Imperial College London
Alternate

ProfessorDemetriosPapageorgiou

Faculty of Natural Sciences, Department of Mathematics

Chair in Applied Maths and Mathematical Physics
 
 
 
//

Contact

 

+44 (0)20 7594 8369d.papageorgiou Website

 
 
//

Location

 

750Huxley BuildingSouth Kensington Campus

//

Summary

 

Publications

Citation

BibTex format

@article{Papaefthymiou:2015:6/1607,
author = {Papaefthymiou, ES and Papageorgiou, DT},
doi = {6/1607},
journal = {Nonlinearity},
pages = {1607--1631},
title = {Vanishing viscosity limits of mixed hyperbolic–elliptic systems arising in multilayer channel flows},
url = {http://dx.doi.org/10.1088/0951-7715/28/6/1607},
volume = {28},
year = {2015}
}
Download

RIS format (EndNote, RefMan)

TY  - JOUR
AB  - This study considers the spatially periodic initial value problem of 2 × 2 quasilinearparabolic systems in one space dimension having quadratic polynomialflux functions. These systems arise physically in the interfacial dynamicsof viscous immiscible multilayer channel flows. The equations describethe spatiotemporal evolution of phase-separating interfaces with dissipationarising from surface tension (fourth-order) and/or stable stratification effects(second-order). A crucial mathematical aspect of these systems is the presenceof mixed hyperbolic–elliptic flux functions that provide the only source ofinstability. The study concentrates on scaled spatially 2π-periodic solutionsas the dissipation vanishes, and in particular the behaviour of such limitswhen generalized dissipation operators (spanning second to fourth-order) areconsidered. Extensive numerical computations and asymptotic analysis suggestthat the existence (or not) of bounded vanishing viscosity solutions dependscrucially on the structure of the flux function. In the absence of linearterms (i.e. homogeneous flux functions) the vanishing viscosity limit doesnot exist in the L∞-norm. On the other hand, if linear terms in the fluxfunction are present the computations strongly suggest that the solutions existand are bounded in the L∞-norm as the dissipation vanishes. It is foundthat the key mechanism that provides such boundedness centres on persistentspatiotemporal hyperbolic–elliptic transitions. Strikingly, as the dissipationdecreases, the flux function becomes almost everywhere hyperbolic except ona fractal set of elliptic regions, whose dimension depends on the order of theregularized operator. Furthermore, the spatial structures of the emerging weaksolutions are found to support an increasing number of discontinuities (measurevaluedsolutions) located in the vicinity of the fractally distributed ellipticregions. For the unscaled problem, such spatially oscillatory solution
AU  - Papaefthymiou,ES
AU  - Papageorgiou,DT
DO  - 6/1607
EP  - 1631
PY  - 2015///
SN  - 0951-7715
SP  - 1607
TI  - Vanishing viscosity limits of mixed hyperbolic–elliptic systems arising in multilayer channel flows
T2  - Nonlinearity
UR  - http://dx.doi.org/10.1088/0951-7715/28/6/1607
UR  - http://hdl.handle.net/10044/1/26512
VL  - 28
ER  -
Download