12.6 Motion of an Object in a Viscous Fluid

Motion of an Object in a Viscous Fluid

When an object moves through a fluid, the fluid pushes back. How hard it pushes, and in what way, depends on whether the flow around the object is smooth or chaotic. The Reynolds number is the tool that tells you which type of flow to expect, and that distinction determines which drag equation you'll use and how the object reaches terminal speed.

Reynolds Number: Calculation and Interpretation

The Reynolds number ($Re$) is a dimensionless quantity that compares inertial forces (the fluid's tendency to keep moving) to viscous forces (the fluid's internal resistance to flow). That ratio tells you whether flow will be orderly or chaotic.

$Re = \frac{\rho v D}{\mu}$

• $\rho$ = fluid density (kg/m³)
• $v$ = object's velocity relative to the fluid (m/s)
• $D$ = characteristic length, such as diameter for a sphere (m)
• $\mu$ = dynamic viscosity of the fluid (Pa·s)

You can also write this using kinematic viscosity $\nu = \mu / \rho$:

$Re = \frac{vD}{\nu}$

How to interpret the result:

• $Re < 2300$: Laminar flow. Viscous forces dominate, so the flow is smooth and predictable. Think of a tiny sphere sinking slowly through honey.
• $2300 < Re < 4000$: Transitional flow. The flow is unstable and can flicker between laminar and turbulent behavior.
• $Re > 4000$: Turbulent flow. Inertial forces dominate, producing chaotic motion with eddies and vortices. Think of wind swirling around a building.

These threshold values (2300 and 4000) apply specifically to flow in pipes. For objects moving through open fluid, the transition Reynolds number can differ, but the general idea is the same: low $Re$ means smooth, high $Re$ means chaotic.

Laminar vs. Turbulent Flow

Laminar flow means the fluid moves in parallel layers that slide past each other without mixing.

• The velocity profile is parabolic: fastest at the center, zero at the boundary (the no-slip condition).
• Drag force is directly proportional to velocity. For a sphere, this is given by Stokes' law:

$F_D = 6\pi \mu R v$

where $R$ is the sphere's radius and $v$ is its speed. Because drag scales linearly with $v$, doubling the speed doubles the drag.

• Biological example: blood flow through capillaries, where vessels are tiny and flow speeds are low, keeping $Re$ well below 2300.

Turbulent flow means the fluid layers mix chaotically, forming eddies and vortices.

• The velocity profile is more uniform across the flow, with a thin boundary layer near the object's surface where velocity drops to zero.
• Drag force is proportional to the square of velocity:

$F_D = \frac{1}{2} C_D \rho A v^2$

where $C_D$ is the drag coefficient (depends on shape and $Re$), $\rho$ is fluid density, and $A$ is the object's cross-sectional area. Here, doubling the speed quadruples the drag.

• Everyday example: air resistance on a car at highway speed, or water rushing through a large pipe.

Terminal Speed in Viscous Fluids

An object falling through a fluid accelerates until the net downward force reaches zero. At that point, it moves at a constant terminal speed ($v_t$). Three forces are in play:

• Gravity pulling down: $F_g = mg$
• Buoyancy pushing up: $F_B = \rho_f g V$, where $\rho_f$ is the fluid density and $V$ is the object's volume
• Drag pushing up (opposing motion): either the Stokes or turbulent drag expression

At terminal speed, the forces balance:

$F_D = F_g - F_B = (\rho_o - \rho_f) g V$

where $\rho_o$ is the object's density. Notice that buoyancy effectively reduces the "weight" the drag force must support. If the object is less dense than the fluid, it won't sink at all.

Finding terminal speed in laminar flow (Stokes' regime):

Set Stokes' drag equal to the net downward force:

$6\pi \mu R v_t = (\rho_o - \rho_f) g V$

For a sphere with $V = \frac{4}{3}\pi R^3$, you can solve for:

$v_t = \frac{2(\rho_o - \rho_f) g R^2}{9\mu}$

This is useful for small, slow-moving objects like sediment particles settling in water or red blood cells sinking in plasma (the basis of the ESR medical test).

Finding terminal speed in turbulent flow:

Set the turbulent drag equal to the net downward force:

$\frac{1}{2} C_D \rho_f A v_t^2 = (\rho_o - \rho_f) g V$

Solving for $v_t$:

$v_t = \sqrt{\frac{2(\rho_o - \rho_f) g V}{C_D \rho_f A}}$

A skydiver in free fall is a classic example. With arms and legs spread (larger $A$, higher $C_D$), terminal speed is roughly 55 m/s. In a head-down dive (smaller $A$), it can exceed 70 m/s.

Shear Stress and Bernoulli's Principle

Two additional ideas connect to how objects interact with viscous fluids:

Shear stress is the force per unit area that acts parallel to a fluid surface. When fluid flows past an object, shear stress at the surface creates the boundary layer, the thin region where the fluid velocity transitions from zero (at the surface) to the free-stream speed. This is the physical origin of viscous drag.

Bernoulli's principle states that where fluid speed increases, pressure decreases (for steady, incompressible flow along a streamline). This helps explain why a spinning baseball curves: the ball drags air faster on one side, lowering the pressure there and creating a net sideways force. The same principle contributes to lift on an airplane wing, though the full explanation also involves the deflection of airflow downward.