Absence Of Surface Tension Research Articles

Hydromagnetic instability of streaming fluids in porous medium

The instability of the plane interface between two uniform, superposed, electrically conducting and counter-streaming fluids through a porous medium is considered in the presence of a horizontal magnetic field. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if perturbations in the direction of streaming are ignored. For perturbations in all other directions there exists instability for a certain wavenumber range. The instability of this system is postponed by the presence of magnetic field. The magnetic field and surface tension are able to suppress this Kelvin-Helmholtz instability for small wavelength perturbations and the medium porosity reduces the stability range given in terms of a difference between the streaming velocities and the Alfven velocity.

The Universal Instability of the Separation Boundary Between a Non-Hydrostatically Stressed Elastic Crystal and its Melt

ABSTRACTIn the absence of surface tension and external force fields, the equilibrium between a hydrostatically stressed crystal and its melt is neutral with respect to the perturbations associated with particle transfer from one region of the boundary into another. However, under the action of arbitrary small nonhydrostatic components of the stress field in the elastic crystal, the neutral equilibrium is transformed to an unstable equilibrium [1]. This instability is very general in nature; for example, for it to be seen the liquid media need only to be able to dissolve the solid phase or in some way to assist the transport of particles along the crystal's surface. In contrast, the surface tension, roughly speaking, stabilizes the shape of the interphase boundary but it cannot suppress the instability generated by the nonhydrostatic components of the stress field in the region of sufficiently long perturbations. Until now the basic instability mechanism discussed here seems to have escaped the attention of theorists. This mechanism allows one to look in a completely new way at a broad range of phenomena. We discuss tentative manifestations and role of this instability in low temperature physics, in materials science, in theory of crystal growth, and, in particular, in theory of epitaxy and of the Stranski-Krastanow pattern of growth of thin crystalline films.

The deformation of a liquid drop moving normal to a plane wall

The deformation of a viscous drop moving under the action of gravity normal to a plane solid wall is studied. Under the assumption of creeping flow, the motion is studied as a function of the viscosity ratio between the drop and the suspending fluid, of surface tension, and of the initial drop configuration. Using the boundary integral formulation, the flow inside and outside the drop is represented in terms of a combined distribution of a single-layer and a double-layer potential of Green functions over the drop surface. The densities of these distributions are identified with the discontinuity in the interfacial surface stress, and with the interfacial velocity. The problem is formulated as a Fredholm integral equation of the second kind for the interfacial velocity which is solved by successive iterations. It is found that in the absence of surface tension, a drop moving away from the wall obtains an increasingly prolate shape, eventually ejecting a trailing tail. Depending on the initial drop configuration, ambient fluid may be entrained into the drop along or away from the axis of motion. Surface tension prevents the formation of the tail allowing the drop to maintain a compact shape throughout its evolution. The deformation of the drop has little effect on its speed of rise. A drop moving towards the wall obtains an increasingly oblate shape. In the absence of surface tension, the drop starts spreading in the radial direction reducing into a thinning layer of fluid. This layer is susceptible to the gravitational Rayleigh–Taylor instability. Surface tension restricts spreading, and allows the drop to attain a nearly steady hydrostatic shape. This is quite insensitive to the viscosity ratio and to the initial drop configuration. The evolution of the thin layer of fluid which is trapped between the drop and the wall is examined in detail, and with reference to film-drainage theory. It is shown that the assumptions underlying this theory are accurate when the surface tension is sufficiently large, and when the viscosity of the drop is of the same or lower order of magnitude as the viscosity of the ambient fluid. The numerical results are discussed with reference to film-drainage asymptotic theories.

Fractal-to-nonfractal crossover for viscous fingers.

We propose a position-space renormalization-group approach to the problem of viscous fingering in the absence of surface tension, with arbitrary viscosity ratio between the injected and displaced fluid. We find there are only two fixed points, the Eden and the diffusion-limited aggregation (DLA) points. The Eden point, which corresponds to a compact cluster with nonfractal surface, is stable in all directions, while the DLA fixed point is a saddle point. Hence if the viscosity of the injected fluid is not zero, the system must eventually cross over to a compact cluster. We also calculate the crossover exponent \ensuremath{\varphi} and crossover radius ${\mathit{R}}_{\ifmmode\times\else\texttimes\fi{}}$, and discuss possible experimental measurements.

