Fluid Flow

Discussion

All things are flowing.

Heraclitis of Ephesus, ca. 500 BCE

continuity equation

For incompressible fluid flow …

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

If the fluid is compressible, then …

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

Notes from The Economist -- "A sverdrup (named for the Norwegian oceanographer and meteorologist Harald Sverdrup) the unit in which ocean currents are measured, is one million cubic metres of water per second. The Gulf Stream, the northern part of a circulation system known as the North Atlantic Gyre, reaches 150 Sverdrups at its peak. On average, it flows at around 100 Sverdrups.

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 probably right.

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

Rearrange

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

The conclusion is most certainly right

P1 + ρgy1 + ½ρv12 = P2 + ρgy2 + ½ρv12

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

q = ½ρv12

applications

The space shuttle and "Max. Q".

The human circulatory system.

[slide]

location area (cm2) speed (cm/s) flow rate (cm3/s)
aorta 3 30 90
arteries 100 0.9 90
capillaries 900 0.1 90
veins 200 0.45 90
vena cava 18 5 90
Blood flow in the systemic circulatory system