Exact solutions for a stream of bubbles in a Hele-Shaw cell by Giovani L. Vasconcelos

Cover of: Exact solutions for a stream of bubbles in a Hele-Shaw cell | Giovani L. Vasconcelos

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Written in English

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Subjects:

  • Bubbles.,
  • Fluid dynamics.

Edition Notes

Book details

Statementby Giovani L. Vasconcelos.
Classifications
LC ClassificationsMicrofilm 94/2621 (Q)
The Physical Object
FormatMicroform
Paginationp. 463-468.
Number of Pages468
ID Numbers
Open LibraryOL1241682M
LC Control Number94628421

Download Exact solutions for a stream of bubbles in a Hele-Shaw cell

Exact solutions are presented for steadily moving bubbles in a Hele–Shaw cell when the effect of surface tension is neglected. These solutions form a three‐parameter family. For specified area, both the speed of the bubble and the distance of its centroid from the channel centerline remain arbitrary when surface tension is ignored.

However, numerical evidence suggests that this twofold Cited by: Exact solutions are presented for steadily moving bubbles in a Hele--Shaw cell when the effect of surface tension is neglected. These solutions form a three-parameter family. For specified area, both the speed of the bubble and the distance of its centroid from the channel centerline remain arbitrary when surface tension is ignored.

Exact solutions are presented for a steady stream of bubbles in a Hele–Shaw cell when the effect of surface tension is neglected. These solutions form a three‐parameter family.

For specified area and distance between bubbles, the speed of the bubble remains arbitrary when surface tension is neglected.

However, numerical and analytical evidence indicates that this arbitrariness is removed Cited by: 9. Exact solutions are presented for the steady motion of a symmetrical bubble through a parallel-sided channel in a Hele-Shaw cell containing a viscous liquid.

The solutions also describe two-dimensional motion in a porous medium, since the two cases are mathematically by: of smooth solutions not expected for arbitrary initial conditions in displacement of more viscous fluid. Stability of steady states and global existence for initial conditions close to a steady state is a more achievable goal.

Without injection or suction or external pressure gradient, a circular bubble in a quiescent fluid is a steady state. connected domains. More recently, Silva & Vasconcelos [14] obtained a solution for a stream of asymmetric bubbles in a Hele-Shaw channel by deploying the Schwarz–Christoffel formula for doubly connected regions.

A few exact solutions for doubly connected time-dependent Hele-Shaw flows have also been obtained [15–18]. The shapes and motion of immiscible bubbles in a Hele–Shaw cell driven by the motion of the surrounding fluid were studied.

Six classes of steady shapes, some of which are remarkable, were observed. Multiple steady states exist over some ranges of parameters and the shape as a function of speed may slow hysteresis.

The observed translational velocities do not agree with available theory. This paper presents new solutions, in analytical form, for the shapes of an assembly of steady co-travelling bubbles in a Hele-Shaw cell. The associated velocity field is also derived.

The assembly can consist of any finite number of bubbles. New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented.

Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. Vasconcelos G () Exact solutions for a stream of bubbles in a Hele-Shaw cell, Proceedings of the Royal Society of London.

Series A: Mathematical and Physical Sciences,(), Exact solutions for a stream of bubbles in a Hele-Shaw cell book publication date: 9-Aug Bubbles and fingers in a Hele-Shaw cell: steady solutions. of bubbles and for a perio dic stream of bubbles with an a of this large class of exact solutions for multiple bubbles, it can.

Exact solutions are presented for a doubly-periodic array of steadily moving bubbles in a Hele-Shaw cell when surface tension is neglected. Vasconcelos reported exact solutions for a finite number of steadily translating bubbles in a Hele-Shaw channel.

He considered symmetrical bubble shapes and, by reducing the problem to a simply connected flow domain, was able to derive Schwarz–Christoffel-type formulae for the conformal mappings determining the bubble interfaces.

Analytical solutions for both a finite assembly and a periodic array of bubbles steadily moving in a Hele-Shaw channel are presented. The particular case of multiple fingers penetrating into the channel and moving jointly with an assembly of bubbles is also analysed.

