Viscosity is calculated by dividing the shear stress by the shear rate. This fundamental relationship is crucial for understanding how fluids resist flow.
The Fundamental Formula
The exact calculation for viscosity ($\eta$) is given by:
$$\eta = \frac{\tau}{\dot{\gamma}}$$
Where:
- $\eta$ (eta) represents viscosity, typically measured in Pascal-seconds (Pa·s).
- $\tau$ (tau) represents shear stress, measured in Pascals (Pa).
- $\dot{\gamma}$ (gamma dot) represents shear rate, measured in reciprocal seconds (s⁻¹).
This formula derives directly from Newton's Law of Viscosity, which establishes the proportionality between shear stress and shear rate for many fluids.
Understanding Key Concepts
To fully grasp the calculation of viscosity, it's essential to understand the terms involved:
What is Viscosity?
Viscosity is a measure of a fluid's resistance to deformation by shear stress or tensile stress. In simpler terms, it describes the "thickness" or "stickiness" of a fluid. For example, honey has a much higher viscosity than water. High-viscosity fluids flow slowly, while low-viscosity fluids flow quickly.
For more information, you can refer to resources on viscosity.
What is Shear Rate?
Shear rate quantifies how quickly adjacent layers of a fluid slide past one another. Imagine a fluid confined between two parallel plates, where one plate is stationary and the other moves at a certain velocity. The shear rate is defined as:
- The velocity of the upper plate (in meters per second) divided by the distance between the two plates (in meters).
Its standard unit is reciprocal seconds [s⁻¹] or [1/s]. A higher shear rate means the fluid layers are moving past each other more rapidly.
Learn more about shear rate.
What is Shear Stress?
Shear stress is the force applied parallel to a surface, divided by the area over which it is applied. In the context of fluids, it's the internal force per unit area that opposes the applied force, causing the fluid to deform or flow. It represents the internal friction within the fluid. Shear stress is typically measured in Pascals (Pa).
Explore further details on shear stress.
The Relationship: Newton's Law of Viscosity
The relationship between these three terms is governed by Newton's Law of Viscosity. According to this law, shear stress is directly proportional to shear rate, with viscosity being the constant of proportionality.
Expressed mathematically, this law is:
$$\tau = \eta \cdot \dot{\gamma}$$
This equation clearly shows that if you know the shear stress applied to a fluid and the resulting shear rate, you can rearrange the formula to solve for viscosity.
Calculating Viscosity: A Practical Example
Let's illustrate with a simple example:
Scenario: Imagine an experiment where a fluid is being tested.
- The shear stress ($\tau$) measured on the fluid is 10 Pascals (Pa).
- The resulting shear rate ($\dot{\gamma}$) observed is 20 reciprocal seconds (s⁻¹).
Calculation:
Using the formula $\eta = \frac{\tau}{\dot{\gamma}}$:
$$\eta = \frac{10 \text{ Pa}}{20 \text{ s}^{-1}}$$
$$\eta = 0.5 \text{ Pa} \cdot \text{s}$$
Thus, the viscosity of this fluid under these conditions is 0.5 Pascal-seconds.
Important Considerations
When calculating viscosity from shear rate, it's important to consider the type of fluid and the conditions:
- Newtonian vs. Non-Newtonian Fluids:
- For Newtonian fluids (like water or simple oils), viscosity remains constant regardless of the shear rate. The calculated viscosity will be a fixed value.
- For Non-Newtonian fluids (such as paint, ketchup, or certain polymers), viscosity changes with shear rate. The calculated value is often referred to as "apparent viscosity" and is specific to the given shear rate.
- Temperature Dependence: Viscosity is highly sensitive to temperature. As temperature increases, the viscosity of most liquids decreases, and vice-versa. Therefore, any viscosity measurement or calculation should ideally specify the temperature at which it was performed.
Key Terms and Units
The following table summarizes the key terms involved in calculating viscosity:
| Term | Symbol | Unit (SI) | Description |
|---|---|---|---|
| Viscosity | $\eta$ (eta) | Pascal-second (Pa·s) | A fluid's resistance to flow |
| Shear Stress | $\tau$ (tau) | Pascal (Pa) | Force per unit area applied parallel to the fluid's surface |
| Shear Rate | $\dot{\gamma}$ (gamma dot) | Reciprocal second (s⁻¹) | The rate at which adjacent layers of a fluid slide past one another |
