Nonconvergence of the capillary stress functional for solutions of the convective Cahn-Hilliard equation. (English) Zbl 1371.35209
Shibata, Yoshihiro (ed.) et al., Mathematical fluid dynamics, present and future. Tokyo, Japan, November 11–14, 2014. Tokyo: Springer (ISBN 978-4-431-56455-3/hbk; 978-4-431-56457-7/ebook). Springer Proceedings in Mathematics & Statistics 183, 3-23 (2016).
Summary: We show that the surface tension term \(-\varepsilon\operatorname{div}(\nabla c^\varepsilon\otimes\nabla c^\varepsilon)\) of the “model H” does generally not converge to the mean curvature functional of the interface as \(\varepsilon\searrow 0\), where \(c^\varepsilon\) is the solution to a convective Cahn-Hilliard equation with mobility constant converging to 0 too fast as \(\varepsilon\searrow 0\). In that case the motion of the interface is dominated by the convection term \(v\cdot\nabla c^\varepsilon\) of the convective Cahn-Hilliard equation.
For the entire collection see [Zbl 1361.76002].
For the entire collection see [Zbl 1361.76002].
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35R35 | Free boundary problems for PDEs |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D45 | Capillarity (surface tension) for incompressible viscous fluids |
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