Theory and experiment on the low-Reynolds-number expansion and contraction of a bubble pinned at a submerged tube tip

1998 ◽  
Vol 356 ◽  
pp. 93-124 ◽  
Author(s):  
HARRIS WONG ◽  
DAVID RUMSCHITZKI ◽  
CHARLES MALDARELLI
Keyword(s):  
Surface Tension ◽  
Reynolds Number ◽  
Numerical Solutions ◽  
Full Range ◽  
Boundary Integral ◽  
Gas Pressure ◽  
Boundary Integral Method ◽  
Liquid Density ◽  
Gravitational Acceleration ◽  
Dynamic Surface

The expansion and contraction of a bubble pinned at a submerged tube tip and driven by constant gas flow rate Q are studied both theoretically and experimentally for Reynolds number Re[Lt ]1. Bubble shape, gas pressure, surface velocities, and extrapolated detached bubble volume are determined by a boundary integral method for various Bond (Bo=ρga2/σ) and capillary (Ca=μQ/σa2) numbers, where a is the capillary radius, ρ and μ are the liquid density and viscosity, σ is the surface tension, and g is the gravitational acceleration.Bubble expansion from a flat interface to near detachment is simulated for a full range of Ca (0.01–100) and Bo (0.01–0.5). The maximum gas pressure is found to vary almost linearly with Ca for 0.01[les ]Ca[les ]100. This correlation allows the maximum bubble pressure method for measuring dynamic surface tension to be extended to viscous liquids. Simulated detached bubble volumes approach static values for Ca[Lt ]1, and asymptote as Q3/4 for Ca[Gt ]1, in agreement with analytic predictions. In the limit Ca→0, two singular time domains are identified near the beginning and the end of bubble growth during which viscous and capillary forces become comparable.Expansion and contraction experiments were conducted using a viscous silicone oil. Digitized video images of deforming bubbles compare well with numerical solutions. It is observed that a bubble contracting at high Ca snaps off.

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.

Generation and Propagation of Long-Crested Finite-Depth Surface Waves: Comparison Between the Corresponding Results From Numerical Models and a Wave Tank

2008 ◽  
Author(s):  
M. Hasanat Zaman ◽  
Wade Parsons ◽  
Okey Nwogu ◽  
Wooyoung Choi ◽  
R. Emile Baddour ◽  
...  
Keyword(s):  
Surface Waves ◽  
Numerical Solutions ◽  
Numerical Models ◽  
Finite Depth ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Surface Elevation ◽  
Ocean Engineering ◽  
Measured Time ◽  
Wave Maker

The evolution of long-crested surface waves subject to side-band perturbations is investigated with two different numerical models: a direct solver for the Euler equations using a non-orthogonal boundary-fitted curvilinear coordinate system and an FFT-accelerated boundary integral method. The numerical solutions are then validated with laboratory experiments performed in the NRC-IOT Ocean Engineering Basin with a segmented wave-maker operating in piston mode. The numerical models are forced by a point measurement of the free surface elevation at a wave probe close to the wave-maker and the numerical solutions are compared with the measured time-series of the surface elevation at a few wave probe locations downstream.

Numerical studies of cusp formation at fluid interfaces in Stokes flow

1998 ◽  
Vol 357 ◽  
pp. 29-57 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Surface Tension ◽  
Stokes Flow ◽  
Boundary Integral ◽  
Numerical Results ◽  
Boundary Integral Method ◽  
Fluid Interfaces ◽  
Two Dimensional ◽  
Numerical Studies ◽  
Insoluble Surfactant ◽  
Cusp Formation

