Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Stokes wave wikipedia , lookup
Bernoulli's principle wikipedia , lookup
Airy wave theory wikipedia , lookup
Reynolds number wikipedia , lookup
Computational fluid dynamics wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Accretion disk wikipedia , lookup
Fluid thread breakup wikipedia , lookup
Magnetorotational instability wikipedia , lookup
10 Class exercise sheet Exercise 10.1: Rotating tube of fluid Consider a cylinder filled with a liquid, rotating with an angular frequency ω about it’s symmetry axis ẑ. We consider the steady state where the fluid is stationary in the rotating frame. The exercise will be done for a fluid element making the calculations simpler 1. ~r is the position vector of a fluid element, write it as a function of ρ and z(ρ). 2. Write the EOM for a fluid element, labeling the force per unit mass on the fluid element as f~. 3. Find the direction and size of f (i.e. n̂(ρ) and |f~|). Remember that the fluid is in the steady state. 4. Find d~r (just the differential of ~r) and explain why n̂ · d~r = 0. 5. Using the last item find the shape of the surface (i.e z(ρ)). 1 Exercise 10.2: Oscilator in a magnetic field (without the field...) A mass is fastened to a frame by springs with elastic constants κ1 and κ2 . The frame rotates in it’s own plane with an angular velocity Ω. The mass moves in the plane of the frame. Show that this problem is equivalent to a 2D harmonic oscillator in a magnetic potential. Hint: Choose the magnetic potential as A = 12 B(−y, x, 0) Exercise 10.3: Compton generator The Compton generator is a simple device to demonstrate the Coriolis force due to the Earth rotation. It consists of a narrow glass tube bend into a closed ring that is completely filled with water and some small particles to observe the motion of the water in the tube. Initially the ring is horizontal and the water is stationary. The ring is then quickly rotated by 180◦ around its East-West diameter and stopped. 1. Find the rotation of the earth ω ~ and position of a particle ~r in the following coordinate system. where W − E is the rotation axis. 2 2. Find the acceleration due to the Coriolis force. What is the tangential acceleration? 3. Average over the ring to find the mean acceleration. This is done because the fluid is incompressible. 4. Finally find the velocity of the water in the tube after the ring is rotated. You can assume that the ring was rotated at a constant angular velocity φ = 2π t, where τ is τ the period of revolution. 3