Calculating Viscosity Using Stokes' Law: An Air Bubble Rising in a Liquid

Understanding the behavior of a bubble rising in a liquid helps us determine the viscosity of the liquid through the application of Stokes' Law. In this article, we explore how to calculate the coefficient of viscosity using the motion of an air bubble through a liquid.

Introduction to the Problem

Consider an air bubble of a 1 cm radius that rises from the bottom portion of a liquid with a density of 1.5 g/cm3 at a constant speed of 0.25 cm/s. We will use Stokes' Law to find the coefficient of viscosity of the liquid.

Application of Stokes' Law

Stokes' Law describes the motion of a small sphere through a viscous fluid and states that the drag force (F_d) acting on the bubble can be expressed as:

F_d 6 pi eta r v

Where:

(eta) is the coefficient of viscosity in poise (P) (r) is the radius of the bubble in cm (v) is the velocity of the bubble in cm/s

Determining the Buoyant Force

The buoyant force (F_b) acting on the bubble must equal the drag force when the bubble rises at a constant speed. The buoyant force can be calculated using Archimedes' principle:

F_b text{Volume of the bubble} times text{Density of the liquid} times g

Given:

(text{Radius of the bubble}, r 1, text{cm}) (text{Density of the liquid}, rho 1.5, text{g/cm}^3) (text{Velocity of the bubble}, v 0.25, text{cm/s})

Calculating the Volume of the Bubble

The volume (V) of a sphere is given by:

V frac{4}{3} pi r^3

Substituting the given radius:

V frac{4}{3} pi (1)^3 frac{4}{3} pi , text{cm}^3

Calculating the Buoyant Force

Assuming (g approx 980, text{cm/s}^2), the buoyant force is:

F_b V cdot rho cdot g left(frac{4}{3} pi right) cdot 1.5 cdot 980, text{g cm/s}^2

Calculating the buoyant force:

F_b approx 1960, text{g cm/s}^2 1960, text{dyn}

Setting Up the Equation

Since the bubble rises at a constant speed, the drag force equals the buoyant force:

6 pi eta r v F_b

Substituting the known values:

6 pi eta 1 cdot 0.25 1960

Solving for the Coefficient of Viscosity

Rearranging the equation to solve for (eta):

(eta frac{1960}{6 pi 0.25})

Calculating the right-hand side:

(eta approx frac{1960}{1.5 pi} approx frac{1960}{4.712} approx 416.5, text{g/cm s} 416.5, text{poise})

Conclusion

The coefficient of viscosity (eta) of the liquid is approximately 416.5 poise. This method allows us to determine the properties of a liquid using the behavior of an air bubble rising through it.

Alternatively, if the result of terminal velocity in fluid dynamics is known, the calculation can be simplified into a linear equation to find the coefficient of viscosity. The coefficient of viscosity in SI units (N s/m2) would be (frac{400}{3}).