PMDC Unit 41 · 10 outcomes

Fluid Dynamics

Unit 5 is fluids that move: a falling raindrop that stops speeding up, drag, laminar vs turbulent flow, Av = constant, and Bernoulli — including why a narrow artery raises speed and drops pressure.

Open the Unit 41 quiz Worked numericals

5.1

Terminal velocity

PMDC: Describe the terminal velocity of an object.

Drop a sphere in a fluid. Weight pulls down. Drag (and upthrust) grow as it speeds up. When those upward forces match the weight, net force is zero, acceleration is zero, and the speed stays at a constant value — terminal velocity $v_{t}$ .

A small raindrop reaches $v_{t}$ quickly. A skydiver does too, once the parachute opens and drag jumps.

Live graph — drag the sliders

Σ F = 0 \Rightarrow a = 0 \Rightarrow v = v_{t}

at $v_{t} : w e i g h t = d r a g$ + upthrust
for Stokes’ sphere (no turbulence): 6 $π$ η r $v_{t}$ = mg − upthrust

Hold these

Terminal velocity is constant, not zero.
Larger drag (bigger area, thicker fluid) means a smaller $v_{t}$ .

5.2

Fluid drag

PMDC: Define and explain the term fluid drag.

Drag is the resistive force a fluid exerts on a body moving through it. It always opposes the motion. Slow and tiny: Stokes’ drag is proportional to speed. Fast and large: drag grows roughly with v² because the body has to shove fluid out of the way.

Live graph — drag the sliders

F_{d} \propto v v = 4 F_{d} \times 4.8

Stokes (laminar, small sphere): $F_{d}$ = 6 $π$ η r v
fast / turbulent: $F_{d}$ ∝ v²

Hold these

Drag is a force, opposite to v.
η is viscosity — ‘thickness’ of the fluid.

5.3

Ideal fluid flow

PMDC: Define steady (streamline / laminar) flow, incompressible flow and non-viscous flow as applied to an ideal fluid.

Steady (streamline, laminar): the fluid at any fixed point always has the same velocity. Layers slide without mixing. Streamlines do not cross.

Incompressible: density does not change. Liquids are close enough. Then volume in equals volume out.

Non-viscous: no internal friction. An ideal fluid is often taken as steady, incompressible and non-viscous — the Bernoulli setup.

Live graph — drag the sliders

steady:

v = v (p os i t i o n)

not v(time)

ideal fluid: steady + incompressible + non-viscous

Hold these

$L amina r = l a y er e d$ and orderly.
Water in a pipe is nearly incompressible. Air is not, if you squeeze it hard.

5.4

Laminar to turbulent

PMDC: Explain that at sufficiently high velocity, the flow of a viscous fluid undergoes a transition from laminar to turbulent conditions.

Turn the tap a little: a smooth thread of water. Open it: the thread breaks into swirls. Fast flow in a viscous fluid becomes turbulent — eddies, mixing, extra drag. The switch depends on speed, pipe size and viscosity (Reynolds number, if you meet it).

Live graph — drag the sliders

R e \propto v v = 3 l amina r

low v → laminar high v → turbulent

Hold these

Turbulence is not ‘more laminar’. It is a different regime.

5.5

Most real flows are turbulent

PMDC: Describe that the majority of practical examples of fluid flow and resistance to motion in a fluid involve turbulent rather than laminar conditions.

Rivers, blood in the aorta, air over a car, a cricket ball in flight — usually turbulent. Laminar is the textbook starting point. The drag you feel in daily life is mostly the v² kind, not Stokes.

Live graph — drag the sliders

R e \propto v v = 3 l amina r

Hold these

Exam pictures may still use laminar + Bernoulli. Real pipes often do not.

5.6

Equation of continuity

PMDC: Describe the equation of continuity Av = constant for the flow of an ideal and incompressible fluid, and solve problems using it.

What goes in must come out. For an incompressible fluid the volume per second is A v. Narrow the pipe, A falls, v must rise so that A v stays the same.

