Problem 40 - Thomas Calculus

Question

Bendixson's criterion The streamlines of a planar fluid flow are the smooth curves traced by the fluid's individual particles. The vectors \(\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}\) of the flow's velocity field are the tangent vectors of the streamlines. Show that if the flow takes place over a simply connected region \(R\) (no holes or missing points) and that if \(M_{x}+N_{y} \neq 0\) throughout \(R\) , then none of the streamlines in \(R\) is closed. In other words, no particle of fluid ever has a closed trajectory in \(R .\) The criterion \(M_{x}+N_{y} \neq 0\) is called Bendixson's criterion for the nonexistence of closed trajectories.

Step-by-Step Solution

Answer

If \( M_x+N_y \neq 0 \) throughout the region, Bendixson's criterion shows streamlines can't be closed.

Step 1: Understand the Flow Representation

In this exercise, we are given the velocity field of a fluid flow \( \mathbf{F} = M(x, y) \mathbf{i} + N(x, y) \mathbf{j} \). Streamlines are curves where the velocity vector is tangent at every point, meaning the direction of the flow is represented by the vectors \( \mathbf{F} \).

Step 2: Define Simply Connected Region

The flow is said to occur over a simply connected region \( R \), which means \( R \) has no holes or missing points. This is crucial because Bendixson's criterion specifically applies to such regions.

Step 3: Apply Bendixson's Criterion

Bendixson's criterion states that if \( M_x + N_y eq 0 \) everywhere in a simply connected region \( R \), then there are no closed trajectories or closed streamlines within that region. This means no particle will trace out a closed loop during its flow path.

Step 4: Verify Condition of Bendixson's Criterion

According to the problem, \( M_x + N_y eq 0 \) holds throughout the entire region \( R \). This condition directly implies that according to Bendixson's criterion, closed streamlines cannot exist.

Step 5: Conclude No Closed Streamlines

Since the condition \( M_x + N_y eq 0 \) is satisfied, Bendixson's criterion confirms that none of the fluid particles in region \( R \) can have a closed trajectory.

Key Concepts

Simply Connected Region

A simply connected region is an essential concept in the study of planar fluid flows. Imagine a flat piece of paper. If you draw a shape on this paper without any holes or gaps, that shape represents a simply connected region.

In more technical terms, a simply connected region is an area in the plane where any loop can be continuously contracted to a single point without leaving the region.

• No holes or missing points are present in the region.
• It allows you to apply certain mathematical theorems comfortably.

Understanding simply connected regions helps simplify complex mathematical problems, particularly in fluid dynamics and vector fields. In the context of Bendixson's criterion, such a region is vital as the criterion is only applicable to these uninterrupted spaces.

Velocity Field

The velocity field of a fluid flow is an integral component of understanding how particles move through a fluid. In mathematical terms, it is represented as a vector field, denoted by \( \mathbf{F} = M(x, y) \mathbf{i} + N(x, y) \mathbf{j} \).

This expression represents the velocity at any point \( (x, y) \) in the plane:

• \( M(x, y) \) refers to the component of the velocity in the \( x \)-direction.
• \( N(x, y) \) is the component of the velocity in the \( y \)-direction.

Each vector \( \mathbf{F} \) at a point describes the speed and direction at which a fluid particle moves. It's vital for modeling fluid dynamics as it provides an accurate depiction of how the fluid behaves at every instance, helping predict flow patterns and behaviors effectively.

Closed Trajectories

Closed trajectories refer to paths where a particle in a fluid returns to its starting point after a complete journey. These loops in a vector field can indicate steady or repetitive motion patterns within the fluid.

However, according to Bendixson's Criterion, if \( M_x + N_y eq 0 \) everywhere in a simply connected region, closed trajectories cannot occur. This is because:

• Any loop path indicates repetitive flow behavior.
• The continuous variation ensured by \( M_x + N_y eq 0 \) disrupts the loop formation.

Therefore, understanding and preventing closed trajectories is crucial in fluid dynamics to avoid stagnant or recurring behaviors, ensuring fluid moves efficiently without cyclical traps in systems or devices.

Streamlines

Streamlines are imaginary lines that flow along with the fluid, essentially mapping out the path taken by particles as they move. With respect to a velocity field, the velocity vector is tangent to the streamline at every single point.

Here is how streamlines work:

• Every streamline shows the flow direction across the entire fluid.
• The density of streamlines indicates fluid speed – closely spaced lines mean high velocity.

They're invaluable for visualizing fluid movement since no fluid actually crosses a streamline. In practical applications, they help engineers and scientists understand fluid patterns in complex systems, offering insights into improving designs or solving flow problems.