Shear stress and shear strain are two important properties of fluids, especially when considering their flow behavior. In the case of Newtonian fluids, the relationship between shear stress and shear strain is given by the equation: shear stress = viscosity * rate of shear strain.

To understand why the correct answer is option 'C' (shear stress is proportional to the rate of shear strain), let's break down the components of this equation and explore the underlying concepts.

1. Shear Stress:
Shear stress is the force per unit area that acts parallel to a surface, causing it to deform. It is a measure of the resistance of a fluid to flow when subjected to an applied force. Shear stress is denoted by the symbol τ (tau) and is expressed in units of Pascal (Pa).

2. Shear Strain:
Shear strain is a measure of the deformation undergone by a fluid when subjected to shear stress. It represents the change in shape or distortion of the fluid due to the applied force. Shear strain is denoted by the symbol γ (gamma) and is a dimensionless quantity.

3. Rate of Shear Strain:
The rate of shear strain represents how quickly the fluid is deforming under the influence of shear stress. It is the time derivative of shear strain and is denoted by the symbol γ_dot (gamma dot). The rate of shear strain is expressed in units of reciprocal time (s^-1).

4. Viscosity:
Viscosity is a measure of a fluid's resistance to flow. It determines how easily a fluid can deform under the influence of shear stress. Newtonian fluids have a constant viscosity, meaning their shear stress is directly proportional to the rate of shear strain. The proportionality constant is the viscosity of the fluid, denoted by the symbol η (eta) and expressed in units of Pascal-second (Pa·s).

Therefore, in the case of a Newtonian fluid, the relationship between shear stress (τ) and rate of shear strain (γ_dot) can be expressed as τ = η * γ_dot. This equation shows that the shear stress acting on the fluid is directly proportional to the rate at which the fluid is deforming. The constant of proportionality is the viscosity of the fluid.

In conclusion, for a Newtonian fluid, the correct answer is option 'C' (shear stress is proportional to the rate of shear strain) because the shear stress acting on the fluid is directly proportional to the rate at which the fluid is deforming under the influence of shear stress.