Viscosity and Stokes' Law

Viscosity

In the previous section on fluid dynamics, we discussed ideal fluids which are non-viscous. However, real fluids exhibit viscosity, which is a measure of the internal friction within the fluid. Viscosity is the property of a fluid that opposes the relative motion between different layers of the fluid. It can be thought of as the "thickness" or "stickiness" of the fluid; highly viscous fluids (like honey or tar) flow slowly, while low viscosity fluids (like water or air) flow easily.

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Viscous Force

When a fluid flows, different layers of the fluid move at different velocities. Viscosity gives rise to internal frictional forces (viscous forces) between these layers that oppose this relative motion. These forces are analogous to the friction between solid surfaces in relative motion.

Consider a layer of fluid flowing over another layer. The faster-moving layer exerts a forward tangential force on the slower-moving layer, while the slower-moving layer exerts a backward tangential force on the faster-moving layer. These forces are viscous forces.

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Coefficient of Viscosity ($\eta$)

To quantify viscosity, we define the coefficient of viscosity (often denoted by $\eta$, sometimes $\mu$). Consider a fluid flowing between two parallel plates, where the bottom plate is stationary and the top plate moves with a velocity $v$. The fluid layers near the bottom plate are almost stationary, while the layers near the top plate move with the velocity of the top plate. This creates a velocity gradient perpendicular to the flow direction.

If the distance between the layers is $dz$ and the velocity difference between them is $dv$, the velocity gradient is $\frac{dv}{dz}$. Experiments show that the tangential viscous force $F_v$ between two layers of area $A$ is directly proportional to the area of the layers and the velocity gradient.

$ F_v \propto A \frac{dv}{dz} $

Introducing the constant of proportionality, the coefficient of viscosity $\eta$:

$ F_v = \eta A \frac{dv}{dz} $

This equation is known as Newton's law of viscosity, and fluids that obey this relationship are called Newtonian fluids (e.g., water, air). Some fluids (non-Newtonian fluids like ketchup, paint) have more complex relationships between shear stress and shear rate.

From the definition $ \eta = \frac{F_v/A}{dv/dz} = \frac{\text{Shear Stress}}{\text{Velocity Gradient}} $, the units of viscosity are Pascal-second (Pa·s) in SI units. $1 \text{ Pa} \cdot \text{s} = 1 \text{ N/m}^2 \cdot \text{s} = 1 \text{ kg/(m} \cdot \text{s})$.

A common CGS unit is the poise (P), named after Jean Léonard Marie Poiseuille. $1 \text{ poise} = 0.1 \text{ Pa} \cdot \text{s}$. Centipoise (cP) is often used, where $1 \text{ cP} = 0.01 \text{ poise} = 10^{-3} \text{ Pa} \cdot \text{s}$. The viscosity of water at 20°C is approximately 1 cP.

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Dependence on Temperature

Viscosity is highly dependent on temperature.

• For liquids, viscosity generally decreases with increasing temperature. This is because the increased thermal energy allows molecules to overcome intermolecular forces more easily and slide past each other.
• For gases, viscosity generally increases with increasing temperature. This is because viscosity in gases arises from the momentum transfer between layers due to random molecular motion. At higher temperatures, molecules move faster and collide more frequently, leading to greater momentum exchange and thus higher viscosity.

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Stokes’ Law ($ F_v = 6\pi\eta r v $)

When a small object moves through a viscous fluid, it experiences a viscous drag force that opposes its motion. For a small sphere moving slowly through a viscous fluid in laminar flow, the viscous drag force is given by Stokes' Law.

Stokes' Law states that the viscous drag force ($F_v$) on a small sphere of radius $r$ moving with a velocity $v$ through a fluid of viscosity $\eta$ is directly proportional to the radius of the sphere, its velocity, and the viscosity of the fluid.

