Question

The drag force, \( F_D \), on a sphere due to a fluid flowing past the sphere is a function of viscosity, \( \mu \), the mass density, \( \rho \), the velocity of flow, \( V \), and the diameter of the sphere, \( D \). Pick the relevant (one or more) non-dimensional parameter(s) pertaining to the above process from the following list.

• A. \( \frac{F_D}{\rho V^2 D^2} \)
• B. \( \frac{\rho F_D}{V^2 D^2} \)
• C. \( \frac{\rho V D}{\mu} \)
• D. \( \frac{\mu \rho}{V D} \)

Hint:

When solving for non-dimensional parameters in fluid dynamics, always start by identifying the physical dimensions of the variables involved. Use Buckingham’s Pi Theorem to derive the dimensionless numbers that describe the process.

Answer 1

The Correct Option is A, C

To solve for the non-dimensional parameters, we need to apply Buckingham’s Pi Theorem, which helps derive dimensionless parameters (Pi terms). Starting with the drag force equation: \[ F_D = F(D, V, \rho, \mu) \] We express the dimensions: \[ F_D = [M L T^{-2}], \quad \rho = [M L^{-3}], \quad V = [L T^{-1}], \quad \mu = [M L^{-1} T^{-1}] \] We substitute these into the drag force equation and apply the Buckingham’s Pi Theorem. After calculating, the resulting non-dimensional parameters are: \[ \pi_1 = \frac{F_D}{\rho V^2 D^2}, \quad \pi_2 = \frac{\rho V D}{\mu} \] Both expressions are dimensionless and represent the non-dimensional parameters related to the drag force on the sphere. 
Hence, the correct answers are (A) and (C).