Dimensional Derivation of Stokes Law

Stokes' law describes the motion of small particles in a viscous fluid medium. It states that the drag force experienced by a small spherical particle moving through a viscous fluid is directly proportional to the velocity of the particle, the viscosity of the fluid, and the radius of the particle. The equation for Stokes' law is:

F = 6πηrv

Where:
- F is the drag force experienced by the particle
- η is the viscosity of the fluid
- r is the radius of the particle
- v is the velocity of the particle

Dimensions of the Variables

To derive the dimensions of the variables in Stokes' law, we can use the principle of dimensional analysis. According to this principle, any physical equation must be dimensionally homogeneous, meaning that the dimensions of the variables on both sides of the equation must be the same.

Dimensions of Force (F)
- The dimensional formula for force is [M L T^-2] (mass x length x time^-2).

Dimensions of Viscosity (η)
- The dimensional formula for viscosity is [M L^-1 T^-1] (mass x length^-1 x time^-1).

Dimensions of Radius (r)
- The dimensional formula for radius is [L] (length).

Dimensions of Velocity (v)
- The dimensional formula for velocity is [L T^-1] (length x time^-1).

Derivation of Dimensions

Using the equation F = 6πηrv, we can determine the dimensions of each variable by equating the dimensions on both sides of the equation.

- Dimension of Force (F) = [M L T^-2]
- Dimension of 6π = Dimensionless constant
- Dimension of Viscosity (η) = [M L^-1 T^-1]
- Dimension of Radius (r) = [L]
- Dimension of Velocity (v) = [L T^-1]

Therefore, the dimensional equation becomes:
[M L T^-2] = [M L^-1 T^-1] [L] [L T^-1]

Final Dimensions

Simplifying the equation, we get:
[M L T^-2] = [M L^2 T^-2]

Comparing the dimensions on both sides, we find:
M = M
L = L
T^-2 = T^-2

Hence, the dimensions of each variable in Stokes' law are consistent, which confirms the dimensional derivation of the equation.

Conclusion

Stokes' law provides a mathematical relationship between the drag force experienced by a small spherical particle moving through a viscous fluid and the variables of velocity, viscosity, and radius. The dimensional derivation of Stokes' law ensures that the equation is consistent and valid, allowing us to understand and analyze the motion of particles in a fluid medium accurately.