Stream and source: Rankine half-body

It is the superposition of a uniform stream of constant speed $U$ and a source of strength $m$.

2D rankine half-body:

$$\phi = Ux + \frac{m}{2\pi }\ln\sqrt {x^2 + y^2} \notag$$

3D Rankine half-body:

$$\phi = Ux - \frac{m}{4\pi \sqrt {x^2 + y^2 + z^2} } \notag$$

1.

Question: find the position $(x_{s},y_{s})$ of the flow field stagnation point for the 2D Rankine half-body.

(a)

Hint: at the stagnation point $\vec{v} = 0$.

(b)

Hint: obtain the expression for $\vec{v}$ and set it equal to $0$, and solve for $x$ and $y$.

2.

Question: evaluate the width $D$ of the 2D rankine half-body.

(a)

Hint: mass conservation

3.

Question: find the position $(x_{s},y_{s},z_{s})$ of the flow field stagnation point for the 3D Rankine half-body.

(a)

Hint: at the stagnation point $\vec{v} = 0$.

(b)

Hint: obtain the expression for $\vec{v}$ and set it equal to $0$, and solve for $x, y$ and $z$.

4.

Question: evaluate the cross-sectional area $A$ of the 3D rankine half-body.

(a)

Hint: mass conservation

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