When a bubble is sandwiched by the two cell walls separated by the distance D of a Hele-Shaw cell, the drag force is found to be given by (9) F ≃ η 2 V R T 2 / D The experimental setup to confirm this is shown in Fig. 3 (a), together with a related setup discussed later. The air drop is slightly elongated, which is characterized by the transverse and longitudinal lengths R T and R L as.

D.G. Crowdy, Theory of exact solutions for the evolution of a fluid annulus in a rotating Hele-Shaw cell, Q. Appl. Math, LX(1), 11–36, (). MathSciNet Google Scholar [19].

We consider the motion of the flattened bubbles which form when air is injected into a viscous fluid contained in the narrow gap between two flat, parallel plates which make up a conventional Hele-Shaw cell, inclined at an angle x to the horizontal.

The motion of bubbles in a Hele–Shaw cell driven by a surrounding fluid or by gravity has been studied. Assuming that the surrounding fluid wets the solid wall and that the bubble surface is rigid due to the surfactant influence, the translational velocity of an elliptic bubble is estimated.

The result indicates that the bubble velocity can decrease by an order of magnitude compared to the. Exact solutions are presented for a steady stream of bubbles in a Hele--Shaw cell when the effect of surface tension is neglected.

These solutions form a three-parameter family. For specified area and distance between bubbles, the speed of the bubble remains arbitrary when surface tension is neglected. Problem 4B Flow near a wall suddenly set in motion (approximate solution) Problem 4D Start-up of laminar flow in a circular tube: Problem 4B Creeping flow around a spherical bubble: Problem 4D Flows in the disk-and-tube system: Problem 4B Use of.

Hele{Shaw’s interest was in visualising stream lines of ow around objects, representing situations in ideal (high Reynolds number) ow, such as modelling air owing around an aerofoil (see Figure). Indeed, one of the initially surprising aspects of ow in a Hele{Shaw cell is that it is mathematically equivalent to ideal ow in two dimensions.

The linear stability of steadily moving bubbles in a Hele–Shaw cell is investigated. It is shown analytically that without the effect of surface tension, the bubbles are linearly unstable with the stability operator having a continuous spectrum.

For small bubbles that are circular, analytical calculations also show that any amount of surface tension stabilizes a bubble. @article{osti_, title = {Infinite stream of Hele--Shaw bubbles}, author = {Burgess, D and Tanveer, S}, abstractNote = {Exact solutions are presented for a steady stream of bubbles in a Hele--Shaw cell when the effect of surface tension is neglected.

These solutions form a three-parameter family. For specified area and distance between bubbles, the speed of the bubble remains arbitrary. dependent exact solutions for multiply-connected Hele-Shaw flows.

In the present work we solve this problem for a doubly-connected geometry and significantly extend the results obtained in [11] by addressing the dynamics of a bubble in a Hele-Shaw cell instead of a finger (which is the singular limit of an infinitely long bubble).

This. () New exact solutions for Hele-Shaw flow in doubly connected regions. Physics of Fluids() Global solutions for a two-phase Hele-Shaw bubble for. Recently, we provided numerical evidence to suggest that the selection problem for a bubble propagating in an unbounded Hele-Shaw cell behaves in an analogous way to other finger and bubble problems in a Hele-Shaw channel; however, the selection of the ratio of bubble speeds to background velocity appears to follow a very different surface.

Download Citation | The Shape Control of a Growing Air Bubble in a Hele--Shaw Cell | The external Hele-Shaw problem when an air bubble is surrounded by oil is considered. It is assumed that the. Solutions volumes should be carefully measured with a graduated cylinder. Add solution completely, to a dry calorimeter.

Don't forget to add the spin bar each time. Set up the calorimeter with the thermometer (0° to 50°C, graduated every °C) supported from a stand so. Comparison with (1) shows that the motion in a Hele-Shaw cell is equivalent to two-dimensional flow in a porous medium of permeability b 2 / We now derive the equations for steady two-dimensional flow produced by a finger along a channel with surface tension effects due to lateral curvature included.

