Table of Contents
Surface Tension, Viscosity and Fluid Flow
1. Introduction
Surface tension, viscosity and fluid flow are important properties of liquids and gases. They explain phenomena such as the formation of liquid droplets, the movement of water through pipes, the floating of small insects on water, the resistance of honey to pouring and the flow of blood through arteries.
These concepts are central to fluid mechanics and have applications in engineering, biology, medicine, meteorology and industrial processes.
2. Surface Tension
2.1 Definition
Surface tension is the property of a liquid surface that causes it to behave as though it were under tension, tending to minimise its surface area.
It arises from cohesive forces between liquid molecules.
Molecules inside a liquid experience attractive forces from neighbouring molecules in all directions. Molecules at the surface experience an imbalance because there are fewer liquid molecules above them.
This produces a net inward attraction and gives the liquid surface its tendency to contract.
2.2 Molecular Explanation
Surface tension is associated with the intermolecular forces of attraction known as cohesion.
A liquid tends to minimise the number of molecules at its surface because surface molecules have higher energy than molecules in the bulk.
Consequently, a liquid tends to adopt a shape with minimum surface area for a given volume.
A sphere has the minimum surface area for a given volume, which helps explain why small freely suspended liquid droplets tend to be spherical.
2.3 Measurement and SI Unit
Surface tension is defined as force per unit length.
[
\gamma=\frac{F}{L}
]
Where:
- (\gamma) = surface tension
- (F) = force acting along the surface
- (L) = length over which the force acts
The SI unit is newton per metre (N/m).
Surface tension can also be described as surface energy per unit area, measured in joules per square metre (J/m²).
For a soap film with two surfaces, the total force may involve both surfaces, depending on the arrangement.
2.4 Factors Affecting Surface Tension
Temperature
Surface tension generally decreases as temperature increases because increased molecular motion weakens the effective influence of cohesive forces at the surface.
It approaches zero at the critical temperature, where the distinction between liquid and gas phases disappears.
Impurities and Dissolved Substances
The effect of impurities depends on their nature and concentration.
For example, detergents and soaps generally reduce the surface tension of water.
Certain dissolved salts can increase the surface tension of water.
2.5 Cohesion and Adhesion
Cohesion refers to the attraction between molecules of the same substance.
Adhesion refers to the attraction between molecules of different substances.
These forces help explain the behaviour of liquids on solid surfaces.
Examples:
- Water adheres to clean glass and tends to wet it.
- Mercury does not wet ordinary glass in the same way because its cohesive forces are relatively strong.
- Water forms a concave meniscus in a clean glass tube.
- Mercury forms a convex meniscus in a glass tube.
2.6 Capillary Action
Capillary action is the movement of a liquid through a narrow tube or porous material due to surface tension and the combined effects of adhesive and cohesive forces.
The height of capillary rise or depression in a narrow tube is approximately:
[
h=\frac{2\gamma\cos\theta}{\rho gr}
]
Where:
- (h) = height of rise or depression
- (\gamma) = surface tension
- (\theta) = contact angle
- (\rho) = liquid density
- (g) = acceleration due to gravity
- (r) = radius of the tube
For water in a clean glass tube, the contact angle is generally less than 90°, so water rises.
For mercury in glass, the contact angle is greater than 90°, so the liquid level is depressed.
Applications of Capillary Action:
- Movement of water through plant tissues
- Absorption of liquid by paper towels
- Ink movement in fountain pens
- Moisture movement through soil
- Wicks in oil lamps and candles
- Medical diagnostic strips
2.7 Everyday Effects of Surface Tension
- Small insects such as water striders can move on water surfaces.
- Small needles may rest on water if placed carefully.
- Soap bubbles form because surface tension helps maintain a thin liquid film.
- Water droplets tend to become spherical.
- Detergents improve wetting by reducing surface tension.
3. Viscosity
3.1 Definition
Viscosity is the property of a fluid that describes its resistance to flow due to internal friction.
A highly viscous liquid flows slowly, while a less viscous liquid flows more easily under comparable conditions.
For example:
- Honey is more viscous than water.
- Glycerine is more viscous than water.
- Engine oil is more viscous than petrol.
Viscosity is important in lubrication, hydraulic systems, blood circulation and industrial fluid transport.
3.2 Dynamic Viscosity
For a Newtonian fluid under simple laminar shear:
[
\tau=\mu\frac{du}{dy}
]
Where:
- (\tau) = shear stress
- (\mu) = dynamic viscosity
- (du/dy) = velocity gradient perpendicular to the flow
The SI unit of dynamic viscosity is pascal-second (Pa·s).
It can also be expressed as N·s/m².
3.3 Kinematic Viscosity
Kinematic viscosity is the ratio of dynamic viscosity to fluid density.
[
\nu=\frac{\mu}{\rho}
]
Where:
- (\nu) = kinematic viscosity
- (\mu) = dynamic viscosity
- (\rho) = fluid density
Its SI unit is square metre per second (m²/s).
3.4 Effect of Temperature on Viscosity
The effect of temperature differs between liquids and gases.
Liquids
The viscosity of most liquids decreases when temperature increases.
This happens because increased molecular motion helps molecules move past one another more easily.
Gases
The viscosity of most gases increases with temperature because molecular motion and momentum transfer between layers increase.
This difference is a common examination topic.
3.5 Newtonian and Non-Newtonian Fluids
Newtonian Fluids
Newtonian fluids have a constant viscosity at a given temperature and pressure, independent of the rate of shear under ordinary conditions.