The instability of a moving viscous drop

The deformation of a moving spherical viscous drop subject to axisymmetric perturbations is considered. The problem is formulated using two different variations of the boundary integral method for Stokes flow, one due to Rallison & Acrivos, and the other based on an interfacial distribution of Stokeslets. An iterative method for solving the resulting Fredholm integral equations of the second kind is developed, and is implemented for the case of axisymmetric motion. It is shown that in the absence of surface tension, a moving spherical drop is unstable. Prolate perturbations cause the ejection of a tail from the rear of the drop, and the entrainment of a thin filament of ambient fluid into the drop. Oblate perturbations cause the drop to develop into a nearly steady ring. The viscosity ratio plays an important role in determining the timescale and the detailed pattern of deformation. Filamentation of the drop emerges as a persistent but secondary mechanism of evolution for both prolate and oblate perturbations. Surface tension is not capable of suppressing the growth of perturbations of sufficiently large amplitude, but is capable of preventing filamentation.

Stability of a compressed gas bubble in a viscous fluid

The stability of the spherical shape of the free surface of a gas bubble compressed by an incompressible fluid as it appears in the inertial confinement fusion problem is considered. (i) The equations derived by Prosperetti [Accad. Naz. Lincei 62, 196 (1977)] generalizing the Plesset equation are recovered in cases when the outer fluid is nonviscous, the flow being not potential, and it is shown that vorticity may change drastically the results of the potential case. (ii) In the case of viscous external fluid, the equations derived by Prosperetti [Q. Appl. Math. 1, 399 (1977)] and other external conditions on a sphere of finite radius are derived. (iii) Assuming that the time scale of the dynamics of the spherical bubble is large with respect to the time scale of the perturbation (frozen assumption), the linear stability of the collapsing bubble is studied numerically. The parameters are here (a) an inertia force (related with acceleration R̈ of the radius of the bubble), (b) the Reynolds number built with the decaying rate of the bubble, (c) surface tension, and (d) the aspect ratio (ratio between the gap width of the viscous fluid and the radius of the bubble). It is shown that the spherical shape is always linearly unstable in the absence of surface tension. In the presence of surface tension, there is a critical inertia parameter value and the most dangerous mode is always stationary. For the case of a large surface tension, the spherical wavenumber l of the most dangerous mode, is low. Finally, it is shown that the Rayleigh–Taylor instability might only be observed for both small aspect ratio and Reynolds number, depending on the surface tension.

Bubbles in the Hele-Shaw cell: Pattern selection and tip perturbations.

We consider the motion of a symmetric finite bubble in a two-dimensional Hele-Shaw cell. In the absence of surface tension, the Saffman-Taylor solution contains two free parameters, U and \ensuremath{\lambda}, for a given bubble area J, where U is the dimensionless speed and \ensuremath{\lambda} is the dimensionless width of the bubble. It is shown that, in the presence of surface tension, a solution does not exist for Ug2 for any bubble area. We also derive the following scaling relations: (a) 2-U\ensuremath{\approxeq}${\ensuremath{\epsilon}}^{2/3}$ for large J; and (b) 2-U\ensuremath{\approxeq}${\ensuremath{\epsilon}}^{2}$ for small J, where \ensuremath{\epsilon} is a small parameter which is proportional to the surface tension. We show that by creating a cusp at the tip of the bubble, one can increase the speed of the bubble Ug2. We present predictions for the shape and speed of the symmetric bubble as a function of the external parameters in the presence of a cusp at the tip. This picture may explain the recent experimental results of Maxworthy, where substantially enhanced velocities were measured for the anomalous bubbles with a tiny bubble at the tip.

HELE-SHAW FREE-BOUNDARY PROBLEMS WITH SUCTION

In the absence of surface tension, the solution of most Hele-Shaw free-boundary problems with suction becomes singular at finite time, say t = t*, when the free boundary develops a cusp. This paper considers the evolution of the free boundary in the presence of small non-zero surface tension for times slightly greater than t*; the suggested structure for the free boundary is that of a smooth part together with thin fingers of air which rapidly penetrate the liquid region.

