Stokes Flow Simulator
Stokes Flow Simulator calculates drag force on a sphere in low Reynolds number flow using viscosity, radius, and velocity, with velocity profile plots.
Stokes Flow Overview
Stokes flow describes the motion of a sphere in a viscous fluid at low Reynolds number, where drag force is given by Stokes’ law:
Stokes’ Law: \( F_d = 6 \pi \mu r v \)
Reynolds Number: \( Re = \frac{\rho v (2r)}{\mu} \)
Where:
- \(F_d\): Drag force (N)
- \(\mu\): Fluid viscosity (Pa·s)
- \(r\): Sphere radius (m)
- \(v\): Sphere velocity (m/s)
- \(\rho\): Fluid density (kg/m³)
- \(Re\): Reynolds number (dimensionless, should be \(< 1\) for Stokes flow)
Example Calculations
Example 1: Water Droplet in Air
\(\mu = 0.001 \, \text{Pa·s}\), \(r = 1 \, \text{mm}\), \(v = 0.01 \, \text{m/s}\), \(\rho = 1.225 \, \text{kg/m}^3\)
Drag Force: \( F_d = 6 \pi \times 0.001 \times 0.001 \times 0.01 \approx 1.885 \times 10^{-7} \, \text{N} \)
Reynolds Number: \( Re = \frac{1.225 \times 0.01 \times (2 \times 0.001)}{0.001} \approx 0.0245 \)
Example 2: Oil Droplet in Water
\(\mu = 0.001 \, \text{Pa·s}\), \(r = 0.5 \, \text{mm}\), \(v = 0.005 \, \text{m/s}\), \(\rho = 1000 \, \text{kg/m}^3\)
Drag Force: \( F_d = 6 \pi \times 0.001 \times 0.0005 \times 0.005 \approx 4.712 \times 10^{-8} \, \text{N} \)
Reynolds Number: \( Re = \frac{1000 \times 0.005 \times (2 \times 0.0005)}{0.001} \approx 5 \)
Example 3: Particle in Glycerin
\(\mu = 1.5 \, \text{Pa·s}\), \(r = 1 \, \text{mm}\), \(v = 0.001 \, \text{m/s}\), \(\rho = 1260 \, \text{kg/m}^3\)
Drag Force: \( F_d = 6 \pi \times 1.5 \times 0.001 \times 0.001 \approx 2.827 \times 10^{-5} \, \text{N} \)
Reynolds Number: \( Re = \frac{1260 \times 0.001 \times (2 \times 0.001)}{1.5} \approx 0.00168 \)
Example 4: Microbead in Water
\(\mu = 0.001 \, \text{Pa·s}\), \(r = 0.1 \, \text{mm}\), \(v = 0.02 \, \text{m/s}\), \(\rho = 1000 \, \text{kg/m}^3\)
Drag Force: \( F_d = 6 \pi \times 0.001 \times 0.0001 \times 0.02 \approx 3.770 \times 10^{-8} \, \text{N} \)
Reynolds Number: \( Re = \frac{1000 \times 0.02 \times (2 \times 0.0001)}{0.001} \approx 4 \)