To address this problem, let’s break it down step-by-step.

Part 1: Verify that $\psi = x^2 - y^2 - 3x - 2y + 2xy$ represents the stream function of an incompressible fluid flow.

For a function $\psi(x, y)$ to represent the stream function of an incompressible flow, it must satisfy the Laplace equation: $\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} = 0.$

Let's calculate $\frac{\partial^2 \psi}{\partial x^2}$ and $\frac{\partial^2 \psi}{\partial y^2}$.

Given: $\psi = x^2 - y^2 - 3x - 2y + 2xy$

1. First partial derivatives:

   • $\frac{\partial \psi}{\partial x} = 2x - 3 + 2y$
   • $\frac{\partial \psi}{\partial y} = -2y - 2 + 2x$

2. Second partial derivatives:

   • $\frac{\partial^2 \psi}{\partial x^2} = 2$
   • $\frac{\partial^2 \psi}{\partial y^2} = -2$

Now, adding these together: $\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} = 2 - 2 = 0$ This satisfies the Laplace equation, so $\psi(x, y)$ can represent the stream function of an incompressible flow.

Part 2: Find the velocity potential $\phi$

For an incompressible flow, the velocity components $u$ and $v$ are given by: $u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}.$

From the derivatives computed above:

• $u = \frac{\partial \psi}{\partial y} = -2y - 2 + 2x$
• $v = -\frac{\partial \psi}{\partial x} = -(2x - 3 + 2y) = -2x + 3 - 2y$

Since $\phi$ is the velocity potential, it must satisfy the Cauchy-Riemann equations: $\frac{\partial \phi}{\partial x} = u \quad \text{and} \quad \frac{\partial \phi}{\partial y} = -v.$

1. Set $\frac{\partial \phi}{\partial x} = -2y - 2 + 2x$.
2. Set $\frac{\partial \phi}{\partial y} = 2x - 3 + 2y$.

By integrating these partial derivatives, we can determine $\phi(x, y)$.

Part 3: Determine the complex potential $f(z) = \phi + i\psi$

The complex potential $f(z)$ can then be formed as: $f(z) = \phi + i \psi.$

This provides a full description of the flow in terms of the complex potential.

Would you like further details on any of these steps, or assistance with integrating to find $\phi$ explicitly?

Related Questions:

1. How does the Laplace equation relate to incompressible fluid flow?
2. What are the implications of the stream function satisfying the Laplace equation?
3. How do we verify if a given function is a valid stream function?
4. What role do the Cauchy-Riemann equations play in fluid dynamics?
5. How does one find the complex potential from a given stream function?

Tip:

For problems involving fluid flow, always start by verifying if the given function satisfies necessary conditions, such as the Laplace equation for incompressibility.