A shallow layer of oil drains from a tilted tray while a light wind holds its free…
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Kinematics, conservation laws and ideal flow
Fluid Dynamics, Ch Interfaces: capillary statics and the stress conditions at a surface
Problem
A shallow layer of oil drains from a tilted tray while a light wind holds its free surface at a small constant slope. The surface height above a fixed reference level is η(x,t)=η0−Rt+sx, where R=2.0×10−3ms−1 is the (uniform) rate at which the mean level falls, s=0.010 is a small, constant, dimensionless slope, and x is measured horizontally along the tray. At every point of the surface the horizontal velocity is uniform, u=0.30ms−1, and there is no motion in y. The surface is a material surface: no oil evaporates or is added.
((a)) Write the kinematic condition satisfied by the vertical velocity w at z=η(x,t).
((b)) Evaluate w at the surface. Say whether the oil at the surface is moving up or down, and whether the surface height at a fixed x is rising or falling.