Geometric Interpretations of Quandle Homology

Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous cycles. The methods are applied to prove that boundary homomorphisms in a homology exact sequence vanish.Comment: 27 Figures 35 page

Film Rupture And Partial Wetting Over Flat Surfaces

Various technical applications require conditions which prevent thin liquid films from breaking into a series of rivulets, leaving the solid surface partly uncovered and lowering the extension of the liquid free interface. What is needed is both a criterion for the stability of the film to identify the minimum flow rate able to ensure the complete wetting of the surface and, after the film rupture, a method to estimate the wet (active) part of the same surface. At low Reynolds and high Weber numbers, the assumptions of a film with uniform thickness and complete wetting of the transfer surface cannot be considered, even approximately, rigorous, hence, leading to unacceptable inaccuracy of simulation results of the transfer performance in that operative region. Accordingly, the inadequacy of previous theoretical models of devices that use falling films as transfer mediums can be ascribed to a major issue, namely the assumption of complete wetting. As they provide a simple variational method to solve complex, multi-variable problems and directly reach a rational explanation of physical phenomena, extremum principles have led to critical results in the theoretical and technical fields. Hamilton’s principle (or principle of least action), Gauss’ principle of least constraint, as well as Oasager’s extremum principle or Prigogine’s principle of minimum entropy production remain central in modern physics and engineering. In this context and with regard to the previously stated technical aim, the principle of minimising the energy of a given stream-wise section of the film is applied in order to model and investigate the film stability. Specifically, a criterion for defining the film stability is established for a rivulet cross-section shape suitable for predicting the transient evolution of the wetting ability under an imposed fluid distribution width. The evolution from uniform film to the stable rivulet configuration is estimated considering the energy of the system under a Lagrangian approach. The Lagrange equation is written with reference to a single generalized wetting coordinate and its time derivative, under the effect of Rayleigh\u27s dissipation function and a generalized force associated to a scalar potential defined as the energy excess with respect to the local energy minimum. This methodology is extended to include the hysteresis behaviour of the contact angle (considering advancing and receding contact angles) and wettability hysteresis when increasing or decreasing mass flow rates are delivered. Finally, a first qualitative and quantitative validation of the results is presented with reference to the visual data captured on a dedicated experimental test section