Stoke's law
Understanding Stoke's Law
Stoke's law describes the force of friction experienced by spherical objects moving through a viscous fluid. It is an important concept in the field of fluid dynamics and is widely used in various scientific and engineering applications.
The Law
Stoke's law is given by the formula:
$$ F_d = 6 \pi \eta r v $$
where:
- $F_d$ is the drag force due to viscosity,
- $\eta$ is the dynamic viscosity of the fluid,
- $r$ is the radius of the spherical object,
- $v$ is the velocity of the object relative to the fluid.
The law assumes that the flow around the object is laminar and that the Reynolds number is low (typically less than 1).
Key Points of Stoke's Law
- Applicability: It applies to spherical objects in a fluid.
- Viscosity: The fluid must have a known viscosity.
- Velocity: The object must have a steady velocity.
- Laminar Flow: The flow around the object must be laminar, not turbulent.
- Reynolds Number: The Reynolds number should be low.
Differences and Important Points
| Aspect | Description |
|---|---|
| Applicability | Only valid for spherical objects in a viscous fluid. |
| Flow Type | Assumes laminar flow around the object. |
| Reynolds Number | Effective for low Reynolds numbers (< 1). |
| Velocity | The object should move with a constant velocity. |
| Viscosity Dependence | The drag force is directly proportional to the fluid's viscosity. |
| Size Dependence | The drag force is directly proportional to the radius of the spherical object. |
| Velocity Dependence | The drag force is directly proportional to the velocity of the object. |
Examples
Example 1: Calculating Drag Force
Suppose a small spherical bead with a radius of $0.001$ m is moving through water with a velocity of $0.01$ m/s. The dynamic viscosity of water at room temperature is approximately $0.001$ Pa·s. Using Stoke's law, we can calculate the drag force:
$$ F_d = 6 \pi \eta r v $$ $$ F_d = 6 \pi (0.001 \text{ Pa·s}) (0.001 \text{ m}) (0.01 \text{ m/s}) $$ $$ F_d = 6 \pi \times 10^{-8} \text{ N} $$ $$ F_d \approx 1.88 \times 10^{-7} \text{ N} $$
Example 2: Terminal Velocity
When an object falls through a fluid, it eventually reaches a terminal velocity where the drag force equals the gravitational force. For a spherical object falling in a viscous fluid, we can use Stoke's law to find this terminal velocity.
The gravitational force is given by:
$$ F_g = mg $$
where $m$ is the mass of the object and $g$ is the acceleration due to gravity. At terminal velocity, $F_d = F_g$, so:
$$ 6 \pi \eta r v_t = mg $$
Solving for the terminal velocity $v_t$:
$$ v_t = \frac{mg}{6 \pi \eta r} $$
If we know the mass, radius, and viscosity, we can calculate the terminal velocity.
Example 3: Experimentally Determining Viscosity
Stoke's law can be used to determine the viscosity of a fluid experimentally. By measuring the terminal velocity of a spherical object of known size and density, we can rearrange Stoke's law to solve for viscosity:
$$ \eta = \frac{mg}{6 \pi r v_t} $$
By conducting experiments with different fluids and spherical objects, we can calculate the viscosity of those fluids.
Conclusion
Stoke's law is a fundamental principle in fluid dynamics that provides a relationship between the drag force on a spherical object and its velocity through a viscous fluid. It is essential for understanding the behavior of particles in fluids and has applications in engineering, geophysics, biophysics, and other fields. However, it is crucial to remember that Stoke's law has limitations and is only accurate under certain conditions, such as low Reynolds numbers and laminar flow.