Fluid Mechanics · Unit 1 · Lesson 9 of 13
How Do We Follow A Tiny Piece Of Fluid?
Drop a single leaf into a river and follow it, or stand on a bridge and watch the water passing under one spot. Which tells you more about the river?
1. Real-World Problem
After an oil spill in a river, an engineer must predict where the oil will be in 2 hours. A second engineer must check whether a flow meter fixed in a pipe reads correctly. One follows the fluid; the other watches one place. Both need precise names for what they are tracking.
2. Predict
Make your guess before you try the experiment. You will see the answer after the next step.
Predict
3. Experiment
Tap anywhere on the flow to release particles and watch the paths they leave. Press Drop an element in Shear flow: the small square leans over while its area stays the same.
Try Around a cylinder and the Flapping flow. In Engineering mode, turn on the dye injector and the streamlines, and compare them with the path of a single particle.
4. Understand
- A fluid particle is a very small amount of fluid: big enough to hold very many molecules (so the continuum model holds), small enough that its properties are uniform throughout. We treat it as a point with mass, velocity, pressure and temperature, and it always contains the same fluid as it moves.
- A fluid element is a small volume of fluid with a chosen shape, usually a tiny box of sides dx, dy and dz, used to write force and mass balances. As it moves it can translate, rotate and deform (stretch or shear).
- In an incompressible flow every fluid element keeps its volume. In a 2D incompressible flow this means each element keeps its area.
Two ways to describe a flow:
| Lagrangian (follow the particle) | Eulerian (watch fixed points) | |
|---|---|---|
| You track | Each particle's position over time | Velocity, pressure at fixed locations |
| Like | Following one leaf down a river | Watching from a bridge |
| Used for | Particle paths, sprays, pollutant spread | Most engineering flow problems |
Three kinds of flow lines:
| Line | What it is | How you see it |
|---|---|---|
| Pathline | The path one particle actually takes over time | A long-exposure photo of one glowing particle |
| Streakline | The line joining all particles that passed through one fixed point | Smoke or dye released continuously from a nozzle |
| Streamline | A line everywhere tangent to the velocity at one instant | Computed; shown by many short arrows |
In steady flow all three coincide. In unsteady flow (the flapping case) they are different, which is why a smoke line is not, in general, the path of a single particle.
5. Derive
Step 1. A particle's velocity is the rate its position changes:
\vec{V} = \frac{d\vec{r}}{dt}r is the particle's position vector (m), t time (s). Over a short time Δt each particle moves by approximately VΔt.
The simulation uses a more accurate, self-adjusting version of the same idea: an adaptive Runge-Kutta method (Dormand-Prince). It takes smaller sub-steps where the flow changes quickly, such as close to the cylinder, and checks its own error at every step.
A sub-step that would end inside the cylinder is rejected and retried with half the size. The result is tested against an independent high-accuracy solution, so the element area stays constant to within 0.1%.
Step 2. The mass of a box-shaped element:
dm = \rho\,dx\,dy\,dzStep 3. A square element of side dx = dy in simple shear u = γ̇ y. Its top edge moves faster than its bottom edge by γ̇ dy, so after time t the top has moved γ̇ dy t further. The lean angle θ satisfies:
\tan\theta = \frac{\dot{\gamma}\,dy\,t}{dy} = \dot{\gamma}\,tThe square becomes a parallelogram with the same base and the same height, so its area stays dx × dy, exactly as a 2D incompressible flow requires.
6. Calculate
Simple shear, γ̇ = 2 per second, square element 1 mm by 1 mm.
- After 0.5 s: tan θ = 2 × 0.5 = 1, so θ = 45°. The top edge has shifted 1 mm.
- Area: still 1 mm².
- If the fluid is water and the element is a 1 mm cube: dm = 1000 × (0.001)³ = 10⁻⁶ kg (1 mg).
7. Apply
- Weather balloons drift with the air: a Lagrangian measurement.
- A fixed anemometer on a mast or a flow meter in a pipe: Eulerian measurements.
- Smoke in a wind tunnel shows streaklines; they match particle paths only when the flow is steady.
- Fuel droplets in an engine spray are tracked one by one (Lagrangian).
- Oil spill and pollutant forecasts follow particles released into a computed Eulerian flow field.
8. Validate
In Engineering mode choose simple shear, γ̇ = 2 per second, drop an element and pause at t = 0.5 s: the lean angle must read 45° and the area 1.000 mm². In the cylinder flow the speed at the top of the cylinder must read 2U, the classical ideal-flow result. In the flapping flow, turn on the dye injector and release one particle from the same point: the streakline and pathline must differ. Where the model does not apply: behind a real cylinder the flow separates into a wake, and in compressible flow element volumes change.
Easy To Mix Up
- A fluid particle is not a molecule. It contains very many molecules.
- A fluid element is a tool for writing equations; it is not a physical object in the fluid.
- A smoke or dye line is a streakline. It equals a pathline only in steady flow.
- Constant area is a property of 2D incompressible flow. In 3D incompressible flow, volume is what stays constant.
9. Quiz
Four quick questions: a concept, a calculation and an interpretation. Each answer comes with an explanation. Score 3 or more to mark this lesson completed.
Check Your Understanding
Four questions: a concept, a calculation and an interpretation. Every answer comes with an explanation. Score 3 of 4 to mark the lesson completed.
Every analysis starts with one line drawn on paper: the boundary. What is inside, what is outside, and what crosses it?
Go to Lesson 10: Where Does Your System End? →Read More
- Cengel and Cimbala, "Fluid Kinematics" (Lagrangian and Eulerian descriptions, flow patterns, motion and deformation of fluid elements).
- NPTEL, Introduction to Fluid Mechanics (Prof. Suman Chakraborty, IIT Kharagpur): Weeks 5 and 6.