The downstream flow beyond an obstacle

This paper is concerned with theoretical predictions, given the upstream conditions from a rigid obstacle of arbitrary shape, of the downstream flow beyond the obstacle for an incompressible inviscid fluid sheet under the action of gravity. The fluid sheet flows upstream over a level bottom, continues to flow over (or under) an obstacle leading to a downstream region over a level bottom. In the absence of surface tension, a nonlinear st., ady-state solution of the problem is used to predict the downstream values of the free-surface wave height for the full range of the far-upstream Froude number. The general results obtained are then applied to a special case of fluid flowing over a stationary hump leading to a supercritical flow far downstream and detailed numerical comparison is made with available experimental results, with very good agreement.

The separation of viscous jets

A viscous jet is not usually observed to separate from a sharp edge in the manner expected theoretically. In the present paper the separation of a creeping jet emerging from a tube with a rounded exit is considered. As a separation criterion, in the absence of surface tension, we propose that the traction normal to the nozzle surface drops to zero at the separation point. Boundary-element calculations then show a behavior that agrees with experimental data and with previous finite-element computations. They also permit the Michael condition to be observed at separation, so that the discrepancy between finite-element calculations and theory is removed.

A finite difference study of the stretching and break-up of filaments of polymer solutions

The stretching and break-up of a viscoelastic filament pulled by a constant weight is studied numerically by a finite difference method. The results show the following tendencies: (1) Newtonian filaments, even in the absence of surface tension, show a rapid increase in elongation at one particular point (they “break” there). (2) The addition of a viscoelastic polymer prevents, or at least delays, the break-up, even if it makes only a small difference to shear viscosity. (3) Surface tension accelerates break-up, but even in the presence of surface tension elasticity has a stabilizing effect.

Shear-flow instability at the interface between two viscous fluids

We consider the linear stability of the cocurrent flow of two fluids of different viscosity in an infinite region (the viscous analogue of the classical Kelvin-Helmholtz problem). Attention is confined to the simplest case, Couette flow, and we solve the problem using both numerical and asymptotic techniques. We find that the flow is always unstable (in the absence of surface tension). The instability arises at the interface between the two fluids and occurs for short wavelengths, when viscosity rather than inertia is the dominant physical effect.

Note on the effect of surface tension on water waves at an inertial surface

This note is to show that in the presence of surface tension small progressive waves can always exist in water having an inertial surface composed of uniformly distributed floating particles, in contrast to the known result in the absence of surface tension that precludes propagation under a surface that is too heavy.

The instability of streaming fluids in a porous medium

The instability of the plane interface between two uniform, superposed, and streaming fluids through porous media is considered. The configuration is taken to be bottom-heavy. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if perturbation in the direction of streaming are ignored, whereas for perturbations in all other directions there exists instability for a certain wavenumber range. The surface tension is able to suppress this Kelvin–Helmholtz instability for small wavelength perturbations and the medium porosity reduces the stability range given in terms of a difference in streaming velocities. For the top-heavy configurations, the surface tension stabilizes a certain wavenumber range.

Note on the reflexion of water waves at a wall in the presence of surface tension

AbstractIn this note we examine the influence of surface tension on surface waves incident against a fixed vertical plane wall. The motion is time harmonic and is determined by making the initial assumption that the free-surface slope at the wall is prescribed. From the unique solution obtained for the velocity potential, the parameter involved in this specification can be determined, for small laboratory-scale waves at least, using some longstanding experimental results on meniscus behaviour at a moving contact line. The effect of surface tension is to produce a motion wherein reflexion from the wall is not complete and there is a local disturbance, in contrast to the classical standing-wave motion in the absence of surface tension.

