Chap 12 Creeping Flow Dimensionless NavierStokes equation Dimensionless

Chap. 12: Creeping Flow Dimensionless Navier-Stokes equation • Dimensionless variables: Creeping flow approximation • Re << 1: Useful for polymer melt flows with high viscosity • 0 in full Navier-Stokes eq. • Nonlinear N-S eq. linear N-S. eq. Fluid Mechanics, KU, 2007

Flow between rotating disks • H << R • Cylindrical coordinates • Analytic solution in creeping flow limit Q) Velocity & Torque ? Fluid Mechanics, KU, 2007

• EOMs (Navier-Stokes) Creeping flow ! Fluid Mechanics, KU, 2007

• Boundary conditions: • Velocity field: (v r over the entire lower plate) • Stress: • Differential force: • Differential torque: • Torque: Parallel-disk viscometer Fluid Mechanics, KU, 2007

Fluid Mechanics, KU, 2007

Chap. 13: The Lubrication Approximation • Simplification that applies to flow btw. “nearly parallel” surfaces • Fundamental to polymer processing Intuitive development Slowly converging walls (half-angle: ) (Chap. 8, Plane-Poiseuille flow) Fluid Mechanics, KU, 2007

Fluid Mechanics, KU, 2007

Example) Simple coating flows Blade coating: Sufficiently small blade angle lubrication approximation Exponential Spreader Bar Lubrication approximation (N-S equations) h(x) U Varying combination of Poiseuille and Couette in channel, or film, of nonuniform height. Pressure rises in most of a converging section and passes through a maximum… h 0 x Fluid Mechanics, KU, 2007

Position at maximum pressure: Separation point: Fluid Mechanics, KU, 2007