Power-Law Slip Profile of the Moving Contact Line in Two-Phase Immiscible Flows

Large scale molecular dynamics (MD) simulations on two-phase immiscible flows show that associated with the moving contact line, there is a very large $1/x$ partial-slip region where $x$ denotes the distance from the contact line. This power-law partial-slip region is verified in large-scale adaptive continuum simulations based on a local, continuum hydrodynamic formulation, which has proved successful in reproducing MD results at the nanoscale. Both MD and continuum simulations indicate the existence of a universal slip profile in the Stokes-flow regime, well described by $v^{slip}(x)/V_w=1/(1+{x}/{al_s})$, where $v^{slip}$ is the slip velocity, $V_w$ the speed of moving wall, $l_s$ the slip length, and $a$ is a numerical constant. Implications for the contact-line dissipation are discussed.Comment: 13 pages, 3 figure

Quantum groupoids

We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid {\cal D}\otimes_{\hbar} U_{\hbar}(\frakg), which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.Comment: 48 pages, typos and minor mistakes corrected, references updadted. Comm. Math. Physics, (to appear