Short Answer:

The variation of pressure with depth in a fluid means that the pressure increases as the depth increases. This happens because the deeper point in a fluid has to support the weight of all the fluid above it. The relationship between pressure and depth is linear and is expressed as P = P₀ + ρgh, where P is the pressure at depth h, P₀ is the surface pressure, ρ is the fluid density, and g is the acceleration due to gravity.

In simple words, the deeper you go into a fluid, the greater the pressure becomes. This principle is very important in designing tanks, dams, submarines, and pipelines that can safely resist fluid pressure at different depths.

Detailed Explanation :

Variation of Pressure with Depth

The variation of pressure with depth is a fundamental concept in fluid mechanics. It describes how pressure increases in a fluid as we move deeper below the surface. This variation occurs because fluids have weight, and each layer of fluid exerts a force on the layers below due to its weight.

In a fluid at rest, there are no shear stresses — only normal stresses (pressure) act perpendicular to surfaces. As depth increases, the pressure at a point is caused by the weight of the fluid column directly above that point. Therefore, the deeper the point, the more fluid weight it supports, and hence, the greater the pressure.

This relationship helps engineers determine the pressure at different depths in fluids, which is crucial in the design of tanks, reservoirs, and hydraulic structures.

Derivation of Pressure Variation with Depth

To derive the relationship between pressure and depth, let us consider a fluid at rest in a container.

Let:

= density of the fluid (kg/m³),
= acceleration due to gravity (9.81 m/s²),
= depth of the fluid column (m),
= pressure at the free surface of the fluid (N/m²),
= pressure at a depth h below the surface (N/m²).

Now, consider a small fluid element in the shape of a vertical cylinder of:

cross-sectional area (A) and
height (dh) as shown in the fluid column.

The fluid element is in equilibrium under three forces:

Pressure force on the top surface, (acts downward),
Pressure force on the bottom surface, (acts upward),
Weight of the fluid element, (acts downward).

Since the fluid is at rest, the net force on the element must be zero.

Hence,

Simplifying,

This equation shows that the rate of increase of pressure with depth is equal to the specific weight (ρg) of the fluid.

Integrating between the surface and depth h:

This is the hydrostatic equation, which gives the variation of pressure with depth in a static fluid.

Interpretation of the Equation

From the equation :

is the pressure at the free surface (atmospheric pressure if the surface is open).
is the additional pressure due to the fluid column of height h.
The pressure increases linearly with depth (h).

Therefore, the pressure at the bottom of a container is higher than the pressure at the top. This relationship is independent of the shape or area of the container; it depends only on the depth, density, and gravity.

Key Points about Pressure Variation

Linear Relationship:
The increase in pressure with depth is directly proportional to depth h.
- If h doubles, pressure also doubles.
Independent of Container Shape:
Pressure at a given depth is the same regardless of whether the container is wide, narrow, or inclined.
Dependent on Fluid Density:
Heavier fluids (like mercury) produce higher pressure for the same depth than lighter fluids (like water).
Pressure Direction:
Pressure always acts perpendicular (normal) to the surface of the container or object in contact with the fluid.

Example Calculation

Let’s find the pressure 10 m below the surface of a water tank open to the atmosphere.

Given:

,
,
,
(atmospheric pressure).

Then,

Hence, the absolute pressure at 10 m depth is 199.4 kPa, while the gauge pressure (pressure above atmospheric) is 98.1 kPa.

Graphical Representation

If we plot pressure (P) on the vertical axis and depth (h) on the horizontal axis, the relationship between pressure and depth is a straight line starting from surface pressure .
The slope of the line equals . This means that pressure increases uniformly as we go deeper.

Applications of Pressure Variation with Depth

Design of Dams and Reservoir Walls:
Pressure increases linearly with depth, so dam walls are designed thicker at the bottom to resist higher pressure.
Submarine and Ship Design:
The pressure variation helps in calculating the pressure on submarine hulls at different ocean depths.
Fluid Storage Tanks:
Engineers use the pressure-depth relationship to design tank walls that can withstand pressure at the bottom.
Hydraulic Systems:
Hydraulic devices use pressure variation principles to lift or press objects.
Barometers and Manometers:
The height of the fluid column in a barometer or manometer is determined by pressure variation with depth.

Factors Affecting Pressure Variation

Depth (h): Pressure increases linearly with depth.
Density (ρ): Denser fluids exert more pressure for the same depth.
Gravitational Acceleration (g): Higher gravity increases pressure at a given depth.
External Pressure (P₀): If the surface pressure changes, the pressure at depth also changes.

Practical Example

In underwater diving, the pressure on a diver increases by about 1 atmosphere (101.3 kPa) for every 10 meters of water depth. At 30 meters deep, a diver experiences about 4 atmospheres (one from air and three from water). Hence, diving equipment and submarines must be designed to withstand such pressures.

Conclusion

In conclusion, the variation of pressure with depth shows that pressure in a static fluid increases linearly with depth due to the weight of the fluid above. The relationship is expressed as . This principle is essential in fluid mechanics for understanding how pressure acts in stationary fluids and for designing structures like tanks, dams, and submarines that must safely resist fluid pressure at different depths. The concept ensures accurate pressure estimation in various engineering and scientific applications.