Dimensionless Numbers in Fluid Mechanics Explained

Dimensionless numbers are crucial in fluid mechanics as they help in comparing different physical phenomena and scaling results from models to full-scale applications. They represent the ratio of different forces acting within a fluid flow. This section details the matching of specific dimensionless numbers with the types of forces they represent.

Matching Fluid Forces with Dimensionless Numbers

The question requires matching the dimensionless numbers in List-I with the dominant forces they represent in List-II.

Euler's Number and Pressure Forces

Euler's number (Eu) is typically defined as the ratio of pressure forces to inertial forces within a fluid. It is used in situations involving pressure differences and flow. Therefore, Euler's number corresponds to Pressure force.

Froude's Number and Gravity Forces

Froude's number (Fr) is important in flows where gravity forces are significant, such as open channel flow or wave motion. It represents the ratio of inertial forces to gravitational forces. Thus, Froude's number is associated with Gravity force.

Mach Number and Compressibility Forces

The Mach number (M) compares the speed of an object or fluid flow to the speed of sound within that fluid. It is a critical parameter in analyzing compressible flows, where the density changes significantly due to pressure variations. It relates inertial forces to compressibility forces. Therefore, Mach number corresponds to Compressibility force.

Weber Number and Surface Tension Forces

The Weber number (We) is used in situations where interfacial effects are important, particularly in multiphase flows or atomization processes. It represents the ratio of inertial forces to surface tension forces. Hence, Weber number relates to Surface Tension.

Correct Matching Summary

Based on the analysis of each dimensionless number:

• (A) Euler's number matches with (I) Pressure force.
• (B) Froude's number matches with (II) Gravity force.
• (C) Mach number matches with (IV) Compressibility force.
• (D) Weber number matches with (III) Surface Tension.

The correct option that reflects this matching is (A) - (I), (B) - (II), (C) - (IV), (D) - (III).