Explain various methods for measurement of dynamic viscosity of a fluid.

Determination of Dynamic Viscosity:

• Falling ball method

• Rotating cylinder viscometer 

Falling ball method:

This method is used to measure viscosity by balancing the force..

If a body is dropped, starting from rest, in the atmosphere, it will accelerate due to gravity. As the speed of body increases, the body will experience a drag force, which tends to oppose the motion of the body. The drag force increases with a velocity of the fall and after certain time when velocity approaches terminal velocity, the upward drag....Show More

Determination of Dynamic Viscosity:

• Falling ball method

• Rotating cylinder viscometer 

Falling ball method:

This method is used to measure viscosity by balancing the force..

If a body is dropped, starting from rest, in the atmosphere, it will accelerate due to gravity. As the speed of body increases, the body will experience a drag force, which tends to oppose the motion of the body. The drag force increases with a velocity of the fall and after certain time when velocity approaches terminal velocity, the upward drag would just balances the downward weight force such that the net force acting on the body will be zero.

Figure below shows the small rigid sphere of radius r and density `rho_s` falling freely from rest under the  action of gravity through a liquid of density `rho_l` and viscosity `mu`. After certain time, it attains the equilibrium state.

Now, at equilibrium state,

Stokes had experimentally found that,

Drag force=`6pi mu r V_t`

Weight= upthrust+ drag force

`4/3pi*r^3*rho_s*g=4/3pi*r^3*rho_l*g+6pi mu r V_t`

Solving for `mu`, we get,

`mu=2/9*(r^2)/(V_t)*(rho_s-rho_l)g`

This equation can be used to calculate the dynamic viscosity of a sphere. This is valid for laminar flow with Reynolds number less that 0.1.

Rotating cylinder viscometer:

Rotating cylinder is used to measure viscosity by balancing the torque.

Rotational viscometers use the idea that the torque required to turn an object in a fluid is a function of the viscosity of that fluid. They measure the torque required to rotate a disk or bob in a fluid at a known speed.

Figure below shows the experimental setup. Let the outer cylinder is rotating with constant angular velocity `omega` whereas the inner cylinder is stationary and equipped with torque measuring devices. Let `r_1` and `r_2`be the radius of outer and inner cylinder respectively. Let t be the thickness of fluid separating two cylinders.

Now, thickness of oil film is,`t=(r_1-r_2)`

The shear stress is,

`tau=mu*(du)/(dy)=mu*(2 pi*N*r_1)/(60t)`

Where, N is the revolution per minute of outer cylinder.

Now, viscous force is, `F=tau*A=mu*(2 pi*N*r_1)/(60t)*2pi*r_2*L`

Where, L is the length of cylinder.

Thus, the viscous torque is given by,

`T=F*r_2=mu*(2 pi*N*r_1)/(60t)*2pi*r_2*L*r_2`

or,`T=1/15*mu*(pi)^2*r_1*(r_2)^2*L*N`

or,`mu=(15t)/((pi)^2*r_1*(r_2)^2*L*N)`

This equation can be used to calculate the dynamic viscosity.