Poiseuille Flow Calculator 

Poiseuille’s Law (Flow Rate): \\( Q = \frac{\pi R^4 \Delta P}{8 \mu L} \\)

Velocity Profile: \\( v(r) = \frac{\Delta P (R^2 – r^2)}{4 \mu L} \\)

Where:

• \\(Q\\): Volumetric flow rate (m³/s)
• \\(v(r)\\): Velocity at radial distance \\(r\\) (m/s)
• \\(\Delta P\\): Pressure drop (Pa)
• \\(R\\): Pipe radius (m)
• \\(L\\): Pipe length (m)
• \\(\mu\\): Dynamic viscosity (Pa·s)

Example 1: Water in Small Pipe
\\(\Delta P = 1000 \, \text{Pa}\\), \\(R = 5 \, \text{mm}\\), \\(L = 1 \, \text{m}\\), \\(\mu = 0.001 \, \text{Pa·s}\\)
Flow Rate: \\( Q = \frac{\pi (0.005)^4 \times 1000}{8 \times 0.001 \times 1} \approx 2.454 \times 10^{-7} \, \text{m}^3/\text{s} \\)
Max Velocity (at \\(r = 0\\)): \\( v(0) = \frac{1000 \times (0.005)^2}{4 \times 0.001 \times 1} \approx 0.00625 \, \text{m/s} \\)

Example 2: Oil in Larger Pipe
\\(\Delta P = 5000 \, \text{Pa}\\), \\(R = 10 \, \text{mm}\\), \\(L = 2 \, \text{m}\\), \\(\mu = 0.1 \, \text{Pa·s}\\)
Flow Rate: \\( Q = \frac{\pi (0.01)^4 \times 5000}{8 \times 0.1 \times 2} \approx 9.817 \times 10^{-6} \, \text{m}^3/\text{s} \\)
Max Velocity: \\( v(0) = \frac{5000 \times (0.01)^2}{4 \times 0.1 \times 2} \approx 0.625 \, \text{m/s} \\)

Example 3: Blood in Capillary
\\(\Delta P = 2000 \, \text{Pa}\\), \\(R = 0.1 \, \text{mm}\\), \\(L = 0.01 \, \text{m}\\), \\(\mu = 0.004 \, \text{Pa·s}\\)
Flow Rate: \\( Q = \frac{\pi (0.0001)^4 \times 2000}{8 \times 0.004 \times 0.01} \approx 1.963 \times 10^{-12} \, \text{m}^3/\text{s} \\)
Max Velocity: \\( v(0) = \frac{2000 \times (0.0001)^2}{4 \times 0.004 \times 0.01} \approx 0.125 \, \text{m/s} \\)

Example 4: Air in Duct
\\(\Delta P = 100 \, \text{Pa}\\), \\(R = 50 \, \text{mm}\\), \\(L = 5 \, \text{m}\\), \\(\mu = 1.8 \times 10^{-5} \, \text{Pa·s}\\)
Flow Rate: \\( Q = \frac{\pi (0.05)^4 \times 100}{8 \times 1.8 \times 10^{-5} \times 5} \approx 0.00136 \, \text{m}^3/\text{s} \\)
Max Velocity: \\( v(0) = \frac{100 \times (0.05)^2}{4 \times 1.8 \times 10^{-5} \times 5} \approx 694.44 \, \text{m/s} \\)