Dimensional analysis in fluid mechanics: What is the product of the Reynolds number (Re), the Froude number (Fr), and the Mach number (Ma), and how is it best described?

Introduction / Context:

This question tests conceptual understanding of the principal dimensionless numbers used in fluid mechanics—Reynolds (Re), Froude (Fr), and Mach (Ma). These arise from balancing inertia against viscous, gravitational, and elastic (compressibility) effects, respectively. Students often memorize definitions, but deeper mastery includes recognizing what combinations do or do not correspond to widely adopted named groups.

Given Data / Assumptions:

• Re = (rho * v * L) / mu (inertial/viscous).
• Fr = v / sqrt(g * L) (inertial/gravity).
• Ma = v / a (inertial/elastic; a is the speed of sound).
• All three are dimensionless for any consistent units.

Concept / Approach:

Each number isolates one physical effect versus inertia. Named numbers like Weber (We = rho * v^2 * L / sigma) or Euler (Eu = Δp / (rho * v^2)) have direct ties to surface tension and pressure forces. By contrast, the product Re * Fr * Ma multiplies three independent ratios. The product remains dimensionless, but there is no ubiquitous textbook name or standard use for it because it conflates three distinct balances into one value that lacks a clear single-physics interpretation.

Step-by-Step Solution:

Verification / Alternative check:

Checking standard references shows We involves surface tension sigma, Eu involves pressure differences, Ca involves viscosity versus surface tension (not to be confused with Cauchy). Cauchy number in compressible flow equals Ma^2, which is not the given product.

Why Other Options Are Wrong:

• Weber number requires sigma and depends on v^2, not v^3.
• Euler number depends on pressure ratio, absent here.
• Cauchy number equals Ma^2, not Re * Fr * Ma.
• Galileo number (also called Gravitational number) involves nu and g differently.

Common Pitfalls:

• Assuming any product of familiar numbers must itself be a named parameter.
• Confusing similarly named “Ca” (Capillary number) with Cauchy number.

Final Answer:

It forms a dimensionless group without a standard, widely used name