scholarly journals Scaling and Self-Similarity in an Unforced Flow of Inviscid Fluid Trapped Inside a Viscous Fluid in a Hele-Shaw Cell

2006 ◽  
Vol 96 (4) ◽  
Author(s):  
Arkady Vilenkin ◽  
Baruch Meerson ◽  
Pavel V. Sasorov
Keyword(s):  
Viscous Fluid ◽  
Inviscid Fluid ◽  
Self Similarity ◽  
Hele Shaw Cell

scholarly journals Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

2019 ◽  
Vol 877 ◽  
pp. 1063-1097 ◽  
Author(s):  
Liam C. Morrow ◽  
Timothy J. Moroney ◽  
Scott W. McCue
Keyword(s):  
Viscous Fluid ◽  
Linear Prediction ◽  
Applied Mathematics ◽  
Viscous Fingering ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Inviscid Fluid ◽  
Interfacial Instabilities ◽  
Hele Shaw Cell ◽  
Injection Rates

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.

Numerical simulations of sink flow in the Hele-Shaw cell with small surface tension

1997 ◽  
Vol 8 (6) ◽  
pp. 533-550 ◽  
Author(s):  
E. D. KELLY ◽  
E. J. HINCH
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Viscous Fluid ◽  
Numerical Simulations ◽  
Boundary Integral Equation ◽  
Numerical Algorithm ◽  
Boundary Integral ◽  
Inviscid Fluid ◽  
Small Surface ◽  
Hele Shaw Cell

The motion of an initially circular drop of viscous fluid surrounded by inviscid fluid in a Hele-Shaw cell withdrawn from an eccentric point sink is considered. Using a numerical algorithm based on a boundary integral equation, the solution for small, finite surface tension is observed. It is found that the zero-surface-tension formation of a cusp is avoided, and instead a narrow finger of inviscid fluid forms, which then rapidly propagates towards the sink. The scaling of the finger in the sink vicinity is determined.

The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid

1958 ◽  
Vol 245 (1242) ◽  
pp. 312-329 ◽  
Keyword(s):  
Porous Medium ◽  
Viscous Fluid ◽  
Normal Modes ◽  
Oil Fields ◽  
Viscous Oil ◽  
Dense Fluid ◽  
Two Dimensional Flow ◽  
Finger Width ◽  
Infinite Set ◽  
Hele Shaw Cell

When a viscous fluid filling the voids in a porous medium is driven forwards by the pressure of another driving fluid, the interface between them is liable to be unstable if the driving fluid is the less viscous of the two. This condition occurs in oil fields. To describe the normal modes of small disturbances from a plane interface and their rate of growth, it is necessary to know, or to assume one knows, the conditions which must be satisfied at the interface. The simplest assumption, that the fluids remain completely separated along a definite interface, leads to formulae which are analogous to known expressions developed by scientists working in the oil industry, and also analogous to expressions representing the instability of accelerated interfaces between fluids of different densities. In the latter case the instability develops into round-ended fingers of less dense fluid penetrating into the more dense one. Experiments in which a viscous fluid confined between closely spaced parallel sheets of glass, a Hele-Shaw cell, is driven out by a less viscous one reveal a similar state. The motion in a Hele-Shaw cell is mathematically analogous to two-dimensional flow in a porous medium. Analysis which assumes continuity of pressure through the interface shows that a flow is possible in which equally spaced fingers advance steadily. The ratio λ = (width of finger)/(spacing of fingers) appears as the parameter in a singly infinite set of such motions, all of which appear equally possible. Experiments in which various fluids were forced into a narrow Hele-Shaw cell showed that single fingers can be produced, and that unless the flow is very slow λ = (width of finger)/(width of channel) is close to ½, so that behind the tips of the advancing fingers the widths of the two columns of fluid are equal. When λ = ½ the calculated form of the fingers is very close to that which is registered photographically in the Hele-Shaw cell, but at very slow speeds where the measured value of λ increased from ½ to the limit 1.0 as the speed decreased to zero, there were considerable differences. Assuming that these might be due to surface tension, experiments were made in which a fluid of small viscosity, air or water, displaced a much more viscous oil. It is to be expected in that case that λ would be a function of μ U/T only, where μ is the viscosity, U the speed of advance and T the interfacial tension. This was verified using air as the less viscous fluid penetrating two oils of viscosities 0.30 and 4.5 poises.

Dynamics of thin vortex rings

2008 ◽  
Vol 609 ◽  
pp. 319-347 ◽  
Author(s):  
IAN S. SULLIVAN ◽  
JOSEPH J. NIEMELA ◽  
ROBERT E. HERSHBERGER ◽  
DIOGO BOLSTER ◽  
RUSSELL J. DONNELLY
Keyword(s):  
Viscous Fluid ◽  
Vortex Ring ◽  
Long Range ◽  
Direct Measurement ◽  
Vortex Rings ◽  
Inviscid Fluid ◽  
Core Size ◽  
Physical Pendulum ◽  
Range Study

As part of a long-range study of vortex rings, their dynamics, interactions with boundaries and with each other, we present the results of experiments on thin core rings generated by a piston gun in water. We characterize the dynamics of these rings by means of the traditional equations for such rings in an inviscid fluid suitably modifying them to be applicable to a viscous fluid. We develop expressions for the radius, core size, circulation and bubble dimensions of these rings. We report the direct measurement of the impulse of a vortex ring by means of a physical pendulum.