Mathematically, the formula is:

$ F_v = 6\pi\eta r v $

where:

• $F_v$ is the magnitude of the viscous drag force (in Newtons, N).
• $\eta$ is the coefficient of viscosity of the fluid (in Pa·s).
• $r$ is the radius of the sphere (in metres, m).
• $v$ is the speed of the sphere relative to the fluid (in m/s).

This formula is valid under certain conditions:

• The sphere is small.
• The speed $v$ is low (so the flow around the sphere is laminar, i.e., Reynolds number is low, $Re = \frac{\rho v (2r)}{\eta} \ll 1$).
• The fluid medium is infinite in extent (no boundaries close to the sphere).

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Terminal Velocity

When a solid object falls through a viscous fluid (like air or water), it is acted upon by three forces:

1. Downward gravitational force (weight): $W = mg = \rho_{object} V_{object} g$.
2. Upward buoyant force: $F_B = \rho_{fluid} V_{object} g$ (assuming full submersion/constant volume).
3. Upward viscous drag force: $F_v = 6\pi\eta r v$ (for a sphere).

Initially, when the object is dropped from rest, its velocity is zero, so the viscous force is zero. As the object accelerates downwards, its speed increases, and the viscous force also increases. Eventually, the viscous force and buoyant force together become equal in magnitude to the gravitational force. At this point, the net force on the object is zero, and its acceleration becomes zero. The object then falls with a constant maximum velocity called the terminal velocity ($v_T$).

At terminal velocity, the forces are balanced:

$ W = F_B + F_v $

$ \rho_{object} V_{object} g = \rho_{fluid} V_{object} g + 6\pi\eta r v_T $

For a sphere of radius $r$, $V_{object} = \frac{4}{3}\pi r^3$.

$ \rho_{object} \left(\frac{4}{3}\pi r^3\right) g = \rho_{fluid} \left(\frac{4}{3}\pi r^3\right) g + 6\pi\eta r v_T $

$ \left(\rho_{object} - \rho_{fluid}\right) \frac{4}{3}\pi r^3 g = 6\pi\eta r v_T $

Solve for $v_T$:

$ v_T = \frac{(\rho_{object} - \rho_{fluid}) \frac{4}{3}\pi r^3 g}{6\pi\eta r} $

$ v_T = \frac{(\rho_{object} - \rho_{fluid}) \frac{4}{3} r^2 g}{6\eta} $

$ v_T = \frac{2 (\rho_{object} - \rho_{fluid}) r^2 g}{9 \eta} $

This formula gives the terminal velocity of a sphere falling through a viscous fluid. The terminal velocity depends on the densities of the object and fluid, the radius of the sphere (squared), gravity, and the viscosity of the fluid. Larger, denser objects fall faster in a viscous medium (have higher terminal velocity) than smaller, less dense objects. Higher fluid viscosity leads to lower terminal velocity.

(Image Placeholder: A sphere falling downwards in a fluid. Downward arrow for Weight W. Upward arrow for Buoyant Force FB. Upward arrow for Viscous Force Fv. At terminal velocity, W = FB + Fv.)

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Applications of Viscosity and Stokes' Law

• Motion of small particles in fluids: Stokes' Law is used to calculate the settling speed of small particles (like dust or sedimentation) in fluids, important in processes like sedimentation, centrifugation, and air pollution studies.
• Viscometry: Viscosity of fluids is measured using viscometers. Some viscometer designs are based on Stokes' Law (e.g., falling ball viscometer).
• Cloud droplets and Raindrops: The terminal velocity of tiny cloud droplets is very small due to air viscosity, keeping them suspended. Raindrops grow larger, increasing their radius and terminal velocity according to Stokes' Law (until they get large enough for flow to become turbulent, where drag force depends on $v^2$).
• Blood Flow: The viscosity of blood affects its flow through blood vessels and contributes to blood pressure.
• Lubrication: Lubricants are viscous fluids used to reduce friction between moving solid surfaces.

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