Dimensional Analysis - Transport Phenomena and Rate Processes - Metallurgical Engineering Previous Year Questions

A ladle containing molten steel is being discharged. The relevant forces are listed in Column I. Match them with their corresponding expressions in Column II. Column I: [P] Pressure force, [Q] Inertial force, [R] Gravity force, [S] Viscous force. Column II: [1] μUL, [2] ρgL³, [3] ρU²L², [4] PL². μ = viscosity, U = characteristic velocity, L = characteristic length, g = acceleration due to gravity, P = pressure.

Analysis of a flow phenomenon in a system requires the following variables:
i. Pressure [M L⁻¹ T⁻²]
ii. Velocity of the fluid [L T⁻¹]
iii. Size of the system [L]
iv. Density of the fluid [M L⁻³]
v. Viscosity of the fluid [M L⁻¹ T⁻¹]

According to Buckingham Pi theorem (dimensional analysis) what is the number of independent DIMENSIONLESS variables needed to describe this system?

Terminal rise velocity of a spherical shaped solid in a liquid obeys the following functional relationship: \(U = f(d, W, \mu, \rho)\) Where, \(U\) is the terminal rise velocity, \(d\) is the diameter of the solid, \(W\) is the apparent weight of the solid, \(\mu\) is the viscosity of liquid and \(\rho\) is the density of liquid. According to Buckingham Π theorem, the number of independent dimensionless variables needed to describe the phenomenon is ____________.