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.

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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.

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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.

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Methodology / Steps to Find Streamline Equation

1. Obtain Velocity Components \( u(x,y,z), v(x,y,z), w(x,y,z) \).
2. Write the Differential Equation of the streamline:
   dx/u = dy/v = dz/w
3. Integrate the equation with respect to space variables to find the streamline equation.
4. Apply boundary or initial conditions to solve for constants of integration.
5. Interpret the resulting equation to understand the flow pattern.

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Example Problem Outline (from lecture)

• Given velocity field:
  u = 3x² - y, v = …
• Set up streamline Differential Equation:
  dx/u = dy/v

Category

Educational

Video