Viscous flows in corner regions: Singularities and hidden eigensolutions

Abstract

Numerical issues arising in computations of viscous flows in corners formed by a liquid-fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the `moving contact-line problem' and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite-element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh-dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh-independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite element method in the context of the present problem are discussed.Comment: Submitted to the International Journal for Numerical Methods in Fluid

Viscous flow in domains with corners: Numerical artifacts, their origin and removal

“…One may expect, that the situation we consider in this paper, where the Navier-slip (i.e. Robin-type) condition is applied on the solid surface instead of a Dirichlet condition, would require merely an extension of the methods used in Sprittles and Shikhmurzaev [20]. Intriguingly, we will see that this conjecture is incorrect: the problem considered requires the development of an entirely new computational tool.…”

“…(10) the pressure is logarithmically singular as the corner is approached and, strictly speaking, one should look to incorporate special 'singular elements' there to capture this behaviour [5]. The platform we have developed has the option of incorporating these singular elements, and they are investigated in Sprittles and Shikhmurzaev [20]. In the results presented here, we have opted against using them in order to simplify our exposition.…”

“…The singular element method, used in Sprittles and Shikhmurzaev [20], falls into the aforementioned class of remedies. This method works when the computed solution closely approximates the asymptotic result in all but a small region near the corner where a singularity is present and hence a code with finite values assigned to every node fails to approximate the solution that tends to infinity.…”

“…This method works when the computed solution closely approximates the asymptotic result in all but a small region near the corner where a singularity is present and hence a code with finite values assigned to every node fails to approximate the solution that tends to infinity. Incorporating singular elements allows the pressure at the corner to be 'tied-up', using the asymptotic form of the solution, and in Sprittles and Shikhmurzaev [20] the method was generalized to account for the influence of eigensolutions with singular pressure behaviour. However, in the present work, the eigensolution derived in Section 3 is not singular and, contrary to what the asymptotics predicts, the code clearly tries to approximate a function that is not only singular but also multivalued at the corner.…”

“…In Sprittles and Shikhmurzaev [20], the standard finite element method was seen to produce such mesh-dependent oscillatory solutions for the case in which the velocity on the solid is an explicit function of distance from the contact line, as opposed to satisfying the Navier condition. We demonstrated that these numerical issues occurred because the code was attempting to approximate singularities that are inherent in the model and that they may be resolved by using the wellknown approach of incorporating specially designed elements adjacent to the corner which allow singular variables to be approximated.…”