Numerical studies are performed addressing the development of regions of high curvature and the spontaneous occurrence of cusped interfacial shapes in two-dimensional and axisymmetric Stokes flow. In the numerical simulations, the velocity field is computed using a boundary-integral method, and the evolution of the concentration of an insoluble surfactant over an evolving interface is computed using an implicit finite-volume method. Three configurations are considered in detail, and the results are used to elucidate three different aspects of cusp formation. In the first series, the deformation of a two-dimensional bubble immersed in a family of straining flows devised by Antanovskii, and of an axisymmetric bubble immersed in an analogous family of flows devised by Sherwood, are examined. The numerical results indicate that highly elongated and cusped two-dimensional shapes, and pointed or cusped axisymmetric shapes, are unstable and should not be expected to occur in practice. In the second series of studies, the role of an insoluble surfactant on the transient deformation of bubbles subject to the Antanovskii or Sherwood flow is investigated. Under certain conditions, the reduced surface tension at the tips raises the local curvature to high values and causes the ejection of a sheet or column of gas by means of tip streaming. In the third series of studies, the coalescence of a polygonal formation of five viscous columns of a fluid placed in an arrangement that differs only slightly from one proposed recently by Richardson is examined. The numerical results confirm Richardson's predictions that transient cusps may occur at a finite time in the presence of surface tension. The underlying physical mechanism is discussed on the basis of reversibility of surface-driven Stokes flow and with reference to the regularity of the motion driven by negative surface tension. Replacing the inviscid ambient gas with a slightly viscous fluid whose viscosity is as low as one hundredth the viscosity of the cylinders suppresses the cusp formation.

Instability of two-layer creeping flow in a channel with parallel-sided walls

1997 ◽  
Vol 351 ◽  
pp. 139-165 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Surface Tension ◽  
Reynolds Number ◽  
Stokes Flow ◽  
Capillary Number ◽  
Travelling Waves ◽  
Boundary Integral ◽  
Creeping Flow ◽  
Two Dimensional ◽  
Two Fluids ◽  
Axial Pressure Gradient

The evolution of the interface between two viscous fluid layers in a two-dimensional horizontal channel confined between two parallel walls is considered in the limit of Stokes flow. The motion is generated either by the translation of the walls, in a shear-driven or plane-Couette mode, or by an axial pressure gradient, in a plane-Poiseuille mode. Linear stability analysis for infinitesimal perturbations and fluids with matched densities shows that when the viscosities of the fluids are different and the Reynolds number is sufficiently high, the flow is unstable. At vanishing Reynolds number, the flow is stable when the surface tension has a non-zero value, and neutrally stable when the surface tension vanishes. We investigate the behaviour of the interface subject to finite-amplitude two-dimensional perturbations by solving the equations of Stokes flow using a boundary-integral method. Integral equations for the interfacial velocity are formulated for the three modular cases of shear-driven, pressure-driven, and gravity-driven flow, and numerical computations are performed for the first two modes. The results show that disturbances of sufficiently large amplitude may cause permanent interfacial deformation in which the interface folds, develops elongated fingers, or supports slowly evolving travelling waves. Smaller amplitude disturbances decay, sometimes after a transient period of interfacial folding. The ratio of the viscosities of the two fluids plays an important role in determining the morphology of the emerging interfacial patterns, but the parabolicity of the unperturbed velocity profile does not affect the character of the motion. Increasing the contrast in the viscosities of the two fluids, while keeping the channel capillary number fixed, destabilizes the interfaces; re-examining the flow in terms of an alternative capillary number that is defined with respect to the velocity drop across the more-viscous layer shows that this is a reasonable behaviour. Comparing the numerical results with the predictions of a lubrication-flow model shows that, in the absence of inertia, the simplified approach can only describe a limited range of motions, and that the physical relevance of the steadily travelling waves predicted by long-wave theories must be accepted with a certain degree of reservation.

