Mechanical Properties of Fluids

Fluids are substances that can flow and do not have a fixed shape. They include liquids and gases. Here, we might ask what it means for a substance to flow?

The property of flow refers to the characteristics of an object to show no resistance to the shear stress.

• Shear stress can change the shape of a solid keeping its volume fixed. But solids resist shear stress. They show some strength when shear stress is applied to them.
• Fluids offer very little resistance to shear stress; their shape changes with the application of very small shear stress. The shearing stress of fluids is about a million times smaller than that of solids.

Properties of Fluids

The fluid flow is governed by certain properties of a fluid such as Viscosity, density and pressure over it.

Density of a Fluid

Density is the mass per unit volume of a substance. It is represented by the Greek symbol ρ.

ρ = MassVolume (1 Litre)

The density of water at 4oC (277 K) is 1000 kg m–3. 

Often it is more sensible to compare the density of a fluid in comparison to water for better understanding. For this, the term relative density is often found to be more useful.

Relative Density

The relative density of a substance is the ratio of its density to the density of water at 4oC. For example, the relative density of aluminium is 2.7. Its density is 2.7 × 103 kg m–3.

It is a dimensionless positive scalar quantity, as it is only a ratio. 

Pressure 

Pressure is the force exerted per unit area, defined as:

P= FA

Where P is pressure, F is force, and A is the area.

Pascal’s Law: 

It states that the pressure applied to an enclosed fluid is transmitted undiminished to all parts of the fluid.

Examples: Hydraulic breaking and Hydraulic lifting.

Hydrostatic Pressure: 

Hydrostatic Pressure is the Pressure exerted by a fluid at rest. It is given by the relation:

P = ρgh

Where: 

• ρ is the density of the fluid, 
• g is the acceleration due to gravity, and 
• h is the height of the fluid column.

Atmospheric Pressure: 

Atmospheric Pressure is the pressure exerted by the weight of the Earth’s atmosphere, approximately 1.01 × 10⁵ Pa at sea level. 

Buoyancy

Every object submerged in a fluid experiences some force against the direction of Gravity. The upward force exerted by a fluid on a submerged object is the Buoyant Force. For example, we weigh less when submerged underwater. 

The amount of force that a person would experience is governed by the Archimedes Principle. 

Archimedes’ Principle: 

It states that the buoyant force is equal to the weight of the fluid displaced by the object: 

F = ρfluid  . g . Vdisplaced

Where 

• ρfluid  is the density of the fluid, 
• Vdisplaced ​ is the volume of fluid displaced and 
• g is the acceleration due to Gravity.

Further, ρfluid . V can be understood as the mass of the liquid displaced. Why?

Because ρ = m/V; 

Therefore, m = ρfluid V, i.e. the mass of the fluid displaced. Thus the F = mfluid.g

Understanding Archimedes’ principle

Underwater, every object experiences a pressure equivalent to ρgh from all directions. Now assume that an object with an area ‘A’ and its height Δh. The overall upward force experienced by such an object would be upward pushing force (minus) downward pushing force.

F = (Upward Pressure X Area) – (Downward Pressure X A)

• • F = (Upward Pressure – Downward Pressure) X A

• F = [ρg (h+ Δh) – ρgh] X A

• F = [(h+ Δh) – h] X ρgA

• F = Δh X ρgA

Since Δh X A = V (Volume of the object), the force on the object would become: 

• F = ρfluid  X g X V 

This is the relation given by the Archimedes.

Apparent Weight: 

The weight of an object in a fluid is less than its weight in air due to the buoyant force. 

As you might remember from our discussion in the Gravitation chapter, our weight is given by the relation, F = mg. Where ‘m’ is our mass and ‘g’ is the acceleration due to gravity.

Now if the object is weighed while being submerged under a fluid, its apparent weight would reduce.

• F = mobject.g – mfluid.g
• F = (mobject  – mfluid).g

This is the apparent weight of an object submerged inside a fluid.

Viscosity

Viscosity is the measure of a fluid’s resistance to flow. It describes internal friction between fluid layers. The coefficient of viscosity (η) is a constant that quantifies this resistance. It is different for different types of liquids, as shown in the table.

Units of Viscosity: The SI unit is Pascal-seconds (Pa·s) or N·s/m².

Poiseuille’s Law: 

It describes the flow rate of a viscous fluid through a pipe:

Where Q is the flow rate, r is the radius of the pipe, ΔP is the pressure difference, η is the viscosity, and L is the length of the pipe.

Types of Flows

The fluid flow can be of two types: 

• Streamline Flow: A type of fluid flow where every particle follows a smooth path, and the flow lines do not cross. Also called laminar flow.
• Turbulent Flow: Irregular, chaotic flow where eddies and swirls form, often at higher velocities or with larger pipes.

