A stream function (ψ) is given by ψ=2x-5y. Calculate the velocity components, and magnitude and direction of resultant velocity.
Stream Function Analysis
Problem Statement
Given the stream function:
ψ = 2x – 5y
Determine:
- Velocity components (u, v)
- Resultant velocity magnitude
- Flow direction
1. Velocity Components
u = -∂ψ/∂y = 5
v = ∂ψ/∂x = 2
u = 5 m/s →, v = 2 m/s ↑
2. Resultant Velocity
R = √(u² + v²) = √(5² + 2²)
R = √29 = 5.385 m/s
3. Flow Direction
θ = tan⁻¹(v/u) = tan⁻¹(2/5)
θ = 21.80° NE
Flow Characteristics
Key features of this flow field:
- Uniform flow field (constant velocity everywhere)
- Straight parallel streamlines
- Satisfies continuity equation (∂u/∂x + ∂v/∂y = 0)
- Irrotational flow (∇×V = 0)
- Represents ideal fluid flow with no viscosity
- Used as building block for complex potential flows
- Demonstrates fundamental stream function properties
