GEF 2500 Problem set 2

GEF 2500 Problem set 2
Consider a straight cylinder, filled with a viscous and homogenous fluid with density ,
and rotating about the z-axis with constant angular velocity k . A relative axis system
  
xyz is defined, rotating with the cylinder. The unit vectors are (i , j , k ) .
1) After a long time, the velocity vrel relative to the coordinate system is zero
everywhere. Using that the fluid is viscous explain why this is the case.
2) Show that the equation of motion valid in the relative axis system is
 
 2 ( xi  yj )   p  gk
(Hint: Start with equation 1.1.4 and use the expressions for the Newtonian forces
you find in the compendium, section 1.1-1.5)
3) The measures of the cylinder are shown on the figure on the next page.
a) Assume that the air pressure just above the surface is zero everywhere. Use
the equation derived in exercise 2 to determine the shape of the surface
   ( x, y) . Note that we are not yet interested in the particular vertical
position of the surface. You should start with the vertical component of (1),
solve for the pressure using the boundary condition at the surface, and then
solve for  .
(Hint: Find the pressure as a function of z and  , then use the horizontal
components of (1) to find an expression for    ( x, y) )
b) Use conservation of mass to determine the vertical position of the surface.
4) The cylinder is now situated a distance R0 from the centre of a table rotating
about the axis z’ with angular velocity k . The cylinder is at rest with respect to
the rotating table (see figure).
a) Another term must be added to the equation of motion. Which?
b) Determine the shape of the surface    ( x, y).