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8.2.5Fluids

Bernoulli’s Equation — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 10 parts Lesson 11 of 14 1 quiz

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8.2.5 Bernoulli’s Equation — AP PHYSICS 1 notes: Use Bernoulli's equation to relate pressure, speed and height at two points along a streamline in an ideal fluid.Assumptions — the fluid is incompressible, has no viscosity (no friction) and moves smoothly without turbulence.Conseq…

  • Consequence
  • Streamline (laminar) flow
  • Turbulent flow
  • Energy per unit volume
  • Work–energy view
  • Failure cases

Section 1 of 10Conditions for ideal flow

1Assumptions
  • Assumptions the fluid is incompressible, has no viscosity (no friction) and moves smoothly without turbulence.
2Consequence
  • Consequence no energy is dissipated and the flow rate is constant.
3Streamline (laminar) flow
  • Streamline (laminar) flow smooth, frictionless movement in lines pointing in the direction of flow;
    • close to real for slow, low-viscosity flow.
4Turbulent flow
  • Turbulent flow streamlines cross and form vortices;
    • typical for high speed or high viscosity, where Bernoulli fails.

Section 2 of 10Energy conservation for a fluid

Energy per unit volume
  • Energy per unit volume pressure energy P, gravitational potential energy ρgy and kinetic energy ½ρv2, each in J/m3 = Pa.
Work–energy view
  • Work–energy view the net work done by pressure forces on a small volume equals its gain in kinetic plus potential energy.
Failure cases
  • Failure cases a pump or turbine adding or removing energy, viscosity, turbulent or time-dependent flow.

Section 3 of 10The equation

P1 + ρgy1 + ½ρv12
P2 + ρgy2 + ½ρv22
along a streamline in steady ideal flow.
  • P is absolute pressure in Pa (or gauge pressures against the same reference).
  • ρ in kg/m3, g = 9.8 m/s2.
  • y is height above an arbitrary reference level; only differences matter.
  • v is the fluid speed in m/s.
Static fluid
  • Static fluid with v = 0 it reduces to P2 − P1 = ρg(y1 − y2), i.e. P = P0 + ρgh.
Bar charts
  • Bar charts the combined height of the three Bernoulli bars is equal at both locations.

Section 4 of 10Horizontal flow: fast means low pressure

1y1 = y2
  • P1 + ½ρv12 = P2 + ½ρv22.
2Result
  • Result where the speed is greater the pressure is lower.
3With continuity
  • With continuity the narrow section is faster and has the lowest pressure.
4Net force
  • Net force the pressure drop supplies the forward force that speeds the fluid up on entering the narrow part,
    • and the pressure rise beyond slows it again.

Section 5 of 10Changing height

Changing height
Rising pipe
  • Rising pipe the fluid gains potential energy, paid for by pressure or kinetic energy.
Constant diameter (same v)
  • Constant diameter (same v) pressure at the higher point is lower by ρgΔy.
Pumping water up a building
  • Pumping water up a building needs extra pressure at the bottom.

Section 6 of 10Applications

Aeroplane wing
  • Aeroplane wing faster air above gives lower pressure above and a net upward force.
Shower curtain
  • Shower curtain moves inward because fast air inside lowers the pressure there.
Venturi meter
  • Venturi meter a pipe narrows and the measured pressure drop reveals the speed.
Atomiser (perfume spray)
  • Atomiser (perfume spray) fast air across a tube lowers the pressure and draws liquid up.
Storm roof
  • Storm roof fast wind above lowers the pressure above while still air inside keeps higher pressure below.

Section 7 of 10Problem-solving routine

1
Pick two points on the same streamline.
2
Choose a reference height.
3
Use continuity to relate the two speeds.
4
Write the full equation and cancel equal terms.
5
Solve for the unknown and check units are Pa.

Section 8 of 10More from the specification

Weight
  • Gravitational force on a mass.
Newton
  • Unit of force, kg m s-2.
Density
  • Mass per unit volume.
Velocity
  • Rate of change of displacement.
Length
  • Resistance is directly proportional to the length of a wire.
Buoyant force
  • Upward force equal to the weight of displaced fluid.
Newton meter
  • A spring-based instrument for measuring force.

Section 9 of 10Key terms

TermMeaning
Bernoulli equationP + ρgy + ½ρv2 is constant along a streamline.
Absolute pressureTotal pressure including atmospheric.
StreamlinePath of a fluid parcel in steady flow.
Dynamic pressureThe term ½ρv2.
Reference heightChosen zero level for y.

Section 10 of 10Exam technique

1
Applying Bernoulli between points not on the same streamline or with a pump between them.
2
Forgetting the ½ or not squaring the speed.
3
Thinking higher speed gives higher pressure.
4
Using different reference heights for the two points.
5
Using kPa or atm instead of Pa when adding to ρgy and ½ρv2.

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