The toroidal bubble

1968 ◽  
Vol 32 (1) ◽  
pp. 97-112 ◽  
Author(s):  
T. J. Pedley
Keyword(s):  
Surface Tension ◽  
Viscous Fluid ◽  
Energy Equation ◽  
Bubble Surface ◽  
Inviscid Fluid ◽  
Initial Instant ◽  
Bubble Volume ◽  
Toroidal Bubble ◽  
History Of ◽  
Impulse Equation

It has been observed by Walters & Davidson (1963) that release of a mass of gas in water sometimes produces a rising toroidal bubble. This paper is concerned with the history of such a bubble, given that at the initial instant the motion is irrotational everywhere in the water. The variation of its overall radius a with time may be predicted from the vertical impulse equation, and it should be possible to make the same prediction by equating the rate of loss of combined kinetic and potential energy to the rate of viscous dissipation. This is indeed seen to be the case, but not before it is recognized that in a viscous fluid vorticity will continually diffuse out from the bubble surface, destroying the irrotationality of the motion, and necessitating an examination of the distribution of vorticity. The impulse equation takes the same form as in an inviscid fluid, but the energy equation is severely modified. Other results include an evaluation of the effect of a hydrostatic variation in bubble volume, and a prediction of the time which will have elapsed before the bubble becomes unstable under the action of surface tension.

The motion of a viscous drop sliding down a Hele-Shaw cell

1986 ◽  
Vol 163 ◽  
pp. 59-67 ◽  
Author(s):  
Kalvis M. Jansons
Keyword(s):  
Surface Tension ◽  
Viscous Fluid ◽  
Contact Angle Hysteresis ◽  
Leading Edge ◽  
Governing Equations ◽  
Drop Shape ◽  
Viscous Drop ◽  
Special Case ◽  
Hele Shaw Cell ◽  
Effect Of Surface

The motion of a viscous drop in a vertical Hele-Shaw cell is studied in a limit where the effect of surface tension through contact-angle hysteresis is significant. It is found that a rectangular drop shape is a possible steady solution of the governing equations, although this solution is unstable to perturbations on the leading edge. Even though the unstable edge is one where a viscous fluid is moving into a less viscous fluid, in this case air, this is shown to be a special case of the well-known Saffman—Taylor instability. An experiment is performed with an initially circular drop in which it is observed that the drop shape becomes approximately rectangular except at the leading edge, where it becomes rounded and sometimes has a ragged appearance.A drop sliding down a vertical Hele-Shaw cell is an example of a system where the action of surface tension is not always one of smoothing, since in this case it leads to the formation of right-angle corners at the back of the drop (rounded only slightly on the lengthscale of the gap thickness of the cell).

Motion of inviscid fluid tongues and bubbles in a Hele-Shaw cell occupied by a viscoelastic fluid

2005 ◽  
Vol 40 (1) ◽  
pp. 103-109
Author(s):  
V. M. Entov ◽  
S. S. Kolganov ◽  
N. V. Kolganova
Keyword(s):  
Viscoelastic Fluid ◽  
Inviscid Fluid ◽  
Hele Shaw Cell

The hydrodynamic stability of helium ll between rotating cylinders. I

1957 ◽  
Vol 241 (1224) ◽  
pp. 9-28 ◽  
Keyword(s):  
Viscous Fluid ◽  
Hydrodynamic Instability ◽  
Present Theory ◽  
Inviscid Fluid ◽  
Rotating Cylinders ◽  
Two Fluids ◽  
Constant Vorticity ◽  
A Value ◽  
Good Accord ◽  
The Difference

The hydrodynamic instability of helium ll between rotating cylinders is investigated on two assumptions regarding the mutual friction force, F, between the normal and the super­fluid components of the liquid. On both assumptions F is proportional to the constant vorticity which prevails in the stationary state and to the difference in the velocities between the two fluids; however, on one assumption the effect of F is confined entirely to the transverse plane, while on the other it is allowed to be isotropic (with respect to the difference in the velocities). The hydrodynamic problem is solved for the case when the two cylinders (of radii R 1 and R 2 ) are rotated in the same direction and ( R 2 — R 1 ) ≪ ½ ( R 2 + R 1 ). It follows from the theory that when ∂( r 2 Ω)/∂ r < 0 (where Ω denotes the angular velocity and r the distance from the axis) the flow becomes (eventually) unstable along two branches: the first of these is the normal (Taylor) instability of a viscous fluid inhibited by its coupling with an inviscid fluid, and the second is the (Rayleigh) instability of the superfluid inhibited, in turn, by its coupling with a viscous fluid. Further, in all cases the critical Taylor number of instability (suitably defined) becomes asymptotic to a relation which is equivalent to Γ 2 = ½ ( R 2 2 — R 2 1 )R 2 1 , where Γ is the coupling constant. From an experiment of Kolm & Herlin’s (1956), to which the present theory appears applicable, a value of Γ = 0·52 is deduced; this is in very good accord with the value Γ = 0·55 which Hall & Vinen (1956 α ) have deduced from an unrelated experiment.

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