where κ is a coefficient the nature of which remains to be determined.
If we suppose the liquid between ab and cd divided into layers as shown in fig. 2, it will be clear that the stress R acts, at each dividing face, forwards in the direction of motion if we consider the upper layer, backwards if we consider the lower layer. Now suppose the original thickness of the layer T increased to nT; if the bounding plane in its new position has the velocity nV, the shearing at each dividing face will be exactly the same as before, and the resistance must therefore be the same. Hence,
R = κ′ω (nV).
(2)
But equations (1) and (2) may both be expressed in one equation if κ and κ′ are replaced by a constant varying inversely as the thickness of the layer. Putting κ = μ/T, κ′ = μ/nT,
R = μωV/T;
or, for an indefinitely thin layer,
R = μωdV/dt,
(3)
an expression first proposed by L. M. H. Navier. The coefficient μ is termed the coefficient of viscosity.