Question

The velocity potential function \((\varphi)\) given below represents which one of the following? \[ \varphi = 5x - 12y \]

• A. Doublet
• B. Irrotational vortex
• C. Source
• D. Uniform flow

Hint:

Differentiate \(\varphi\) with respect to \(x\) and \(y\) to get the velocity components, then see if they are constant.

Answer 1

The Correct Option is D

Step 1: Find the velocity components from the potential function.
For 2D potential flow, velocity components come from the velocity potential as
\[ u = \frac{\partial \varphi}{\partial x}, \qquad v = \frac{\partial \varphi}{\partial y} \]
Given \(\varphi = 5x - 12y\),
\[ u = \frac{\partial}{\partial x}(5x-12y) = 5, \qquad v = \frac{\partial}{\partial y}(5x-12y) = -12 \]

Step 2: Interpret the result.
Both \(u\) and \(v\) come out as plain constants, they do not depend on \(x\) or \(y\) at all.
This means the velocity vector \((u,v) = (5,-12)\) is exactly the same at every point in the flow field, in both magnitude and direction.
A flow field with a constant velocity vector everywhere is, by definition, a uniform flow, with speed
\[ V = \sqrt{5^2+12^2} = \sqrt{25+144} = \sqrt{169} = 13 \]
directed at an angle \(\theta = \tan^{-1}(-12/5)\) below the x-axis.

Step 3: Why the other options are wrong.
A source has \(\varphi = \frac{\Lambda}{2\pi}\ln r\), which depends on the distance \(r\) from the origin, not a linear function of \(x\) and \(y\).
An irrotational (line) vortex has \(\varphi = \frac{\Gamma}{2\pi}\theta\), depending on the angle \(\theta\), again not linear in \(x,y\).
A doublet has \(\varphi = \frac{\kappa \cos\theta}{2\pi r}\), which decays with distance and is singular at the origin, unlike the given \(\varphi\), which is smooth and defined everywhere.

Final Answer:
Since \(\varphi\) is a simple linear function of \(x\) and \(y\), giving constant velocity components everywhere, it represents a uniform flow. \[ \boxed{\text{Option (D)}} \]