Explanation:

A hydrodynamic bearing is a type of bearing that relies on a fluid film to support the rotating shaft. The bearing characteristic number, also known as the Sommerfeld number, is a dimensionless parameter that determines the performance of the hydrodynamic bearing. It is defined as the ratio of the viscous forces to the inertial forces in the fluid film.

The bearing characteristic number depends on several factors, including the dimensions of the bearing and the operating conditions. It is calculated using the following formula:

Bearing Characteristic Number = (Viscosity of the fluid) * (Shaft speed) * (Bearing length) / (Load on the bearing) * (Fluid film thickness)

Now, let's discuss why the correct answer is option C, which states that the bearing characteristic number in a hydrodynamic bearing depends on cost-effective design.

1. Cost-effective design:
- The bearing characteristic number plays a crucial role in determining the design parameters of the hydrodynamic bearing.
- By optimizing the bearing characteristic number, engineers can design a bearing that is cost-effective.
- A higher bearing characteristic number leads to a thinner fluid film, which reduces the amount of fluid required and hence, lowers the cost of the bearing.
- On the other hand, a lower bearing characteristic number requires a thicker fluid film, increasing the cost of the bearing.

2. Real-life situation:
- While it is true that the bearing characteristic number should be closer to real-life situations, this is not the most important factor in determining the bearing design.
- The bearing characteristic number is primarily used to determine the load-carrying capacity and stability of the hydrodynamic bearing.
- It ensures that the bearing can withstand the applied loads and operate reliably under normal operating conditions.
- While it is important to consider real-life situations, the primary focus is on the performance and safety of the bearing.

3. Safer design:
- While a higher bearing characteristic number may indicate a safer design in terms of load-carrying capacity, it does not necessarily guarantee overall safety.
- Other factors such as material selection, lubrication, and proper maintenance also contribute to the safety of the bearing.
- The bearing characteristic number is just one of many parameters that need to be considered in the design process.

4. No other assumption is possible:
- This statement is incorrect because the bearing characteristic number depends on various assumptions such as the fluid film being fully developed, laminar flow, and constant properties of the fluid.
- These assumptions are necessary to simplify the calculations and provide a reasonable estimate of the bearing performance.

In conclusion, the correct answer is option C because the bearing characteristic number in a hydrodynamic bearing plays a significant role in determining the cost-effective design. By optimizing this parameter, engineers can design a bearing that provides reliable performance while minimizing the cost of production.

B