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

Bernoulli's equation is based on conservation of energy.

For a given specific energy there are two alternate depths.

Re = 500 < 2000, so the flow is laminar.

Total static energy is the sum of pressure head and potential (datum) head.

For a vertical rectangular lamina with top edge at the free surface, the centre of pressure lies at $2h/3$ from the surface.

For pipes in parallel the total flow equals the sum of flows in each pipe.

$G = \frac{7.85}{9.81} \approx 0.8$.

For a rectangular channel at critical flow, hydraulic depth equals critical depth: $y_c = D$.

$C_d = C_c \times C_v$.

Reynolds Number and Classification of Pipe Flow

Use in classifying flow

Reynolds number ($Re$) is a dimensionless number that expresses the ratio of inertia forces to viscous forces in a flowing fluid. For flow through a circular pipe it is used to predict whether the flow is laminar, transitional or turbulent:

Thus by simply computing $Re$ from the fluid properties, velocity and pipe diameter, an engineer can decide the flow regime, which in turn fixes the friction-factor relation and head-loss calculation.

Derivation of the expression

Reynolds number is defined as the ratio of inertia force to viscous force:

Inertia force $= \text{mass} \times \text{acceleration} = \rho \,(\text{Volume}) \times \dfrac{v}{t}$

Taking a characteristic length $L$ (so volume $\propto L^3$ and $v t \propto L$):

Viscous force $= \tau \times \text{area} = \mu \dfrac{dv}{dy} \times A$. Taking $\dfrac{dv}{dy} \propto \dfrac{v}{L}$ and $A \propto L^2$:

Therefore:

For a pipe the characteristic length is the diameter $D$, so:

where $\rho$ = density, $v$ = mean velocity, $D$ = pipe diameter, $\mu$ = dynamic viscosity and $\nu = \mu/\rho$ = kinematic viscosity.

Derivation of the Darcy–Weisbach Equation

Consider a steady, uniform, incompressible flow through a pipe of diameter $D$ and length $L$ between sections 1 and 2.

Notation: $A$ = cross-sectional area, $P$ = wetted perimeter, $v$ = mean velocity, $f_0$ = frictional resistance (shear) per unit area of wetted surface, $h_f$ = head loss due to friction, $w = \rho g$ = specific weight.

Step 1 — Forces acting on the fluid element

Writing the equation of motion for the fluid between the two sections (flow is uniform, so velocity and momentum do not change; net force = 0):

• Pressure force at section 1 $= p_1 A$
• Pressure force at section 2 $= p_2 A$
• Frictional resistance along the pipe wall $= f_0 \times (\text{surface area}) = f_0 \, (P \cdot L)$

Force balance (taking the pipe horizontal so weight component along flow is zero, or absorbing the datum term into the head):

Step 2 — Relate pressure drop to head loss

Applying Bernoulli's equation between 1 and 2 for a pipe of constant diameter (so $v_1 = v_2$) and same datum:

Substituting into the force balance:

Since $\dfrac{A}{P} = m$ (hydraulic mean depth), this is $h_f = \dfrac{f_0}{\rho g}\cdot\dfrac{L}{m}$.

Step 3 — Express the wall shear in terms of velocity

Experiment shows the frictional resistance per unit area is proportional to the square of velocity:

where $f'$ is a non-dimensional coefficient of friction. Hence:

Step 4 — Apply to a circular pipe

For a circular pipe of diameter $D$:

Therefore:

Defining the Darcy friction factor $f = 4f'$:

This is the Darcy–Weisbach equation for the head loss due to friction (major loss) in a pipe, where $f$ is the Darcy friction factor, $L$ the pipe length, $D$ the diameter, $v$ the mean velocity and $g$ the acceleration due to gravity.