৪৯তম বিসিএস ⎯ ফলিত রসায়ন [৫৪১]

ক) Liquid water at 25°C → Incompressible for practical purposes. Its density changes less than 0.1% even under large pressure variations, so it is treated as incompressible in engineering.

খ) Air at atmospheric pressure → Compressible. All gases exhibit significant density changes when pressure or temperature varies, making compressibility important in fluid dynamics calculations.

গ) Steam at 1 atm → Compressible because it is in the gaseous phase (water vapor). Its properties follow the ideal gas approximation, so density changes with pressure and temperature.

ঘ) Supercritical CO₂ → Compressible because its density is highly sensitive to pressure and temperature changes, especially near the critical point.

Air at atmospheric pressure is the correct answer, because it is the most common example of a compressible fluid.

An ideal fluid is a theoretical construct in fluid mechanics used to simplify analysis.
It has the following properties:

No viscosity (zero internal friction, so no energy loss due to shear).

Incompressible (constant density, regardless of pressure changes).

Non-conductive for heat (no thermal energy transfer internally).

This type of fluid does not exist in nature, but it is used in models like Bernoulli’s equation and potential flow theory for simplicity.

** Euler’s Equation of Motion describes the relationship between pressure, velocity, and elevation in an inviscid (non-viscous) fluid under steady flow.

It is derived from Newton’s Second Law (F = ma) applied to a fluid element, which is essentially the principle of conservation of linear momentum.

** Conservation of mass gives the continuity equation, not Euler’s equation.

** Conservation of energy leads to Bernoulli’s equation, which is obtained by integrating Euler’s equation under specific assumptions.

** Hydrostatic law applies to fluids at rest, not to moving fluids.

Euler’s equation of motion applies to an ideal fluid, which is assumed to be:
✔ Inviscid (no viscosity → viscous forces are zero)
✔ Non-conductive (no heat transfer)
✔ Continuous and non-rotational in some cases

The equation is derived from Newton’s Second Law (Conservation of Momentum), but during the derivation, viscous shear forces are neglected because the fluid is assumed to have zero viscosity.

Other forces considered in Euler’s equation:

Pressure forces → included

Gravity (body forces) → can be included as a term

Inertial forces → included

Bernoulli’s equation is derived under the following assumptions:

Incompressible fluid – density remains constant.

Inviscid flow – viscosity is negligible.

Steady flow – fluid properties at any point do not change with time.

Along a streamline – applies to a single streamline unless extended for an irrotational field.

The option “Viscous flow” contradicts the assumption because Bernoulli neglects viscous effects

Bernoulli’s equation assumes:
✔ Incompressible fluid
✔ Inviscid flow
✔ Steady state
✔ Along a streamline

(ক): Horizontal pipe with no friction → satisfies assumptions.
(খ): Nozzle with large reservoir under steady flow → valid (low-speed, incompressible).
(গ): High-speed air (Mach > 1) → compressibility becomes significant, so Bernoulli’s equation in its simple form does not apply.
(ঘ): Fits all assumptions → valid.

The original Bernoulli’s equation assumes ideal fluid (inviscid, no friction), but in real systems, there are energy losses due to:
Pipe friction
Fittings and bends
Valves

To include these losses, Bernoulli’s equation is modified as:
p₁/ρg + v₁²/2g + z₁ = p_2/ρg + v_2²/2g + z² + h_L

Here, h_L​ = head loss (energy loss per unit weight of fluid).

Discharge coefficient (Cd) is a correction factor applied to theoretical discharge calculations to match actual discharge in real-world conditions.
Ideal Bernoulli-based formulas assume incompressible, non-viscous, no-energy-loss flow, which is rarely true in practice.

Cd accounts for:
✔ Viscosity effects → causing friction losses.
✔ Energy losses → due to turbulence, friction, and separation at constrictions.
✔ Non-ideal behavior → deviations from theoretical assumptions like perfectly streamlined flow.

Orifice meter → Causes a large permanent pressure drop because of sharp-edged restriction and flow separation.

Venturi meter → Designed with a smooth converging and diverging section, which minimizes energy losses due to reduced turbulence and gradual expansion → lowest head loss among differential pressure meters.

Pitot tube → Measures velocity head locally but does not typically measure total discharge directly.

Rotameter → Works on variable area principle; head loss is relatively small, but Venturi is superior for high accuracy with minimal loss in pipelines.

Head loss comparison:
Venturi meter < Rotameter < Pitot tube < Orifice meter (highest).

Pitot tube principle is based on Bernoulli’s equation, when a moving fluid is brought to rest at the stagnation point in the tube, the pressure rises to stagnation pressure.
The Pitot tube compares this stagnation pressure with the static pressure of the flowing fluid using a differential manometer.

Venturi meter: Works on Bernoulli’s principle (pressure difference).

Rotameter: Correct; free-floating float in a tapered tube.

Orifice meter: Works on differential pressure through an orifice plate.

Pitot tube: Measures stagnation and static pressure difference.