Given the stream function $\psi(x, y) = xy(x^2 - y^2)$ for a two-dimensional fluid flow, we want to find the corresponding velocity components and the complex potential.

1. Velocity Components:

In a two-dimensional incompressible flow, the velocity components $u$ and $v$ are given by:

$u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}$

For $\psi(x, y) = xy(x^2 - y^2)$:

• Calculate $u$ (velocity in the x-direction):

$u = \frac{\partial \psi}{\partial y} = \frac{\partial}{\partial y} \left[ xy(x^2 - y^2) \right]$ $u = x \cdot \frac{\partial}{\partial y} \left[ y(x^2 - y^2) \right]$ $u = x \cdot \left[ (x^2 - y^2) - 2y^2 \right]$ $u = x(x^2 - 3y^2)$

• Calculate $v$ (velocity in the y-direction):

$v = -\frac{\partial \psi}{\partial x} = -\frac{\partial}{\partial x} \left[ xy(x^2 - y^2) \right]$ $v = -y \cdot \frac{\partial}{\partial x} \left[ x(x^2 - y^2) \right]$ $v = -y \cdot \left[ (x^2 - y^2) + 2x^2 \right]$ $v = -y(3x^2 - y^2)$

2. Complex Potential:

The complex potential $W(z)$ is given by:

$W(z) = \phi(x, y) + i\psi(x, y)$

where $\phi(x, y)$ is the velocity potential and $\psi(x, y)$ is the stream function.

To find $W(z)$, we start by finding the complex velocity $w(z) = u - iv$:

$w(z) = x(x^2 - 3y^2) + iy(3x^2 - y^2)$

Finding the Complex Potential $W(z)$:

To find $W(z)$, we integrate $w(z)$ with respect to $z = x + iy$:

$W(z) = \int w(z) \, dz$

This will give us:

$W(z) = \int \left[x(x^2 - 3y^2) + i y(3x^2 - y^2)\right] \, dz$

Given that $z = x + iy$, we can try to express the components in terms of $z$ and $\overline{z}$ (the complex conjugate). However, this process can be complex and may require specific techniques based on the form of $w(z)$.

Summary:

• The velocity components are: $u = x(x^2 - 3y^2), \quad v = -y(3x^2 - y^2)$
• The complex potential $W(z)$ can be found by integrating the complex velocity $w(z) = u - iv$.

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Would you like more detailed steps in calculating the complex potential, or do you have any questions?

Here are some related questions for deeper understanding:

1. How can we express the complex potential in terms of $z$ and $\overline{z}$?
2. What are the boundary conditions typically applied when solving for the complex potential?
3. Can we find streamline patterns using the stream function $\psi(x, y)$?
4. How does the non-linearity in the stream function affect the flow?
5. What are the physical implications of the derived velocity components?

Tip: In fluid dynamics, the stream function and velocity potential are powerful tools to visualize and analyze fluid flows, especially in the context of potential flow theory.