Problem 64 - Calculus: Early Transcendental Functions

Question

Sketch the vector field \(\mathbf{F}=\left\langle\frac{1}{1+x^{2}}, 0,0\right\rangle .\) If this represents the velocity field of a fluid and a paddle wheel is placed in the fluid at various points near the origin, explain why the paddle wheel would not start spinning. Compute \(\nabla \times \mathbf{F}\) and label the fluid flow as rotational or irrotational. How does this compare to the motion of the paddle wheel?

Step-by-Step Solution

Answer

The fluid's vector field is sketch based on the function \(\mathbf{F}=\left\langle\frac{1}{1+x^{2}}, 0,0\right\rangle\). Due to these vectors, a paddle wheel placed in this flow would not start spinning because there is no tangential component in the flow to impart rotation. The curl of this vector field calculated as \(\nabla \times \mathbf{F} = 0\), further indicating the irrotational nature of the flow, consistent with the observed behavior of the paddle wheel.

Step 1: Sketch vector field

The given vector field has all vectors in the x-direction whose magnitude depends on the x-coordinate and is independent of the y and z coordinates. To sketch this vector field, create a 3-dimensional graph with x-axis raging from -3 to 3 since \(1 / 1+x^2\) varies over this interval. Draw arrows representing the vectors given by the function at various points along x-axis.

Step 2: Analyze the motion of the paddle wheel

If the vector field represents the velocity of a fluid flow, then the motion of the paddle wheel is determined by the behaviour of the vectors in the vicinity of the wheel's position. Since all vectors in this case have zero y and z components, there would be no tangential motion to make the paddle wheel start spinning, regardless of where we place it.

Step 3: Calculate the curl of the vector field

The curl of a vector field \(\mathbf{F}\) is a vector that indicates the rotation of the field. It is calculated using the cross product with the gradient operator \(\nabla\), so \(\nabla \times \mathbf{F}\). Given that \(\mathbf{F}\) has components \(f_x = \frac{1}{1+x^{2}}, f_y = 0, f_z = 0\), the curl can be found by computing \(i j k \ \frac{d}{dx} \frac{d}{dy} \frac{d}{dz} \ f_x f_y f_z\), yielding \(\nabla \times \mathbf{F} = 0\)

Step 4: Labeling the fluid flow

Since the curl of the vector field equals to zero, this indicates that the flow is irrotational. As there is no rotation in the fluid flow, no spinning motion would be transferred to the paddle wheel.

Key Concepts

Curl of a Vector Field

When discussing vector fields, the concept of curl plays a critical role, especially in fluid dynamics. Imagine wading through water and swishing your hand in a circular motion. The swirling action you observe can be precisely described using curl. In mathematics, the curl of a vector field, denoted as \(abla \times \mathbf{F}\), is a vector that describes the infinitesimal rotation of the field. The formula to calculate the curl involves the cross product of the del operator \(abla\) and the vector field \(\mathbf{F}\). For example, consider the vector field \(\mathbf{F} = \left\langle \frac{1}{1+x^2}, 0, 0 \right\rangle\). Here, the curl computation involves taking partial derivatives and organizing them into a determinant: \[i j k \ \frac{d}{dx} \frac{d}{dy} \frac{d}{dz} \ f_x f_y f_z\] In this vector field, the resulting curl is zero, \(abla \times \mathbf{F} = 0\). This zero curl means there is no rotational component in the field, making it irrotational. This is why a paddle wheel placed within this field will not spin. The lack of rotational flow translates to a stable environment where rotational forces are absent.

Irrotational Flow

Irrotational flow describes a specific type of fluid movement where there is no rotation in any part of the flow. This absence of rotational movement implies that any paddle or rotor placed within such a flow wouldn't spin on its own due to the flow alone. Irrotational flow is characterized by having a curl of zero. As discussed with the vector field \(\mathbf{F} = \left\langle \frac{1}{1+x^2}, 0, 0 \right\rangle\), the curl \(abla \times \mathbf{F} = 0\) confirms that the flow is irrotational.

• Irrotational flows are commonly studied for their ability to simplify the analysis of fluid behavior.
• They occur naturally in situations where viscosity, or internal fluid friction, has minimal influence.

In practical terms, irrotational flow often denotes an idealized or theoretical scenario, but its principles are also foundational in understanding many real-world fluid dynamics applications. Thus, positing a paddle wheel in such flow will result in no spinning, as there's no curl to provide a rotational moment to get the wheel moving.

Fluid Dynamics

Fluid dynamics is the study of how fluids (liquids and gases) move and behave. It's a highly relevant field across different areas, from engineering to environmental science. In fluid dynamics, the understanding of vector fields forms the basis to analyze and predict fluid motion.When examining fluid with vector fields, like \(\mathbf{F} = \left\langle \frac{1}{1+x^2}, 0, 0 \right\rangle\), you delve into how every point in the fluid has a specific velocity vector that describes which direction and at what speed the fluid flows at that point:

• Velocity Field: Describes the speed and direction at every point.
• Streamlines: Lines that illustrate the path taken by fluid particles.

In this scenario, since there is no y or z component in the vector field, the flow is completely uniform in the x-direction.One significant aspect of fluid dynamics is understanding whether a flow is irrotational or rotational. This distinction helps in designing devices or predicting natural occurrences like weather patterns. In our exercise, the fluid flow is irrotational, confirming that no rotational energy would be conferred upon any object placed in it, such as a paddle wheel. This fundamental understanding aids in myriad applications like aviation and weather forecasting, bridging abstract mathematical concepts to tangible physical phenomena.