Show that the following stream function represents an irrotational flow. ψ=6x-4y+7xy+9

Stream Function Analysis

Problem Statement

Show that the following stream function represents an irrotational flow:

ψ = 6x – 4y + 7xy + 9

1. Determine the Velocity Components

For a stream function ψ(x,y), the velocity components are defined as:

u = -∂ψ/∂y
v = ∂ψ/∂x

Given:
ψ = 6x – 4y + 7xy + 9

Step 1.1: Calculate u = -∂ψ/∂y
u = -∂(6x – 4y + 7xy + 9)/∂y
u = -(0 – 4 + 7x + 0)
u = -(-4 + 7x)
u = 4 – 7x

Step 1.2: Calculate v = ∂ψ/∂x
v = ∂(6x – 4y + 7xy + 9)/∂x
v = 6 + 7y

The velocity components are:
u = 4 – 7x
v = 6 + 7y

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 = ∂(6 + 7y)/∂x = 0

Step 2.2: Calculate ∂u/∂y
∂u/∂y = ∂(4 – 7x)/∂y = 0

Step 2.3: Calculate vorticity ωz
ωz = 1/2(∂v/∂x – ∂u/∂y)
= 1/2(0 – 0)
= 0

Since ωz = 0, the flow is irrotational.

3. Verify the Continuity Equation

Although not required to prove irrotationality, we can also verify the continuity equation:

∂u/∂x + ∂v/∂y = 0

Step 3.1: Calculate ∂u/∂x
∂u/∂x = ∂(4 – 7x)/∂x = -7

Step 3.2: Calculate ∂v/∂y
∂v/∂y = ∂(6 + 7y)/∂y = 7

Step 3.3: Verify the continuity equation
∂u/∂x + ∂v/∂y = -7 + 7 = 0

The continuity equation is also satisfied, confirming this is a physically possible fluid flow.

Conclusion

We have shown that the given stream function:

ψ = 6x – 4y + 7xy + 9

1. Yields velocity components:
u = 4 – 7x
v = 6 + 7y

2. Has zero vorticity (ωz = 0), confirming the flow is irrotational.

3. Satisfies the continuity equation (∂u/∂x + ∂v/∂y = 0), confirming it represents a physically possible fluid flow.

Physical Interpretation

This irrotational flow described by the stream function has the following characteristics:

The stream function ψ = 6x – 4y + 7xy + 9 defines streamlines in the flow where ψ = constant
Since the flow is irrotational, a velocity potential function φ also exists for this flow
The velocity field has a linear distribution with components u = 4 – 7x and v = 6 + 7y
Fluid elements in this flow undergo translation and deformation but do not rotate
This flow satisfies both irrotationality and continuity conditions, making it an important case in fluid mechanics
Bernoulli’s equation can be applied between any two points in this irrotational flow field