The velocity potential (ϕ) is given by ϕ=x^2-y^2. Find the velocity components in x and y direction. Also show that ϕ represents a possible case of fluid flow.
Potential Flow Analysis
Problem Statement
For the velocity potential given by:
ϕ = x² – y²
Determine:
- Velocity components u and v
- Verify if ϕ represents a valid fluid flow
1. Velocity Components
u = -∂ϕ/∂x = -2x
v = -∂ϕ/∂y = 2y
u = -2x, v = 2y
2. Validity Check (Laplace Equation)
∇²ϕ = ∂²ϕ/∂x² + ∂²ϕ/∂y² = 0?
∂²ϕ/∂x² = 2
∂²ϕ/∂y² = -2
2 + (-2) = 0
Satisfies Laplace equation → Valid potential flow
Physical Significance
Key characteristics of this flow:
- Hyperbolic streamlines: x² – y² = constant
- Represents a 2D incompressible, irrotational flow
- Stagnation point at origin (0,0)
- Symmetrical about both coordinate axes
- Models flow near corner regions
- Basis for more complex potential flow solutions
- Demonstrates fundamental relationship between potential and velocity fields