Introductory Fluid Mechanics L13 p1 - Stream Function - 2D Incompressible Flow

Summary

This lecture segment focuses on the stream function in two-dimensional incompressible flow. It explains how the stream function, denoted by Psi (ψ), mathematically describes streamlines and satisfies the continuity equation. The velocity components are expressed in terms of partial derivatives of the stream function, and it is shown that the stream function remains constant along a streamline. The lecture also demonstrates the relationship between velocity and streamlines, emphasizing the role of the stream function in fluid mechanics for analyzing flow behavior.

Takeaways

• 📘 The lecture focuses on the concept of the stream function in two-dimensional incompressible flow.
• 🧮 The stream function is used to mathematically describe streamlines in fluid mechanics, denoted by the symbol Ψ.
• 📏 The U component of velocity is the partial derivative of the stream function with respect to Y, and the V component is the negative partial derivative of the stream function with respect to X.
• ⚙️ The stream function is valid for two-dimensional flow only and satisfies the continuity equation.
• 🔄 For two-dimensional incompressible flow, the continuity equation simplifies to ∂u/∂x + ∂v/∂y = 0.
• ✅ Substituting the stream function expressions for U and V into the continuity equation proves that the stream function satisfies the continuity requirement.
• 🔗 The stream function remains constant along a streamline, meaning the derivative of the stream function along the streamline is zero.
• ✏️ Streamlines are always tangential to the local velocity vector, and a vector in the direction of the streamline leads to the derivative of the stream function being zero.
• 🔄 Along a streamline, the stream function is constant, but it may change when moving from one streamline to another.
• 📊 The key properties of the stream function are that it satisfies continuity and remains constant along a streamline.

Q & A

What is the purpose of the stream function in fluid mechanics?

The stream function is used to mathematically describe streamlines in a two-dimensional incompressible flow. It helps represent the velocity field, ensuring that the flow satisfies continuity.

How are the velocity components expressed using the stream function?

In two-dimensional incompressible flow, the U component of velocity is the partial derivative of the stream function with respect to Y, and the V component is the negative partial derivative of the stream function with respect to X.

What is the continuity equation for two-dimensional incompressible flow?

For two-dimensional incompressible flow, the continuity equation is expressed as: partial u/partial x + partial v/partial y = 0.

How does the stream function satisfy the continuity equation?

When the velocity components U and V, expressed in terms of the stream function, are substituted into the continuity equation, the result is 0 = 0, meaning the stream function satisfies the continuity condition.

What is the significance of the stream function along a streamline?

Along a streamline, the stream function remains constant. This means that the derivative of the stream function along a streamline is zero, indicating no change in the stream function along the path of the streamline.

What does the cross-product between velocity and a differential element along a streamline imply?

The cross-product between the velocity vector and a differential element along a streamline results in zero, implying that the element is tangential to the streamline and follows the flow direction.

Why is the stream function constant along a streamline?

The stream function is constant along a streamline because the velocity vector is tangential to the streamline, meaning no component of flow crosses the streamline, and thus the stream function remains unchanged.

What happens to the stream function when moving from one streamline to another?

When moving from one streamline to another, the stream function takes a different constant value. This is because the stream function is only constant along a specific streamline but varies between different streamlines.

Why is the stream function only valid for 2D flow?

The stream function is designed for two-dimensional incompressible flows, where the velocity components can be fully represented using the partial derivatives of a single scalar function. In 3D flow, the velocity field is more complex and requires different mathematical tools.

What is the geometric interpretation of a streamline in fluid flow?

A streamline represents a path that is always tangential to the local velocity vector of the flow at every point. Along this path, particles of the fluid travel in the direction of the flow without crossing streamlines.