An object moving through a viscous fluid behaves similarly to a stationary object experiencing a moving fluid stream. For example, when riding a bicycle at 10 m/s on a windless day, the sensation of air resistance is the same as if you were stationary and experiencing a 10 m/s wind. In both cases, the interaction between the fluid and the object depends on the relative motion between the two.

Just like in tubes or vessels, the flow around a moving object can be laminar, turbulent, or transitional. To predict the flow regime, we use a form of the Reynolds number that applies to moving objects. This version is defined as:

where:

• [latex]\rho[/latex] is the fluid density,
• [latex]v[/latex] is the speed of the object relative to the fluid,
• [latex]L[/latex] is a characteristic length of the object (such as its diameter),
• [latex]\eta[/latex] is the fluid’s viscosity.

This dimensionless quantity helps determine whether the flow around the object is smooth (laminar) or chaotic (turbulent). General guidelines include:

• If [latex]{N'}_{R} 1[/latex]: Flow around the object is generally laminar—common for small, smooth objects like blood cells or pollen in plasma.
• If [latex]{N'}_{R}[/latex] is between 1 and 10: Transitional flow may begin, influenced by surface roughness and fluid properties. A turbulent wake may form behind the object, while the front remains laminar.
• If [latex]{N'}_{R}[/latex] is between 10 and [latex]10^6[/latex]: Flow can oscillate unpredictably between laminar and turbulent, depending on object shape and surface texture.
• If [latex]{N'}_{R} > 10^6[/latex]: Flow is fully turbulent, even at the surface of the object.

Laminar flow around small biological particles is common in physiological systems—for instance, with red blood cells in capillaries or tiny aerosols in respiratory airflow. In contrast, larger or faster-moving objects (e.g., medical tools in fluid environments or falling drops) may experience turbulent flow, affecting how forces like drag and lift act on them.

See Figure 90.1 for an illustration of flow transitions around an object.

A key consequence of viscosity is the resistive force known as viscous drag, denoted by [latex]{F}_{\text{V}}[/latex]. This drag force opposes the motion of an object moving through a fluid and differs from dry friction—it typically depends on the object’s speed and the characteristics of the fluid.

In Figure 90.1, we see how viscous drag changes under different flow conditions. When the Reynolds number [latex]{N'}_{R}[/latex] is less than 1 (laminar flow), viscous drag is directly proportional to speed. As [latex]{N'}_{R}[/latex] increases into the range of 10 to [latex]10^6[/latex], the drag force becomes roughly proportional to the square of speed. This stronger dependence explains why even a mild headwind can substantially increase drag for cyclists or swimmers. For [latex]{N'}_{R} > 10^6[/latex], the flow is entirely turbulent, and drag increases dramatically, behaving nonlinearly.

For slow, smooth motion through a viscous fluid—such as a small particle moving in plasma—we can use the special case known as Stokes’ law, which relates the viscous drag force on a small sphere to the viscosity of the fluid and the sphere’s size and speed:

Here, [latex]R[/latex] is the radius of the sphere, [latex]\eta[/latex] is the fluid viscosity, and [latex]v[/latex] is the object’s speed. This relationship explains how larger objects and more viscous fluids both lead to greater drag. These principles are used in medical and laboratory applications, such as estimating blood cell sedimentation rates or designing flow-based diagnostic devices.

Terminal Speed in Biological Systems

As an object falls through a viscous fluid, it experiences increasing viscous drag, which eventually balances the downward gravitational force. At this point, the object stops accelerating and continues falling at a constant speed known as the terminal speed. This phenomenon is observed in many biological and medical contexts—for example, sedimentation of blood cells in a centrifuge or the descent of respiratory droplets through air.

Terminal speed depends on three forces: the object’s weight [latex]w[/latex], the upward viscous drag [latex]{F}_{\text{V}}[/latex], and the upward buoyant force [latex]{F}_{\text{B}}[/latex]. These forces are illustrated in Figure 90.2. The terminal speed is highest when the object has a high density and small size and is falling through a low-viscosity fluid.

Understanding terminal speed is useful in clinical and laboratory settings. For example, centrifuges artificially increase effective gravity to speed up sedimentation of cells or particles. This technique is used in separating plasma from blood cells, analyzing urine, or preparing lab samples.

Section Summary

• When an object moves through a fluid, the Reynolds number is given by:
  [latex]{N'}_{\text{R}} = \frac{\rho v L}{\eta}[/latex]
• For [latex]{N'}_{R} 1[/latex], flow is laminar and drag is proportional to speed.
• For [latex]{N'}_{R} > 10^6[/latex], flow is turbulent and drag increases dramatically.
• Terminal speed occurs when drag and buoyant forces balance the object’s weight, resulting in constant velocity.