Fluid Kinematics MCQ with answer and solution

Concept

The derivative of φ gives velocity in the same direction. The derivative of ψ gives velocity at 90° to that direction.

Formula

u = -∂φ/∂x, v = -∂φ/∂y ; u = -∂ψ/∂y, v = ∂ψ/∂x

Step-by-step solution

1. Velocity potential φ is defined so that V_s = -∂φ/∂s for any direction s. 2. So the negative derivative of φ in any direction gives the velocity in that same direction. Statement (I) is true. 3. Stream function ψ gives u = -∂ψ/∂y and v = ∂ψ/∂x. 4. Here the derivative with respect to y gives u (the x-velocity), and the derivative with respect to x gives v (the y-velocity). 5. So the derivative of ψ in a direction gives the velocity perpendicular (90°) to that direction, not along it. Statement (II) is false. 6. Answer is C.

Explanation

The negative derivative of the velocity potential in any direction gives the velocity in that same direction. The derivative of the stream function in a direction gives the velocity component at right angles to that direction. So Statement (I) is true and Statement (II) is false.

Why the other options are wrong

A and B: Statement (II) is false, because ψ gives the velocity perpendicular to the chosen direction. D: Statement (I) is true by the definition of velocity potential.

Shortcut

φ: same direction. ψ: perpendicular direction.

Common mistake

Thinking that φ and ψ both give velocity in the direction of differentiation.