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.

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.

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.

Cusp formation for time-evolving bubbles in two-dimensional Stokes flow

2000 ◽  
Vol 412 ◽  
pp. 227-257 ◽  
Author(s):  
MICHAEL SIEGEL
Keyword(s):  
Surface Tension ◽  
Stokes Flow ◽  
Capillary Number ◽  
Broad Class ◽  
Finite Amplitude ◽  
Bubble Surface ◽  
Two Dimensional ◽  
Variable Surface ◽  
Cusp Formation

Analytical and numerical methods are applied to investigate the transient evolution of an inviscid bubble in two-dimensional Stokes flow. The evolution is driven by extensional incident flow with a rotational component, such as occurs for flow in a four-roller mill. Of particular interest is the possible spontaneous occurrence of a cusp singularity on the bubble surface. The role of constant as well as variable surface tension, induced by the presence of surfactant, is considered. A general theory of time- dependent evolution, which includes the existence of a broad class of exact solutions, is presented. For constant surface tension, a conjecture concerning the existence of a critical capillary number above which all symmetric steady bubble solutions are linearly unstable is found to be false. Steady bubbles for large capillary number Q are found to be susceptible to finite-amplitude instability, with the dynamics often leading to cusp or topological singularities. The evolution of an initially circular bubble at zero surface tension is found to culminate in unsteady cusp formation. In contrast to the clean flow problem, for variable surface tension there exists an upper bound Qc for which steady bubble solutions exist. Theoretical considerations as well as numerical calculations for Q > Qc verify that the bubble achieves an unsteady cusped formation in finite time. The role of a nonlinear equation of state and the influence of surface diffusion of surfactant are both considered. A possible connection between the observed behaviour and the phenomenon of tip streaming is discussed.

Numerical studies of singularity formation at free surfaces and fluid interfaces in two-dimensional Stokes flow

1997 ◽  
Vol 331 ◽  
pp. 145-167 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Numerical Study ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Analytic Structure ◽  
Two Dimensional ◽  
Thin Film Lubrication ◽  
Transient Deformation ◽  
Lubrication Equation

We consider the analytic structure of interfaces in several families of steady and unsteady two-dimensional Stokes flows, focusing on the formation of corners and cusps. Previous experimental and theoretical studies have suggested that, without surface tension, the interfaces spontaneously develop such singular points. We investigate whether and how corners and cusps actually develop in a time-dependent flow, and assess the stability of stationary cusped shapes predicted by previous authors. The motion of the interfaces is computed with high resolution using a boundary integral method for three families of flows. In the case of a bubble that is subjected to the family of straining flows devised by Antanovskii, we find that a stationary cusped shape is not likely to occur as the asymptotic limit of a transient deformation. Instead, the pointed ends of the bubble disintegrate in a process that is reminiscent of tip streaming. In the case of the flow due to an array of point-source dipoles immersed beneath a free surface, which is the periodic version of a flow proposed by Jeong & Moffatt, we find evidence that a cusped shape indeed arises as the result of a transient deformation. In the third part of the numerical study, we show that, under certain conditions, the free surface of a liquid film that is levelling under the action of gravity on a horizontal or slightly inclined surface develops an evolving corner or cusp. In certain cases, the film engulfs a small air bubble of ambient fluid to obtain a composite shape. The structure of a corner or a cusp in an unsteady flow does not have a unique shape, as it does at steady state. In all cases, a small amount of surface tension is able to prevent the formation of a singularity, but replacing the inviscid gas with a viscous liquid does not have a smoothing effect. The ability of the thin-film lubrication equation to produce mathematical singularities at the free surface of a levelling film is also discussed.

A boundary integral method for the simulation of two-dimensional particle coarsening

1986 ◽  
Vol 1 (2) ◽  
pp. 117-144 ◽  
Author(s):  
G. B. McFadden ◽  
P. W. Voorhees ◽  
R. F. Boisvert ◽  
D. I. Meiron
Keyword(s):  
Integral Method ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Particle Coarsening
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