Question

1.(b) (6%) Find the dimensionless parameter that measures the viscous effect relative to the gravitational effect.

Answer 1

In fluid dynamics, certain dimensionless numbers are used to describe the effect of one force relative to the other.

Archimedes number (Ar) is a  dimensionless number ( ratio of two forces), which is the ratio of gravitational forces to viscous forces in a fluid. It is used to determine the motion of fluids with a difference in their densities. Archimedes number finds its application in studying/calculating gravitational settling of particles in fluids.

${\mathrm {Ar}}={\frac {gL^{3}\rho _{\ell }(\rho -\rho _{\ell })}{\mu ^{2}}}$

• g is the acceleration due to gravity m/s2
• $\rho _{\ell }$ stands for the density of the fluid, kg/m3
• $\rho$ stands for the density of the body, kg/m3
• $\mu$ stands for the dynamic viscosity, kg/ms
• stands for the characteristic length of the body, m