# Unit of Coefficient of Viscosity

## What is coefficient of viscosity?

The unit of coefficient of viscosity can be understood by defining the coefficient of viscosity. The coefficient of viscosity is defined for two parallel layers of liquid as the tangential force required to maintain a unit velocity gradients between these layers.

The other way of defining the coefficient of viscosity is as the ratio of shear stress to the velocity gradient of the fluid. Mathematical representation is:

$$\eta =\frac{Fr}{Av}$$

Where,

𝛈: coefficient of viscosity

F: tangential force

A: area

v: velocity

r: the distance between the layers

## Dimensional formula of coefficient of viscosity

Mathematical representation is:

$$\eta =\frac{Fr}{Av}$$

Dimensional formula of force F: M1L1T-2

Dimensional formula of area A: M0L2T0

Dimensional formula of distance r: M1L1T0

Dimensional formula of velocity v: M0L1T-1

$$[\eta] =\frac{\left [ M^{1}L^{1}T^{-2} \right ]\left [ M^{0}L^{1}T^{0} \right ]}{\left [ M^{0}L^{2}T^{0} \right ]\left [ M^{0}L^{1}T^{-1} \right ]}=\left [ M^{1}L^{-1}T^{-1} \right ]$$

Dimensional formula of coefficient of viscosity η: M1L-1T-1

### Unit of coefficient of viscosity

Following is the unit of coefficient of viscosity in different systems:

• SI unit: Ns.m-2
• CGS unit: poise

### Coefficient of viscosity of liquid

The coefficient of viscosity of a liquid is defined as the viscous force acting per unit area between two adjacent layers of a liquid such that the velocity gradient is normal to the direction of flow of the liquid. Mathematically it is given as:

$$\eta =\frac{F}{A(\frac{dv}{dx})}$$

Where,

F: viscous force

A: unit area

$$\frac{dv}{dx}$$ : velocity gradient

SI unit of coefficient of viscosity of the liquid is kg m-1 s-1 which is similar to the unit of viscosity.

### Coefficient of viscosity of water

Using Poiseuille’s law, liquid flow through a capillary tube of a uniform cross-section, the coefficient of viscosity of water can be measured.

$$V=\frac{\pi Pr^{4}}{8\eta L}$$ $$\eta = \frac{\pi Pr^{4}}{8VL}$$

Where,

P: pressure difference between the two ends

L: length of the capillary tube

r: inner radius of the capillary

V: volume of the liquid

𝛈: coefficient of viscosity

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