Problem 14 - 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. \(G(z)=e^{z}\)

Step-by-Step Solution

Answer

The stream function is \(\psi(x, y) = e^{x} \sin y\). The velocity field is \(\mathbf{V}(x, y) = \langle e^{x} \cos y, -e^{x} \sin y \rangle\). The boundary \(y = 0\) is a streamline.

Step 1: Express the Complex Potential

The given complex potential is \(G(z) = e^{z}\). Express it in terms of real and imaginary components: \(z = x + iy\),\[G(z) = e^{x+iy} = e^{x}(\cos y + i \sin y) = \phi(x, y) + i \psi(x, y)\]. The real part \(\phi(x, y) = e^{x} \cos y\) is the velocity potential, and the imaginary part \(\psi(x, y) = e^{x} \sin y\) is the stream function.

Step 2: Verify Boundary as Streamline

For the boundary to be a streamline, \(\psi(x, y)\) must be constant on the boundary. Consider the boundary condition \(y = 0\), then \(\psi(x, 0) = e^{x} \sin 0 = 0\). The boundary \(y = 0\) is a streamline since \(\psi(x, y)\) is constant along it.

Step 3: Determine the Velocity Vector Field

The velocity components are derived from the partial derivatives of the complex potential. Compute the partial derivatives: \(u = \frac{\partial \phi}{\partial x} = e^{x} \cos y\) and \(v = -\frac{\partial \phi}{\partial y} = -e^{x} \sin y\). Thus, the velocity vector field \(\mathbf{V}(x, y)\) is \(\mathbf{V}(x, y) = \langle e^{x} \cos y, -e^{x} \sin y \rangle\).

Step 4: Sketch the Streamlines

Use the stream function \(\psi(x, y) = e^{x} \sin y\) to sketch the streamlines. For several constant values \(k\), set \(e^{x} \sin y = k\) and plot \(y = \arcsin \left(\frac{k}{e^{x}}\right)\). For example, for \(k = 0.5\) and \(k = 1\), plot these curves to represent different streamlines. Use a graphing utility to visualize them.

Key Concepts

Stream Function

In fluid dynamics, the stream function is a mathematical tool that helps describe fluid flow. It provides valuable insight into the motion of fluid elements within a flow field. For an incompressible, two-dimensional flow, the stream function \(\psi(x, y)\) can be defined such that the velocity components of the fluid are related to the derivatives of \(\psi(x, y)\). This means:

• The velocity component in the x-direction, \(u\), is equal to the partial derivative of the stream function with respect to \(y\): \( u = \frac{\partial \psi}{\partial y} \).
• The velocity component in the y-direction, \(v\), is the negative partial derivative of the stream function with respect to \(x\): \( v = -\frac{\partial \psi}{\partial x} \).

The stream function, then, is directly linked to the behavior of streamlines, which are paths traced by fluid particles. By knowing the stream function, one can calculate and sketch the flow field's streamlines. In our exercise, \(\psi(x, y) = e^{x} \sin y\) is the stream function. By setting this function equal to a constant value, we can study and visualize different streamlines, confirming that the boundary \(y = 0\) is a streamline itself because the stream function there remains constant.

Velocity Vector Field

The velocity vector field \(\mathbf{V}(x, y)\) gives a complete description of the velocity distribution within the flow field. It indicates both the magnitude and direction of the velocity at any point. In mathematical terms, the velocity vector field for a two-dimensional flow is represented by:\[ \mathbf{V}(x, y) = \langle u(x, y), v(x, y) \rangle \]Where \(u(x, y)\) and \(v(x, y)\) are the velocity components in the x and y directions, respectively.To find these components from a given complex velocity potential \(G(z)\), we use the derivatives:

• The x-component \(u\) is the partial derivative of the velocity potential \(\phi(x, y)\) with respect to \(x\): \(u = \frac{\partial \phi}{\partial x} = e^{x} \cos y\).
• The y-component \(v\) is the negative partial derivative of the velocity potential with respect to \(y\): \(v = -\frac{\partial \phi}{\partial y} = -e^{x} \sin y\).

As a result, the velocity vector field in our exercise is \(\mathbf{V}(x, y) = \langle e^{x} \cos y, -e^{x} \sin y \rangle \). This field describes the speed and direction of fluid movement across the region \(R\).

Streamlines

Streamlines are an important concept that helps to visualize the flow of fluid. They represent the path that a fluid element follows in a steady flow, showing the direction in which a fluid parcel travels over time. The key property of streamlines is that they are tangent to the velocity vector field at every point, which means they offer a snapshot of the instantaneous velocity of the flow.For a given stream function \(\psi(x, y)\), streamlines can be found by setting\[ \psi(x, y) = k \]where \(k\) is a constant. This condition implies that the value of the stream function does not change along a streamline, allowing us to plot them easily.For our specific example with \(\psi(x, y) = e^{x} \sin y\), streamlines can be drawn by solving \(y = \arcsin\left(\frac{k}{e^{x}}\right)\). Different values of \(k\) yield different streamlines. Streamlines thus map out the pattern of flow and enable us to visualize the circulation of fluid within the flow field. Using a graphing utility can further enhance the visualization, making it clearer how fluid particles move through the region.