Determining the Viscosity Coefficient of a Liquid with a Given Density

In this article, we delve into the process of determining the coefficient of viscosity for a liquid with a specific density using a scientific approach based on Stokes Law.

Introduction to Viscosity and Stokes Law

Viscosity is a property of fluids that quantifies the fluid's resistance to flow. For a spherical bubble rising through a liquid, the dynamic viscosity (coefficient of viscosity) can be calculated using Stokes Law. The law describes the drag force acting on a sphere moving through a viscous medium.

Understanding the Relationship

The formula for the drag force (Fd) acting on a sphere moving through a fluid is given by:

Fd 6πηrv

where:

η is the dynamic viscosity coefficient of the fluid, r is the radius of the sphere (in this case, the bubble), and v is the velocity of the sphere.

For a bubble rising at a constant speed, the drag force equals the buoyant force minus the weight of the bubble. However, we can simplify the analysis by focusing on the drag force, given that we are asked to disregard the density of air.

Given Data and Calculations

The given data includes:

Density of the liquid, ρ 1750 kg/m3 Radius of the bubble, r 10 cm 0.1 m Speed of the bubble, v 3.5 mm/s 0.0035 m/s

Calculation Method

Using Stokes Law, we can rearrange the formula to solve for η:

η Fd / (6πr v)

The drag force (Fd) can be expressed as the weight of the displaced liquid minus the weight of the bubble. The buoyant force (Fb) is given by:

Fb ρgV

where V is the volume of the bubble, which can be calculated as:

V (4/3)πr3

The weight of the bubble (W) can be neglected, so we can equate the buoyant force to the drag force:

Fd Fb ρgV

Substituting for V gives:

Fb ρg ((4/3)πr3 )

Thus:

η (ρg ((4/3)πr2 )) / (6 v)

With the acceleration due to gravity g ≈ 9.81 m/s2, we substitute the known values into the equation:

η (1750 kg/m3 × 9.81 m/s2 × ((4/3) × 0.12)) / (6 × 0.0035 m/s)

Step-by-Step Calculation

Let's perform the calculation step by step:

Calculate 0.12 0.01 Calculate (4/3) × 0.01 0.013333 Calculate 1750 × 9.81 × 0.013333 ≈ 230.3 N

Substituting back into the viscosity formula:

η 230.3 / (6 × 0.0035)

Calculating the denominator:

6 × 0.0035 0.021

Thus:

η ≈ 230.3 / 0.021 ≈ 10976.19 Pa.s

Conclusion

The coefficient of viscosity of the liquid is approximately:

η ≈ 10976 Pa.s

This process showcases how to determine the dynamic viscosity of a liquid using Stokes Law and given data, providing a clear and detailed approach to understanding and applying these principles in practical scenarios.