A Theory of Exact Solutions for Plane Viscous Blobs

Crowdy, Darren; Tanveer, S.

doi:10.1007/s003329900052

Cited by 22 publications

(25 citation statements)

References 8 publications

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“…For flow driven by a single point source (sink) of strength Q > 0 (Q < 0) at the origin, it is straightforward to find the evolution equations for M k (t), from (3.8) (see [10]; the discussion there parallels [8], and similar results also arise in [48]). The case γ > 0 leads to a difficult system of nonlinear differential equations, but the ZST problem is simple, leading to an infinite system of conserved quantities exactly as for Hele-Shaw:…”

Section: Conserved Quantities (Stokes Flow Moments)mentioning

confidence: 92%

“…The moments provide a compact formulation for problems with a polynomial mapping function, but for a general rational mapping function they lead to an infinite system of coupled equations, which is an unnecessary complication (indeed, the question of whether the moments completely specify the motion in such cases is unclear, although it seems likely). However, as noted in [10], the quantity ζ k in (3.20) can be replaced by an arbitrary function of ζ, and a procedure for constructing conserved quantities of this type for rational maps is described in [8].…”

Section: Conserved Quantities (Stokes Flow Moments)mentioning

confidence: 99%

“…The moments are 37) though the integrand in (4.37) can be generalised to any analytic function ofζ; generalisations of this kind may again be useful for rational maps (cf. [8]). We may, as before, derive the result…”

Section: The Momentsmentioning

confidence: 99%

“…There is no doubt that Stokes flow leads to the more complicated structure in the zero-surfacetension case, but set against that is the fact that it is possible to construct solutions with non-zero surface tension; would that this were the case for the Hele-Shaw problem! We have also outlined some ways in which the theory for Stokes flow can be extended; we have not mentioned the many complications that arise when the fluid domain is multiply connected, but [6] describes some general results for multiply connected flows in n dimensions, while [8] and [46] give explicit solutions using complex variable methods. Among the many other fascinating possibilities for future investigation, we single out in particular the idea of a weak solution, in which the free boundary to a zero-surfacetension problem has a persistent singularity.…”

Section: General Remarksmentioning

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: 92%

Section: Conserved Quantities (Stokes Flow Moments)mentioning

confidence: 99%

Section: The Momentsmentioning

confidence: 99%

Section: General Remarksmentioning

confidence: 99%

See 2 more Smart Citations

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

Cummings¹,

Howison²,

King

1999

Eur. J. Appl. Math.

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“…These are integrals over a two-dimensional domain whereas the conserved integrals that we discuss here are along a one-dimensional path within a two-dimensional domain. For further discussion of this and related conservation integrals, see Richardson (1972Richardson ( , 1992, Tanveer & Vasconcelos (1995), Cummings, King & Howison (1997), Crowdy & Tanveer (1998) and Crowdy (1999).…”

Section: 2mentioning

confidence: 99%

Exact results with the J-integral applied to free-boundary flows

Amar

Rice

2002

J. Fluid Mech.

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We apply the J-integral to free-boundary flows in a channel geometry such as viscous fingering or blob injection in Hele-Shaw cells, void propagation in electromigration, and injection of air bubbles into inviscid liquids. The theory of that and related conservation integrals, developed in elasticity, is outlined in a way that is applicable to fluid mechanics problems. Depending on the boundary conditions, for infinite bubbles in Laplacian fields we are able to use the J-integral to predict finger width if such solutions exist or to predict that there are no solutions. For finite sized bubbles, bounds can sometimes be derived. In the case of Hele-Shaw flows, in which solutions appear as a continuum, finger width cannot be constrained, but we do obtain a new derivation and generalization of Richardson moment conservation. Applications to vortex motion are also outlined briefly.

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A Theory of Exact Solutions for Annular Viscous Blobs

Crowdy

Tanveer

1998

J. Nonlinear Sci.

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A Theory of Exact Solutions for Plane Viscous Blobs

Cited by 22 publications

References 8 publications

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

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

Exact results with the J-integral applied to free-boundary flows

A Theory of Exact Solutions for Annular Viscous Blobs

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