Asymmetric fluid-structure dynamics in nanoscale imprint lithography

2∂pzµ u = + C1z+ C2. [2.11]∂x2The boundary conditions due to the no slip condition is the speed of the surface,so on,and on = 0z = hu = U1z u = U2where U 1 and U 2 are the two surface speeds.Substituting these into equation 2.11 produces C2= µ U2and( U −U)1 2 ∂phC1= µ − . The velocity in the x direction at any point in z in theh ∂x2film is given byThe velocity gradient is2z( z − zh) + ( U1−U2) U2∂pu = + . [2.12]2 µ ∂xh∂u ∂p⎛ h ⎞ 1= ⎜ z − ⎟ + ( U1 −U2) . [2.13]∂zµ ∂x⎝ 2 ⎠ hThe integral ∫ h udz equals q x , the flow rate in the x direction per unit width of y.0Integrating equation 2.12 gives3 22∂p⎛ z z h ⎞zqx= ( U1U2) U2z2 x⎜ − + − +3 2⎟. [2.14]µ ∂ ⎝ ⎠ 2h0Putting in the limits and simplifying, the result is3h ∂phq x= − + ( U1+ U2) . [2.15]12 µ ∂x2Following the same procedure for y it is easily found that3h ∂pq y= − + +212 µ ∂y( V1V ) 2hh. [2.16]25