Introduction:
Incompressible fluids are those fluids whose density remains constant regardless of changes in temperature and pressure. Examples of incompressible fluids include water and oil. Thermodynamic properties, on the other hand, are characteristics of a fluid that describe its behavior under different conditions of temperature and pressure. These properties include density, viscosity, specific heat, and thermal conductivity, among others.

Explanation:
The statement that thermodynamic properties do not play an important role in the behavior of incompressible fluids at varying temperature and pressure is incorrect. The correct answer is option 'B' - No.

Importance of Thermodynamic Properties:
Thermodynamic properties play a crucial role in understanding and predicting the behavior of fluids, even if they are incompressible. Here's why:

1. Density:
Although incompressible fluids have a constant density, changes in temperature and pressure can affect their density indirectly. For example, as the temperature of a fluid increases, its volume may expand, leading to a decrease in density. Similarly, changes in pressure can also influence the density of the fluid. Density is important when calculating the buoyant force, pressure gradients, and flow rates.

2. Viscosity:
Viscosity is the measure of a fluid's resistance to flow. Incompressible fluids still exhibit viscosity, which can vary with temperature and pressure. The viscosity of a fluid affects its flow behavior, such as laminar or turbulent flow, and it plays a significant role in determining friction losses in pipes and channels.

3. Specific Heat:
Specific heat is the amount of heat energy required to raise the temperature of a substance by a certain amount. While incompressible fluids have a constant specific heat, it is essential to consider this property when analyzing heat transfer processes involving these fluids. Specific heat is used to calculate the heat exchanged during processes like heating or cooling.

4. Thermal Conductivity:
Thermal conductivity is the property that describes the ability of a material to conduct heat. Incompressible fluids still possess thermal conductivity, which is essential in analyzing heat transfer phenomena. It helps determine the rate at which heat is transferred through the fluid and can significantly impact the temperature distribution in a system.

Conclusion:
In conclusion, thermodynamic properties are indeed important in the behavior of incompressible fluids at varying temperature and pressure. These properties provide insights into how the fluid will behave under different conditions and are vital for accurate analysis and prediction of fluid flow, heat transfer, and other related processes. Thus, option 'B' - No is the correct answer.