New exact solutions of an idealized unsteady single-phase Hele-Shaw problem of air-bubble motion in a slot-type channel are constructed under the assumption of bubble symmetry relative to the central axis of the channel.

Qualitative features of the interface evolution, which distinguish this case from the earlier considered cases of Hele-Shaw flow with different geometry, are detected. Step-by-step Textbook Solutions Work. Learn how to solve your math, science, engineering and business textbook problems instantly.

Chegg's textbook solutions go far behind just giving you the answers. We provide step-by-step solutions that help you. Mathematical formulation for the steady Hele–Shaw Bubble Problem. A Hele–Shaw cell is a pair of long parallel plates of width 2 a, separated by a small gap b, with b ≪ a.

We consider a finite-sized bubble steadily translating with velocity U in a Hele–Shaw cell containing a viscous fluid which moves with velocity V far upstream and. A new class of exact solutions is reported for an infinite stream of identical groups of bubbles moving with a constant velocity [ital U] in a Hele-Shaw cell when surface tension is neglected.

It is suggested that the existence of these solutions might explain some of the complex behavior observed in recent experiments on rising bubbles in a. Jaquard & Seguier () presented an exact solution describing the growth of a small circular bubble into fingers with h = 4, but this depends on the initial shape being circular; an initial elliptical shape would give h + 4.

To our knowledge, there is still no adequate mathematical theory for the width of fingers in a Hele Shaw cell. Complex bubble dynamics in a vertical Hele-Shaw cell Phys. Flu 共 兲 Downloaded 18 Feb to Redistribution subject to AIP license or copyright, see.

The general form of the solution is obtained for the problem of the development of a bubble in a Hele-Shaw channel. This solution belongs to the class of parameterized solutions characterized by the property that the partial derivatives of the mapping function are rational in the auxiliary plane.

A generalized problem of the system dynamics is written for the free parameters and its complete. Dynamics of configuration of rising bubble at Bg ¼ 1 and Bm ¼ The gravity force is from right to left.

Initial configuration of bubble is given by the formula r ¼ 1+( cos j+ Viscous bubbles in a Hele-Shaw cell are studied by two-dimensional theory.

The motion is caused by buoyancy and/or a pressure gradient driving a uniform basic flow in the surrounding fluid. A formula for the velocity of a steady bubble is derived, involving the ratio between the added mass of the bubble and the displaced fluid mass.

We report experimental observations of the dynamics of bubbles ascending in a Hele-Shaw cell filled with water. The bubbles are generated by injecting a continuous stream of air through a capillary of 70 μm of diameter at the bottom of the cell.

Changing the air flow, bubbles are formed at rates of Hz, Hz and Hz, with diameters of approximately 3 mm. existence of solutions in a two phase Hele-Shaw flow with surface tension for near-circular sufficiently smooth initial interface shape. Further, the circular bubble is shown to be asymptotically stable to all sufficiently smooth initial perturbation.

Introduction. The displacement of a more viscous fluid by a less viscous one in a Hele-Shaw. cell or particle should move through the laser beam at a given moment.

To accomplish this, the sample is injected into a stream of sheath fluid within the flow chamber. The flow chamber in a benchtop cytometer is called a flow cell and the flow chamber in a stream-in-air cytometer is called a nozzle tip.

The design of the flow. Depending on the initial direction of the elliptic deformation of the bubble the pear-like or bent dumbbell shapes have been observed. In order to investigate the shapes of the bubble rising in the vertical Hele–Shaw cell we have carried out the linear stability analysis of the circular shape of the bubble which is steady solution of the problem.When subjected to higher pressure, these cavities, called "bubbles" or "voids", collapse and can generate a shock wave that is strong very close to the bubble, but rapidly weakens as it propagates away from the bubble.

Cavitation is a significant cause of wear in some engineering contexts. Collapsing voids that implode near to a metal surface cause cyclic stress through repeated implosion.

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