Coalescence of Two Equal Cylinders: Exact Results for Creeping Viscous Plane Flow Driven by Capillarity

Hopper, Robert

doi:10.1111/j.1151-2916.1984.tb19692.x

Cited by 130 publications

(154 citation statements)

References 7 publications

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“…In other words, the formulation is essentially in the ζ-plane, and indeed, we could define 20) which can also be obtained by using Green's theorem on the definition (3.18). By contrast, the Hele-Shaw result was obtained quite independently of any considerations of the ζ-plane.…”

Section: Conserved Quantities (Stokes Flow Moments)mentioning

confidence: 99%

“…Stokes flows with free surfaces, on the other hand, have until recently received much less attention, but there has been much more activity following the demonstration by Hopper [20,21] and Richardson [41] that explicit unsteady solutions could be constructed (steady solutions were given earlier in [4,17,37,39,49]). A bibliography listing more than 600 papers in both areas can be found at the web address http://www.maths.ox.ac/~howison/ .…”

Section: Introductionmentioning

confidence: 99%

“…Maps of this form have been used to describe viscous sintering [20,21,44]. The map (4.41) contains 2N unknowns; N equations for these can be written down straight away using our results for singularity dynamics.…”

mentioning

confidence: 99%

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Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Cummings¹,

Howison²,

King

1999

Eur. J. Appl. Math.

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We discuss the application of complex variable methods to Hele-Shaw flows and twodimensional Stokes flows, both with free boundaries. We outline the theory for the former, in the case where surface tension effects at the moving boundary are ignored. We review the application of complex variable methods to Stokes flows both with and without surface tension, and we explore the parallels between the two problems. We give a detailed discussion of conserved quantities for Stokes flows, and relate them to the Schwarz function of the moving boundary and to the Baiocchi transform of the Airy stress function. We compare the results with the corresponding results for Hele-Shaw flows, the principal consequence being that for Hele-Shaw flows the singularities of the Schwarz function are controlled in the physical plane, while for Stokes flow they are controlled in an auxiliary mapping plane. We illustrate the results with the explicit solutions to specific initial value problems. The results shed light on the construction of solutions to Stokes flows with more than one driving singularity, and on the closely related issue of momentum conservation, which is important in Stokes flows, although it does not arise in Hele-Shaw flows. We also discuss blow-up of zero-surface-tension Stokes flows, and consider a class of weak solutions, valid beyond blow-up, which are obtained as the zero-surface-tension limit of flows with positive surface tension.

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Section: Conserved Quantities (Stokes Flow Moments)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Cummings¹,

Howison²,

King

1999

Eur. J. Appl. Math.

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“…The two-dimensional viscous sintering problem has received a revival of theoretical interest following the work of Hopper [11], who analysed this unit problem and showed that it admits exact solutions in the sense that the full free boundary problem can be reduced to the integration of a set of coupled nonlinear ordinary differential equations. In a natural extension of Hopper's result, Richardson [19] extended the class of exact solutions to the case of two unequal cylinders coalescing.…”

Section: Introductionmentioning

confidence: 99%

“…For two equal cylinders, Hopper [11] showed the fluid domain under the sinter dynamics can be described by means of a conformal map from a unit ζ-disc of the form…”

Section: Introductionmentioning

confidence: 99%

Viscous sintering of unimodal and bimodal cylindrical packings with shrinking pores

Crowdy

2003

Eur. J. Appl. Math

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Calculations, based on exact solutions, of the viscous sintering of simple packings with unimodal and bimodal distributions of particle sizes with shrinking pores are performed. The case of square unit cells is considered in detail. It is found that, for the most part, pore shrinkage times have very weak dependence on the precise details of the pore shape and that accurate estimates of total pore shrinkage times can be obtained based only on knowledge of the initial relative density (green density) and unit cell size. An exception is found when the particle packing is loose, the enclosed pore having a large perimeter-to-area ratio. By studying two different square packings of equal-sized particles it is shown that the looser packing can exhibit dilation, rather than densification, in the early stage of sintering.

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Analysis of capillary‐driven viscous flows during the sintering of ceramic powders

Martínez-Herrera

Derby

1994

AIChE Journal

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Coalescence of Two Equal Cylinders: Exact Results for Creeping Viscous Plane Flow Driven by Capillarity

Cited by 130 publications

References 7 publications

Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Two-dimensional Stokes and Hele-Shaw flows with free surfaces

Viscous sintering of unimodal and bimodal cylindrical packings with shrinking pores

Analysis of capillary‐driven viscous flows during the sintering of ceramic powders

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