To analyze the two-dimensional fluid flow where the stream function is given by $\psi(x, y) = xy(x^2 + y^2)$, we need to find the corresponding velocity field and the complex potential.

1. Velocity Components:

In a two-dimensional incompressible flow, the velocity components $u$ (in the x-direction) and $v$ (in the y-direction) can be derived from the stream function $\psi(x, y)$ as follows:

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

Given $\psi(x, y) = xy(x^2 + y^2)$:

• Calculate $u$:

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

• Calculate $v$:

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

2. Complex Potential:

The complex potential $W(z)$ is a function of the complex variable $z = x + iy$, and it can be expressed as:

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

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

To find $\phi(x, y)$, we use the fact that the complex velocity $w(z)$ (which is the derivative of the complex potential) is given by:

$w(z) = u - iv$

From our earlier results:

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

To find $W(z)$, we integrate $w(z)$ with respect to $z$:

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

Integrating term by term can be complex, but we can approximate the potential $\phi(x, y)$ and then form the complete complex potential.

However, the steps are not straightforward without further assumptions or simplifications. Typically, we analyze specific boundary conditions or symmetry in the problem to simplify the integration. In the absence of additional data, a general form might be difficult to express without more detailed analysis.

Summary:

• The velocity components are: $u = x(x^2 + 3y^2), \quad v = -y(3x^2 + y^2)$
• The complex potential $W(z)$ would require integrating the complex velocity, which can involve more advanced techniques or assumptions.

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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 do you calculate the velocity potential $\phi(x, y)$ in fluid dynamics?
2. What are the physical interpretations of the stream function and potential function in fluid flows?
3. Can you explain how boundary conditions affect the determination of the complex potential?
4. What are the implications of having a non-linear stream function like $xy(x^2 + y^2)$?
5. How does the complex potential relate to the flow patterns in a fluid?

Tip: Understanding how to derive velocity components from a stream function is crucial in fluid dynamics, especially when dealing with potential flow theory.