How do you calculate viscous drag?

Viscous drag, also known as fluid friction, is typically calculated for small, spherical objects moving slowly through a fluid using Stokes' Law. This law provides a direct method to determine the resistive force exerted by a fluid on an object due to its viscosity.

The exact formula for calculating viscous drag under these specific conditions is:

F = 6πηrv

Where:

• F represents the viscous drag force, measured in Newtons (N).
• η (eta) is the coefficient of viscosity of the fluid, indicating its resistance to flow. It is measured in Newton seconds per square meter (N s m⁻²) or Pascal seconds (Pa s).
• r is the radius of the object, measured in meters (m). This formula applies specifically to spherical objects.
• v is the velocity of the object relative to the fluid, measured in meters per second (ms⁻¹).

Understanding Viscous Drag

Viscous drag is a type of fluid resistance that arises from the internal friction within a fluid (viscosity) and the friction between the fluid and the surface of a moving object. Unlike pressure drag (form drag), which is caused by pressure differences around an object, viscous drag is dominant at low speeds and for objects with large surface areas relative to their volume.

When is Stokes' Law Applicable?

Stokes' Law is an idealization that accurately describes viscous drag under very specific conditions:

• Small Spherical Objects: The formula is derived for perfectly spherical particles.
• Laminar Flow (Low Reynolds Number): The fluid flow around the object must be smooth and undisturbed, with no turbulence. This typically occurs at very low velocities. The Reynolds number, a dimensionless quantity, helps determine if flow is laminar (Re < 1 for Stokes' Law).
• Homogeneous Fluid: The fluid must be uniform in its properties.
• Infinite Fluid Medium: The fluid must extend far enough in all directions so that boundaries do not affect the flow around the object.

For larger objects, higher velocities, or non-spherical shapes, the calculation becomes more complex, often involving drag coefficients and the Reynolds number, as other forms of drag (like pressure drag) become significant.

Factors Affecting Viscous Drag

Several key factors influence the magnitude of viscous drag:

• Fluid Viscosity (η): The more viscous a fluid, the greater the drag force. For example, moving an object through honey requires more force than moving it through water due to honey's higher viscosity.
• Object Size (r): Viscous drag is directly proportional to the radius of the object. Larger objects experience greater drag.
• Object Velocity (v): The faster an object moves through a fluid, the greater the viscous drag.
• Object Shape: While Stokes' Law applies to spheres, object shape is crucial for other scenarios. Streamlined shapes reduce drag, while blunt shapes increase it. For viscous drag, surface area in contact with the fluid is particularly important.
• Fluid Temperature: Viscosity of most liquids decreases with increasing temperature, thus reducing viscous drag. Gases, however, typically become more viscous with increasing temperature.

Practical Insights and Examples

Understanding viscous drag is critical in various fields:

• Sedimentation: Stokes' Law is used to determine the settling velocity of small particles (like silt or clay) in water, which is vital in environmental science and wastewater treatment.
• Fluid Mechanics Design: Engineers consider viscous drag when designing lubrication systems, pipelines, and microfluidic devices to optimize flow and minimize energy loss.
• Biophysics: The movement of microorganisms in bodily fluids, or the flow of blood through capillaries, is influenced by viscous forces.
• Aerodynamics/Hydrodynamics (at low speeds): While often dominated by pressure drag, viscous effects are crucial for boundary layer behavior and skin friction, especially for very small objects or at very low speeds.

Example Calculation:

Imagine a tiny spherical dust particle with a radius (r) of 5 micrometers (5 x 10⁻⁶ m) falling through air at a constant velocity (v) of 0.002 ms⁻¹. If the dynamic viscosity of air (η) at that temperature is approximately 1.8 x 10⁻⁵ Pa s.

Using Stokes' Law:
F = 6πηrv
F = 6 π (1.8 x 10⁻⁵ N s m⁻²) (5 x 10⁻⁶ m) (0.002 ms⁻¹)
F ≈ 3.39 x 10⁻¹² N

This tiny force represents the viscous drag acting on the dust particle.

Summary of Viscous Drag Calculation

The following table summarizes the key components for calculating viscous drag using Stokes' Law: