Water is driven along a horizontal channel between two wide parallel fixed plates a…
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity
Fluids and Continuum Physics, Ch The Navier-Stokes equations and exact viscous solutions
Problem
Water is driven along a horizontal channel between two wide parallel fixed plates a distance h=2.0 mm apart. Take y measured from the mid-plane, so the plates are at y=±h/2. The pressure falls by 200 Pa for every metre along the channel, so the driving pressure gradient is −∂p/∂x=G=200 Pa m−1, uniform along the channel. The water is Newtonian and incompressible with μ=1.0×10−3 Pa s and ρ=1000 kg m−3. Assume the flow is steady, smooth, layered and unidirectional, u=(u(y),0,0), that the gap is much smaller than the plate width so edge effects may be neglected, and that the no-slip condition holds at both plates.
((a)) Reduce the Navier-Stokes equation for this flow, solve it with the no-slip conditions to obtain u(y), and evaluate the speed on the mid-plane.
((b)) Find the volume flow rate per unit width of channel.
((c)) The pressure gradient is switched on suddenly at t=0 on water at rest. Using the chapter's estimate δ≈νt for the thickness of the layer into which vorticity has diffused from a wall, estimate the time after which the vortical layers from the two walls meet at the mid-plane, that is the time scale on which the parabolic profile is established.