Viscosity

The significance of the parameter $\mu$ , appearing in the previous expression for the stress tensor, can be seen from the form taken by the relation (1.25) in the special case of simple shearing motion. With $\partial v_1/\partial x_2$ as the only non-zero velocity derivative, all of the components of $d_{ij}$ are zero apart from the shear stresses

$$d_{12} = d_{21} = \mu\,\frac{\partial v_1}{\partial x_2}.$$ (1.27)

Thus, $\mu$ is the constant of proportionality between the rate of shear and the tangential force per unit area when parallel plane layers of fluid slide over one another. This constant of proportionality is generally referred to as viscosity. It is a matter of experience that the force acting between layers of fluid undergoing relative sliding motion always tends to oppose the motion, which implies that $\mu>0$ .

The viscosities of dry air and pure water at $20^\circ\,{\rm C}$ and atmospheric pressure are about $1.8 \times 10^{-5}\, {\rm kg/(m\,s)}$ and $1.0\times 10^{-3}\, {\rm kg/(m\,s)}$ , respectively. In neither case does the viscosity exhibit much variation with pressure. However, the viscosity of air increases by about $0.3$ percent, and that of water decreases by about 3 percent, per degree Centigrade rise in temperature (Batchelor 2000).