Solve Complex Potential: Streamfunction from Velocity Potential

In summary, the conversation discusses finding the streamfunction of a complex potential F(z) with constants U and a, and z=x+iy. The method involves separating the real and imaginary parts of the equation and using the identity k*e^(i*theta) = k*(cos(theta) + i*sin(theta)). The streamfunction is then found to be k*sin(theta). The speaker expresses gratitude for the helpful response.
  • #1
Lucus
4
0
I need to find the streamfunction of the complex potential
F(z)=U(z^2 + 4a^2)^1/2, where U and a are constants and z=x+iy. I can't figure out how to separate the real and imaginary parts in order to isolate the streamfunction from the velocity potential. Thanks in advance for any help!
 
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  • #2
Just write the equation as

U*R^(1/2)*e^(i*theta/2)

Just write out z and you will find R and theta.

then use the identity k*e^(i*theta) = k*(cos(theta) + i*sin(theta))

that way you separate the potential = k*cos(theta)
and the streamfunction is k*sin(theta)

Sorry that I didn't use latex, it can get confusing
let me know if this helped you out or not
Jaap
 
  • #3
Thank you, Jaap de Vries, for responding to my question. Your answer is very helpful. Thanks again!
 

1. What is the relationship between the streamfunction and velocity potential in a complex potential flow?

The streamfunction and velocity potential are both mathematical functions that describe the flow of a fluid in a complex potential flow. The streamfunction, denoted by Ψ, is a scalar function that describes the flow in the direction perpendicular to the flow. The velocity potential, denoted by φ, is a scalar function that describes the flow in the direction of the flow.

2. How do you solve for the streamfunction from a given velocity potential in a complex potential flow?

The streamfunction can be solved for by taking the derivative of the velocity potential with respect to one of the coordinates, typically the one perpendicular to the flow. This derivative is then multiplied by -1 to get the streamfunction. In mathematical terms, Ψ = -∂φ/∂y or Ψ = ∂φ/∂x, depending on the direction of the flow.

3. Can the streamfunction and velocity potential be used to calculate other flow properties in a complex potential flow?

Yes, the streamfunction and velocity potential can be used to calculate other flow properties such as the velocity components, pressure, and vorticity in a complex potential flow. These calculations involve taking derivatives of the streamfunction and velocity potential and using them in various equations.

4. How are the boundary conditions applied when solving for the streamfunction from a given velocity potential in a complex potential flow?

The boundary conditions for the streamfunction and velocity potential are applied through the use of the Cauchy-Riemann equations, which relate the derivatives of these functions to each other. These equations are used to ensure that the streamfunction and velocity potential satisfy the boundary conditions of the flow.

5. What are some real-world applications of solving for the streamfunction and velocity potential in a complex potential flow?

Solving for the streamfunction and velocity potential in a complex potential flow has many real-world applications, including the design and analysis of airfoils, propellers, and other aerodynamic shapes. It is also widely used in the study of fluid mechanics and can help in understanding and predicting the behavior of fluids in various situations, such as in weather patterns and ocean currents.

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