Introduction:
Hydrodynamic lubrication is a mechanism that occurs when two surfaces in relative motion are separated by a thin fluid film. This film of lubricant reduces friction and wear between the surfaces. The film thickness plays a crucial role in determining the effectiveness of the lubrication.

Explanation:
The film thickness in hydrodynamic lubrication is influenced by various factors, including speed. Let's understand why the correct answer is option 'B' - the film thickness increases with an increase in speed.

1. Reynolds Equation:
To understand the film thickness in hydrodynamic lubrication, we need to consider the Reynolds equation. This equation describes the pressure distribution in the lubricant film and is given by:

δh/δx = (ηU) / (6μ)

Where:
- δh/δx is the change in film thickness with respect to the relative distance between the surfaces.
- η is the viscosity of the lubricant.
- U is the velocity of the sliding surface.
- μ is the dynamic viscosity of the fluid.

2. Speed and Pressure:
As the speed increases, the velocity of the sliding surface (U) also increases. According to the Reynolds equation, the change in film thickness (δh/δx) is directly proportional to the velocity (U) of the sliding surface. Therefore, an increase in speed leads to an increase in film thickness.

3. Increased Lubricant Flow:
When the speed increases, there is a greater flow of lubricant between the surfaces. This increased flow results in a thicker film of lubricant being present. The lubricant is forced into the space between the surfaces, creating a hydrodynamic wedge that separates them.

4. Effective Lubrication:
A thicker film of lubricant provides better separation between the surfaces, reducing the contact and friction between them. This improves the effectiveness of the lubrication and helps in preventing wear and damage to the surfaces.

5. Limitations:
It is important to note that there are certain speed limits beyond which the hydrodynamic lubrication mechanism may not be effective. At very high speeds, the lubricant film may not be able to withstand the pressure and may break down, leading to metal-to-metal contact and increased friction.

Conclusion:
In hydrodynamic lubrication, the film thickness increases with an increase in speed. This is due to the relationship between the velocity of the sliding surface and the change in film thickness described by the Reynolds equation. A thicker film of lubricant provides better separation between the surfaces, reducing friction and wear. However, there are limits to the effectiveness of hydrodynamic lubrication at very high speeds.

A