Fluid Mechanics
6.0 Viscosity
6.0 Viscosity
It is the property of the fluid by virtue of which an internal frictional comes into play when the fluid is in motion in the form of layers having relative motion.
It opposes the relative motion of the different layers. Viscosity is also called as fluid friction or viscous drag of fluid.
Consider a viscous flow of liquid between two layers. The bottom layer is stationary and the top layer moves with velocity $\overrightarrow v $
The liquid layers slide smoothly over one another. The flow speed of intermediate layers of liquid decreases uniformly from top to bottom as shown in the figure.
According to Newton, the viscous force $F$ between the two layers depends on,
1. Area of contact $$F \propto A$$
2. Velocity gradient $\left( {\frac{{dv}}{{dy}}} \right)$ between the layers $$F \propto \frac{{dv}}{{dy}}$$
On combining the above two equations we get, $$F = - \eta A\frac{{dv}}{{dy}}$$ where,
$\eta $: Coefficient of viscosity of liquid
$A$: Area of each layer
$\left( {\frac{{dv}}{{dy}}} \right)$: Velocity gradient
-ve sign shows that viscous force is acting in a direction opposite to the flow of a liquid
6.1 Coefficient of viscosity
Coefficient of viscosity of a liquid is defined as the tangential force required to maintain unit velocity gradient between two parallel layers of liquid each of unit area.
It is denoted by the symbol $\eta $.
Note:
- It depends upon the nature of the liquid
- The dimensional formula for $\eta $ is $\left[ {{M^1}{L^{ - 1}}{T^{ - 1}}} \right]$
- The SI unit of $\eta $ is Poiseuille $(PI)$ or $Pa-s$ or $Nm^{-2}s$
- The CGS unit of $\eta $ is $dyne/cm^2$ called $poise$. Also, $$1\;PI = 10\;poise$$
- The value of viscosity for ideal fluid is zero
- Viscosity is due to the transport of momentum
- Viscosity of liquids is much greater than that of gases
- Viscosity of a liquid decreases with rise in temperature
- Viscosity of a gas increases with rise in temperature $$\eta \propto \sqrt T $$
- Viscosity of gas increases with increase in pressure but the viscosity of water decreases
- Viscosity of gases is independent of pressure
6.2 Poiseuille's equation
The rate of steady flow $(Q)$ of a liquid i.e. volume of the liquid flowing per second at a steady rate through a horizontal capillary tube of length $l$, radius $r$ under a pressure difference at its end is given by, $$Q = \frac{{\pi P\;{r^4}}}{{8\eta l}}$$ This equation is known as Poiseuille's equation.
Poiseuille's equation can also be written as, $$\begin{equation} \begin{aligned} Q = \frac{P}{{\left( {\frac{{8\eta l}}{{\pi {r^4}}}} \right)}} \\ Q = \frac{P}{R} \\\end{aligned} \end{equation} $$ where,
$P$: Pressure difference at its end
$R$: Liquid resistance
$\eta $: Coefficient of viscosity
The current $I$ flowing through a resistance $R$ having potential difference $V$ across its end is given by, $$I = \frac{V}{R}$$
On comparing the above two equations, we understand that the problem of series and parallel combination of pipes can be solved in a similar manner as is done in case of electric circuit.
Note:
- Potential difference is replaced by the pressure difference $\left( {\Delta P} \right)$
- The electric resistance is replaced by the liquid resistance
- The electric current $I$ is replaced by volume flow rate $Q$ or $$\frac{{dV}}{{dt}}$$
6.2.1 Series combination
When two capillaries of length $l_1$ & $l_2$ and radii $r_1$ & $r_2$ respectively are joined in series through which the liquid is flowing, then the effective resistance $R_S$ is given by, $$\begin{equation} \begin{aligned} {R_S} = {R_1} + {R_2} \\ {R_S} = \frac{{8\eta {l_1}}}{{\pi r_1^4}} + \frac{{8\eta {l_2}}}{{\pi r_2^4}} \\\end{aligned} \end{equation} $$
6.2.2 Parallel combination
When two capillaries of length $l_1$ & $l_2$ and radii $r_1$ & $r_2$ respectively are joined in parallel through which the liquid is flowing, then the 
effective resistance $R_P$ is given by, $$\begin{equation} \begin{aligned} \frac{1}{{{R_P}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} \\ \frac{1}{{{R_P}}} = \frac{{\pi r_1^4}}{{8\eta {l_1}}} + \frac{{\pi r_2^4}}{{8\eta {l_2}}} \\\end{aligned} \end{equation} $$
