Problem 11 - 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. \(G(z)=\sin z\)

Step-by-Step Solution

Answer

The streamlines are \( \cos x \sinh y = C \), and the velocity field is \( \mathbf{V}(x, y) = (\cos x \cosh y, \sin x \sinh y) \).

Step 1: Express the Complex Potential

Given the complex potential function as \( G(z) = \sin z \), recall that the complex potential is generally of the form \( G(z) = \phi(x, y) + i \psi(x, y) \), where \( \phi \) is the velocity potential and \( \psi \) is the stream function.

Step 2: Identify Components of Complex Potential

For \( G(z) = \sin z = \sin(x + iy) \), use the identity \( \sin(a + b) = \sin a \cosh b + i \cos a \sinh b \). This gives us \( G(z) = \sin x \cosh y + i \cos x \sinh y \). Therefore, \( \phi(x, y) = \sin x \cosh y \) and \( \psi(x, y) = \cos x \sinh y \).

Step 3: Verify the Boundary of R as a Streamline

The boundary of \( R \) being a streamline implies \( \psi(x, y) = C \) on the boundary for some constant \( C \). This means that for points on the boundary, \( \cos x \sinh y = C \), verifying that the boundary satisfies the definition of a streamline.

Step 4: Compute the Velocity Vector Field

The velocity vector \( \mathbf{V} \) is given by \( \mathbf{V}(x, y) = abla G = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y} \right) \). Calculating these, \( \frac{\partial \phi}{\partial x} = \cos x \cosh y \) and \( \frac{\partial \phi}{\partial y} = \sin x \sinh y \). Thus, \( \mathbf{V}(x, y) = \left( \cos x \cosh y, \sin x \sinh y \right) \).

Step 5: Sketch the Streamlines

Streamlines of the flow are determined by the equation \( \psi(x, y) = \text{constant} \). This corresponds to \( \cos x \sinh y = C \). Using a graphing utility, sketch the curves for different constants \( C \). These curves show the paths followed by particles in the flow over time, illustrating how the stream function is constant along any streamline.

Key Concepts

Stream Function

The concept of a stream function is integral to understanding fluid dynamics. It provides a scalar field
such that its contours represent streamlines in fluid flow. For two-dimensional, incompressible flow,
the stream function is defined such that its partial derivative with respect to a spatial coordinategives the velocity component perpendicular to that coordinate.

• The stream function \( \psi(x, y) \) is often introduced with the complex potential function, where \( G(z) = \phi(x, y) + i \psi(x, y) \), \( \phi \) being the velocity potential and \( \psi \) the stream function.
• For the complex potential \( G(z) = \sin(x + iy) \), the stream function is given by \( \psi = \cos x \sinh y \).
• Streamlines are the path traced by particles in a fluid flow and are also represented by the stream function \( \psi(x, y) = \text{constant} \).

The stream function simplifies analyzing the flow patterns, signaling which direction particles move without tracing each particle individually. Additionally, because the stream function \( \psi \) is constant along a streamline, it directly simplifies the evaluation of fluid flow boundaries, as shown in exercise part (a).

Velocity Vector Field

The velocity vector field provides a way to visualize the speed and direction of fluid at every
point in the flow. It is a vector function \( \mathbf{V}(x, y) \), whose components are derived from the gradient of the velocity potential \( \phi \).

• The expression for the velocity vector field is derived from \( abla \phi \), where \( \phi(x, y) \) is the real part of the complex potential.
• In the given exercise, for \( G(z) = \sin(x + iy) \), the components of the velocity field are \( \frac{\partial \phi}{\partial x} \) and \( \frac{\partial \phi}{\partial y} \).
• These derivative calculations lead to the velocity vector \[ \mathbf{V}(x, y) = (\cos x \cosh y, \sin x \sinh y) \].

Understanding the velocity vector field is crucial because it conveys how fluid moves through space, and how fast it travels. It provides engineers and scientists with a powerful tool to predict dynamics of fluids in varying circumstances.

Streamlines

Streamlines offer an intuitive way to map out the flow of fluids in any medium. By definition,
they are lines that are always tangent to the velocity vector field.

• In a fluid flow represented by a velocity vector field \( \mathbf{V}(x, y) \), streamlines show paths taken by fluid particles.
• The equation representing streamlines is \( \psi(x, y) = \text{constant} \), resulting directly from the stream function.
• For example, in the exercise where \( G(z) = \sin(x + iy) \), the streamlines are described by the equation \( \cos x \sinh y = C \).

Visualizing streamlines helps identify the pattern of motion in a fluid system, which is fundamental in fields such as meteorology, oceanography, and aerodynamics. By examining the curvature and density of streamlines, insights into flow speed and direction are easily garnered.

Complex Potential Function

The complex potential function combines the velocity potential and the stream function into a single
complex function. It simplifies the study of potential flows, particularly in two-dimensional incompressible flows.

• The complex potential is expressed as \( G(z) = \phi(x, y) + i \psi(x, y) \), linking two key aspects of flow: velocity and direction.
• In the exercise example \( G(z) = \sin z \), it equates to \( \sin(x + iy) \), breaking down into \( \phi(x, y) = \sin x \cosh y \) (real part) and \( \psi(x, y) = \cos x \sinh y \) (imaginary part).
• This form allows for straightforward analysis and solution derivation for flow characteristics in regions where complex analysis is applicable.

The use of complex potentials allows the unification of flow variables into a single framework, providing streamlined computational methods and deeper insight into the behavior and stability of fluid flows. This capability is advantageous in many applications involving fluid dynamics.