University of Warsaw Advanced Hydrodynamics
Faculty of Physics Selected Topics in Fluid Mechanics
Summer Semester 2019/20
Exercise Sheet 2
1. Spinning up a rotating lawn sprinkler. Use the global angular momentum balance to calculate the time evolution of angular velocity Ω(t) of a rotating lawn sprinkler after the water pressure is turned on. Consider a sprinkler constructed with a small number n of horizontal arms of length R mounted on a common pivot. An arm of a lawn sprinkler is shown in the figure below. Each arm is a tube carrying water from the pivot toward a nozzle with outlet area A. The water leaves the nozzle with constant vector velocity U along the normal to the nozzle exit. The moment of inertia I of the whole sprinkler plus water is known. Assume that friction in the pivot and air resistance on the arms are negligible.
2. Damping of sound waves. The motion of a viscous and compressible fluid is descri- bed by
ρ ∂u
∂t + u · ∇u
= −∇p + η∇2u +
ζ +η
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∇(∇ · u),
∂ρ
∂t + ∇ · (ρu) = 0,
where η is the shear viscosity and ζ is the volume viscosity.
(a) Assume a small perturbation of the equilibrium state (ρeq, peq, ueq = 0),
ρ(r, t) = ρeq+ ρ0(r, t), p(r, t) = peq+ p0(r, t), u(r, t) = ueq+ u0(r, t), and derive linear governing equations for the perturbation variables ρ0and u0. Use the relation
∇p = ∂p
∂ρ
s
∇ρ.
(b) Assume a solution in the form of a plane wave propagating in the z-direction, ρ0(r, t) = ˆρ exp[i(kz − ωt)], u0(r, t) = ˆu exp[i(kz − ωt)],
(c) Identify the transverse and longitudinal modes. Show that the transverse modes are diffusive and evaluate the diffusion coefficient.
(d) Derive the dispersion relation w(k) for the longitudinal mode and determine the amplitude attenuation of the sound wave.
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