Quasi-steady deformation of a two-dimensional bubble placed within a potential viscous flow

Antanovskiĭ, L. K.

doi:10.1007/bf00989523

Cited by 14 publications

(29 citation statements)

References 16 publications

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“…Hence from (3.6) we see that 2) and this holds together with the force balance condition (3.7). The left-hand side here is easily rewritten in terms of ζ; for the right-hand side we note that, with ζ = e iθ ,…”

Section: Details Of the Theory For Stokes Flowmentioning

confidence: 73%

“…It is defined by K(ζ) = ζ/(1+ζ) 2 , and maps |ζ| < 1 onto the whole complex plane, minus the semi-infinite line segment (−∞, −1/4]. For this simple case (4.57) gives the exact relation between ζ andζ in terms of τ .…”

Section: The Momentsmentioning

confidence: 99%

“…For times t ≥ t * the weak solution theory of §4.6 gives b(t) ≡ 1 (the cusp persists); and in terms of theζ-theory we havef (ζ, t) = f (ζ, t). Thus f (ζ, t) = a(t) ζ − ζ 2 2 ,…”

Section: The Momentsmentioning

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: Details Of the Theory For Stokes Flowmentioning

confidence: 73%

Section: The Momentsmentioning

confidence: 99%

See 1 more Smart Citation

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

Cummings¹,

Howison²,

King

1999

Eur. J. Appl. Math.

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“…Assuming the pores are not too close together, the net effect of all other pores in the 'inner' region close to any given pore is to induce an irrotational, time-dependent, straining flow which can affect both the rate of pore shrinkage and the evolution of its shape. The advantage of this lies in the fact that a wide class of exact solutions for an isolated compressible pore in a polynomially singular, time-dependent, potential straining flow are known (Crowdy 2003b;Tanveer & Vasconcelos 1995;Antanovskii 1994) which can be used to exactly solve the 'inner problem' for the free-surface evolution of the pore. At the same time, microstructural information (such as the rate of pore area shrinkage) derived from solution of the inner problem gives the macroscopic parameters needed in the outer flow approximation.…”

Section: Discussionmentioning

confidence: 99%

“…A formula for k N (t) is given below. The inner problem can be solved exactly by generalizing the exact solutions of Antanovskii (1994) to the case of an arbitrarily compressible bubble. This can be done in the same way that Crowdy (2003b) generalized the exact solutions of Tanveer & Vasconcelos (1995) to the case of compressible bubbles.…”

Section: Higher-order Modelsmentioning

confidence: 99%

An elliptical-pore model for late-stage planar viscous sintering

Crowdy

2004

J. Fluid Mech.

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A simple 'elliptical-pore model' of the shrinkage of compressible pores in late-stage planar viscous sintering is proposed. The model is in the spirit of matched asymptotics and relies on splitting the flow into an 'inner' and 'outer' problem. The inner problem in the vicinity of any given pore involves solving for its free-surface evolution exactly using complex-variable methods. The outer flow due to all other pores is assumed to be given by an assembly of point sinks/sources. As a test of the model, the evolution of a singly infinite periodic row of compressible pores is considered in detail. The effectiveness of the simple model is tested by comparison with a full numerical simulation. A novel boundary integral method based on Cauchy potentials and conformal mapping is used. In the case of pores with constant pressure, it is found that pores shrink faster than if in isolation. Compressible pores obeying the ideal gas law are also studied and are found to tend to a quasi-steady non-circular state. A higher-order model is also presented and compared with numerical simulations of the viscous sintering of a doubly periodic array of pores in Stokes flow.

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A Theory of Exact Solutions for Plane Viscous Blobs

Crowdy

Tanveer

1998

J. Nonlinear Sci.

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Quasi-steady deformation of a two-dimensional bubble placed within a potential viscous flow

Cited by 14 publications

References 16 publications

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

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

An elliptical-pore model for late-stage planar viscous sintering

A Theory of Exact Solutions for Plane Viscous Blobs

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