other in the sense of the underlying Hamiltonian dynamics. In the absence of both gravity and surface tension, the residues of $\unicode[STIX]{x1D6F1}_{w}(w,t)$ are also the constants of motion while non-zero gravity $g$ ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ at each poles position reveals the existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is $4N$ for zero gravity and $4N-1$ for non-zero gravity. For the second-order poles we found $6N$ motion integrals for zero gravity and $6N-1$ for non-zero gravity. We suggest that the existence of these non-trivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics.

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