MATH$0015$ Week $5$ Problem Sheet $2$

A stationary circular cylinder with boundary $x^2+y^2=a^2$ is in a two$-$dimensional irrotational flow field whose velocity has Cartesian components $(u,v)$ such that

$u-U(1+ky)0$ and $v-$Ukx$0,as,\frac{x^2+y^2}{a^2}$

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

Find a streamfunction for the flow.

For

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

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

Describe the form $of$ the flow when

Your streamline sketches should show the points $of$ intersection $of$ any streamlines terminating $on$ 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$\mathrm{.}0,$ show also that the velocity vanishes $at$ two points $on$ the cylinder and that these lie $iny>0.$

Carefully sketch representative streamlines for the flow when $0.$

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

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

Describe the form $of$ the flow when

Your streamline sketches should show the points $of$ intersection $of$ any streamlines terminating $on$ 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$\mathrm{.}0,$ show that the greatest value $of$ the fluid speed $on$ the cylinder $is2U(1+ka).$

For $0,$ show also that the velocity vanishes $at$ two points $on$ the cylinder and that these lie $iny>0.$

Carefully sketch representative streamlines for the flow when $0.$

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

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

Describe the form $of$ the flow when

Your streamline sketches should show the points $of$ intersection $of$ any streamlines terminating $on$ 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.