Irrotational Flow Analysis
Problem Statement
The velocity components in a two-dimensional flow are:
v = -8xy² + 8/3x³
1. Verify the Continuity Equation
For a two-dimensional flow to be physically possible, it must satisfy the continuity equation:
∂u/∂x + ∂v/∂y = 0
Given:
u = 8x²y – 8/3y³
v = -8xy² + 8/3x³
Step 1.1: Calculate ∂u/∂x
∂u/∂x = ∂(8x²y – 8/3y³)/∂x = 16xy
Step 1.2: Calculate ∂v/∂y
∂v/∂y = ∂(-8xy² + 8/3x³)/∂y = -16xy
Step 1.3: Verify the continuity equation
∂u/∂x + ∂v/∂y = 16xy + (-16xy) = 0
2. Check for Irrotationality
A flow is irrotational if the vorticity (or rotation) is zero. In a two-dimensional flow, this means:
ωz = 1/2(∂v/∂x – ∂u/∂y) = 0
Step 2.1: Calculate ∂v/∂x
∂v/∂x = ∂(-8xy² + 8/3x³)/∂x = -8y² + 8x²
Step 2.2: Calculate ∂u/∂y
∂u/∂y = ∂(8x²y – 8/3y³)/∂y = 8x² – 8y²
Step 2.3: Calculate vorticity ωz
ωz = 1/2(∂v/∂x – ∂u/∂y)
= 1/2[(-8y² + 8x²) – (8x² – 8y²)]
= 1/2[-8y² + 8x² – 8x² + 8y²]
= 1/2[0]
= 0
Conclusion
We have shown that the given velocity field:
v = -8xy² + 8/3x³
2. Has zero vorticity (ωz = 0), confirming the flow is irrotational.
Physical Interpretation
This irrotational flow has the following characteristics:
- Since the flow is irrotational, a velocity potential function φ exists such that u = ∂φ/∂x and v = ∂φ/∂y
- The flow satisfies conservation of mass as indicated by the continuity equation
- Fluid elements in this flow translate and deform, but do not rotate
- This type of flow is important in many engineering applications, such as aerodynamics and hydrodynamics
- Due to its irrotational nature, Bernoulli’s equation can be applied between any two points in the flow field
