Question

The complex position vectors of two parallel interacting equal fluid vortices moving with their axes of rotation always perpendicular to the \(z\)-plane are \(z_{1}\) and \(z_{2}\). The equations governing their motions are $$ \frac{d z_{1}^{*}}{d t}=-\frac{i}{z_{1}-z_{2}}, \quad \frac{d z_{2}^{*}}{d t}=-\frac{i}{z_{2}-z_{1}} $$ Deduce that (a) \(z_{1}+z_{2}\), (b) \(\left|z_{1}-z_{2}\right|\) and (c) \(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\) are all constant in time, and hence describe the motion geometrically.

Short answer

The quantities \(z_1 + z_2\), \(|z_1 - z_2|\), and \(|z_1|^2 + |z_2|^2\) are constant in time, indicating the vortices maintain their distance and their center moves linearly while they rotate around it.

Step-by-step solution

Differentiate the sum of the vortices

Let \( z_1 \) and \( z_2 \) be the complex position vectors of the vortices. First, find the time derivative of their sum: \( z_1 + z_2 \). The time derivative is: \[ \frac{d}{dt}(z_1 + z_2) = \frac{d z_1}{dt} + \frac{d z_2}{dt} \].

Utilize the given differential equations

Replace \( \frac{d z_1}{dt} \) and \( \frac{d z_2}{dt} \) using the conjugate and the given equations: \[ \left(\frac{d z_1}{dt}\right)^* = -\frac{i}{z_1 - z_2} \] \[ \left(\frac{d z_2}{dt}\right)^* = -\frac{i}{z_2 - z_1} \].

Identify that (a) is constant

Conjugating the sum and noting the imaginary terms cancel out: \[ \left(\frac{d z_1}{dt} + \frac{d z_2}{dt}\right)^* = 0 \] Thus, \( z_1 + z_2 \) is constant, so (a) is true.

Differentiate the distance between vortices

Next, find the time derivative of the distance \( |z_1 - z_2|^2 \): \[ \frac{d}{dt} \left| z_1 - z_2 \right|^2 = \frac{d}{dt} ((z_1 - z_2)(z_1 - z_2)^*) \].

Apply the product rule

Using the product rule: \[ = (z_1 - z_2) \left(\frac{d z_1}{dt} - \frac{d z_2}{dt}\right)^* + (z_1 - z_2)^* \left(\frac{d z_1}{dt} - \frac{d z_2}{dt}\right). \]

Simplify the derivative

Substitute from the differential equations: \[ = (z_1 - z_2) \left(0\right) + (z_1 - z_2)^* (0) = 0 \]. Hence, \( |z_1 - z_2|^2 \) is constant, proving (b).

Differentiate the sum of the magnitudes

Now, find the time derivative of \( |z_1|^2 + |z_2|^2 \). Note: \[ \frac{d}{dt} (|z_1|^2 + |z_2|^2) = \frac{d}{dt}(z_1 z_1^* + z_2 z_2^*). \]

Utilize the chain rule

Using the chain rule: \[ = z_1 \frac{d z_1^*}{dt} + z_1^* \frac{d z_1}{dt} + z_2 \frac{d z_2^*}{dt} + z_2^* \frac{d z_2}{dt}. \]

Substitute from the given differential equations

Substitute differential equations: \[ z_1 \left(-\frac{i}{z_1 - z_2}\right) + z_1^* \left(0\right) + z_2 \left(-\frac{i}{z_2 - z_1}\right) + z_2^* \left(0\right) = 0. \]

Conclude that the sum is constant

Since the derivative is zero, \( |z_1|^2 + |z_2|^2 \) is constant. This proves (c).

Describe the motion geometrically

Given the above constants, geometrically the motion means the vortices rotate around their center point maintaining distance, and their center of mass follows a straight-line path.

Key concepts

fluid vortices

Fluid vortices are regions within a fluid where the flow revolves around an axis line, which can be straight or curved. In the context of this exercise, we are dealing with two such vortices represented by complex position vectors, \( z_1 \) and \( z_2 \).

Vortices in fluids are crucial for understanding various physical phenomena, such as turbulence and rotational fluid dynamics. These vortices move according to specific rules governing their motion, often described by differential equations.
They play a significant role in aerodynamics, weather systems, and even in biological contexts like blood flow.

differential equations

Differential equations are mathematical equations that relate functions to their derivatives. They are fundamental in describing how physical quantities change.

In the given exercise, the motion of the fluid vortices is described by the following differential equations: $$ \frac{d z_{1}^{*}}{d t}=-\frac{i}{z_{1}-z_{2}}, \frac{d z_{2}^{*}}{d t}=-\frac{i}{z_{2}-z_{1}} $$
Here, the symbols \( \frac{d z_1}{dt} \) and \( \frac{d z_2}{dt} \) denote the time derivatives of \( z_1 \) and \( z_2 \), respectively. These equations describe how the positions of the vortices change over time based on their current positions.
Solving these differential equations helps us understand the behavior of the fluid vortices and predict their future positions.

complex analysis

Complex analysis is the field of mathematics that studies functions of complex numbers.
In this exercise, we represent the positions of two fluid vortices as complex numbers.

The complex position vectors \( z_1 \) and \( z_2 \) not only simplify the calculations but also provide a powerful way to interpret the motion of the vortices.
The use of complex numbers allows us to capture both magnitude and direction in a single expression, facilitating the analysis of their interactions.
For example, finding the sum \( z_1 + z_2 \) or the magnitude \( |z_1 - z_2| \) provides insights into the vortices' combined and relative motion.

conservation laws

Conservation laws in physics describe quantities that remain constant over time within a closed system.

In fluid dynamics, one often deals with conservation of mass, momentum, and energy.

In the context of this exercise, we demonstrate the conservation of specific quantities related to the two interacting fluid vortices:

• The sum of the position vectors, \( z_1 + z_2 \).
• The distance between the vortices, \( |z_1 - z_2| \).
• The sum of the squares of the magnitudes, \( |z_1|^2 + |z_2|^2 \).

These conserved quantities help us understand that the vortices move in such a way that their overall configuration remains stable over time. Geometrically, it implies that the vortices rotate around their combined center with a constant separation and aligned motion.