Asymmetric fluid-structure dynamics in nanoscale imprint lithography

Appendix A: Axisymmetric ProblemThe Reynolds equation for the geometry of a flat, circular plateapproaching a flat surface is given as∂ ⎛ 3 ∂p⎞ 1 ∂ ⎛ 3 ∂p⎞ dh⎜rh⎟ + ⎜h⎟ = 12µr . [A.1]∂r⎝ ∂r⎠ r ∂θ⎝ ∂θ⎠ dtIf the plate is parallel as it approaches the surface, then for a drop fluidwith radius r b , axial symmetry exists and the pressure is only a function of theradius. Thus in the case of an axisymmetric squeeze film, the Reynolds equationis given asIntegrating givesDividing through,and sincedpdr≠ ∞when = 0and a further integration givesddr⎛⎜rh⎝3dp ⎞⎟ = 12µ rdr ⎠dhdt. [A.2]3 dp 2 dhrh = 6µ r + A . [A.3]dr dtdpdr6 r dh A= µ + , [A.4]33h dt rhr then A = 0 . Hence,dpdr6µ r dh= , [A.5]3h dt23µ r dhp = + B . [A.6]3h dt103