To measure the viscosity of a fluid experimentally, we consider the variation of shear stress with respect to the rate of shear strain.

Shear stress is a measure of the force per unit area that is applied parallel to the surface of a material, causing it to deform. On the other hand, shear strain is a measure of the deformation per unit length that occurs in a material as a result of this applied force. By studying the relationship between shear stress and shear strain, we can determine the viscosity of a fluid.

- Shear Stress and Shear Strain Relationship:
When a fluid is subjected to shear stress, it starts to deform and flow. The relationship between shear stress (τ) and shear strain rate (γ̇) is defined by the equation τ = μγ̇, where μ is the viscosity of the fluid. This equation is known as Newton's law of viscosity.

- Experimental Setup:
In order to measure the viscosity of a fluid experimentally, a viscometer is used. A viscometer is a device that applies a known shear stress to the fluid and measures the resulting shear strain rate. Various types of viscometers are available, such as capillary viscometers, rotational viscometers, and falling sphere viscometers.

- Capillary Viscometer Example:
Let's consider a capillary viscometer, which consists of a narrow tube with a small diameter. The fluid is allowed to flow through the capillary tube under the influence of gravity or an applied pressure. The flow rate of the fluid is measured, and the pressure drop across the capillary is determined.

- Calculation of Shear Strain Rate:
The shear strain rate (γ̇) in the capillary viscometer can be calculated using the equation γ̇ = (2V)/(πr^2l), where V is the volume flow rate, r is the radius of the capillary, and l is the length of the capillary.

- Determining Viscosity:
By measuring the pressure drop and the corresponding flow rate, the shear stress (τ) can be determined. Dividing the shear stress by the shear strain rate (τ/γ̇) gives the viscosity (μ) of the fluid.

In conclusion, the variation of shear stress with respect to the rate of shear strain is considered in order to experimentally measure the viscosity of a fluid. By analyzing this relationship using a viscometer, the viscosity of the fluid can be determined.