%0 Unpublished Work %1 citeulike:5872293 %A Sprittles, James E. %A Shikhmurzaev, Yulii D. %D 2009 %K 76d05-incompressible-navier-stokes-equations %T Viscous flows in corner regions: Singularities and hidden eigensolutions %U http://arxiv.org/abs/0906.3462 %X Numerical issues arising in computations of viscous flows in corners formedby a liquid-fluid free surface and a solid boundary are considered. It is shownthat on the solid a Dirichlet boundary condition, which removes multivaluednessof velocity in the `moving contact-line problem' and gives rise to alogarithmic singularity of pressure, requires a certain modification of thestandard finite-element method. This modification appears to be insufficientabove a certain critical value of the corner angle where the numerical solutionbecomes mesh-dependent. As shown, this is due to an eigensolution, which existsfor all angles and becomes dominant for the supercritical ones. A method ofincorporating the eigensolution into the numerical method is described thatmakes numerical results mesh-independent again. Some implications of theunavoidable finiteness of the mesh size in practical applications of the finiteelement method in the context of the present problem are discussed.

@unpublished{citeulike:5872293, abstract = {{Numerical issues arising in computations of viscous flows in corners formedby a liquid-fluid free surface and a solid boundary are considered. It is shownthat on the solid a Dirichlet boundary condition, which removes multivaluednessof velocity in the `moving contact-line problem' and gives rise to alogarithmic singularity of pressure, requires a certain modification of thestandard finite-element method. This modification appears to be insufficientabove a certain critical value of the corner angle where the numerical solutionbecomes mesh-dependent. As shown, this is due to an eigensolution, which existsfor all angles and becomes dominant for the supercritical ones. A method ofincorporating the eigensolution into the numerical method is described thatmakes numerical results mesh-independent again. Some implications of theunavoidable finiteness of the mesh size in practical applications of the finiteelement method in the context of the present problem are discussed.}}, added-at = {2017-06-29T07:13:07.000+0200}, archiveprefix = {arXiv}, author = {Sprittles, James E. and Shikhmurzaev, Yulii D.}, biburl = {https://www.bibsonomy.org/bibtex/2dfc22e3bf2feb5dbda0872e23d12185f/gdmcbain}, citeulike-article-id = {5872293}, citeulike-attachment-1 = {sprittles_09_viscous_33757.pdf; /pdf/user/gdmcbain/article/5872293/33757/sprittles_09_viscous_33757.pdf; 0a7b482b9ed9490d46b385d86b58831875fa1546}, citeulike-linkout-0 = {http://arxiv.org/abs/0906.3462}, citeulike-linkout-1 = {http://arxiv.org/pdf/0906.3462}, comment = {(private-note)advertised by SGM 2009-10-02}, day = 18, eprint = {0906.3462}, file = {sprittles_09_viscous_33757.pdf}, howpublished = {arXiv:0906.3462}, interhash = {59776332dabf7336ed45a8999d944188}, intrahash = {dfc22e3bf2feb5dbda0872e23d12185f}, keywords = {76d05-incompressible-navier-stokes-equations}, month = jun, posted-at = {2009-10-02 02:39:37}, priority = {2}, timestamp = {2017-06-29T07:13:07.000+0200}, title = {{Viscous flows in corner regions: Singularities and hidden eigensolutions}}, url = {http://arxiv.org/abs/0906.3462}, year = 2009 }