Live graph — drag the sliders

A_{1} v_{1} = A_{2} v_{2} v_{2} = 1 \times (4/2)^{2} = 4 m/s

$A v = co n s t an t$
$A_{1} v_{1} = A_{2} v_{2}$
$A = π r^{2} \Rightarrow v \propto 1/ r^{2}$

Hold these

Halve the radius, speed becomes four times, not two.
This is volume per second. Mass per second is $ρ$ A v.

5.7

Continuity as conservation of mass

PMDC: Identify that the equation of continuity is a form of the principle of conservation of mass.

If density is constant, saving volume is saving mass. No fluid is created or destroyed in the pipe. Continuity is mass conservation written for a flow.

Live graph — drag the sliders

ρ

v = co n s t Q = 18

(units of r²v)

$ρ$ A $v = co n s t an t$ (mass per second)
if $ρ$ constant: A $v = co n s t an t$

Hold these

Compressible gas: $ρ$ can change, so A v need not stay put.

5.8

Bernoulli in blood physics

PMDC: Interpret and apply Bernoulli’s effect in blood physics.

A narrowing (stenosis) in an artery: speed up (continuity), pressure down (Bernoulli). The wall can be sucked inward, which narrows it more — a medical warning. Faster blood at a constriction is not ‘higher pressure’.

Live graph — drag the sliders

v = Q / A_{t o t a l} n = 6 \Rightarrow v \times 1/6

narrow $v esse l \Rightarrow v$ ↑ ⇒ P ↓

Hold these

Fast flow, low pressure. That is the trap in blood-physics questions.

5.9

Bernoulli’s equation for a horizontal tube

PMDC: Derive Bernoulli’s equation for the case of a horizontal tube of flow.

Along a horizontal streamline the height does not change, so the g h terms cancel. Work done by pressure difference goes into changing kinetic energy. The result: P + ½ $ρ$ v² is constant.

Full Bernoulli (if height also changes) keeps the $ρ$ g h term.

Live graph — drag the sliders

Δ P = \frac{1}{2} \times 1000 \times (4^{2} - 2^{2}) = 6000 Pa

horizontal: P + ½ $ρ v^{2} = co n s t an t$
general: P + ½ $ρ$ v² + $ρ$ g $h = co n s t an t$

Hold these

Needs steady, incompressible, non-viscous flow along a streamline.
Higher v, lower P — if h is the same.

5.10

Pressure from different flow rates

PMDC: Describe that a pressure difference can arise from different rates of flow of fluid (Bernoulli’s effect).

Two regions, same height, different speeds. The faster side has lower pressure. That is why a breeze can lift a roof, why a perfume spray works, and why a Venturi meter infers speed from a pressure drop.

Live graph — drag the sliders

Δ P = \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2}) = 30000 Pa

$Δ P = \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2}) (h or i z o n t a l)$

Hold these

Do not say ‘faster fluid pushes more’. It pulls less (lower P).

Unit 41 formula sheet

Write these until they are cheaper than thinking, then open the quiz.

at $v_{t} : w e i g h t = d r a g$ + upthrust
for Stokes’ sphere (no turbulence): 6 $π$ η r $v_{t}$ = mg − upthrust
Stokes (laminar, small sphere): $F_{d}$ = 6 $π$ η r v
fast / turbulent: $F_{d}$ ∝ v²
ideal fluid: steady + incompressible + non-viscous
low v → laminar high v → turbulent
$A v = co n s t an t$
$A_{1} v_{1} = A_{2} v_{2}$
$A = π r^{2} \Rightarrow v \propto 1/ r^{2}$
$ρ$ A $v = co n s t an t$ (mass per second)
if $ρ$ constant: A $v = co n s t an t$
narrow $v esse l \Rightarrow v$ ↑ ⇒ P ↓
horizontal: P + ½ $ρ v^{2} = co n s t an t$
general: P + ½ $ρ$ v² + $ρ$ g $h = co n s t an t$
$Δ P = \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2}) (h or i z o n t a l)$

Open the Unit 41 quiz