Recall that, in a flow problem, if the flow pattern is described by an analytic function w=ϕ+iψ
, then w
 is called the complex potential,  ϕ
  the velocity potential and ψ
 the stream function.

Now in a flow problem if the stream function is given by ψ=x2−y2
, what will be its velocity potential ?

[Hint: Guessing the complex potential may help, but one can do it otherwise]

Question 3Answer

a.
2xy


b.
2ixy


c.
−2xy


d.
−2ixy

Given that the stream function is provided as \( \psi = x^2 - y^2 \), we want to find the corresponding velocity potential \( \phi \), or the real part of the complex potential \( w = \phi + i\psi \).

### Steps to Find the Velocity Potential:

1. **Complex potential definition**: The complex potential \( w = \phi + i\psi \), where \( \phi \) is the velocity potential and \( \psi \) is the stream function.
   
2. **Cauchy-Riemann equations**: Since \( w = \phi + i\psi \) is analytic, it must satisfy the Cauchy-Riemann equations:
   \[
   \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}
   \]
   
3. **Given stream function**: \( \psi = x^2 - y^2 \)

4. **First Cauchy-Riemann equation**: 
   \[
   \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}
   \]
   Calculating \( \frac{\partial \psi}{\partial y} \):
   \[
   \frac{\partial \psi}{\partial y} = \frac{\partial}{\partial y}(x^2 - y^2) = -2y
   \]
   Therefore, 
   \[
   \frac{\partial \phi}{\partial x} = -2y
   \]

5. **Second Cauchy-Riemann equation**: 
   \[
   \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}
   \]
   Calculating \( \frac{\partial \psi}{\partial x} \):
   \[
   \frac{\partial \psi}{\partial x} = \frac{\partial}{\partial x}(x^2 - y^2) = 2x
   \]
   Therefore, 
   \[
   \frac{\partial \phi}{\partial y} = -2x
   \]

6. **Solving for \( \phi \)**: Integrating both partial derivatives, we obtain \( \phi \).
   - From \( \frac{\partial \phi}{\partial x} = -2y \), integrate with respect to \( x \):
     \[
     \phi(x, y) = -2xy + f(y)
     \]
   - From \( \frac{\partial \phi}{\partial y} = -2x \), integrate with respect to \( y \):
     \[
     \phi(x, y) = -2xy + g(x)
     \]

   Since the function of integration \( f(y) \) or \( g(x) \) must be a constant, the velocity potential is:
   \[
   \phi(x, y) = -2xy
   \]

Thus, the correct answer is:

### **Option (c): \( -2xy \)**

---

Would you like further details on any of the steps, or have any other questions?

Here are 5 related questions to explore further:
1. How are the Cauchy-Riemann equations used in fluid flow problems?
2. What is the physical significance of the velocity potential \( \phi \)?
3. How can one derive the streamlines from the stream function?
4. What conditions must a complex potential \( w \) satisfy to describe a flow?
5. Can the method of complex potential be applied to non-laminar flows?

**Tip:** In flow problems, the Cauchy-Riemann equations ensure that the velocity field satisfies continuity, implying an incompressible flow.

In a flow problem if the stream function is given by ψ = x^2 − y^2, what will be its velocity potential?

Solve for the velocity potential when the stream function is given as ψ = x^2 − y^2 using Cauchy-Riemann equations. This fluid dynamics problem explores complex potential theory, stream functions, and velocity potential for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation