In this work we study computationally the dynamics of a liquid bridge formed between a two-dimensional trapezoidal cavity, which represents an axisymmetric cell or a plane groove engraved in a roll, and a moving plate. The flow is a model of the liquid transfer process in gravure printing systems. The considered plate kinematics represents the actual motion of a roll-to-roll system, which includes extension, shear and rotation relative to the cavity. The fluid flow is modelled by solving the Stokes equations, discretized with the finite element method; the evolving free surfaces are accommodated by employing a pseudosolid mesh deforming algorithm. The results show that as the roll radius is reduced, thus increasing the lateral and rotational motions of the top plate relative to the cavity, a larger volume of liquid is transferred to the plate. However, due to lateral displacement of the contact lines, special care must be taken concerning the wettability properties of the substrate to avoid errors in the pattern fidelity. The predictions also show a strong nonlinear behaviour of the liquid fraction extracted from a cavity as a function of the capillary number. At high capillary numbers the fluid dynamics is mainly controlled by the extensional motion due to the strong contact line pinning. However, at low values of the capillary number, the contact lines have higher mobility and the liquid fraction primarily depends on the lateral and rotational plate velocity. These mechanisms tend to drag the fluid outside the cavity and increase the liquid fraction transferred to the plate, as has been observed in experiments.
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