Problem 14 - A First Course in Fluid Dynamics

Question

A source and a sink of equal and opposite strengths \(\pm m\) lie at \(r=0, z=\mp a\) in a cylindrical system and a uniform stream \(U\) flows parallel to the \(z\)-axis at infinity. Write down the stream function \(\Psi\) and find the cquation for the dividing streamline. Find equations for the length and width of the Rankine body so formed and examinc the limit as \(a \rightarrow 0\) with \(m a\) held constant.

Step-by-Step Solution

Answer

The stream function is given by combining contributions from uniform flow and the source-sink pair. The Rankine body has length \(4a\) and width \(\frac{m}{2\pi U}\), which simplifies as \(a \to 0\).

Step 1: Identify Locations of Source and Sink

The problem states that there is a source at \(z = -a\) and a sink at \(z = a\). Both are located along the \(z\)-axis and have strengths \(\pm m\).

Step 2: Determine Stream Function for Source and Sink

For a point source, the stream function in cylindrical coordinates is given by \(\Psi_s = \frac{m}{4\pi} \ln(r^2 + (z + a)^2)\). For a sink, it is \(\Psi_{sink} = -\frac{m}{4\pi} \ln(r^2 + (z - a)^2)\).

Step 3: Write Stream Function for Uniform Flow

The uniform flow in the \(z\) direction has a stream function \(\Psi_U = Ur\).

Step 4: Combine Stream Functions

The total stream function \(\Psi\) is the sum of the individual parts: \[ \Psi = \Psi_U + \Psi_s + \Psi_{sink} = Ur + \frac{m}{4\pi} \ln(r^2 + (z + a)^2) - \frac{m}{4\pi} \ln(r^2 + (z - a)^2). \]

Step 5: Find Dividing Streamline Equation

The dividing streamline represents the separation between two flow regions and is defined by \(\Psi = 0\). So, set the expression for \(\Psi\) to zero and solve for the equation of the streamline. This yields: \[ Ur + \frac{m}{4\pi} \left( \ln(r^2 + (z + a)^2) - \ln(r^2 + (z - a)^2) \right) = 0. \]

Step 6: Simplify Dividing Streamline Equation

After simplifying the equation, we find: \[ Ur = \frac{m}{4\pi} \ln \left( \frac{r^2 + (z+a)^2}{r^2 + (z-a)^2} \right). \]

Step 7: Determine Length and Width of Rankine Body

By analyzing the equation, the length of the Rankine body is \(4a\) and the maximum width, perpendicular to the \(z\)-axis, is found by setting \(z=0\) and solving for \(r\). The resulting equation shows the width is \(\frac{m}{2\pi U}\).

Step 8: Examine Limit as \(a \rightarrow 0\)

As \(a\rightarrow 0\), the terms involving \(z\) in expressions simplify, and the length approaches zero while the width expression modifies to reflect a concentrated point source of strength \(ma\).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are an essential mathematical tool used in problems involving fluid dynamics.
They help simplify the complexity of three-dimensional problems by using a system based on

• Radius: The distance from a point to the axis of symmetry (often the z-axis).
• Angular Displacement: The rotation around the axis, denoted as theta (\( \theta \)).
• Height: The distance along the axis, usually represented by z.

By adopting these coordinates, a point can be expressed as \((r, \theta, z)\), where \(r\) is the radial distance, \(\theta\) is the angular position, and \(z\) is the axial height.
This set-up is particularly beneficial in fluid flow problems involving symmetry and axis-aligned phenomena, as they naturally align with the physical scenario.

Stream Function

The concept of a stream function plays a critical role in fluid dynamics, particularly in visualizing fluid flow in two-dimensional, incompressible, and irrotational fields.
Simply put, the stream function, \( \Psi \), enables us to describe the flow without directly dealing with velocity, allowing calculation of potential flow patterns.

• A helpful property of \( \Psi \) is that its lines, or contours, represent paths followed by particles in the flow.
• Due to this, locations where \( \Psi \) is constant indicate streamlines, which help deduce flow direction.

For point sources and sinks, the stream function can be expressed as \( \Psi = \frac{m}{4\pi} \ln(r^2 + (z \pm a)^2) \) in cylindrical coordinates.
Such expressions justly simplify visualizing complex flow patterns.

Rankine Body

A Rankine body is an interesting fluid dynamic concept referring to the shape formed around a source and sink in an otherwise uniform flow.
This results from the superposition of stream functions, creating a streamlined body (often ellipsoidal) that divides flow into different regions.

• In this problem, the structure determined by calculations is that the Rankine body is of length \(4a\).
• The maximum width finds its source when you solve for the radial component, given by \(\frac{m}{2\pi U}\).

The synthesis of source, sink, and a uniform flow creates a unique shape, where streamlines separate the inside 'fluid body' from the outside 'free stream'.
This concept aids in understanding the impact of local fluid sources and sinks in a flow field.

Uniform Flow

Uniform flow describes a scenario in fluid dynamics where the velocity at every point in the fluid remains constant both in magnitude and direction.
It simplifies complex fluid dynamics problems by establishing a consistent base flow, which can be further modified by additional elements such as sources, sinks, and obstacles.

• In the problem scenario, a uniform stream depicted by \( U \) flows parallel to the z-axis.
• Such flow becomes the backdrop against which the influence of additional components like the axial source and sink are analyzed.

The impact of a uniform flow is crucial as it enables evaluation of the resultant effect of other flow perturbations.
In this case, the uniform flow helps establish the primary flow direction and aids in determining the unique characteristics of the resultant streamlined body.