Highly nonlinear standing water waves with small capillary effect

1998 ◽  
Vol 369 ◽  
pp. 253-272 ◽  
Author(s):  
WILLIAM W. SCHULTZ ◽  
JEAN-MARC VANDEN-BROECK ◽  
LEI JIANG ◽  
MARC PERLIN
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Standing Waves ◽  
High Amplitude ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Ripple Formation ◽  
Node Distribution ◽  
Highly Nonlinear ◽  
Periodic Standing Waves

We calculate spatially and temporally periodic standing waves using a spectral boundary integral method combined with Newton iteration. When surface tension is neglected, the non-monotonic behaviour of global wave properties agrees with previous computations by Mercer & Roberts (1992). New accurate results near the limiting form of gravity waves are obtained by using a non-uniform node distribution. It is shown that the crest angle is smaller than 90° at the largest calculated crest curvature. When a small amount of surface tension is included, the crest form is changed significantly. It is necessary to include surface tension to numerically reproduce the steep standing waves in Taylor's (1953) experiments. Faraday-wave experiments in a large-aspect-ratio rectangular container agree with our computations. This is the first time such high-amplitude, periodic waves appear to have been observed in laboratory conditions. Ripple formation and temporal symmetry breaking in the experiments are discussed.

The flow of a liquid film along a periodic wall

1988 ◽  
Vol 188 ◽  
pp. 275-300 ◽  
Author(s):  
C. Pozrikidis
Keyword(s):  
Surface Tension ◽  
Liquid Film ◽  
Numerical Procedure ◽  
Surface Profile ◽  
Rotating Disk ◽  
Boundary Integral ◽  
Creeping Flow ◽  
Boundary Integral Method ◽  
Free Surface Profile ◽  
Gravity Driven Flow

The creeping flow of a liquid film along an inclined periodic wall of arbitrary geometry is considered. The problem is formulated using the boundary-integral method for Stokes flow. This method is extended to two-dimensional flows involving free surfaces, and is implemented in an iterative numerical procedure. Detailed calculations for flow along a sinusoidal wall are perfomed. The free-surface profile is studied as a function of flow rate, inclination angle, wave amplitude, and surface tension, and is compared with previous asymptotic solutions. The results include streamline patterns, velocity profiles and wall-shear-stress distributions, and establish criteria for flow reversal. For specified wall geometry, the asymptotic behaviour for very small flow rates is shown to be a strong function of surface tension. It is demonstrated that these results are valid in a qualitative sense for general wall geometries. The analogy between gravity-driven flow and the flow of a liquid layer on a rotating disk (spin coating) is also discussed.

Marangoni effects on the motion of an expanding or contracting bubble pinned at a submerged tube tip

1999 ◽  
Vol 379 ◽  
pp. 279-302 ◽  
Author(s):  
HARRIS WONG ◽  
DAVID RUMSCHITZKI ◽  
CHARLES MALDARELLI
Keyword(s):  
Surface Tension ◽  
Dynamic Surface Tension ◽  
Bond Number ◽  
Boundary Integral ◽  
Dynamic Surface ◽  
Constant Flow Rate ◽  
Tangential Flow ◽  
Marangoni Effects ◽  
Gibbs Elasticity ◽  
Made In

This work studies the motion of an expanding or contracting bubble pinned at a submerged tube tip and covered with an insoluble Volmer surfactant. The motion is driven by constant flow rate Q into or out of the tube tip. The purpose is to examine two central assumptions commonly made in the bubble and drop methods for measuring dynamic surface tension, those of uniform surfactant concentration and of purely radial flow. Asymptotic solutions are obtained in the limit of the capillary number Ca→0 with the Reynolds number Re=o(Ca−1, non-zero Gibbs elasticity (G), and arbitrary Bond number (Bo). (Ca=μQ/a2σc, where μ is the liquid viscosity, a is the tube radius, and σc is the clean surface tension.) This limit is relevant to dynamic-tension experiments, and gives M→∞, where M=G/Ca is the Marangoni number. We find that in this limit the deforming bubble at each instant in time takes the static shape. The surfactant distribution is uniform, but its value varies with time as the bubble area changes. To maintain a uniform distribution at all times, a tangential flow is induced, the magnitude of which is more than twice that in the clean case. This is in contrast to the surface-immobilizing effect of surfactant on an isolated translating bubble. These conclusions are confirmed by a boundary integral solution of Stokes flow valid for arbitrary Ca, G and Bo. The uniformity in surfactant distribution validates the first assumption in the bubble and drop methods, but the enhanced tangential flow contradicts the second.

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