Engineering
Hagen-Poiseuille Laminar Flow formula: R^4 radius law & hydraulic resistance
Calculate ideal fully developed laminar flow, hydraulic resistance and a Reynolds-number applicability check for a straight round tube.
Radius-to-the-fourth scaling within the laminar model
Assume steady, fully developed, incompressible Newtonian flow in a straight rigid circular tube with no slip. Radius is internal radius in metres; dynamic viscosity is in Pa·s, length in metres and signed pressure drop in pascals. The formula is not a clinical model of arteries or an infusion-device specification.
B6 supplies fluid density for a Reynolds-number check using the predicted mean velocity and diameter2R. This exercise rejects predicted Re>=2000 as outside its selected laminar range. Passing that check alone does not establish fully developed flow or all other assumptions.
The revised example uses100Pa and density1000kg/m³: predicted Re400 and flow1.2566mL/s. The previous1000Pa input would predict Re4000, inconsistent with the assumed laminar regime. Halving radius at fixed pressure and other parameters divides modeled flow by16.
| Capillary Parameter | Cell | SI Input | Flow Output |
|---|---|---|---|
| Internal Radius R (m) | B2 | 0.002 m (2 mm) | R^4 = 1.6E-11 m^4 |
| Pressure Drop dP (Pa) | B3 | 100 Pa | Driving pressure head |
| Viscosity mu & Length L | B4 = 0.001 Pa*s, B5 = 0.5 m | Assumed constant viscosity, 0.5 m tube | 8 * mu * L = 0.004 |
| Flow Rate Q (mL/s) | Poiseuille Law | 0.4 * PI() = 1.2566 mL/s | At radius1mm:0.0785mL/s |
Hagen-Poiseuille laminar flow rate in mL/s (cm^3/s)
Multiply m³/s by1,000,000 to obtain mL/s. The sign follows the chosen pressure-drop direction.
=IF(OR(COUNT(B2:B6)<>5,B2<=0,B4<=0,B5<=0,B6<=0),"Invalid inputs",IF(B6*ABS(B3)*B2^3/(4*B4^2*B5)>=2000,"Outside laminar model",ROUND(1000000*PI()*B2^4*B3/(8*B4*B5),4)))Hydraulic resistance in Pa·s/m³
This expression returns resistance, not pressure. Multiply unrounded resistance by a target flow in m³/s to get the required pressure difference, then check applicability again.
=IF(OR(COUNT(B2:B6)<>5,B2<=0,B4<=0,B5<=0,B6<=0),"Invalid inputs",ROUND((8 * B4 * B5) / (PI() * (B2 ^ 4)), 0))Check predicted Reynolds number
Use the magnitude of velocity for Reynolds number; flow direction does not change the regime check.
=IF(OR(COUNT(B2:B6)<>5,B2<=0,B4<=0,B5<=0,B6<=0),"Invalid inputs",ROUND(B6*ABS(B3)*B2^3/(4*B4^2*B5),2))Try the example
Use the original values or change them to test your understanding.