Problem 43 - Advanced Engineering Mathematics

Question

Find the streamlines of the flow associated with the given complex function. $$ f(z)=1 / \bar{z} $$

Step-by-Step Solution

Answer

Streamlines are hyperbolas given by \( y = \frac{k}{x} \).

Step 1: Identify complex components

The complex function is given as \( f(z) = \frac{1}{\overline{z}} \). Here, \( z = x + iy \) and \( \overline{z} = x - iy \). Substituting \( z \) in the function gives us: \[ f(z) = \frac{1}{x - iy} \].

Step 2: Compute modulus and imaginary part

The modulus squared of \( \overline{z} \) is \( x^2 + y^2 \), so \( \left(\frac{1}{\overline{z}}\right) = \frac{x}{x^2 + y^2} + i\left(-\frac{y}{x^2 + y^2}\right) \). This shows us the velocity components: \( u = \frac{x}{x^2 + y^2} \), \( v = -\frac{y}{x^2 + y^2} \).

Step 3: Determine streamlines equation

Streamlines satisfy \( u \frac{dx}{ds} + v \frac{dy}{ds} = 0 \), creating the ordinary differential equation \( \frac{dx}{dy} = \frac{u}{v} = -\frac{x}{y} \).

Step 4: Solve for streamlines

We solve the differential equation \[ \frac{dy}{dx} = -\frac{y}{x} \] by separation of variables: \( \frac{dy}{y} = -\frac{dx}{x} \). Integrating both sides results in \( \ln|y| = -\ln|x| + C \) or equivalently \( y = \frac{k}{x} \) where \( k = e^C \).

Key Concepts

Streamlines

Streamlines are imaginary lines within a fluid flow that represent the path followed by tiny particles as they move along. Every streamline is tangential to the velocity of the fluid at every point along its path.
In other words, if you were to release a tiny particle into the flow, its path would match the streamline exactly. Streamlines are crucial because they help visualize complex fluid flows.

• Streamlines never intersect. If they did, it would imply two different velocities at the same point, which is not possible.
• In steady flow, streamlines remain constant over time.

Understanding their pattern helps in predicting the behavior of the flow, essential in both natural phenomena and engineering applications.

Complex Functions

Complex functions, like the one given in the problem, consist of complex numbers, typically of the form \( z = x + iy \), where \( x \) and \( y \) are real numbers. Such functions are an essential concept in complex analysis, a branch of mathematics that studies functions of complex variables.
Complex analysis provides powerful methods and results, with applications ranging from physics to engineering.

• The function \( f(z) = 1 / \overline{z} \) involves the conjugate of a complex number. The conjugate \( \overline{z} \) of \( z = x + iy \) is \( x - iy \).
• Using the conjugate can reveal relationships between fluid flow characteristics, as it transforms complex arithmetic into analyzable components.

This specific function helps in modeling flow patterns due to its rotational symmetry and scaling properties.

Velocity Components

Velocity components are derived from the real (\( u \)) and imaginary (\( v \)) parts of a complex function, representing the flow's movement in the \( x \) and \( y \) directions, respectively. They are essential in understanding the flow dynamics in a 2D plane.
To find these components in the given problem, we decompose the function \( f(z) = \frac{1}{x - iy} \) into:

• \( u = \frac{x}{x^2 + y^2} \)
• \( v = -\frac{y}{x^2 + y^2} \)

These components indicate the respective influence of each axis on the flow's velocity.
Studying these components gives insight into how fluid particles accelerate or decelerate, and these parameters are vital for designing systems involving fluid transport or predicting weather patterns.

Differential Equations

Differential equations are mathematical equations that relate variables to their derivatives. In the context of streamlines, they are used to find equations describing the path of fluid flow.
For streamlines: the equation \( \frac{dx}{dy} = \frac{u}{v} \) arises from equating the dot product of velocity components and differentials to zero.
This transforms into the separable differential equation \( \frac{dy}{dx} = -\frac{y}{x} \), simple yet powerful in capturing the essence of the flow pattern via solution methods like separation of variables.

• Solving \( \frac{dy}{y} = -\frac{dx}{x} \) allows integration leading to \( \ln|y| = -\ln|x| + C \), simplifying to \( y = \frac{k}{x} \).

Such equations are vital in physics and engineering to model phenomena like thermal conduction, wave propagation, and fluid dynamics.