Fluid Flow

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

All things are flowing -- Heraclitus ca. 500 BCE

continuity equation

When a fluid flows is an incompressible manner …

φ =	V	= Av = constant	⇒	A1v1 = A2v2
	t

If the fluid is compressible, then …

I =	m	= ρAv = constant	⇒	ρ1A1v1 = ρ2A2v2
	t

bernoulli's equation

Bernoulli's equation is based on the law of conservation of energy; the increased kinetic energy of a fluid is offset by a reduction of the "static energy" associated with pressure. The fluid is assumed incompressible and inviscid (that is, the fluid does not generate drag).

Something like this is right, but this derivation must certainly be wrong. (It has a glaring algebra error. Can you spot it?)

W			=	ΔE
PΔV			=	ΔU	+	ΔK
P2V2	−	P1V1	=	U2 − U1	+	K2 − K1

Rearrange

P1V1	+	U1	+	K1	=	P2V2	+	U2	+	K2
P1V1	+	mgh1	+	½ mv12	=	P2V2	+	mgh2	+	½ mv12
P1V1	+	mgh1	+	½ mv12	=	P2V2	+	mgh2	+	½ mv12
V1		V1		V1		V2		V2		V2
P1	+	ρgh1	+	½ ρv12	=	P2	+	ρgh2	+	½ ρv12

The conclusion is right

P₁ + ρgh₁ + ½ ρv₁² = P₂ + ρgh₂ + ½ ρv₁²

The third term in this equation is the dynamic pressure (q).

q = ½ ρv₁²

The space shuttle and "Max. Q".

Summary

• Continuity Equation - conservation of mass for fluids (mass per volume = mass density)

  Δm	=	ρΔV	=	ρAΔs	= ρAv = constant
  Δt		Δt		Δt

• Bernoulli's Equation - conservation of energy for fluids (energy per volume = energy density)

  P1 + ½ ρv12 + ρgh1 = P2 + ½ ρv22 + ρgh2

Problems

practice

1. Write something.
   • Answer it.
2. Write something else.
   • Answer it.
3. Write something different.
   • Answer it.
4. Write something completely different.
   • Answer it.

conceptual

1. Explain why blood corpuscles tend to flow down the center of blood vessels and not along the edges.

numerical

1. A 15 cm radius air duct is used to replenish the air of a room 10 m × 6.0 m × 3.0 m every 10 min. How fast does the air flow in the duct?
2. If the wind blows at 25 m/s over a house, what is the net force on an 8 m × 20 m roof?
3. Water at a pressure of 3.3 atm at street level flows into an office building at a speed of 0.50 m/s through a pipe 5.0 cm in diameter. The pipe tapers down to 2.5 cm diameter by the top floor, 25 m above. Calculate the flow velocity and pressure in such a pipe on the top floor. Ignore viscosity.

Resources

• aerodynamics
  • Henri Marie Coanda, Aeronautics Learning Laboratory for Science Technology, and Research (ALLSTAR), NASA
  • Automatic Shut-Off Dispensing Nozzle, Paul Wilder, US Patent 4453578 (12 January 1983)

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