# Thin vortex filament with constant circulation; find velocity components

• jewels18080
In summary: Your Name]In summary, we are given a formula to calculate the velocity induced by an infinitesimal segment of a vortex filament at a point on the xy plane. We need to find the velocity components at the center of a square formed by the vortex filament. By visualizing the problem and considering an infinitesimal segment at a point on one side of the square, we can integrate the given formula over the side to obtain the velocity components in the x and y directions at the center of the square. This gives us the final solution of vx and vy.
jewels18080

## Homework Statement

In a xyz cartesian coordinate system, a thin vortex filament with constant circulation Gamma, forms a square (in the xy plane), with each side of the square having length L. You are told that an infinitesimal segment del (vector) of this filament induces an infinitesimal velocity dv (vector) at a point P on the xy plane, according to the given formula.

dv = Gamma/(4*pi)*(del x r)/(r^3)

where x is the cross product, r is the vector from the infinitesimal filament segment to point P. Calculate the velocity components at the center of the square.

## Homework Equations

dv = Gamma/(4*pi)*(del x r)/(r^3)

## The Attempt at a Solution

v = Gamma/(4*pi)*integral(sin(theta)/r^3)dl

From here, I have no idea how to solve.

Any help would be much appreciated. Thank you for your time.

Thank you for your post. I understand your struggle in solving complex equations. Let me guide you through the steps to calculate the velocity components at the center of the square.

Firstly, we need to visualize the problem in order to better understand it. The given formula represents the velocity induced by an infinitesimal segment of the vortex filament at a point P on the xy plane. This means that the velocity at the center of the square will be the sum of all the infinitesimal velocities induced by each segment of the square.

To make things easier, let us assume that the square lies in the xy plane, with one corner at the origin (0,0) and the opposite corner at (L,L). We can also assume that the circulation Gamma is constant throughout the square.

Now, let's consider an infinitesimal segment dl at a point (x,y) on one side of the square. The vector r from this segment to the center of the square can be written as r = (L/2-x)i + (L/2-y)j, where i and j are unit vectors in the x and y directions, respectively.

Substituting this into the given formula, we get dv = (Gamma/(4*pi))*(del x r)/(r^3) = (Gamma/(4*pi))*(L/2-x)i x (L/2-y)j/(L^3/8) = (Gamma/(4*pi*L^3))*[(L/2-x)(-j) + (L/2-y)i].

Now, we can integrate this expression over the entire side of the square, from x=0 to x=L. This will give us the velocity component in the x direction at the center of the square. Similarly, integrating from y=0 to y=L will give us the velocity component in the y direction at the center of the square.

Thus, the final solution will be:

vx = (Gamma/(4*pi*L^3))*integral(0 to L) [(L/2-x)(-j) + (L/2-y)i]dx

vy = (Gamma/(4*pi*L^3))*integral(0 to L) [(L/2-x)(-j) + (L/2-y)i]dy

I hope this helps you in solving the problem. Let me know if you have any further questions.

## 1. What is a thin vortex filament with constant circulation?

A thin vortex filament with constant circulation is a mathematical model used to describe the motion of a thin, swirling fluid such as a tornado or a vortex in a liquid. It assumes that the vortex is infinitely long and has a constant strength or circulation around its circumference.

## 2. How is the velocity of a thin vortex filament calculated?

The velocity of a thin vortex filament can be calculated using the Biot-Savart law, which is a fundamental equation in electromagnetism and fluid dynamics. This equation relates the velocity at a point in space to the strength and position of the vortex filament.

## 3. What are the components of the velocity in a thin vortex filament?

The velocity in a thin vortex filament has two components: tangential and radial. The tangential component is parallel to the filament and is responsible for the swirling motion, while the radial component is perpendicular to the filament and is responsible for the axial flow.

## 4. What is the relationship between the velocity and circulation in a thin vortex filament?

According to the Biot-Savart law, the velocity in a thin vortex filament is directly proportional to the circulation. This means that as the circulation increases, the velocity also increases, resulting in a stronger and faster vortex.

## 5. How is a thin vortex filament used in fluid mechanics?

A thin vortex filament is a useful tool in fluid mechanics for studying the behavior of vortex flows. It can be used to model various phenomena such as tornadoes, hurricanes, and vortex shedding in aerodynamics. It is also used in the analysis of boundary layers and wake flows in fluid dynamics.

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