Question

In steady, inviscid, incompressible flow, superpose a uniform stream of speed \(U\) along \(+x\) (left \(\to\) right) with a source of strength \(\Lambda\) at the origin. Which statement about the stagnation point's location is NOT true?

• A. It is located to the left of the origin
• B. It moves closer to the origin for increasing \(\Lambda\), while \(U\) is held constant
• C. It moves closer to the origin for increasing \(U\), while \(\Lambda\) is held constant
• D. It is located along the \(x\)-axis

Hint:

For uniform flow \(+\) source, the stagnation point sits on the upstream side \((x < 0)\) at \(x_s = -\Lambda/(2\pi U)\). Check “closer/farther” questions directly from this formula.

Answer 1

The Correct Option is B

Step 1: Velocity on the \(x\)-axis for a source + uniform flow. For a source of strength \(\Lambda\) at the origin, the velocity magnitude is \(u_r = \dfrac{\Lambda}{2\pi r}\) radially outward. On the \(x\)-axis, the velocity is purely \(x\)-directed: \[ u_{\text{source}}(x) = \begin{cases} \dfrac{\Lambda}{2\pi x}, & x > 0 \quad (\text{points } +x) \\ -\dfrac{\Lambda}{2\pi |x|}, & x < 0 \quad (\text{points } -x) \end{cases} \] The uniform stream contributes \(+U\) (towards \(+x\)) everywhere. 

Step 2: Stagnation condition. A stagnation point must lie on the \(x\)-axis by symmetry (so \(v=0\)). On the \(\boldsymbol{x < 0}\) side, the two contributions can cancel: \[ U - \frac{\Lambda}{2\pi |x_s|} = 0 \quad \Rightarrow \quad |x_s| = \frac{\Lambda}{2\pi U}, \qquad x_s = -\frac{\Lambda}{2\pi U}. \] Thus the stagnation point is to the left of the origin, on the \(x\)-axis. 

Step 3: Parametric dependence. \(|x_s| = \dfrac{\Lambda}{2\pi U}\). Hence: Increasing \(\Lambda\) (with \(U\) fixed) \(\Rightarrow |x_s|\) increases (moves farther left), not closer. Increasing \(U\) (with \(\Lambda\) fixed) \(\Rightarrow |x_s|\) decreases (moves closer). Therefore statement (B) is not true. \[ \boxed{\text{(B) is NOT true}} \]