Viscosity

Figure 22.6: Two solid plates separated by a distance $d$, the gap being filled by a viscous fluid. The lower plate is stationary and the upper plane is moving to the right at speed $v_p = v(d) = Sd$. The fluid is sheared, with the fluid moving according to $v(y) = Sy$. The fluid velocity matches that of the plates where the fluid touches the plates. The upper plate experiences a drag force $F_{drag} = - \mu S A$ where $\mu$ is the viscosity of the fluid and $A$ is the area of the plate.

If two objects are not in physical contact but are separated by a thin layer of fluid (i. e., a liquid or a gas), there is still a frictional or viscous drag force between the two objects but its behavior is different. Figure 22.6 tells the story: The viscous drag force in this case is

$$F_{drag} = - \mu S A$$ (23.15)

where $S = v_p /d$ is the shear in the fluid, $A$ is the area of the plates, and $\mu$ is the viscosity of the fluid. (Don't confuse this parameter with the static and dynamic coefficients of friction!) The parameter $v_p$ is the velocity of the top plate with respect to the bottom plate and $d$ is the separation between the plates.

Viscosity has the dimensions mass per length per time. The most common unit of viscosity is the Poise: $1 \mbox{ Poise} = 1 \mbox{ g} \mbox{ cm}^{-1} \mbox{ s}^{-1}$. The viscosity of water varies from $0.0179 \mbox{ Poise}$ at $0^{\circ} \mbox{ C}$ to $0.0100 \mbox{ Poise}$ at $20^{\circ} \mbox{ C}$ to $0.0028 \mbox{ Poise}$ at $100^{\circ} \mbox{ C}$. The viscosity of water thus decreases with increasing temperature, which is typical of liquids. In contrast, the viscosity of a gas is independent of the density of the gas and is proportional to the square root of its absolute temperature. The viscosity of a gas thus increases with temperature, in contrast to the viscosity of a liquid. For air at $20^{\circ} \mbox{ C}$, the viscosity is $1.81 \times 10^{-4} \mbox{ Poise}$.

Thin layers of oil between moving parts are commonly used in machinery to reduce friction, since the resulting viscous drag is generally much less than the corresponding kinetic friction which would occur if the parts were in direct contact. The ways in which the layer of oil is maintained between moving parts are fascinating, but beyond the scope of this course.