Which of the following is a fundamental property of the relationship between Equipotential lines and Streamlines in a 2D irrotational flow?
- AThey are always parallel to each other.
- BThey intersect at an angle of 45°.
- CThey are orthogonal (intersect at 90°) at every point.Correct
- DThey only exist in turbulent flow.
Solution
Concept
Orthogonality of equipotential lines and streamlines
Formula
(dy/dx)_φ × (dy/dx)_ψ = -1
Step-by-step solution
1. In 2D plane irrotational flow of an ideal fluid, velocity potential φand stream function ψsatisfy Laplace's equation. 2. The slope of an equipotential line (φ = c) is (dy/dx)_φ = -u / v. 3. The slope of a streamline (ψ = c) is (dy/dx)_ψ = v / u. 4. The product of their slopes is (-u/v) × (v/u) = -1, proving they intersect at right angles (orthogonal at 90°). Answer: C
Explanation
In 2D irrotational flow of an ideal fluid, lines of constant velocity potential (equipotential lines, φ = constant) and lines of constant stream function (streamlines, ψ = constant) form an orthogonal flow net. By Cauchy-Riemann conditions, the product of their slopes at any point of intersection equals -1, proving they are mutually orthogonal everywhere.
Why the other options are wrong
A) Equipotential lines and streamlines are mutually perpendicular, never parallel. B) The intersection angle is exactly 90°, not 45°. D) Flow nets formed by potential and stream lines apply to laminar irrotational flow, not turbulent flow.
Shortcut
Equipotential lines and streamlines intersect at 90° (orthogonal flow net, m1 × m2 = -1).
Common mistake
Believing streamlines and equipotential lines are parallel or intersect at 45°.
Correct answer: C) They are orthogonal (intersect at 90°) at every point.