Fluid Flow: MDCAT Physics notes
Fluid Flow notes for MDCAT: ideal fluids, laminar (streamline) vs turbulent flow, critical velocity, incompressibility and the equation of continuity.
Types of fluid flow
A fluid in motion can flow in two broad ways.
Laminar (streamline) flow
Every particle passing a given point follows the same path, called a streamline. The fluid moves in layers, and adjacent layers slide smoothly past each other without mixing. Streamlines never cross. Laminar flow occurs at low speeds.
Turbulent flow
The motion is irregular and chaotic, with eddies and whirlpools; layers mix. Turbulence sets in when the flow speed exceeds a certain value called the critical velocity. For a transition from laminar to turbulent flow, the velocity must be greater than the critical velocity. Because most everyday flows are fast, most kinds of fluid flow are turbulent.
| Feature | Laminar | Turbulent |
|---|---|---|
| Speed | Below critical velocity | Above critical velocity |
| Layers | Slide smoothly, do not mix | Mix irregularly |
| Path of particles | Fixed streamlines | Eddies, unpredictable |
| Example | Slow flow of honey, blood in small vessels | Smoke rising higher up, rapids in a river |
Ideal fluid
To analyse flow simply, we assume an ideal fluid that is:
- Non-viscous: no internal friction.
- Incompressible: its density is constant everywhere.
- In steady (laminar) flow: velocity, density and pressure at each point do not change with time.
- Irrotational: no angular momentum about any point.
Liquids are nearly incompressible; gases can be treated as incompressible only at low speeds.
Equation of continuity
For steady flow of an incompressible fluid through a pipe of varying cross-section, the mass entering per second equals the mass leaving per second:
$$\rho A_1 v_1 = \rho A_2 v_2 \quad\Rightarrow\quad A_1v_1 = A_2v_2 = \text{constant}$$
$Av$ is the volume flow rate ($\text{m}^3\text{s}^{-1}$). The equation is a statement of conservation of mass. Where the pipe narrows, the fluid speeds up: speed is inversely proportional to area.
Example: water enters a pipe of area $4\ \text{cm}^2$ at $1\ \text{m s}^{-1}$ and exits a section of area $1\ \text{cm}^2$. Then $v_2 = 4 \times 1/1 = 4\ \text{m s}^{-1}$. For a circular pipe, $A \propto d^2$, so halving the diameter makes the speed four times.
Everyday uses: narrowing a garden hose nozzle makes water squirt faster; a river flows faster where it is narrow.
Key formulas
- $A_1v_1 = A_2v_2$
- Volume flow rate $= Av$; mass flow rate $= \rho Av$
- Circular pipe: $v \propto 1/d^2$
Common MDCAT traps
- Turbulence needs a speed greater than critical velocity, not equal or slightly less.
- In laminar flow layers slide past each other; they do not mix.
- Incompressible means density is constant, not mass, pressure or force.
- Most real flows are turbulent because of high velocities, not zero viscosity.
Quick revision
- Streamlines never intersect.
- Continuity equation = conservation of mass.
- Narrow pipe: higher speed.
- Ideal fluid: non-viscous, incompressible, steady, irrotational.