Weak solution for the Hele-Shaw problem: Viscous shocks and singularities

Lee, S. Y.; Teodorescu, Răzvan; Wiegmann, P.

doi:10.1134/s0021364010140043

Cited by 6 publications

(12 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A main feature of the supercritical regime is the appearance of one-dimensional arcs (we call them whiskers) that point out of the droplet. In the context of [22] these whiskers are interpreted as pressure shock waves. Appearance of the whiskers in the supercritical regime was also observed in [4] (for complex orthogonal polynomials with some exponential weight).…”

Section: Introductionmentioning

confidence: 99%

The supercritical regime in the normal matrix model with cubic potential

Kuijlaars

Tovbis

2015

Advances in Mathematics

View full text Add to dashboard Cite

The normal matrix model with a cubic potential is ill-defined and it develops a critical behavior in finite time. We follow the approach of Bleher and Kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a Hermitian form. This reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime, namely that the zeros of the polynomials tend to a number of segments (the motherbody) inside a domain (the droplet) that attracts the eigenvalues in the normal matrix model.In the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the Boutroux condition. The Boutroux condition determines a system of contours Σ 1 , consisting of the motherbody and whiskers sticking out of the domain. We find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain.In the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3 × 3 matrix valued Riemann-Hilbert problem. It follows that the zeros of the orthogonal polynomials tend to Σ 1 , with the exception of at most three spurious zeros.

show abstract

Section: Introductionmentioning

confidence: 99%

The supercritical regime in the normal matrix model with cubic potential

Kuijlaars

Tovbis

2015

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…However, such an equation can lead to the appearance of final time singularities. To prevent these singularities and to regularize the problem, a term containing the surface tension is usually introduced into the equation describing Laplacian growth (indeed, the other methods for regularization exist, for example, in the combustion problem, the effect of diffusion as a perturbation might be considered [9], in comressible fluid, a singular cusp-like finger triggers shockslines of discontinuity for pressure [35,36]). Unfortunately, in the presence of such a surface tension term, obtaining an analytical solution in the form of poles becomes impossible.…”

Section: Introductionmentioning

confidence: 99%

Laplacian Growth Without Surface Tension in Filtration Combustion: Analytical Pole Solution

Kupervasser

2015

Pole Solutions for Flame Front Propagation

View full text Add to dashboard Cite

Filtration combustion (FC) is described by Laplacian growth without surface tension. These equations have elegant analytical solutions that replace the complex integro-differential motion equations by simple differential equations of pole motion in a complex plane. The main problem with such a solution is the existence of finite time singularities. To prevent such singularities, nonzero surface tension is usually used. However, nonzero surface tension does not exist in FC, and this destroys the analytical solutions. However, a more elegant approach exists for solving the problem. First, we can introduce a small amount of pole noise to the system. Second, for regularization of the problem, we throw out all new poles that can produce a finite time singularity. It can be strictly proved that the asymptotic solution for such a system is a single finger. Moreover, the qualitative consideration demonstrates that a finger with 1 2 of the channel width is statistically stable. Therefore, all properties of such a solution are exactly the same as those of the solution with nonzero surface tension under numerical noise. The solution of the Saffman-Taylor problem without surface tension is similar to the solution for the equation of cellular flames in the case of the combustion of gas mixtures. V C 2014 Wiley Periodicals, Inc. Complexity 21: 31-42, 2016 FIGURE 1 E[perimental (a) and numerical (b) concentration distributions in displacement viscous fluid by a less viscous one from Hele-Shaw cell. Fluids are miscible, flow direction is from left to right ½33-35.

show abstract

“…Conversely, the Whitham equations can be interpreted as the evolution of Riemann surfaces ‡ In Refrs. [1,2] we described hydrodynamics of shocks initiated by the most generic cusp singularity, referred to as a (2,3)-cusp. In local Cartesian coordinates aligned with a cusp axis, the shape of a critical finger is y 2 ∼ x 3 , as in Fig.…”

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

“…Such solutions of ill-defined non-linear differential equations are known as weak solutions [9].In Refrs. [1,2] we developed the weak solution of Hele-Shaw problem, which we believe to be a solution of the problem of Hele-Shaw singularities. Within this solution, a singularity triggers a branching graph of shocks or cracks propagating through the viscous fluid.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Viscous shocks in Hele–Shaw flow and Stokes phenomena of the Painlevé I transcendent

Lee

Teodorescu

Wiegmann

2011

Physica D: Nonlinear Phenomena

Self Cite

View full text Add to dashboard Cite

In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers [1,2] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painlevé I equation. We argue that the Painlevé I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation shocks appear as Stokes level-lines of the Painlevé linear problem.Harmonic measure is the probability of a Brownian excursion starting at a marked point (in this case it is a drain) to exit the domain through that boundary element. The probability density of Brownian excursion is the Poisson excursion kernel, equal to the gradient |∇p| of a harmonic function p with a source at a marked point and Dirichlet boundary condition.A well-known stochastic realization of the Darcy law is DLA -diffusion-limited aggregation [4]. It is realized through Brownian excursions of particles with a small size emanating at a constant rate from a distant point, with absorbing boundary, such that the stopping-set cluster grows. The Darcy law emerges at vanishing particle size, → 0.Computer experiments with DLA [4] show that a boundary, initially featureless, very quickly develops into a branching graph with a width controlled by the size of one particle . This signals that the limit → 0 is impossible.Darcy's law (1) also indicates that Hele-Shaw flow tends to a microscale. Non-linear differential equation (1) is ill-defined. A smooth initial boundary first evolves into a fingering pattern, then fingers at a finite, critical time develop cusp singularities [5,6,7,8]. At that point the differential form of the Darcy law stops making sense. This phenomenon constitutes the major problem in the field.Unlike in fluid dynamics, DLA processes possess a dimensional parameter -the particle size . It plays the role of a minimal area, which regularizes singularities emerging in fluid dynamics.Comparing Hele-Shaw flows to DLA suggests that, after a singularity is reached, the flow should feature shocks -a graph of curved lines where pressure (and velocity) suffer discontinuities. Such solutions of ill-defined non-linear differential equations are known as weak solutions [9].In Refrs. [1,2] we developed the weak solution of Hele-Shaw problem, which we believe to be a solution of the problem of Hele-Shaw singularities. Within this solution, a singularity triggers a branching graph of shocks or cracks propagating through the viscous fluid. Shocks at small Reynolds number is a new phenomena, not yet being observed experimentally. We refer them as viscous shocks. An emerging pattern of viscous shocks is reminiscent DLA patterns.

show abstract

Weak solution for the Hele-Shaw problem: Viscous shocks and singularities

Cited by 6 publications

References 20 publications

The supercritical regime in the normal matrix model with cubic potential

The supercritical regime in the normal matrix model with cubic potential

Laplacian Growth Without Surface Tension in Filtration Combustion: Analytical Pole Solution

Viscous shocks in Hele–Shaw flow and Stokes phenomena of the Painlevé I transcendent

Contact Info

Product

Resources

About