Laminar Flow

 Laminar flow.jpg

Properties of laminar flow of gas or fluids:

Smooth unobstructed flow of gas through a tube of relatively uniform diameter

Few directional changes

Slow, steady flow through straight smooth, rigid, large caliber, cylindrical tube

Outer layer flow slower than center due to friction, results in discrete cylindrical layers, or streamlines

Double flow by doubling pressure as long as the flow pattern remains laminar.

 

 Poiseuille_abstraction.JPG

a) A tube showing the imaginary lamina. b) A cross section of the tube shows the lamina moving at different speeds. Those closest to the edge of the tube are moving slowly while those near the center are moving quickly.

 

Poiseuille's Law

Relates factors that determine laminar flow

Indicates degree of resistance to fluid flow through a tube

The resistance, (or pressure gradient), to fluid flow through a tube is directly related to the length, flow, and viscosity (as length, flow, or viscosity increases the resistance will also increase) ; and inversely (as radius increases, resistance decreases) related to the radius of the tube to the fourth power. (doubling radius increases gas flow by factor of 16)

If r = 3 then r4 = 81

If r = 6 (radius doubles) then r4 = 1296 (same as multiplying 81 x 16)

If r = 1.5 (radius reduced to half original value) then r4 = 5.06 (1/16th of original value)

This is very significant in disease states that cause narrowing of the airways because it can significantly increase the amount of work the patient has to do to get air into their lungs - particularly in children whose airways are small to start with! Conditions that can cause airways to narrow include asthma, croup, and edema caused by inflammation.

 

Variables and constants of Poiseuille's Law:

Variables can be rearranged to solve for different components:

To solve for viscosity the equation is arranged:

 

To solve for flowrate the equation is arranged:

 

To solve for driving pressure the equation is arranged:

 

In general, the relationships between the variables is:

(a decrease in flowrate will result in a decrease in resistance, and decrease driving pressure)

 

(a decrease in the radius of the tube, will cause an increase in the resistance to flow - exponentially by a power of 4, if driving pressure is constant, flow will decrease; to maintain flow will need to increase pressure)

 

(an increase in the length of the tube will result in an increase in resistance, if driving pressure is unchanged then flow will decrease; to maintain constant flow pressure will have to be increased)

 

(if all other factors are held constant, and increase in the driving pressure will increase flowrate

 

Key Points:

The more viscous a fluid, the greater the pressure gradient required to cause it to move through a given tube.

The resistance offered by a tube is directly proportional to its length; the pressure required to achieve a given flow through a tube must increase in direct proportion to the length of the tube.

Because the resistance of the flow is inversely proportional to the fourth power of the radius, small decreases in the radius of a tube cause profound decreases in the flow of the fluid through the tube.