# Vector Fields

Vector Field of a Fluid
Fields: Algebra and Calculus
Image Created By: Direct Imaging
Website: [1]

Vector Field of a Fluid

The vector field shown here represents the velocity of a fluid. Each vector represents the fluid's velocity at the point the arrow begins.

# Basic Description

Figure 1: As a ball moves through the air, its velocity can be represented by an arrow; Click to enlarge
When a ball is thrown, the ball has an ever changing position and velocity as it moves through the air. We can show its velocity at a moment in time with a vector, which can be pictured as an arrow. The length of the arrow corresponds to how fast the ball is moving, and the direction the arrow points corresponds to the direction the ball is moving, as shown in Figure 1.

However, a fluid is more complicated, because different parts of the fluid move at different speeds and in different directions. So instead of using a single arrow, we must use a large number of arrows to get an idea of the fluid's motion, as in this page's main image. Each arrow represents the velocity of the fluid at the point the arrow begins. The direction of the vector shows which way the fluid is moving, and the vector's length shows how fast the fluid is moving. The collection of vectors is known as a vector field.

The vector field in the page's main image is calmer around the edges, as nearby vectors point in the same general direction, while in the center is more turbulent, as nearby vectors point towards each other.

Visit this site for an applet that demonstrates behavior of objects in vector fields. Use the "plot vector field" option and try drawing different objects in the fluid using the "flow" tab.

# A More Mathematical Explanation

Note: understanding of this explanation requires: *Linear Algebra

A vector field can be thought of as a function from $\mathbb{R}^k \rightarrow \mathbb{R}^k$ where [...]

A vector field can be thought of as a function from $\mathbb{R}^k \rightarrow \mathbb{R}^k$ where the domain consists of points in space and the range consists of resultant vectors from those points.

Figure 2: Rotational vector field: click to enlarge

For example, one such field would be $F(x,y) = (-y,x)$ which is a 'rotational field', graphed in Figure 2. At point (2,3) we find the vector (-3,2), at point (0,1), we find the vector (-1,0), and so on. An object placed in this fluid will naturally rotate about the origin.

Vector Fields are useful in analyzing fluids because different parts of the fluid have different velocities. As one can see in the main image, a fluid's velocity can be modeled with a vector field, with each vector representing the velocity of the fluid at that point.

Figure 3: Flow line through a vector field: click to enlarge

The vector field of a fluid can be used to find the path an object will ideally follow if placed in the fluid. At each point, the vector field shows us the direction that the fluid is moving, and an object in the fluid will naturally follow this motion. As the object moves through the fluid it will follow the vector it finds at each new point. The path the object traces out is known as a flow line, as shown in Figure 3.

Analytically a flow line is a parametrized path $\vec{x(t)}$ which satisfies the differential equation $F(\vec{x(t)}) = \frac{d(\vec{x(t)})}{dt}$. This equation means the object's velocity at any position is equal to the direction and magnitude of the field at that point.

Using the idea of flow lines, we can understand more about the fluid in this page's main image. The center portion of the fluid is more turbulent, as vectors point towards each other and overlap. An object in this part of the fluid would travel erratically as it is pushed back and forth. Meanwhile the parts of the fluid surrounding the center are calmer, and are directed more uniformly. An object at the very bottom of the image would flow rightward, following all of the rightward-pointing vectors.