Scale-invariant regime in Rayleigh-Taylor bubble-front dynamics

The best matching procedure described in Chapter 4 is equivalent to the introduction of a principal fibre bundle in configuration space. Essentially one introduces a one-dimensional gauge connection on the time axis, which is a representation of the Euclidean group of rotations and translations (or, possibly, the similarity group which includes dilatations). To accommodate temporal relationalism, the variational principle needs to be invariant under reparametrizations. The simplest way to realize this in point–particle mechanics is to use Jacobi’s reformulation of Mapertuis’ principle. The chapter concludes with the relational reformulation of the Newtonian N-body problem (and its scale-invariant variant).

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Singularities

10.1093/oso/9780198805021.003.0012 ◽

2018 ◽

Author(s):

S. G. Rajeev

Keyword(s):

Stokes Equations ◽

Negative Answer ◽

Three Dimensions ◽

Picard Iteration ◽

Navier Stokes ◽

Scale Invariant ◽

Navier Stokes Equations ◽

L2 Norm ◽

Two Dimensional Flow ◽

Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

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Wetting front dynamics in an isotropic porous medium

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.16 ◽

2012 ◽

Vol 694 ◽

pp. 399-407 ◽

Cited By ~ 10

Author(s):

Yulii D. Shikhmurzaev ◽

James E. Sprittles

Keyword(s):

Porous Medium ◽

Incompressible Liquid ◽

Porous Matrix ◽

Viscous Incompressible Liquid ◽

Pore Scale ◽

Wetting Front ◽

Model Case ◽

New Approach ◽

Threshold Phenomena ◽

Front Dynamics

AbstractA new approach to the modelling of wetting fronts in porous media on the Darcy scale is developed, based on considering the types (modes) of motion the menisci go through on the pore scale. This approach is illustrated using a simple model case of imbibition of a viscous incompressible liquid into an isotropic porous matrix with two modes of motion for the menisci, the wetting mode and the threshold mode. The latter makes it necessary to introduce an essentially new technique of conjugate problems that allows one to link threshold phenomena on the pore scale with the motion on the Darcy scale. The developed approach (a) makes room for incorporating the actual physics of wetting on the pore scale, (b) brings in the physics associated with pore-scale thresholds, which determine when sections of the wetting front will be brought to a halt (pinned), and, importantly, (c) provides a regular framework for constructing models of increasing complexity.

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