Viscous Forces Become Dominant for Small Particles in Fluid Media

The motion of a mass in a fluid, such as air or water, eventually changes from iner-tial to diffusive as the mass of the moving object is reduced. Newton's Laws (inertial) are a good starting point for the motions of artillery shells and baseballs, even though these masses move through a viscous medium, the atmosphere. The first corrections for air resistance are usually velocity-dependent drag forces. A completely different approach has to taken for the motion of a falling leaf or for the motion of a microscopic mass in air or in water.

The most relevant property of the medium is the viscosity r], defined in terms of the force F = rjvA/z necessary to move a flat surface of area A parallel to an extended surface at a spacing z and relative velocity v in the medium in question. The unit of viscosity rj is the Pascal-second (one Pascal is a pressure of 1 N/m2). The viscosity of air is about 0.018 x 10-3 Pa ■ s, while the value for water is about 1.8 x 10-3 Pa • s. The traditional unit of viscosity, the Poise, is 0.1 Pa - s in magnitude.

The force needed to move a sphere of radius R at a velocity v through a viscous medium is given by Stokes' Law,

This is valid only for very small particles and small velocities, under conditions of streamline flow such that the Reynolds number NReynoids is less than approximately 2000. NReynoids> which is dimensionless, is defined as JVReynoids = 2.Rpi;/77> where R is the radius, p the mass density, v the velocity and rj the viscosity.

2.5 Viscous Forces Become Dominant for Small Particles in Fluid Media

The fall, under the acceleration of gravity g, of a tiny particle of mass m in this regime is described, following Stokes' Law, by a limiting velocity obtained by setting F (from equation 2.6) equal to mg. This gives v=mg/6mjR. (2.8)

As an example, a particle of 10 jam radius and density 2000 kg/m3 falls in air at about 23 mm/s, while a 15 nm particle of density 500 kg/m3 will fall in air at about 13 nm/s. In the latter case one would expect random jostling forces/(t) on the particle by impacts with individual air molecules (Brownian motion) to be present as well as the slow average motion. Newton's laws of motion as applied to the motion of artillery shells are not useful in such cases, nor for any cases of cells or bacteria in fluid media.

An appropriate modification of Newton's Second Law for such cases is the Lange-vin equation [3]

Fext +f(t) = [4jzpR3/3]d2x/dt2 + 6jtrjR dx/dt. (2.9)

This equation gives a motion x(t) which is a superposition of drift at the terminal velocity (resulting, as above, from the first and last terms in the equation) and the stochastic diffusive (Brownian) motion represented by f(t).

In the absence of the external force, the diffusive motion can be described by

where

is the diffusivity of the particle of radius R in a fluid of viscosity rj at temperature T. Consideration of the exponential term allows one to define the "diffusion length" as xims=(4Dt)1/2. (2.12)

Returning to the fall of the 15 nm particle in air, which exhibits a drift motion of 13 nm in one second, the corresponding diffusion length for 300 K is xrms=2D1/2 = 56jim. It is seen that the diffusive motion is dominant in this example.

The methods described here apply to slow motions of small objects where the related motion of the viscous medium is smooth and not turbulent. The analysis of diffusion is more broadly applicable, for example, to the motion of electrons in a conductor, to the spreading of chemical dopants into the surface of a silicon crystal at elevated temperatures, and to the motion of perfume molecules through still air.

In the broader but related topic of flying in air, a qualitative transition in behavior is observed in the vicinity of 1 mm wingspan. Lift forces from smooth flow over air foil surfaces, which derive from Bernoulli's principle, become small as the scale is reduced. The flight of the bumblebee is not aerodynamically possible, we are told, and the same conclusion applies to smaller flying insects such as mosquitoes and gnats. In these cases the action of the wing is more like the action of an oar as it is forced against the relatively immovable water. The reaction force against moving the viscous and massive medium is the force that moves the rowboat and also the force that lifts the bumblebee.

No tiny airplane can glide, another consequence of classical scaling. A tiny airplane will simply fall, reaching a terminal velocity that becomes smaller as its size is reduced.

These considerations only apply when one is dealing with small particles in a liquid or a gas. They do not apply in the prospect of making smaller electronic devices in silicon, for example.

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