Reynolds Number (Re)

Reynolds number is a dimensionless quantity that helps predict the flow regime of a fluid —whether it is laminar or turbulent.

Reynolds Number (Re): It is the ratio of inertial forces to viscous forces in a fluid flow.

It is given by the following relation:

Where ρ is the fluid density, v is the velocity, D is the characteristic length (like the pipe diameter), and η is the viscosity.

Flow Regimes based on Reynolds Number

Reynolds number helps determine the nature of fluid flow in pipes, around objects, and in various engineering applications.

• Laminar Flow: Occurs when Re<2000, characterized by smooth and orderly flow.
• Turbulent Flow: Occurs when Re>4000, characterized by chaotic and irregular flow.

Between 2000 and 4000, the flow may transition between laminar and turbulent.

Continuity Equation:

States that for an incompressible fluid, the mass flow rate remains constant: 

A1​v1​ = A2​v2​

where A is the cross-sectional area and v is the velocity of the fluid.

This equation implies that if the cross-sectional area of a pipe decreases, the fluid velocity must increase, and vice versa.

Bernoulli’s Equation

Bernoulli’s Principle states that for an incompressible, non-viscous fluid, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline: 

P + 12.ρ.v2 + ρgh = constant

Where P is the pressure, ρ is the density, v is the velocity, g is the acceleration due to gravity, and h is the height.

This relation is nothing but the conservation of energy itself. Let’s consider the Equation of conservation of energy. This equation applies to incompressible fluids. Since most liquids like water are practically incompressible, this equation would apply to them. 

F.d + 12mv2 + mgh = Constant

Here ‘F.d’ is the energy due to a pushing force experienced by an object, ‘12mv2‘ is the kinetic energy of the fluid in the packet, and ‘mgh’ is the potential energy, as we have considered in the Work and Energy Chapter.

Now, let’s divide this equation with the volume of the packet. For an incompressible Fluid like water, we can assume that Volume would remain constant despite the application of pressure.

12. dV + 12.mV.v2 + mV.gh = constant

•  F. dA.d + 12.ρ.v2 + ρgh = FA + 12.ρ.v2 + ρgh = constant

• P + 12.ρ.v2 + ρgh = constant

Applications of Bernauli’s Theorem

Bernoulli’s principle explains phenomena such as the lift on an aeroplane wing and the working of a venturi meter.

• Aeroplane Lift: The air moves faster over the curved top surface of the wings, creating lower pressure compared to the bottom. This pressure difference results in an upward force, lifting the plane.
• Venturi Effect in Pipelines: In a narrowing pipe, fluid velocity increases and pressure decreases as per Bernoulli’s principle. This is utilized in devices like Venturi tubes, where pressure difference can measure flow rates, commonly used in engineering and fluid mechanics.
• Carburettors: In an internal combustion engine, carburettors use Bernoulli’s principle to mix air and fuel. As air speeds up through a narrow section of the carburettor, pressure drops, sucking fuel into the airflow to create the proper air-fuel mixture for combustion.

Surface Tension:

Surface tension is the force per unit length acting along the surface of a liquid, which causes the surface to behave like a stretched elastic membrane. It is responsible for phenomena such as the formation of droplets and the rise of liquids in capillaries.

Definition: The force acting along the surface of a liquid that tends to minimize the surface area.

Surface Tension (T) = FL

Where F is the force acting along the surface and L is the length over which the force acts.

The SI unit of surface tension is Newton per meter (N/m).

Effects of Surface Tension

Surface tension causes liquids to minimize their surface area, leading to phenomena like:

• Formation of spherical droplets.
• The rise of liquid in narrow tubes (capillary action).
• Insects walking on water.

What are Mechanical Properties of Fluids?

Mechanical Properties of Fluids deal with the behavior of liquids and gases (fluids) when forces are applied on them.

Which key concepts are covered under Mechanical Properties of Fluids?

Key concepts include pressure, buoyancy, Pascal’s Law, Archimedes’ Principle, viscosity, surface tension, and Bernoulli’s Principle.

Why is this topic important for UPSC?

Questions on Mechanical Properties of Fluids often appear in UPSC Prelims related to environment, oceanography, hydraulics, and atmospheric pressure.

What is Bernoulli’s Principle?

Bernoulli’s Principle states that where the speed of a fluid increases, its pressure decreases, and vice versa. It is used in aircraft wings and spray guns.

Where are Mechanical Properties of Fluids applied in daily life?

Applications include water supply systems, dams, submarines, hydraulic lifts,