# (p.258) Appendix A The Applicability of Stokes’ Law

# (p.258) Appendix A The Applicability of Stokes’ Law

In the transcription of the 1911–1912 lectures of H.A. Lorentz (1921) on kinetic problems, there is a section in which Lorentz considers the validity of Stokes’ law for the description of Brownian motion. Since this appears to be very little known the present author’s translation of the relevant section is presented here.^{40}

# §7. Investigation of the validity of Stokes’ law for Brownian motion

One can now ask the question, what the resistance would be if the sphere had an arbitrarily changing motion. This is important for Brownian motion. One must, however, restrict oneself to low velocities, for in the calculation the terms **u** *∂* **u**/*∂x* and so forth^{41} were neglected. If one knew the velocity as a function of time for the duration of the experiment, then one could develop this in a Fourier series and apply the result (24).^{42} Through this, the resistance will then be known. However, this is not well adapted to a general discussion.

The formula of Stokes may not be applied to Brownian motion; the motion is much too fast for that. In (24), the term *R(nρ*/2*µ*)^{1/2} may be neglected with respect to 1 when the period, *T*, is large compared to *πρ* R^{2} /*µ*. Let *πρ* R^{2}/*µ* = *θ*. Then, for very slow vibrations, where the period *T* is large with respect to *θ*, Stokes law may be applied, and this holds also for other motions in which the velocity changes but little in the time *θ*.

As an example of this, we may check whether Stokes’ law may be applied to the decay of the motion of a sphere as a consequence of the friction in the fluid. If *m* is the mass and *v* the velocity of the sphere, and we apply Stokes’ law, then

so that

*v* drops to 1/*e* of *v* _{0} after the time *m*/6 *πµ*R. If the density of the sphere is *ρ* _{1}then this time is
(p.259)

If *ρ* and *ρ* _{1} are comparable with each other, then we see that *θ* and this time *τ* are of the same order of magnitude. Stokes’ law may then not be applied to the decay of the motion.

For granules which undergo Brownian motion,*θ* becomes very small. If *R*= 5 × 10^{-5},*µ* = 18 × 10^{-5} then *θ* = *π* × 25 × 10^{-5}/18 × 10^{-5} × *ρ* = 4 × 10^{-5} *ρ* approximately.

Certainly the motion of suspended particles will change very strongly in this time so that the details of the Brownian motion cannot be calculated with Stokes’ law. In many other cases, however, the motion will change little in the time *θ* and the law can be applied.^{43}

## Notes:

(40) This material is presented with the permission of Koninklijke Brill NV, Leiden, The Netherlands

(41) (Translator’s note) This refers to the nonlinear terms in the Navier–Stokes equations for the fluid.

(42)
Equation (24) of the original reads ‘Thus the resistance is — 6 *π*R*w* _{bol}(1+R√(*ηρ*/2*µ*)).’ *R* is the sphere radius, *µ* the viscosity, *ρ* the fluid density, *η* the vibrational frequency, and *w _{bol}* =

*w*

_{0}exp(

*int*).