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  • 1
    Book
    Book
    Multiscale finite element methods /; 4 (2009)
    New York, NY :Springer,
    Details
    Title: Multiscale finite element methods /; 4
    Author: Efendiev, Yalchin
    Contributer: Hou, Thomas Y.
    Publisher: New York, NY :Springer,
    Year of publication: 2009
    Pages: XII, 234 S. : , Ill., graph. Darst.
    Series Statement: Surveys and Tutorials in the Applied Mathematical Sciences 4
    ISBN: 978-0-387-09495-3 , 978-0-387-09496-0
    Type of Medium: Book
    Language: English
    URL: http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1163.65080&format=complete
    URL: lizenzfrei
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  • 2
    Electronic Resource
    Electronic Resource
    Numerical study of Hele-Shaw flow with suction (1999)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 11 (1999), S. 2471-2486 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: We investigate numerically the effects of surface tension on the evolution of an initially circular blob of viscous fluid in a Hele-Shaw cell. The blob is surrounded by less viscous fluid and is drawn into an eccentric point sink. In the absence of surface tension, these flows are known to form cusp singularities in finite time. Our study focuses on identifying how these cusped flows are regularized by the presence of small surface tension, and what the limiting form of the regularization is as surface tension tends to zero. The two-phase Hele-Shaw flow, known as the Muskat problem, is considered. We find that, for nonzero surface tension, the motion continues beyond the zero-surface-tension cusp time, and generically breaks down only when the interface touches the sink. When the viscosity of the surrounding fluid is small or negligible, the interface develops a finger that bulges and later evolves into a wedge as it approaches the sink. A neck is formed at the top of the finger. Our computations reveal an asymptotic shape of the wedge in the limit as surface tension tends to zero. Moreover, we find evidence that, for a fixed time past the zero-surface-tension cusp time, the vanishing surface tension solution is singular at the finger neck. The zero-surface-tension cusp splits into two corner singularities in the limiting solution. Larger viscosity in the exterior fluid prevents the formation of the neck and leads to the development of thinner fingers. It is observed that the asymptotic wedge angle of the fingers decreases as the viscosity ratio is reduced, apparently towards the zero angle (cusp) of the zero-viscosity-ratio solution. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.870112
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  • 3
    Electronic Resource
    Electronic Resource
    Dynamic generation of capillary waves (1999)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 11 (1999), S. 1042-1050 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: We investigate the dynamic generation of capillary waves in two-dimensional, inviscid, and irrotational water waves with surface tension. It is well known that short capillary waves appear in the forward front of steep water waves. Although various experimental and analytical studies have contributed to the understanding of this physical phenomenon, the precise mechanism that generates the dynamic formation of capillary waves is still not well understood. Using a numerically stable and spectrally accurate boundary integral method, we perform a systematic study of the time evolution of breaking waves in the presence of surface tension. We find that the capillary waves originate near the crest in a neighborhood, where both the curvature and its derivative are maximum. For fixed but small surface tension, the maximum of curvature increases in time and the interface develops an oscillatory train of capillary waves in the forward front of the crest. Our numerical experiments also show that, as time increases, the interface tends to a possible formation of trapped bubbles through self-intersection. On the other hand, for a fixed time, as the surface tension coefficient τ is reduced, both the capillary wavelength and its amplitude decrease nonlinearly. The interface solutions approach the τ=0 profile. At the onset of the capillaries, the derivative of the convection is comparable to that of the gravity term in the dynamic boundary condition and the surface tension becomes appreciable with respect to these two terms. We find that, based on the τ=0 wave, it is possible to estimate a threshold value τ0 such that if τ≤τ0 then no capillary waves arise. On the other hand, for τ sufficiently large, breaking is inhibited and pure capillary motion is observed. The limiting behavior is very similar to that in the classical KdV equation. We also investigate the effect of viscosity on the generation of capillary waves. We find that the capillary waves still persist as long as the viscosity is not significantly greater than surface tension. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.869975
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