JEE Physics Mechanical Properties Of Fluids Notes Set 07

Viscosity and the Coefficient of Viscosity

Viscosity is the internal friction of a fluid: the property by which layers moving at different speeds exert tangential forces on each other. For layers separated by a distance y with relative speed u, Newton's law gives \( F=\eta A\frac{du}{dy} \), where \( \frac{du}{dy} \) is the velocity gradient and \( \eta \) the coefficient of viscosity.
The SI unit of \( \eta \) is N s/m² (Pa s, also called decapoise). In CGS the unit is the poise, and \( 1\ \text{Pa s}=10\ \text{poise} \). The dimensions are \( [ML^{-1}T^{-1}] \).
Viscous force differs from solid friction: it depends on the area of contact and on the velocity gradient, it is zero when there is no relative motion, and it does not depend on the normal pressure in the way solid friction does.
Worked example: a plate moving over a thin oil film is held at constant speed by a force F. Then \( F=\eta A\frac{v}{d} \), so thicker films or slower plates need less force.

Effect of Temperature on Viscosity

The viscosity of a liquid decreases rapidly as the temperature rises, because the molecules gain energy and escape from their neighbours more easily. The viscosity of a gas increases with temperature, because faster molecules transfer momentum between layers more effectively (roughly \( \eta\propto\sqrt T \)). Viscosity of a liquid increases with pressure only slightly, with water being a notable exception at low temperature.

Stokes' Law and Terminal Velocity

A sphere of radius r moving slowly at speed v through a fluid of viscosity \( \eta \) feels a retarding force \( F=6\pi\eta rv \), proportional to radius, speed and viscosity.
A sphere released in a long column of fluid accelerates until weight equals buoyancy plus viscous drag. The speed then stays constant, the terminal velocity \( v_T=\frac{2r^2(\rho-\sigma)g}{9\eta} \), where \( \rho \) is the density of the sphere and \( \sigma \) that of the fluid. Terminal velocity is proportional to \( r^2 \) and to \( \rho-\sigma \), and inversely proportional to \( \eta \).
If \( \rho<\sigma \) (an air bubble in a liquid) the velocity is upward with the same formula using \( \sigma-\rho \). In vacuum there is no drag, so there is no terminal velocity. Before the terminal speed is reached the net force falls gradually to zero and the speed increases at a decreasing rate; the viscous force grows from zero to equal the effective weight.
Worked example: eight identical drops of terminal velocity 5 cm/s coalesce into one drop. The radius doubles, so the new terminal velocity is \( 4\times5=20 \) cm/s.
The rate of heat produced at terminal velocity is \( Fv\propto r\cdot v^2\propto r^5 \).