Dimensional Analysis and Similitude Questions and Answers

Answer: A

Pressure has dimensions \(ML^{-1}T^{-2}\).

Pressure is force per unit area. Force is mass times acceleration, \(MLT^{-2}\), and area is \(L^{2}\):

\[ [p] = \frac{MLT^{-2}}{L^{2}} = ML^{-1}T^{-2} \]

\(MLT^{-2}\) is force itself, \(MLT^{-1}\) is momentum, and \(ML^{-1}T^{-1}\) is dynamic viscosity (Pa s).

Answer: C

The dimension of dynamic viscosity is \(ML^{-1}T^{-1}\).

From Newton's law of viscosity, \(\tau = \mu\,du/dy\):

\[ \begin{aligned} [\tau] &= \frac{MLT^{-2}}{L^{2}} = ML^{-1}T^{-2} \\ [du/dy] &= T^{-1} \\ [\mu] &= ML^{-1}T^{-1} \end{aligned} \]

This matches the SI unit Pa.s, which is kg.m^-1.s^-1. \(MLT^{-2}\) is the dimension of force, and \(L^{2}T^{-1}\) is kinematic viscosity, \(\nu = \mu/\rho\), which has no mass dimension. \(L^{-2}T^{2}\) corresponds to no standard fluid property.

Answer: B

The correct matching is (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii).

• Mach number is the square root of inertia force over elastic (compressibility) force, \(V/c\) (a-ii).
• Froude number is the square root of inertia force over gravity force, \(V/\sqrt{gL}\) (b-i).
• Nusselt number is convective over conductive heat transfer, \(hL/k\) (c-iv).
• Prandtl number is momentum diffusivity over thermal diffusivity, \(\nu/\alpha\) (d-iii).

Option C swaps Nusselt and Prandtl, and options A and D give the Mach number the Prandtl number's ratio, so only option B has all four right.