Einstein's "On the effect on the motion of a liquid of a very small sphere suspended in it"

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Einstein's "On the effect on the motion of a liquid of a very small sphere suspended in it"

Introduction

The success of a molecular kinetic theory of gases enabled scientists to actually make molecular measurements based on observed physical phenomena. However, formulation of a similar theory for liquids proved to be much less easy. This probably motivated Einstein to look at a slightly different problem, a liquid with particles dissolved in it where one might be able to separately treat the solvent as continuous, the solute discrete, and ultimately deduce useful properties for both. In "On the effect on the motion of a liquid of a very small sphere suspended in it", Einstein considered the change in heat dissipation of a liquid in the vicinity of solute particles.


Hydrodynamic Equations

Consider a homogeneous incompressible liquid with viscocity <math>k</math>. In the pure solvent (without solute), The velocity field in the vicinity of an arbitrary point <math>(x_0,y_0,z_0)</math> is given by

<math>\displaystyle (u_0,~v_0,~w_0) = (A \xi,~ B \eta,~ C \zeta) \quad \text{ where } \quad \xi = x - x_0, ~\eta = y - y_0, ~\zeta = z - z_0,

</math>

<math>\text{and} ~ A + B + C = 0 \quad \text{since the fluid is incompressible.}</math>