ANS:D - Weber

The ratio of inertial forces to surface tension forces is called the Weber number. Here’s an explanation of each term:

• Euler number: This term is not typically associated with the ratio of inertial forces to surface tension forces. The Euler number usually refers to different concepts in fluid dynamics, such as the Euler equation or Euler's number in mathematical contexts.
• Froude number: This number relates inertial forces to gravitational forces and is used to determine the type of flow (subcritical, critical, or supercritical) in open-channel flow and similar applications. It is defined as Fr=VgL\text{Fr} = \frac{V}{\sqrt{gL}}Fr=gL​V​, where VVV is the velocity of the flow, ggg is the acceleration due to gravity, and LLL is a characteristic length.
• Mach number: This number compares the flow velocity to the speed of sound in the fluid medium and is crucial in aerodynamics and compressible flow. It is defined as Ma=Va\text{Ma} = \frac{V}{a}Ma=aV​, where VVV is the flow velocity and aaa is the speed of sound.
• Weber number: The Weber number (We) compares the inertial forces (due to flow) to the surface tension forces in a fluid system. It is defined as We=ρV2Lσ\text{We} = \frac{\rho V^2 L}{\sigma}We=σρV2L​, where ρ\rhoρ is the density of the fluid, VVV is the characteristic velocity, LLL is the characteristic length, and σ\sigmaσ is the surface tension of the fluid.

Therefore, the correct answer to the question regarding the ratio of inertial forces to surface tension forces is the Weber number. The Weber number (We) is a dimensionless number that compares the inertial forces of a fluid flow to the surface tension forces present at the interface of the fluid.

Definition and Formula:

The Weber number is defined as: We=ρV2Lσ\text{We} = \frac{\rho V^2 L}{\sigma}We=σρV2L​ where:

• ρ\rhoρ is the density of the fluid,
• VVV is the characteristic velocity of the flow,
• LLL is a characteristic length scale (such as the diameter of a droplet or the width of a liquid jet),
• σ\sigmaσ is the surface tension of the fluid.

Interpretation:

1. Inertial Forces vs. Surface Tension Forces: The Weber number quantifies the balance between the kinetic energy (inertial forces) of the flow and the cohesive forces (surface tension) that tend to minimize the surface area of the fluid.
2. Flow Regimes:
   • We < 1: In this regime, surface tension forces dominate over inertial forces. Fluid behavior is more influenced by surface tension, and small droplets or thin films tend to form stable shapes.
   • We > 1: Here, inertial forces dominate over surface tension forces. The fluid tends to overcome surface tension effects, leading to breakup or atomization of droplets, splashing, or jet formation.
3. Applications:
   • Fluid Dynamics: Used in fluid mechanics to understand the breakup of droplets, behavior of liquid jets, and splash dynamics.
   • Industrial Processes: Relevant in industries such as spray drying, inkjet printing, fuel injection in engines, and chemical processes involving mixing and dispersion of fluids.
   • Natural Phenomena: Applies to phenomena like raindrop formation, wave breaking, and atomization in ocean spray.
4. Limitations: The Weber number assumes incompressible, Newtonian fluids with constant properties. It may not accurately represent fluid behavior in cases involving highly viscous fluids or non-Newtonian fluids.

In summary, the Weber number is a fundamental parameter in fluid dynamics that characterizes the competition between inertial forces and surface tension forces, influencing the behavior of fluids in a variety of natural and industrial processes.