How to Calculate the Coefficient of Viscosity with Examples
The concept of viscosity is essential for understanding how different fluids flow and resist motion. Take a simple example: if you place drops of water and honey on a slanting surface, you’ll notice that water flows quickly while honey moves slowly. This difference is due to viscosity—honey is more viscous than water, meaning it has more internal resistance to flow.
Viscosity is defined as the resistance offered by different layers of a fluid as they move over one another. The force needed to slide one layer past another depends on the internal friction between layers.
What is the Coefficient of Viscosity?
The coefficient of viscosity quantifies the internal friction within a fluid. It is defined as the ratio of the shearing stress to the velocity gradient between fluid layers.
Mathematically, the coefficient of viscosity (denoted by η – Greek letter eta) is given by:
η = F · d / (A · v)
- F = Tangential force applied (in Newtons)
- d = Distance between the two layers (in meters)
- A = Area of each layer (in square meters)
- v = Relative velocity between the layers (in meters/second)
This formula comes from observing that, practically, there will always be some frictional force preventing one layer from sliding easily over the next.
Understanding the Viscous Gradient
The viscous or velocity gradient is the difference in velocity between adjacent layers, divided by the distance between them (v/x or du/dy). It measures how rapidly the fluid's velocity changes from one layer to another.
Units and Dimensions of Coefficient of Viscosity
In SI units, the coefficient of viscosity is measured in pascal-second (Pa·s) or Newton-second per square meter (N·s·m-2). In CGS units, it is measured in dyne-second per cm2, known as poise.
1 poise = 0.1 Pa·s
| System | Unit | Symbol |
|---|---|---|
| SI | Pascal-second | Pa·s or N·s·m-2 |
| CGS | Poise | g·cm-1·s-1 |
| MKS | Kg·m-1·s-1 | Kg·m-1·s-1 |
The dimensional formula for coefficient of viscosity is [M L-1 T-1].
Coefficient of Viscosity – Practical Calculation and Example
Suppose two large plates are separated by a thin liquid film of thickness d. If you apply a tangential force F to keep the top plate moving with velocity v, over area A, the coefficient of viscosity can be calculated:
η = F · d / (A · v)
This equation is fundamental for solving numerical problems on viscosity.
Viscosity of Water and Common Fluids
Each liquid has a characteristic coefficient of viscosity. For water at room temperature (25°C):
| Fluid | Viscosity (SI Units) | Viscosity (CGS/Poise) |
|---|---|---|
| Water (25°C) | 8.90 × 10-4 Pa·s | 0.0091 poise |
| Honey | Much higher than water | — |
| Glycerin | Higher than water | — |
The viscosity of gases is generally lower than liquids.
Poiseuille’s Law for Experimental Determination
The coefficient of viscosity for liquids can be measured using Poiseuille’s Law. If a liquid flows through a capillary tube of length l and radius r, the rate of flow V is related to the pressure difference P and viscosity η by:
V = (π P r4) / (8 η l)
Rearranged to solve for η:
η = (π P r4) / (8 V l)
Key Differences: Viscosity vs. Density
| Property | Viscosity | Density |
|---|---|---|
| Definition | Resistance to flow due to internal friction | Mass per unit volume |
| Effect | Affects speed of flow (thicker fluids flow slower) | Affects buoyancy and weight |
| Unit | Pa·s (SI) | Kg·m-3 (SI) |
Solving Numericals: Step-by-Step Approach
- List all known quantities (F, d, A, v).
- Make sure all values are in SI units.
- Use the formula: η = F · d / (A · v).
- Substitute the values and solve step-by-step.
- Mention the answer with the correct unit (Pa·s).
Example: If F = 2 N, d = 0.002 m, A = 0.25 m2, and v = 0.1 m/s,
η = (2 × 0.002) / (0.25 × 0.1) = 0.004 / 0.025 = 0.16 Pa·s.
