Poiseuille's Law and Reynolds Number

Poiseuille's law describes laminar flow through a horizontal tube, defining flow rate as the ratio of pressure difference to flow resistance. The law shows that flow rate increases with greater pressure differences and decreases with higher resistance. Resistance depends on fluid viscosity, tube length, and tube radius raised to the fourth power, making tube diameter a critical factor in determining flow.

Tube radius has a dramatic effect on flow rate because resistance is inversely proportional to the fourth power of the radius. A slight decrease in radius causes a significant reduction in flow rate. To maintain the same flow rate through a narrower tube, the pressure difference must increase substantially, demonstrating the nonlinear relationship between tube dimensions and flow.

Viscosity directly affects flow resistance; more viscous fluids experience greater resistance and flow more slowly through tubes at constant temperature and pressure. Flow resistance is directly proportional to the coefficient of viscosity. In non-viscous fluids, resistance is zero and all fluid layers move at the same velocity, whereas viscous fluids show velocity gradients with fastest flow at the center.

The Reynolds number is a dimensionless value that classifies flow behavior. A Reynolds number below 2,000 indicates laminar flow, while above 3,000 indicates turbulent flow. Between 2,000 and 3,000, flow is unstable and may show chaotic behavior, oscillating randomly between laminar and turbulent states due to small variations in conditions.

In the transition zone between Reynolds numbers 2,000 and 3,000, tiny variations in factors like surface roughness or obstructions can cause exaggerated, nonlinear effects on the flow system. The flow initially remains laminar but can become turbulent due to disturbances, causing it to oscillate randomly between laminar and turbulent states, demonstrating chaotic behavior.

Fluids flow from high pressure to low pressure regions. The flow rate is directly proportional to the pressure difference between two points in a horizontal tube. Greater pressure differences produce higher flow rates, while smaller pressure differences result in slower flow, provided all other factors like tube dimensions and fluid viscosity remain constant.

Flow resistance encompasses all factors affecting flow rate except pressure difference itself. Resistance is directly proportional to tube length and fluid viscosity, and inversely proportional to the fourth power of tube radius. These factors combine to determine how much resistance a fluid encounters when flowing through a tube under a given pressure gradient.