Viscosity

Viscosity is the property of fluid by virtue of which a tangential viscous force or backward dragging force acts between different layers of it and it tries to stop the motion of a fluid. The internal friction force or viscous force exists due to relative motion of layers of a fluid.

Newton's Formula of Viscosity

Let us consider a liquid flowing over a solid surface. The bulk of a liquid is divided into different horizontal layers increases as we move away from the surface as shown in the figure.

Let us consider two layers PQ and RS of a liquid at a distance x and x + dx from the surface moving with velocity v +dv respectively. As these exists relative motion between different layers of a moving liquid. So,the backward viscous force acts tangentially to it. Newton studied the viscous force acting between two layers of liquid assuming it flow to be laminated and found that

1. It is directly proportional to the area of layers in contact.
   $$\text {i.e.} F \propto A \dots (i)$$
2. It is directly proportional to the velocity gradient between the layers
   $$\begin{align*} F &\propto A \frac {dv}{dx}\\ F &= -\eta A \frac {dv}{dx} \\ \end{align*}$$
3. where the constant of proportionality'η' is known as the coefficient of viscosity of a liquid.

Here negative sign shows that viscous always acts opposite to the motion of liquid.

Dimensional Formula of Coefficient of Viscosity

We have,

\begin{align*}\eta &= \frac {F}{A\frac {dv}{dx}}\\ \eta &= \frac {MA}{A\frac {dv}{dx}} \\ &= \frac {[MLT^{-2}]}{\frac {[L^2][T^{-1]}}{[L]}}\\ &=[ML^{-1}T^{-1}] \end{align*}

Hence, coefficient of viscosity of a liquid is defined as the viscus drag or viscus force acting per unit area of the layerhaving unit velocity gradient perpendicular to the direction of the flow of the liquid .

Unit of Coefficient of Viscosity

$$a = m^{x^{rty}}$$

$$G = \frac {6.64 \times 10 ^{-11} \times M}{R}$$

CGS unit of η is Poise (P)

\begin{align*} &= \frac {1 \text {dyne}}{1cm^2 \times \left (\frac {1cms^{-1}}{cm} \right )} \\ &= 1 dyne\: sec\: cm^{-5} \\ &= \text {poise} \\ \end{align*}

SI-unit of η is Deca poise

\begin{align*} \eta &= N\: sec \: m^{-2} \\ &= 10^5 dyne \: sec \: \frac {1}{10^4 \: cm^{-2}} \\ &= 10\: \text {poise} \\ &= \text {decapoise} \\ 1 \: \text {decapoise} &= 10 \: \text {poise} \\ \end{align*}