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8.2.4Fluids

Continuity Equation — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 10 of 14 1 quiz

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Lesson summary

8.2.4 Continuity Equation — AP PHYSICS 1 notes: Apply the continuity equation A1v1 = A2v2 to find fluid speeds in pipes of changing cross-section.Steady flow — fluid cannot pile up or disappear, so the mass entering one end each second leaves the other.Incompressible fluid — mass…

  • Consequence
  • V/t = Av = constant
  • Circular pipes
  • Inverse proportion
  • Hose with thumb over end
  • Streamlines

Section 1 of 9Conservation of mass in a pipe

1Steady flow
  • Steady flow fluid cannot pile up or disappear, so the mass entering one end each second leaves the other.
2Incompressible fluid
  • Incompressible fluid mass conservation is volume conservation, so V/t is constant at every position.
3Consequence
  • Consequence a river squeezing through a gorge speeds up to pass the same volume per second.

Section 2 of 9The equation

V/t
Av = constant
  • the product of cross-sectional area and speed is the same at every point.
A1v1
A2v2
  • the continuity equation for two points in the same pipe.
  • A in m2 (any consistent unit works for ratios), v in m/s.
Circular pipes
  • A ∝ d2, so v2 = v1(d1/d2)2; speed is inversely proportional to the square of the radius.

Section 3 of 9Narrow means fast

Inverse proportion
  • Inverse proportion halving the area doubles the speed;
    • halving the diameter quarters the area and quadruples the speed.
Hose with thumb over end
  • Hose with thumb over end reduces A so v rises while Q is nearly unchanged, and the spray travels farther.
Streamlines
  • Streamlines bunched streamlines in the narrow section mean higher speed.
Graphs
  • Graphs v against A for fixed Q is a hyperbola;
    • v against 1/A is a straight line through the origin with gradient Q.

Section 4 of 9Branches and junctions

Junction
  • Junction flow rate in equals total flow rate out, Qin = Q1 + Q2 + ….
Equation
  • Amainvmain = A1v1 + A2v2 + ….
Total area
  • Total area blood slows in capillaries because their total area is huge, so the sum of branch areas enters the equation.

Section 5 of 9Limits of the model

Limits of the model
Requirements
  • Requirements incompressible fluid and no gain or loss of fluid.
Gases at high speed, leaks or extra inlets
  • Gases at high speed, leaks or extra inlets use mass flow rates ρAv instead.
Pressure
  • Pressure continuity gives no information about pressure; Bernoulli is needed.

Section 6 of 9Using continuity

Qualitative answer
  • "A decreases, Av is constant, so v increases."
Two-section problems
  • Two-section problems v2 = v1(A1/A2) without finding Q.
With Bernoulli
  • With Bernoulli first find the unknown speed from continuity, then substitute both speeds into Bernoulli for the pressure difference.
Pressure drop
  • Pressure drop a pipe narrowing to half its diameter gives four times the speed, so ½ρv2 grows sixteen-fold.
Everyday cases
  • Everyday cases wide slow rivers and fast narrow gorges, siphons, fire hoses, sprinklers, blood vessels.

Section 7 of 9More from the specification

1Gauge pressure
  • Pressure above atmospheric, ρgh.
2Potential energy
  • Energy stored by the configuration of a system.
3Weight
  • Gravitational force on a mass.
4Newton
  • Unit of force, kg m s-2.
5Density
  • Mass per unit volume.
6Velocity
  • Rate of change of displacement.
7Length
  • Resistance is directly proportional to the length of a wire.
8Buoyant force
  • Upward force equal to the weight of displaced fluid.
9Ideal fluid
  • Incompressible fluid with no viscosity.
10Newton meter
  • A spring-based instrument for measuring force.

Section 8 of 9Key terms

TermMeaning
Continuity equationA1v1 = A2v2; conserved flow rate in steady incompressible flow.
IncompressibleDensity does not change.
JunctionPlace where a pipe splits or joins; flow rates add.
NozzleNarrowed outlet that increases fluid speed.
Flow rateQ = Av, in m3/s.

Section 9 of 9Exam technique

1
Concluding a narrower pipe carries less fluid per second — Q is the same everywhere in steady flow.
2
Using diameters in the ratio instead of squaring them for areas.
3
Adding speeds at a junction instead of flow rates.
4
Applying A1v1 = A2v2 to a pipe with a leak or extra inlet.
5
Assuming faster fluid means higher pressure.

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