Fluid Flow

Glenn Elert

Fluid Flow

Discussion

All things are flowing.

Heraclitis of Ephesus, ca. 500 BCE

flow rates

mass flow rate

I =	m
	t

volume flow rate

φ =	V
	t

continuity equation

If the fluid is compressible, then we use the mass flow rate…

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

For incompressible fluid flow, we use the volume flow rate…

φ =	V	= Av = constant	⇒	A₁v₁ = A₂v₂
	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			=	∆E
− P∆V			=	∆U	+	∆K
P₁V₁	−	P₂V₂	=	(U₂ − U₁)	+	(K₂ − K₁)

Rearrange and assume the fluid is effectively incompressible (i.e., that it's volume remains constant as it flows from one region to another).

P₁V₁	+	U₁	+	K₁	=	P₂V₂	+	U₂	+	K₂

P₁V₁	+	mgh₁	+	½mv₁²	=	P₂V₂	+	mgh₂	+	½mv₂²

P₁V₁	+	mgh₁	+	½mv₁²	=	P₂V₂	+	mgh₂	+	½mv₂²
V₁	+	V₁	+	V₁	=	V₂	+	V₂	+	V₂

P₁	+	ρgh₁	+	½ρv₁²	=	P₂	+	ρgh₂	+	½ρv₂²

The conclusion is most certainly right

P₁ + ρgy₁ + ½ρv₁² = P₂ + ρgy₂ + ½ρv₂²

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

q = ½ρv₁²

applications

The space shuttle and "Max. Q".

The human circulatory system.

Blood flow in the systemic circulatory system
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