Match List-I with List-II on fluid types and their rheological... | Agrigyan

Both statements are factually incorrect:

• Statement (I) is incorrect: In open channel hydraulics, a rectangular channel is most economical (most efficient) when the hydraulic radius is half the depth of flow (\(R = y/2\)), which requires the bed width to be twice the depth of flow (\(b = 2y\)), NOT equal to depth.
• Statement (II) is incorrect: By definition, in a compressible flow, the density of the fluid does not remain constant (density varies with pressure and temperature, \(\rho \ne \text{constant}\)). Density remains constant in an incompressible flow.

Head loss due to friction in pipe flow is calculated using the Darcy-Weisbach formula:

\[h_f = \frac{4 f L v^2}{2 g d}\]

where

\(f = 0.005\) is the Fanning friction factor (or using \(f_D = 4f = 0.02\)): Pipe diameter \(d = 7.5\text{ cm} = 0.075\text{ m}\) Pipe length \(L = 120\text{ m}\) Flow velocity \(v = 1.8\text{ m/s}\) Acceleration due to gravity \(g = 9.81\text{ m/s}^2\) Substituting values: \[h_f = \frac{4 \times 0.005 \times 120 \times (1.8)^2}{2 \times 9.81 \times 0.075} = \frac{2.4 \times 3.24}{1.4715} = \frac{7.776}{1.4715} = 5.2844\text{ m} \approx 5.28\text{ m}\]

Detailed Analysis of Fluid Rheology:

• Apparent Viscosity Decreases with Shear Rate -> (III) Pseudoplastic Fluids (Shear-thinning fluids, e.g., fruit purees, polymer melts).
• Shear Stress Directly Proportional to Rate of Deformation -> (II) Newtonian Fluids (Constant viscosity obeying Newton's law $\tau = \mu \frac{du}{dy}$, e.g., water, milk, mineral oils).
• Yield Stress Followed by Linear Flow -> (IV) Bingham-Plastic Fluids (e.g., toothpaste, tomato paste, drilling muds).
• Viscosity Increases with Shear Rate -> (I) Dilatant Fluids (Shear-thickening fluids, e.g., concentrated cornstarch suspension, quicksand).
• Shear Stress Not Directly Proportional to Deformation Rate -> (V) Non-Newtonian Fluids (General umbrella class of fluids with non-linear or yield stress behavior).

Detailed Analysis of Bernoulli's Equation:

• Principle of Conservation of Energy (Option 2): Bernoulli's equation is the mathematical statement of the law of conservation of mechanical energy for steady, incompressible, frictionless (inviscid) fluid flow along a streamline:
  $$\frac{P}{\rho g} + \frac{v^2}{2g} + z = \text{Constant}$$
  Representing pressure head, velocity head (kinetic energy), and datum head (potential energy) per unit weight of fluid. Hence, Option 2 is correct.
• Conservation of Mass (Option 1): Represented by the Continuity Equation ($\nabla \cdot \vec{v} = 0$).
• Conservation of Momentum (Option 3): Represented by the Navier-Stokes / Euler equation of motion.