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This document contains a fluid dynamics tutorial covering topics such as pathlines, streamlines, and velocity fields. The exercises primarily focus on two-dimensional fluid flows, including:

1. **Exercise 1**: Involves finding the parametric equations of pathlines and streamlines for a fluid with velocity field components \( u = a y \) and \( v = a b t \), where \( a \) and \( b \) are positive constants. Tasks include eliminating the parameter \( t \) and sketching pathlines and streamlines for specific conditions.
  
2. **Exercise 2**: Focuses on a plane flow with constant velocity \( u_0 \) and time-varying component \( v = a t \). The goal is to show incompressibility, derive pathline equations, and examine streamlines and particle movement.
   
3. **Exercise 3**: Analyzes a tornado represented in polar coordinates with a velocity field involving \( u_r = -\frac{a}{r} \) and \( u_\theta = \frac{b}{r} \). The tasks include checking fluid incompressibility, deriving streamlines, and computing pathlines.

4. **Exercise 4**: Explores planar motion in Cartesian coordinates with velocity components \( u = \omega y \) and \( v = -\omega x + a\omega^2 t \), requiring analysis of stationarity, incompressibility, rotation, and particle motion.

Would you like to go through specific exercises in detail, such as solving for pathlines or streamlines in one of the exercises?

Here are some questions based on the content:

1. What is the significance of pathlines and streamlines in fluid flow analysis?
2. How do you find the parametric equations of pathlines for different velocity fields?
3. Can you explain the difference between steady and unsteady flows in fluid dynamics?
4. How does incompressibility affect the behavior of fluids?
5. What is the relationship between velocity and acceleration in polar coordinates for fluid motion?

**Tip**: Streamlines represent the direction of the fluid at any point in time, while pathlines trace the actual trajectory of individual particles.

This document contains a fluid dynamics tutorial covering topics such as pathlines, streamlines, and velocity fields. The exercises primarily focus on two-dimensional fluid flows, including parametric equations of pathlines and streamlines, incompressibility of fluid motion, and analyzing a tornado using polar coordinates.

Explore key concepts in fluid dynamics, including pathlines and streamlines, velocity fields, and incompressibility. This tutorial provides detailed exercises focusing on 2D fluid flow analysis, polar coordinate systems, and velocity field equations. Suitable for advanced undergraduate or graduate students.

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