Problem Statement
Find the kinematic viscosity of an oil having a density of 980 kg/m³ when at a certain point in the oil, the shear stress is 0.25 N/m² and the velocity gradient is 0.3 s⁻¹.
Given Data
- Density, \(\rho = 980 \, \text{kg/m}^3\)
- Shear Stress, \(\tau = 0.25 \, \text{N/m}^2\)
- Velocity Gradient, \(\frac{du}{dy} = 0.3 \, \text{s}^{-1}\)
Solution
1. Calculate the Dynamic Viscosity (\(\mu\))
We start with Newton's law of viscosity and rearrange it to solve for the dynamic viscosity, \(\mu\).
2. Calculate the Kinematic Viscosity (\(\nu\))
Kinematic viscosity is defined as the ratio of dynamic viscosity to density.
3. Convert Kinematic Viscosity to Stokes
The CGS unit for kinematic viscosity is the Stoke (St), where 1 m²/s = 10,000 Stokes.
The kinematic viscosity of the oil is \( \nu \approx 0.00085 \, \text{m}^2/\text{s} \) or \( 8.5 \, \text{Stokes} \).
Explanation of Viscosity Types
It's important to distinguish between the two types of viscosity calculated here:
1. Dynamic Viscosity (\(\mu\)):
Also known as absolute viscosity, this represents a fluid's internal resistance to shear forces. It directly relates the shear stress (\(\tau\)) applied to a fluid to the rate of deformation (the velocity gradient, \(du/dy\)). It's the measure of a fluid's "stickiness" or "thickness" and has units of N·s/m² or Poise.
2. Kinematic Viscosity (\(\nu\)):
Kinematic viscosity is the ratio of dynamic viscosity to the fluid's density (\(\nu = \mu/\rho\)). It measures a fluid's resistance to flow under the influence of gravity. Because it includes density, it describes how easily a fluid flows in relation to its mass. The units are m²/s or Stokes.
Physical Meaning
The kinematic viscosity is a crucial parameter in fluid dynamics, particularly in the Reynolds number, which determines whether a flow is laminar or turbulent. It essentially represents the ratio of viscous forces to inertial forces.
A high kinematic viscosity means that viscous forces are dominant, and the fluid will flow smoothly (laminar flow) and resist motion. A low kinematic viscosity means inertial forces are more significant, and the fluid is more prone to turbulent, chaotic flow.
For example, honey has a very high dynamic viscosity (it's very thick), but if it were also extremely dense, its kinematic viscosity might not be as high as another fluid that is less thick but much lighter. Kinematic viscosity gives a measure of how quickly momentum diffuses through the fluid compared to its bulk motion, which is why it's so important for analyzing flow behavior.
