Fluid Mechanics: Buoyancy, Stability, and Pressure Principles

Posted on Mar 30, 2025 in Communication

Understanding Buoyancy

Buoyancy is the upward force exerted by a fluid (liquid or gas) on an object submerged in it. This force arises from the pressure difference exerted by the fluid on different parts of the object. When an object is immersed in a fluid, the pressure at the bottom of the object is higher than the pressure at the top, causing a net upward force. This buoyant force acts opposite to the force of gravity.

According to Archimedes’ principle, the magnitude of the buoyant force on an object is equal to the weight of the fluid displaced by the object.

Mathematically:

F_b = ρ * g * V

Where:

• F_b is the buoyant force,
• ρ is the density of the fluid,
• g is the acceleration due to gravity,
• V is the volume of fluid displaced.

Centre of Buoyancy

The centre of buoyancy is the point through which the buoyant force acts on a submerged object. It is the centroid of the displaced fluid’s volume, i.e., the center of mass of the displaced fluid. For symmetrical objects, the center of buoyancy coincides with the geometric center of the submerged portion of the object. For irregular objects or when only part of an object is submerged, the center of buoyancy shifts to the centroid of the displaced volume of fluid.

Metacentre and Stability

The metacentre is a point used in the stability analysis of floating bodies. It is the point where the line of action of the buoyant force (acting vertically through the center of buoyancy) intersects the body’s axis when it is slightly tilted or displaced. The metacentre plays a crucial role in determining whether a floating body will remain stable when tilted.

• If the metacentre (M) is above the center of gravity (G) of the body, the body is stable and will return to its upright position after being tilted. This is called positive stability.
• If the metacentre (M) is below the center of gravity (G), the body is unstable and will tip over. This is called negative stability.

Metacentric Height (GM)

The metacentric height (GM) is the vertical distance between the center of gravity (G) of a floating body and the metacentre (M). It is a measure of the stability of a floating body:

GM = M − G

Pressure Calculation in Layered Liquids

Consider a tank with layers of oil and water.

(i) Pressure Intensity at the Interface

The pressure at the interface of the oil and water is caused only by the oil column above the interface.

The density of oil (ρ_oil) can be found from the specific gravity of oil (S_o):

ρ_oil = S_o × ρ_water = 0.9 × 1000 kg/m³ = 900 kg/m³

Now, the pressure at the interface (due to the oil layer only) is:

P_interface = ρ_oil × g × h_oil

Substituting the values (assuming h_oil = 1m and g = 9.81 m/s²):

P_interface = 900 kg/m³ × 9.81 m/s² × 1 m = 8829 Pa

So, the pressure intensity at the interface is 8829 Pa (or 8.83 kPa).

(ii) Pressure Intensity at the Bottom of the Tank

At the bottom of the tank, the pressure is due to both the water column and the oil column above it.

Pressure due to the water column (assuming h_water = 2m):

P_water = ρ_water × g × h_water = 1000 kg/m³ × 9.81 m/s² × 2 m = 19620 Pa

The total pressure at the bottom is the sum of the pressure due to the oil and the pressure due to the water:

P_bottom = P_interface + P_water = 8829 Pa + 19620 Pa = 28449 Pa

Thus, the pressure intensity at the bottom of the tank is 28449 Pa (or 28.45 kPa).

Total Pressure on a Submerged Plate

Given:

• Diameter of the circular plate, d = 1.5 m
• Depth of the center of the plate, h̄ = 3.0 m

Calculations:

1. Area of the plate (A):
   A = (π/4) * d² = (π/4) * (1.5 m)² = 1.767 m²
2. Total Pressure (Force F):
   The total pressure (hydrostatic force) is given by:
   F = ρ * g * A * h̄
   Assuming water (ρ = 1000 kg/m³) and g = 9.81 m/s²:
   F = 1000 kg/m³ × 9.81 m/s² × 1.767 m² × 3.0 m = 52002.81 N
   F = 52002.81 N
3. Position of the Center of Pressure (h*):
   The depth of the center of pressure is given by:
   h* = (I_G / (A * h̄)) + h̄
   Where I_G is the moment of inertia of the area about an axis passing through its centroid and parallel to the free surface. For a circle, I_G = (π * d⁴) / 64.
   I_G = (π * (1.5 m)⁴) / 64 = 0.2485 m⁴
   Now, substitute the values into the h* equation:
   h* = (0.2485 m⁴ / (1.767 m² * 3.0 m)) + 3.0 m
   h* = 0.0468 m + 3.0 m = 3.0468 m
   h* = 3.0468 m

Capillary Rise and Fall Explained

Capillary rise and capillary fall are phenomena that occur due to surface tension, adhesion, and cohesion when a liquid comes into contact with a narrow tube, called a capillary tube. These effects are especially noticeable in liquids like water and mercury.

1. Capillary Rise (Example: Water)

When a capillary tube is inserted into water, the liquid rises above the external level. This is known as capillary rise. The phenomenon can be explained by the following factors:

• Adhesion: This is the attractive force between the liquid (water) molecules and the solid surface of the tube (usually glass). In the case of water, the adhesive forces between the water molecules and the glass are stronger than the cohesive forces between the water molecules themselves.
• Cohesion: This is the attractive force between the molecules of the same substance (in this case, water).

Since the adhesive forces between water and glass are stronger than the cohesive forces between water molecules, water “climbs” the walls of the tube, leading to a concave meniscus (the curved surface of the liquid inside the tube). The liquid continues to rise until the upward adhesive forces are balanced by the weight of the liquid column.

The height of capillary rise (h) is given by the formula:

h = (2 * γ * cosθ) / (ρ * g * r)

Where:

• γ = surface tension of the liquid (water)
• θ = contact angle (for water and clean glass, it is usually small, close to 0°)
• ρ = density of the liquid
• g = acceleration due to gravity
• r = radius of the capillary tube

In the case of water, the contact angle θ is less than 90°, which means the liquid wets the surface, causing it to rise.

2. Capillary Fall (Example: Mercury)

In the case of mercury, the opposite phenomenon occurs, known as capillary fall. When a capillary tube is inserted into mercury, the liquid level inside the tube falls below the external level. This happens because the cohesive forces between mercury molecules are stronger than the adhesive forces between mercury and the glass surface.

• Adhesion: The adhesive forces between mercury and the glass are weaker than the cohesive forces between mercury molecules.
• Cohesion: The cohesive forces between mercury molecules are strong because mercury is a non-polar, metallic liquid, and its molecules tend to stick strongly to each other.

Because cohesion dominates adhesion, mercury molecules pull away from the glass walls, forming a convex meniscus. The strong cohesive forces pull the liquid column downward until the pressure difference balances the surface tension effects, resulting in a lower level inside the tube compared to the outside.