ANS:A - V

In the low Reynolds number region, the drag force on a sphere is proportional to VVV, where VVV represents the velocity of the sphere relative to the fluid.

Explanation:

1. Reynolds Number (Re):
   • Reynolds number ReReRe is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It is defined as: Re=ρ⋅V⋅DμRe = \frac{\rho \cdot V \cdot D}{\mu}Re=μρ⋅V⋅D​ where:
     • ρ\rhoρ is the fluid density,
     • VVV is the velocity of the object relative to the fluid,
     • DDD is a characteristic length (diameter of the sphere in this case),
     • μ\muμ is the dynamic viscosity of the fluid.
2. Low Reynolds Number Region:
   • In the low Reynolds number region (typically Re<1Re < 1Re<1), viscous forces dominate over inertial forces.
   • For a sphere moving through a fluid at low Reynolds numbers, the drag force FdF_dFd​ is proportional to the velocity VVV of the sphere relative to the fluid.
   • Therefore, Fd∝VF_d \propto VFd​∝V.
3. Options Analysis:
   • V2V^2V2: This corresponds to the quadratic dependence observed in the high Reynolds number region (turbulent flow).
   • V4V^4V4: This does not correspond to the drag force characteristics at any Reynolds number range for a sphere.
   • V0.5V^{0.5}V0.5: This does not correctly describe the relationship observed for spheres at low Reynolds numbers.

Conclusion:

Thus, in the low Reynolds number region, the drag force on a sphere is proportional to VVV, where VVV is the velocity of the sphere relative to the fluid. This relationship arises due to the dominance of viscous forces over inertial forces at low Reynolds numbers.