NEB Class 12 Science Fluids Mechanics Question Paper 2082 (Set B) Nepal

Bernoulli's equation relates various energy forms (pressure, kinetic, potential) of a fluid.

Two depths for a given specific force are conjugate (sequent) depths.

Constant rate but velocity changes with the tapering section: steady non-uniform flow.

Energy heads are expressed as energy per unit weight.

Poise is the CGS unit of dynamic viscosity.

The sudden-contraction loss coefficient is $K = \left(\frac{1}{C_c} - 1\right)^2$.

The simple (one-dimensional) continuity equation Q = A·v applies to incompressible fluid with steady flow.

Chezy's $C = \frac{1}{n}R^{1/6}$.

For a rectangular notch, $Q \propto H^{3/2}$, so the percentage error in Q is 1.5 times the percentage error in H.

Definitions

1. Surface tension ($\sigma$): The property of a liquid by virtue of which its free surface behaves like a stretched elastic membrane. It is the force acting per unit length along a line drawn on the free surface, acting perpendicular to that line and tangential to the surface. SI unit: $\text{N/m}$.

2. Viscosity ($\mu$): The property of a fluid by which it offers resistance to the relative (shear) motion between adjacent fluid layers. By Newton's law of viscosity, the shear stress is

where $\dfrac{du}{dy}$ is the velocity gradient. SI unit of dynamic viscosity: $\text{Pa·s}$ (or poise in CGS).

3. Compressibility: The measure of the change in volume (or density) of a fluid under the action of applied pressure. It is the reciprocal of the bulk modulus of elasticity $K$:

It indicates how easily a fluid can be compressed.

4. Newtonian fluid: A fluid in which the shear stress is directly proportional to the velocity gradient (rate of shear strain), i.e. it obeys Newton's law of viscosity $\tau = \mu \dfrac{du}{dy}$ with constant $\mu$. Examples: water, air, thin oils.

5. Real fluid: A fluid that possesses viscosity, surface tension and compressibility, and therefore offers resistance to flow (experiences shear stress) when in motion. All actual fluids existing in nature (water, oil, air) are real fluids, as opposed to the idealized non-viscous "ideal fluid."

Derivation of the Darcy–Weisbach Equation

Consider steady, uniform flow of a fluid through a horizontal pipe of diameter $D$ (cross-sectional area $A$) and length $L$ between sections 1 and 2. Let $h_f$ be the head loss due to friction, $f'$ the frictional resistance per unit wetted area per unit velocity-squared, and $v$ the mean velocity.

Step 1 — Forces acting on the fluid element.

• Pressure force at section 1: $p_1 A$ (driving the flow)
• Pressure force at section 2: $p_2 A$ (opposing)
• Frictional resistance at the pipe wall: $F = f' \times (\text{wetted surface area}) \times v^2 = f' \,(\pi D L)\, v^2$

Step 2 — Apply equilibrium (steady flow, no acceleration).

Step 3 — Relate pressure difference to head loss.

Applying Bernoulli's equation between 1 and 2 for the horizontal pipe of constant diameter ($z_1=z_2$, $v_1=v_2$):

Step 4 — Combine.

Putting $f = \dfrac{f'}{\rho/2} = \dfrac{2f'}{\rho}$ (the dimensionless Darcy friction coefficient), we obtain the Darcy–Weisbach equation:

where $f$ is the coefficient of friction (and $4f$ is the Darcy friction factor). This gives the major (frictional) head loss in a pipe of length $L$ and diameter $D$ carrying a mean velocity $v$.

General Applications of Bernoulli's Principle

Bernoulli's equation, $\dfrac{p}{\rho g} + \dfrac{v^2}{2g} + z = \text{constant}$, states that for steady, incompressible, non-viscous flow the total energy head (pressure + kinetic + potential) is conserved along a streamline. Common applications include:

• Venturimeter – measuring discharge (flow rate) through a pipe.
• Orifice meter and nozzle meter – discharge measurement.
• Pitot tube – measuring flow velocity at a point.
• Aerofoil / aeroplane wing lift – higher velocity over the curved upper surface lowers pressure, producing lift.
• Spinning ball (Magnus effect), atomizer/carburettor (spray), and chimney draught.

Measurement of Discharge Through a Pipe (Venturimeter)

A venturimeter consists of a converging cone, a throat, and a diverging cone fitted in a pipe. Let:

• Section 1 = inlet, area $A_1$, velocity $v_1$, pressure $p_1$
• Section 2 = throat, area $A_2$, velocity $v_2$, pressure $p_2$

Step 1 — Bernoulli's equation (horizontal pipe, $z_1=z_2$):

where $h$ is the pressure-head difference (read from the differential manometer).

Step 2 — Continuity equation:

Step 3 — Substitute and solve for $v_2$:

Step 4 — Theoretical discharge:

Step 5 — Actual discharge (introducing the coefficient of discharge $C_d$ to account for real losses):

Here, for a mercury–water differential manometer, the equivalent head of flowing liquid is

with $x$ the manometer deflection. Thus, by measuring the manometer deflection, the discharge through the pipe is determined using Bernoulli's principle.