Flow

Keypoints

For laminar flow

• Hagen-Poiseuille equation
  i.e. Flow = Pressure x r⁴pi/8nl
• Resistance proportional to viscosity
  * Not affected by density
• Viscosity make laminar flow more likely

For turbulent flow

• R changes with flow
• Resistance affected by density
  * Not affected by viscosity
• Density make turbulent flow more likely

Laminar flow

Principle

• Laminar flow occurs when stream lines of the flow are parallel to the sides of the tube
• Laminar flow proportional to 1/viscosity
• Laminar flow proportional to the 4th power for radius
• Axial flow moves twice as fast as mean velocity
• Longitudinal velocity profile is of a parabola.
• Density of gas is NOT relevant to laminar flow rate

Flow calculation

For pure laminar flow,
--> Pressure (cmH₂O) = Flow (L/sec) x Resistance

where

• Resistance = 8 nl/r⁴pi
  * n = viscosity
  * l = length
  * r = radius
  * unit = cmH₂O.L^-1.sec
  * unit = kPa.L^-1.sec

Hagen-Poiseuille equation

Laminar flow = Pressure x r⁴(pi)/8nl

• Only applies to
  * Newtonian fluids (motion does not affect dynamic viscosity)
  * Steady flow
  * Laminar flow
• Blood is non-Newtonian and viscosity changes with flow

Turbulent flow

Principle

• Resistance is NOT constant
  * R changes with flow
• Pressure proportional to approx. (flow rate)²
• Greater resistance when flow is turbulent, everything else being equal.
• Pressure proportional density of gas
• Pressure inversely proportional to the 5th power of radius
  * i.e. Fanning equation
• Viscosity of gas is NOT relevant to turbulent flow rate

Reynold's number (Re)

Formula

Re = DV x (d/n)

• D = diameter
• V = mean velocity
• d = density
• n = viscosity
• Dimensionless (no units)

Significance

• When Re>3000
  --> predominantly turbulent flow
• When Re<2000
  --> predominantly laminar flow

Thus,

• Ratio of density:viscosity (d:n) are the inherent properties of gas that contribute to Re
• Turbulence more likely with higher velocity and diameter

Additional notes

Conductance

Conductance = 1/resistance
* Unit for conductance = L.cmH₂O^-1.sec^-1

Entrance length

Entrance length - the distance required for laminar flow to be established
= 0.03 x diameter x Re

Viscosity

Viscosity
= ratio of shear stress to shear rate of the fluid

Unit for viscosity
--> dynes/cm²