Question

The value of bulk modulus of a fluid is required to determine

• A. Reynold's number
• B. Froude's number
• C. Mach number
• D. Euler's number

Hint:

Whenever bulk modulus ($K$) or compressibility is mentioned in fluid flow, immediately associate it with the speed of sound and the Mach number ($Ma$).
Mach number is the fundamental parameter governing compressible flows.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks which of the given dimensionless parameters of fluid mechanics requires the bulk modulus of elasticity ($K$) of a fluid to be evaluated.
The bulk modulus characterizes the compressibility of a fluid, representing the pressure increase required to cause a unit relative change in volume.

Step 2: Key Formula or Approach:

• The speed of sound ($c$) in a fluid is directly related to its bulk modulus ($K$) and density ($\rho$) by: \[ c = \sqrt{\frac{K}{\rho}} \]
• The Mach number ($Ma$) is defined as the ratio of flow velocity ($V$) to the local speed of sound ($c$): \[ Ma = \frac{V}{c} = \frac{V}{\sqrt{K/\rho}} \]

Step 3: Detailed Explanation:

• Mach Number ($Ma$):
The Mach number is a measure of the compressibility of a fluid flow.
As shown in the formula, the speed of sound $c$ depends on the bulk modulus of elasticity $K$.
Thus, determining the Mach number requires knowing $K$.

• Reynolds Number ($Re$):
Reynolds number is the ratio of inertia forces to viscous forces: \[ Re = \frac{\rho V L}{\mu} \] It does not depend on compressibility or bulk modulus.

• Froude Number ($Fr$):
Froude number is the ratio of inertia forces to gravity forces: \[ Fr = \frac{V}{\sqrt{gL}} \] It is used in free-surface flows and is independent of bulk modulus.

• Euler Number ($Eu$):
Euler number is the ratio of pressure forces to inertia forces: \[ Eu = \frac{\Delta p}{\rho V^2} \] It does not involve bulk modulus.

Step 4: Final Answer:
Hence, the bulk modulus of a fluid is directly required to determine the Mach number.