Explain velocity gradient and coefficient of viscous and gives their units.

Explanation

Let laminar floe on a horizontal surface as shown in figure


Suppose two layer P and Q at distance x and x+dx from stationary surface.
The velocity difference dv between two layers separated by distance dx.
`(dv)/(dx)` is know as velocity gradient.
Velocity gradient : The difference in velocity between adjacent layers of the fluid is known as a velocity gradient.
Viscous force between two layers depend on following factors .
(1) Depends on the area (A) of two contact surface .
`FpropA` ..... (1)
(2) Velocity gradient between two layers `(dv)/(dx)`
`Fprop(dv)/(dx)` ......(2)
`thereforeFpropA(dv)/(dx)`....(3)
`thereforeF=-etaA(dv)/(dx)` .....(4)
Here `eta` is known as coefficient of viscosity .Its magnitude depend on the kind of fluid and on temperature.
Negative sing indicates viscosity force , is opposite to the velocity of layers of fluid.
As the temperature increase in liquid `eta` will be decrease while in gas it increases.
In equation (4), A=1 unit, `(dv)/(dx)=-1` then `F=eta`.
Definition of coefficient of viscosity :
The viscous force acting per unit surface area of contact and per unit velocity gradient between two adjacent layers in a laminar flow is known as the coefficient of viscosity .
Unit of coefficient of viscosity :
(1) CGS unit of viscosity is dyne `scm^(-2)`
or `gcm^(-1)s^(-1)`. It is called poise. `P_(a)*S`
MKS unit of viscosity is `Nsm^(-2)` or `kgm^(-1)s^(-1)`. It is called decapoise or poisenile.
1 poisenille `(pl)=Nsm^(-2)`
`=(10^(5)"dyne")sxx(10^(2)cm)^(-2)`
`=10" dyne"*s*cm^(-2)`
=10 poise.
(3) Dimensional formula `:[M^(1)L^(-1)T^(-1)]`