Hydrodynamics and Elasticity: Class 9
Viscous flows II
1. Two-fluid motion Consider the flow of two immiscible incompressible viscous liquids with different densities and viscosities, moving between two parallel plates (see figure). The upper plate has a constant speed U = U ˆx. Determined the velocity field in the fluid and the distribution of the component σxy of the stress tensor in the regions 0 ≤ y ≤ h and h ≤ y ≤ 2h. Assume that the interface remains flat during the motion.
ρ 1 , µ 1 ρ 2 , µ 2
U
y x
h
h
Rysunek 1:
2. Stokes problem A half-space filled with a viscous fluid is bounded by a flat infinite surface. Find the evolution of the velocity field caused by a sudden uniform motion of the plane with the velocity
U(t) =
(0 for t < 0,
U ˆx for t ≥ 0. (1)
Hint Assume the flow u(y, t) to be self-similar, that is u = f (ζ), ζ = u
√νt, (2)
where ν = η/ρ is the kinematic viscosity.
Note The problem can also be solved using the Laplace transform.
