Question

For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. $$\mathbf{v}=\langle 0,-z, y\rangle$$

Short answer

Answer: The curl of the velocity field represents the local rotation or "vorticity" of the field. In this case, the curl has a constant value of \(\langle 2, 0, 0 \rangle\), indicating that a particle placed in this field would experience a rotational motion around the x-axis with a constant intensity of rotation.

Step-by-step solution

Write down the three components of the given velocity field

The given velocity field is \(\mathbf{v} = \langle 0, -z, y \rangle\). The three components are: $$v_x = 0$$ $$v_y = -z$$ $$v_z = y$$

Compute the curl of the velocity field

The curl of a vector field \(\mathbf{v} = \langle v_x, v_y, v_z \rangle\) is defined as: $$\nabla \times \mathbf{v} = \langle \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z}, \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x}, \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \rangle$$ We will now compute the curl of our given field: $$\frac{\partial v_z}{\partial y} = \frac{\partial y}{\partial y} = 1$$ $$\frac{\partial v_y}{\partial z} = \frac{\partial (-z)}{\partial z} = -1$$ $$\frac{\partial v_x}{\partial z} = \frac{\partial 0}{\partial z} = 0$$ $$\frac{\partial v_z}{\partial x} = \frac{\partial y}{\partial x} = 0$$ $$\frac{\partial v_y}{\partial x} = \frac{\partial (-z)}{\partial x} = 0$$ $$\frac{\partial v_x}{\partial y} = \frac{\partial 0}{\partial y} = 0$$ Therefore, the curl of the velocity field is $$\nabla \times \mathbf{v} = \langle 1 - (-1), 0 - 0, 0 - 0 \rangle = \langle 2, 0, 0 \rangle$$

Sketch the curl of the velocity field

To sketch the curl, we can view it as a vector field with constant vector \(\langle 2, 0, 0 \rangle\). This means that for any point in space (x, y, z), the value of the curl is a vector pointing in the positive x-direction with magnitude 2. You can visualize this as a field of parallel arrows, all pointing along the x-axis and having the same length.

Interpret the curl

The curl of a vector field represents the local rotation or "vorticity" of the field. In our case, the curl is a constant vector pointing in the positive x-direction with magnitude 2. This means that if a small particle were placed in this velocity field, it would experience a rotational motion around the x-axis due to the local rotation caused by the field. Additionally, the magnitude of the curl indicates the intensity of the rotation. Since the curl is constant, this indicates that the rotation is uniform throughout the entire field.

Key concepts

Vector Calculus

Vector calculus is a vital branch of mathematics focused on vector fields and the operations that can be performed on them. It's an extension of basic calculus into higher dimensions, where each point in space has a vector value rather than just a single scalar value.
One of the key operations in vector calculus is the "curl," which measures the rotation of a vector field. The curl is particularly important in physics and engineering, especially when dealing with fluid dynamics and electromagnetism.
**Curl in Vector Fields** The curl of a vector field \( \mathbf{v} = \langle v_x, v_y, v_z \rangle \) is given by the expression:\[abla \times \mathbf{v} = \left\langle \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z}, \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x}, \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \right\rangle. \]This operation results in another vector field, which indicates how the original field "twists" around any given point. For the velocity field given in the problem, the curl was computed to be \( abla \times \mathbf{v} = \langle 2, 0, 0 \rangle \), signifying a constant twist around the x-axis.

Vorticity

Vorticity is a concept tied closely to the curl of a vector field, particularly in the context of fluid flow. It represents the local spinning motion of a fluid, akin to how a tornado rotates about its center.
In the provided exercise, the curl of the vector field \( \mathbf{v} = \langle 0, -z, y \rangle \) was calculated as \( \langle 2, 0, 0 \rangle \). This vector indicates the direction and intensity of vorticity. Since it's a constant vector pointing in the positive x-direction, any fluid element in this field experiences a uniform rotation about the x-axis.
**Understanding Vorticity**
- **Direction**: The direction of the curl vector indicates the axis of rotation. Here, the x-axis is the rotation axis.- **Magnitude**: The curl's magnitude symbolizes the strength of rotation. A magnitude of 2 suggests a steady turning rate around the x-axis.Such uniform vorticity can indicate the presence of a rotational flow or a mechanism causing consistent turning motion in the fluid.

Vector Field Sketching

Sketching a vector field, especially when focused on its curl and vorticity, enhances understanding by visualizing the direction and strength of vectors in space. The exercise's vector field had a constant curl of \( \langle 2, 0, 0 \rangle \), suggesting all vectors are aligned parallel to the x-axis and of equal length.
**Steps for Sketching the Curl**

• Identify the constant vector of the curl, here \( \langle 2, 0, 0 \rangle \).
• In any sketch, represent each vector of the curl in the field with an arrow originating along the x-axis.
• Ensure all arrows are parallel and of the same size, reflecting the constant nature of the curl.

This method of sketching reveals the behavior of the vector field at a glance. For this specific problem, the uniform vectors in the positive x-direction indicate a steady rotational influence throughout the field. Visualizing can provide a tangible sense of how a particle would behave when introduced into the vector field.