Linearized planing-surface theory with surface tension. Part I: Smooth detachment

AbstractIn the absence of surface tension, the problem of determining a travelling surface pressure distribution that displaces a portion of the free surface in a prescribed manner has been solved by several authors, and this “planing-surface” problem is reasonably well understood. The effect of inclusion of surface tension is to change, in a dramatic way, the singularity in the integral equation that describes the problem. It is now necessary in general to allow for isolated impulsive pressure, as well as a smooth distribution over the wetted length. Such pressure points generate jump discontinuities in free-surface slope. Numerical results are obtained here for a class of problems in which there is a single impulse located at the leading edge of the planing surface and detachment with continuous slope at the trailing edge. These results do not appear to approach the classical results in the limit as the surface tension approaches zero, a paradox that is resolved in Part II, which follows.

Measurement of high frequency capillary waves on steep gravity waves

The properties of high frequency capillary waves generated by steep gravity waves on deep water have been measured with a high resolution laser optical slope gauge. The results have been compared with the steady theory of Longuet-Higgins (1963). Good qualitative agreement is obtained. However, the quantitative predictions of the capillary wave slopes cannot be verified by the data because the theory requires knowledge of an idealized quantity - the crest curvature of the gravity wave in the absence of surface tension - which cannot be measured experimentally.

On forced three-dimensional surface waves in a channel in the presence of surface tension

AbstractIn this paper wave-maker theory including the effect of surface tension is determined for three-dimensional motion of water in a semi-infinite rectangular channel with outgoing surface wave modes allowed for at infinity; the motion is generated by a harmonically oscillating vertical plane wave-maker at the end of the channel and the cases of both infinite and finite constant depth are treated. The solution of the boundary-value problem for the velocity potential is more complicated in the presence of surface tension due mainly to the additional effect of the channel walls at which the normal free surface slopes are prescribed—as also is the slope at the wave-maker—to ensure uniqueness. The simpler three-dimensional solution for a semi-infinite region—obtained long ago by Sir Thomas Havelock in the absence of surface tension for the case of infinite depth—is also noted.

The effect of surface tension on the motion of an isolated fluid body

Consider a body made f rom a uniform, incompressible, linear viscous fluid. The body may be deformed in an arbitrary way, by the application of tractions at its boundary and by the application of body forces, up until some instant of time, but thereafter we suppose it to be isolated in the sense that the applied tractions and body forces are removed and cease to act at all subsequent times. Our aim is to investigate the motion of the body after isolation, or, more specifically, to investigate the effect of the presence or absence of surface tension on the motion after isolation. To this end we prove two results. We prove first of all that if there is no surface tension and if the angular momentum, which must be constant after isolation, does not vanish then either the diameter of the body grows unboundedly or the mean pressure ultimately becomes negative. Since the action of surface tension is to oppose the growth of the surface area of the body we might suspect that this result cannot be generally true when the body has non-vanishing surface tension, at least for values of the angular momentum which are not too large. We shall establish that there is substance to this suspicion by showing, and this is our second result, that if the angular momentum is not too large, in a sense which will be made precise, then there are isolated motions of the body in which the diameter remains constant and the pressure is always positive. The mechanical properties of the body are determined by the constant values M, V,/~, a of its mass, its volume, its viscosity and its surface tension, respectively. I t will be assumed that, in the motions with which we are concerned, the region B(t) , occupied by the body at the time t, is a compact three-dimensional manifoldwith-boundary, whose boundary aB(t) is a class C 2 two-dimensional manifold with unit outward normal n(x, t) at xeaB(t). Notwithstanding our use of the symbol V for the constant total volume, we will use it also to denote the volume measure on B(t). The area measure on aB(t) is denoted by A and A(t) is the area of aB(t). We need certain vector identities (2), (3) and (4) involving the normal n; before we can formulate them we must extend n (., t) to a class C 1 unit vector field defined on a neighbourhood of the boundary. To do this let e > 0 be a positive number and let U~(t) be the open neighbourhood

Fundamental singularities in the theory of water waves with surface tension

In this paper the forms are obtained for the harmonic potential functions describing the fundamental wave-source and multipole singularities which pertain to the study of infinitesimal time-harmonic waves on the free surface of water when the effect of surface tension is included. Line and point singularities are considered for both the cases of infinite and finite constant depth of water. The method used is an extension of that which has been used to obtain these potentials in the absence of surface tension.