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MATH0015 Week 7 Written Assignment 2

A stationary circular cylinder with boundary z2 +y2  = α2 is in a two-dimensional irrotational flow field whose velocity has Cartesian components (k, u) such that

k _ U (1 + ay) → 0   and   u _ Uaz → 0       as    (z2 + y2 )/α2  → 3.

Here U and a are positive constants. The circulation around the cylinder is zero.

1. Find a streamfunction for the flow.

2. For 0  <  aα  <  1,  show that the greatest value of the fluid speed on the cylinder is 2U (1 + aα).

3. For 0 < aα < 1, show also that the velocity vanishes at two points on the cylinder and that these lie in y > 0.

4.  Carefully sketch representative streamlines for the flow when 0 < aα < 1.

5.  Obtain the location of any stagnation points for the flow when aα = 1. Carefully sketch the streamlines in this case.

6.  Obtain the location of any stagnation points for the flow when aα > 1.Carefully sketch the streamlines in this case.

7. Describe the form of the flow when aα < 0.

Your streamline sketches should show the points of intersection of any streamlines termi- nating on the cylinder, with the streamlines meeting the cylinder at the correct angle.  You should also find, and show accurately, the far-field asymptotes of all streamlines.