Fluid Mechanics · Unit 1 · Lesson 6 of 13
Still, Moving Or Pushed: Three Ways To Look At A Fluid
The water in your overhead tank is still. Open a tap and it flows. Shut the tap suddenly and the pipe can bang and shudder. Are these three different subjects?
1. Real-World Problem
A building's overhead tank feeds a pipe that narrows at a fitting. The plumber wants to know three things: the pressure at a tap when nothing flows, how fast water moves through the narrow fitting, and why the pressure there is lower when water flows.
2. Predict
Make your guess before you try the experiment. You will see the answer after the next step.
Predict
3. Experiment
Start on Statics with the valve closed: every gauge reads the same pressure, set by the water height H. Switch to Kinematics: the water speeds up in the narrow part. Then open Dynamics: the pressure there drops while the energy line stays level.
In Engineering mode, change the tank height H and the narrow diameter d₂, and read the flow rate, speeds and gauge pressures.
4. Understand
Fluid mechanics has three branches, depending on what you ask.
| Branch | What it studies | Question it answers | Example |
|---|---|---|---|
| Fluid statics | Fluids at rest | How hard does the fluid push? | Pressure on a dam |
| Fluid kinematics | Motion of fluids, without looking at forces | Where and how fast does it flow? | Speed of water in a narrow pipe |
| Fluid dynamics | Motion of fluids together with the forces and energy involved | What forces and energy are involved? | Force on a pipe bend, pump power |
The banging pipe in the curiosity question is unsteady flow (water hammer), a dynamics problem that Bernoulli's steady equation cannot describe.
5. Derive
Step 1. Statics. Pressure in a still fluid grows with depth (Lesson 2):
p = p_0 + \rho g hp is pressure at depth h (Pa), p₀ the pressure at the surface (Pa), ρ density (kg/m³), g = 9.81 m/s², h depth (m).
Step 2. Kinematics. Volume flow rate is area times average speed. If the flow is steady, the fluid incompressible and nothing is added or lost, the same flow passes every section:
Q = A\,V \qquad A_1 V_1 = A_2 V_2Q is volume flow rate (m³/s), A cross-section area (m²), V average speed (m/s).
Step 3. Dynamics. Bernoulli's equation, which comes from Newton's second law applied along a streamline:
p + \tfrac{1}{2}\rho V^2 + \rho g z = \text{constant}z is height above a reference (m). It holds only when all of these are true: the flow is steady, incompressible and nearly inviscid (friction negligible), the two points lie on the same streamline, and no pump or turbine adds or removes energy between them.
Step 4. Between two sections at the same height (z₁ = z₂):
p_1 + \tfrac{1}{2}\rho V_1^2 = p_2 + \tfrac{1}{2}\rho V_2^2Where speed goes up, pressure goes down. That is what the gauges show in the Dynamics tab.
Step 5. From the tank surface (atmospheric, V ≈ 0, height H above the pipe) to the nozzle exit (atmospheric): Vₑ = √(2gH), Torricelli's result from Lesson 3. Then, using gauge pressures, every section obeys p = ρgH − ½ρV².
6. Calculate
H = 1.0 m, D₁ = 40 mm, d₂ = 25 mm, dₑ = 20 mm.
- Vₑ = √(2 × 9.81 × 1.0) = 4.43 m/s
- Aₑ = π × 0.020² ÷ 4 = 3.14 × 10⁻⁴ m², so Q = 3.14 × 10⁻⁴ × 4.43 = 1.39 × 10⁻³ m³/s = 1.39 L/s
- A₁ = 1.257 × 10⁻³ m², so V₁ = 1.11 m/s; A₂ = 4.91 × 10⁻⁴ m², so V₂ = 2.83 m/s
- Valve closed: every gauge reads ρgH = 1000 × 9.81 × 1.0 = 9.81 kPa
- Valve open: p₁ = 9810 − ½ × 1000 × 1.11² = 9.20 kPa; p₂ = 9810 − ½ × 1000 × 2.83² = 5.79 kPa
7. Apply
- Statics: dams, water tanks, manometers, submarines.
- Kinematics: a thumb on a garden hose makes water squirt faster, because A goes down and V goes up.
- Dynamics: the kick of a fire hose, lift on a wing, the power a pump needs, the venturi in a carburettor that uses low pressure to draw in fuel.
8. Validate
In Engineering mode set H = 1 m and d₂ = 25 mm: readouts must show Q = 1.39 L/s, V₂ = 2.83 m/s, p₁ = 9.20 kPa and p₂ = 5.79 kPa, and every energy bar must total 1.00 m in the head view, or 9.81 kPa in the pressure view. Where this model does not apply: long or rough pipes (friction losses), a partly closed valve (large local loss), a tap closed suddenly (unsteady water hammer), and narrow sections so small that the pressure falls towards the vapour pressure.
Easy To Mix Up
- Kinematics does not ignore physics; it only leaves forces aside for the moment.
- Bernoulli's equation is not a general law. It is a result for steady, incompressible, nearly frictionless flow along a streamline.
- "Fluid dynamics" in everyday use often means all moving-fluid study. In this unit it means motion together with forces and energy.
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.
One engineer designs a canal from experiments; another writes equations for ideal flow. Who is doing hydraulics and who hydrodynamics?
Go to Lesson 7: Hydraulics Or Hydrodynamics: Practice Or Theory? →Read More
- Cengel and Cimbala, "Pressure and Fluid Statics", "Fluid Kinematics" and "Mass, Bernoulli and Energy Equations" (Bernoulli's equation and its limits).
- NPTEL, Introduction to Fluid Mechanics (Prof. Suman Chakraborty, IIT Kharagpur): Weeks 4, 5 and 7.