conservation and continuity

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Presentation transcript:

conservation and continuity Fluid Flow conservation and continuity § 15.6

Volume Flow Rate Volume per time through an imaginary surface perpendicular to the velocity DV/Dt units: m3/s

Volume Flow Rate DV/Dt = v·A if v is constant over A

Mass Flow Continuity Constant mass flow for a closed system Dm = Dt 1 2 r1A1v1 = r2A2v2

Volume Flow Continuity Constant volume flow if incompressible If r1 = r2, DV Dt = 1 2

Poll Question Where is the velocity greatest in this stream of incompressible fluid? Here. Same for both. Can’t tell.

Quick Question Where is the density greatest in this stream of incompressible fluid? Here. Same for both. Can’t tell.

Poll Question Where is the volume flow rate greatest in this stream of incompressible fluid? Here. Same for both. Can’t tell.

Bernoulli’s Equation Energy in fluid flow § 15.7

Incompressible Fluid Continuity condition: constant volume flow rate dV1 = dV2 v1A1 = v2A2

Poll Question Where is the kinetic energy of a parcel greatest in this stream of incompressible fluid? Here. Same for both. Can’t tell.

Changing Cross-Section Fluid speed varies Faster where narrow, slower where wide Kinetic energy changes Work is done!

Ideal Fluid No internal friction (viscosity) No non-conservative work!

Poll Question Where would the pressure be greatest if the fluid were stationary? Here. Same for both. Can’t tell.

Work-Energy Theorem Wnet = DK K1 + Wnet = K2 K1 + U1 + Wother = K2 + U2 Wother = DK + DU dW = dK + dU 16

Work done by Pressure W = F·Ds Work done on fluid at bottom: W1 = p1A1·Ds1 Work done on fluid at top: W2 = –p2A2·Ds2 Total work done on fluid : W = p1A1·Ds1–p2A2·Ds2 = (p1 – p2)DV

Kinetic Energy Change Steady between “end caps” Lower cap: K1 = ½ mv12 Upper cap: K2 = ½ mv22 m = rV DK = 1/2 rV (v22–v12)

Potential Energy Change Steady between “end caps” Lower cap: U1 = mgy1 Upper cap: U2 = mgy2 m = rV DU = rgV (y2–y1)

Put It All Together Wother = DK + DU (p1 – p2)V = 1/2 rV (v22–v12) + rgV (y2–y1) (p1 – p2) = 1/2 r (v22–v12) + r g(y2–y1) p1 + 1/2 rv12 + rgy1 = p2 + 1/2 rv22 + rgy2 This is a conservation equation Strictly valid only for incompressible, inviscid fluid

What Does It Mean? Faster flow  lower pressure Maximum pressure when static pV is energy

Example problem A bullet punctures an open water tank, creating a hole that is a distance h below the water level. How fast does water emerge from the hole?

Torricelli’s Law p1 + 1/2 rv12 + rgy1 = p2 + 1/2 rv22 + rgy2 v22 = 2gh 1/2 rv22 = rg(y2–y1) + (p2–p1) – 1/2 rv12 1/2 rv22 = rgh v22 = 2gh v2 = 2gh look familiar?