Problem 13 - Advanced Engineering Mathematics

Question

A complex velocity potential \(G(z)\) is defined on a region \(R\). (a) Find the stream function and verify that the boundary of \(R\) is a streamline. (b) Find the corresponding velocity vector field \(\mathbf{V}(x, y)\). (c) Use a graphing utility to sketch the streamlines of the flow. \(G(z)=z^{2}+1 / z^{2}\)

Step-by-Step Solution

Answer

Stream function is imaginary part of potential; velocity vector is derived from potential; streamlines plotted using graphing tool.

Step 1: Express the complex velocity potential with real and imaginary parts

The given complex velocity potential is \( G(z) = z^2 + \frac{1}{z^2} \). Let \( z = x + yi \). Using the identity \( z^2 = (x+yi)^2 = x^2 - y^2 + 2xyi \) and \( \frac{1}{z^2} = \frac{1}{(x+yi)^2} \), we need to express \( \frac{1}{(x+yi)^2} \) using its conjugate method to find the real (\( \phi \)) and imaginary (\( \psi \)) parts of \( G(z) \).

Step 2: Differentiate the potential to find the stream function

Differentiate \( G(z) = z^2 + \frac{1}{z^2} \) with respect to \( z \) to get the complex velocity \( V(z) = \frac{dG}{dz} = 2z - \frac{2}{z^3} \). The stream function \( \psi(x, y) \) is associated with the imaginary part of \( G(z) \), thus it's \( \psi(x, y) = 0 \) on the boundary implying the boundary of \( R \) is a streamline. Simplify \( V(z) \) by expressing \( z = x+yi \) to obtain the real-valued stream function.

Step 3: Express the velocity vector field \( \mathbf{V}(x, y) \)

Using \( V(z) = 2z - \frac{2}{z^3} \), substitute \( z = x + yi \) and separate the terms into real and imaginary parts to find \( \mathbf{V}(x, y) = (u(x, y), v(x, y)) \), where \( u(x, y) \) and \( v(x, y) \) represent the components of the velocity vector field.

Step 4: Graph the streamlines

Using a graphing utility, plot the streamlines of the potential flow. The lines where \( \psi(x, y) = C \) are the streamlines, where \( C \) is a constant. Selecting various constants will help generate a family of streamlines that represent the flow field.

Key Concepts

Stream Function

A stream function is a mathematical tool used to describe the flow of an ideal fluid. In two-dimensional flow, the stream function, typically denoted as \( \psi(x, y) \), helps to conveniently describe the flow pattern without directly referring to the velocity components. For any given point \((x, y)\), the value of \(\psi\) remains constant along streamlines, which are essentially the paths that particles in the fluid follow.

In the example you've just seen, the complex potential \(G(z)=z^2+1/z^2\) can be broken down into real and imaginary parts. The imaginary part corresponds to the stream function, \(\psi(x, y)\). The key information here is that at the boundary of the region \(R\), the stream function must be zero, implying that the boundary itself is a streamline.

• The stream function is associated with the imaginary part of a complex potential.
• It represents contours of constant stream function, \(\psi = C\).
• At a boundary streamline, \(\psi = 0\).

Using the stream function helps you see the flow pattern more clearly, as it removes the need to calculate the velocity at each point separately.

Velocity Vector Field

The velocity vector field is a comprehensive representation of fluid speed and direction at any point in space. For a flow in two dimensions, this field is expressed as \( \mathbf{V}(x, y) = (u(x, y), v(x, y)) \), where \( u \) and \( v \) are the velocity components in the \(x\) and \(y\) directions, respectively.

To find this field from the complex potential, you differentiate the potential \( G(z) \) with respect to \( z \) to obtain \( V(z) = 2z - \frac{2}{z^3} \). Upon substituting \( z = x + yi \), this derivation splits into real and imaginary parts. Each part represents one component of the velocity field:

• \( u(x, y) \) correlates with the real part, which indicates changes in the \(x\) direction.
• \( v(x, y) \) aligns with the imaginary part, representing variations in the \(y\) direction.

Evaluating these gives the full velocity vector at every point \((x, y)\), illustrating how each particle in the fluid moves over time.

Streamlines

Streamlines are an illustrative concept in fluid dynamics that show the trajectory that fluid particles will follow, which is vital for visualizing flow patterns. These lines are continuous and tangent to the velocity vector field at any point, meaning that they indicate the direction of a particle's instantaneous movement.

The critical characteristic of streamlines is that they correspond to lines of constant stream function, \( \psi(x, y) = C \). This makes them extremely useful since by setting different constants \(C\), you can generate a family of streamlines that define the flow field:

• Streamlines never intersect, as it would imply multiple velocities at one point.
• They offer a way to see the motion of fluids without tracking individual particles.
• Graphing utilities can assist in sketching these streamlines effectively based on the equation \( \psi(x, y) = C \).

Understanding streamlines can help predict how a flow behaves in complex systems, aiding in design and analysis of fluid systems.