Summary of "Fluid Mechanics | Module 3 | Stream, Path & Streak Lines (Lecture 23)"
Summary of "Fluid Mechanics | Module 3 | Stream, Path & Streak Lines (Lecture 23)"
This lecture focuses on three fundamental concepts in Fluid Mechanics related to visualizing fluid flow: Streamlines, Pathlines, and Streaklines. The instructor explains their definitions, physical meanings, mathematical formulations, and key properties, supported by example problems and conceptual clarifications.
Main Ideas and Concepts
1. Streamlines
- Definition: An imaginary line drawn in a flow field such that the tangent at any point on the line represents the direction of the velocity vector of the fluid particle at that point at a given instant.
- Key Characteristics:
- Streamlines represent the instantaneous flow pattern.
- At any point on a streamline, the velocity vector is tangent to the streamline.
- The flow Velocity Components (u, v, w) define the direction of the streamline.
- Mathematical Representation: The Differential Equation for Streamlines is given by: dx/u = dy/v = dz/w Using Velocity Components, the streamline equation can be derived by integrating the above relation.
- Important Property: Two Streamlines cannot intersect because that would imply two different velocity directions at the same point, which is physically impossible.
- Example Problem: Given a velocity field, the streamline equation can be found by solving the Differential Equation with boundary/initial conditions.
2. Pathlines
- Definition: The actual path traced by a single fluid particle as it moves through the flow field over time.
- Key Characteristics:
- Pathlines show the trajectory of a specific fluid particle.
- They are obtained by tracking the particle’s position at successive times.
- Difference from Streamlines: Streamlines are instantaneous snapshots; Pathlines are time-dependent trajectories. Pathlines can intersect themselves or other Pathlines.
3. Streaklines
- Definition: The locus of all fluid particles that have passed through a particular fixed point in the flow field.
- Key Characteristics:
- Streaklines are formed by continuously injecting dye or marking particles at a fixed point and observing the line formed downstream.
- They represent the instantaneous positions of all particles that passed through the injection point over time.
- Difference from Streamlines and Pathlines: Streaklines are often used in experiments (e.g., Dye Injection). They can differ from Streamlines and Pathlines in unsteady flows.
Additional Important Points
- Instantaneous Picture of Flow: Streamlines represent the flow field at a specific instant (snapshot in time).
- Mathematical Approach:
- Velocity Components are functions of spatial coordinates (x, y, z) and possibly time (t).
- The streamline equation is derived by equating the ratios of infinitesimal displacements to Velocity Components.
- Physical Interpretation: Velocity vector at any point is tangent to the streamline. Streamlines give insight into flow direction but not particle history.
- Non-Intersection of Streamlines: Streamlines cannot cross because velocity direction at a point is unique.
- Pathlines Can Intersect: Since Pathlines trace individual particles over time, they can intersect or loop back.
- Streaklines Can Intersect: Streaklines, formed by particles passing through a point over time, can also intersect.
- Time Intervals in Streaklines: Particles are released at fixed time intervals from a point. The instantaneous positions of these particles at a later time form the streakline.
- Relation in Steady Flow: In steady flow, Streamlines, Pathlines, and Streaklines coincide.
Methodology / Steps to Find Streamline Equation
- Obtain Velocity Components \( u(x,y,z), v(x,y,z), w(x,y,z) \).
- Write the Differential Equation of the streamline: dx/u = dy/v = dz/w
- Integrate the equation with respect to space variables to find the streamline equation.
- Apply boundary or initial conditions to solve for constants of integration.
- Interpret the resulting equation to understand the flow pattern.
Example Problem Outline (from lecture)
- Given velocity field: u = 3x² - y, v = …
- Set up streamline Differential Equation: dx/u = dy/v
Category
Educational
