JNTU B.Tech II Year MECHANICS OF FLUIDS Previous Question Paper (July/August - 2021 - R15)

Manometers are pressure-measuring devices that use the principle of hydrostatic equilibrium in a liquid column to determine pressure. They are broadly classified into two main categories: simple manometers and differential manometers.

1. Simple Manometers: These are used to measure pressure at a single point in a fluid system relative to atmospheric pressure. Common types include:

• Piezometer: A simple glass tube open to the atmosphere at one end and connected to the fluid container at the other. It is suitable only for moderate positive pressures where the fluid is a liquid.
• U-Tube Manometer: Consists of a U-shaped glass tube containing a manometric liquid (usually mercury). One end is connected to the point of pressure measurement, and the other is open to the atmosphere. It can measure both positive and negative (vacuum) pressures.
• Single Column Manometer: A modified U-tube where one limb is made much larger in cross-section than the other. This allows for more precise readings as the movement of the meniscus in the large reservoir is negligible, making the scale reading on the narrow limb sufficient.

2. Differential Manometers: These are used to measure the difference in pressure between two points in a pipe or between two different pipes. They do not measure absolute or gauge pressure directly.

• U-Tube Differential Manometer: Consists of a U-tube connected to two points whose pressure difference is required. The deflection of the heavy manometric fluid indicates the pressure head difference.
• Inverted U-Tube Manometer: Used for measuring pressure differences in liquids. It consists of an inverted U-tube with the upper portion filled with air or a light fluid, which is then trapped to measure the pressure difference between two liquid-filled pipes.

In all cases, the pressure is calculated using the hydrostatic law,

The equilibrium of a body submerged in or floating on a fluid is governed by the principles of buoyancy and the relative positions of the center of gravity and the center of buoyancy. For a body to be in equilibrium, the net vertical force must be zero, meaning the weight of the body must equal the buoyant force.

Submerged Bodies: A body is fully submerged when it is completely surrounded by the fluid. Its equilibrium state depends on the vertical alignment of the center of gravity (G) and the center of buoyancy (B):

• Stable Equilibrium: Occurs when G is located vertically below B. If the body is tilted, the buoyant force and weight create a restoring couple that returns the body to its original position.
• Unstable Equilibrium: Occurs when G is located vertically above B. Any small disturbance creates a couple that increases the displacement, causing the body to overturn.
• Neutral Equilibrium: Occurs when G and B coincide. The body remains at rest in any position to which it is displaced.

Floating Bodies: A body is floating when it is partially submerged in a fluid. The equilibrium is determined by the relationship between the center of gravity (G), the center of buoyancy (B), and the metacenter (M), which is the point about which the body oscillates when tilted:

• Stable Equilibrium: Occurs when the metacenter (M) is above the center of gravity (G). The restoring couple generated by the shift in the center of buoyancy acts to bring the body back to its upright position.
• Unstable Equilibrium: Occurs when the metacenter (M) is below the center of gravity (G). The couple generated by the displacement tends to increase the tilt, causing the body to capsize.
• Neutral Equilibrium: Occurs when the metacenter (M) coincides with the center of gravity (G). The body remains in equilibrium in any tilted position.

The metacentric height (GM) is defined as the distance between the center of gravity and the metacenter. The condition for stability is expressed as:

where BM is the distance from the center of buoyancy to the metacenter, calculated as BM = I/V, with I being the second moment of area of the waterplane section and V being the displaced volume of fluid.