Real-Life Applications
- Design of lubricants for machines and engines
- Blood flow analysis in medical science
- Piping and industrial fluid transport
- Hydraulic system performance
Further Learning & Related Resources
- Viscosity - Complete Concept
- Unit of Viscosity Explained
- Mechanical Properties of Fluids
- Lab: Coefficient of Viscosity
To master this concept, practice solving questions and refer to detailed study notes on viscosity. Understanding the coefficient of viscosity forms a bridge to mastering other fluid mechanics topics and enhances problem-solving confidence in Physics exams.
FAQs on Coefficient of Viscosity: Meaning, Formula, SI Unit & Applications
1. What is meant by the coefficient of viscosity in Physics?
The coefficient of viscosity is a quantitative measure of a fluid's internal resistance to flow when one layer moves over another. It is defined as the tangential force required per unit area to maintain a unit velocity gradient between parallel layers of the fluid. It is a key physical property representing how 'thick' or 'thin' a fluid is.
2. State the SI unit and dimensional formula of coefficient of viscosity.
The SI unit of the coefficient of viscosity is pascal-second (Pa·s) or kg·m-1·s-1. The dimensional formula of the coefficient of viscosity is [M1L-1T-1].
3. How does temperature affect the coefficient of viscosity of a liquid?
Increasing temperature generally decreases the coefficient of viscosity of a liquid.
Explanation:
- At higher temperatures, increased molecular motion reduces intermolecular attractions.
- This causes fluids like water to flow more easily, thus lowering viscosity.
- For liquids: viscosity decreases with increasing temperature.
4. What is the physical significance of a high coefficient of viscosity?
A high coefficient of viscosity means the fluid is very resistant to flow.
This indicates:
- The fluid has high internal friction.
- It acts 'thicker' and moves slowly (e.g., honey, glycerine).
- More force is needed to make the fluid layers slide past one another.
5. Why is the coefficient of viscosity different for liquids and gases?
The coefficient of viscosity differs for liquids and gases due to different molecular mechanisms:
- Liquids: Viscosity arises primarily from intermolecular attraction.
- Gases: Viscosity is mainly due to momentum transfer between molecules.
- As a result, temperature effects and viscosity magnitudes behave differently for liquids and gases.
6. Write the mathematical expression for coefficient of viscosity and explain each term.
The mathematical expression: η = F·d / (A·v)
Where:
- η = Coefficient of viscosity
- F = Tangential force applied
- d = Distance between fluid layers
- A = Area of each layer
- v = Relative velocity between layers
7. What happens to the viscosity of water when impurities are added?
Adding impurities (like dissolved salts or solutes) generally increases the viscosity of water.
This is because:
- Solute particles hinder the motion of water molecules.
- They increase internal friction, making the water flow more slowly.
8. Describe how Poiseuille's law helps determine the coefficient of viscosity experimentally.
Poiseuille’s law relates the rate of flow of a liquid through a narrow tube to the liquid’s viscosity.
Experimentally:
- The volume of fluid flowing per second is measured for known pressure, tube length, and radius.
- Using Poiseuille’s equation, the coefficient of viscosity can be calculated from these measurements.
9. How is the coefficient of viscosity important in real-life applications?
The coefficient of viscosity is crucial in many fields:
- Lubricant design for engines and machinery
- Blood flow analysis in medical diagnostics
- Pipeline and fluid transport industries
- Performance of hydraulic systems
10. Distinguish between viscosity and density of a fluid.
Viscosity is a measure of a fluid’s resistance to flow due to internal friction, while density is the mass per unit volume of a fluid.
- High viscosity: Fluid flows slowly (“thick”).
- High density: Fluid is heavier for the same volume.
They are two independent properties of fluids.
11. What common misconceptions exist about viscosity in liquids and gases?
A common misconception is that all fluids respond to temperature the same way.
In reality:
- For liquids, viscosity decreases as temperature increases.
- For gases, viscosity increases as temperature increases, due to enhanced molecular motion and collisions.
12. List some practical methods for measuring the coefficient of viscosity.
Common methods to measure viscosity include:
- Capillary tube method (using Poiseuille’s law)
- Falling sphere method (using Stokes’ law)
- Rotational viscometers