Examples include:
- Water
- Air
- Many simple oils
Non-Newtonian Fluids
Non-Newtonian fluids do not follow a simple linear relationship between shear stress and shear rate.
Examples include:
- Toothpaste
- Paint
- Blood
- Ketchup
- Cornstarch suspension
Some non-Newtonian fluids become less viscous when stirred or sheared more rapidly, while others become more viscous.
3.6 Importance of Viscosity
Viscosity affects:
- Lubrication of machine components
- Fuel flow in engines
- Blood circulation
- Oil transportation through pipelines
- Movement of groundwater
- Design of pumps and hydraulic equipment
- Manufacturing of paints, cosmetics and food products
4. Fluid Flow
4.1 Definition
Fluid flow is the movement of a liquid or gas from one region to another due to forces such as pressure differences, gravity or applied mechanical energy.
Fluid flow is studied through fluid dynamics.
4.2 Types of Fluid Flow
Steady Flow
At a fixed point, fluid velocity and other flow properties remain constant with time.
Unsteady Flow
Fluid properties at a fixed point change with time.
Laminar Flow
Fluid moves in smooth, orderly layers with limited mixing.
Turbulent Flow
Fluid moves irregularly, with fluctuations, eddies and mixing.
Uniform Flow
At a particular instant, fluid velocity is the same at every point along the direction considered.
Non-Uniform Flow
Velocity varies from one location to another.
4.3 Reynolds Number
The Reynolds number is a dimensionless quantity used to predict whether flow is likely to be laminar or turbulent.
For flow in a circular pipe:
[
Re=\frac{\rho vD}{\mu}
]
Where:
- (\rho) = fluid density
- (v) = average flow velocity
- (D) = pipe diameter
- (\mu) = dynamic viscosity
For typical flow in a straight circular pipe:
- Re below approximately 2300: generally laminar.
- Re between approximately 2300 and 4000: transitional.
- Re above approximately 4000: generally turbulent.
These are approximate engineering guidelines. Actual transition depends on disturbances, pipe conditions and other factors.
4.4 Equation of Continuity
The continuity equation expresses conservation of mass.
For steady incompressible flow:
[
A_1v_1=A_2v_2
]
It means that the volume of fluid passing through each cross-section per unit time remains constant.
Therefore, when a pipe becomes narrower, fluid velocity increases.
4.5 Bernoulli’s Principle
Bernoulli’s principle relates pressure, velocity and height in ideal fluid flow.
[
P+\frac{1}{2}\rho v^2+\rho gh=constant
]
For horizontal flow, where height remains constant:
[
P+\frac{1}{2}\rho v^2=constant
]
Under these ideal conditions, higher flow velocity is associated with lower static pressure along a streamline.
Applications include Venturi meters, flow measurement and certain aerodynamic systems.
4.6 Poiseuille’s Law
Poiseuille’s law describes the volume flow rate of a Newtonian fluid through a long, straight, cylindrical tube under steady laminar flow.
[
Q=\frac{\pi r^4\Delta P}{8\mu L}
]
Where:
- (Q) = volume flow rate
- (r) = radius of the tube
- (\Delta P) = pressure difference
- (\mu) = dynamic viscosity
- (L) = length of the tube
The equation demonstrates that flow rate is highly sensitive to tube radius because it depends on the fourth power of the radius.
If the radius doubles, the ideal flow rate increases by a factor of 16, provided the other variables remain constant.
Poiseuille’s law helps explain why small changes in the radius of blood vessels can significantly affect blood flow.
5. Terminal Velocity
Terminal velocity is the constant maximum speed reached by an object falling through a fluid when the net force on it becomes zero.
During a fall through a fluid, three forces may act:
- Weight acting downward.
- Buoyant force acting upward.
- Drag force acting opposite to the direction of motion.
At terminal velocity:
[
W=F_B+F_D
]
Where:
- (W) = weight
- (F_B) = buoyant force
- (F_D) = drag force
Terminal velocity is important in parachuting, sedimentation, particle settling and fluid engineering.
6. Practical Applications
- Surface tension explains the formation of droplets and soap bubbles.
- Capillary action helps transport water through plants.
- Detergents reduce surface tension and improve cleaning.
- Viscosity determines the suitability of oils for lubrication.
- Reynolds number helps engineers assess flow regimes.
- Poiseuille’s law helps model blood flow through narrow vessels under idealised conditions.
- Bernoulli’s principle is used in flow measurement.
- Terminal velocity helps explain the motion of raindrops and parachutists.
7. Worked Examples
Example 1: Surface Tension
A force of 0.06 N acts along a liquid surface of length 0.2 m. Calculate surface tension.
[
\gamma=\frac{F}{L}
]
[
\gamma=\frac{0.06}{0.2}=0.3,N/m
]
Answer: 0.3 N/m.
Example 2: Dynamic Viscosity
A fluid experiences a shear stress of 4 N/m². Its velocity gradient is 20 s⁻¹. Calculate its dynamic viscosity.
[
\tau=\mu\frac{du}{dy}
]
[
\mu=\frac{4}{20}=0.2,Pa\cdot s
]
Answer: 0.2 Pa·s.
Example 3: Reynolds Number
Water flows through a pipe with an average velocity of 0.5 m/s. The pipe diameter is 0.02 m. Take water density as 1000 kg/m³ and dynamic viscosity as 0.001 Pa·s.
[
Re=\frac{\rho vD}{\mu}
]
[
Re=\frac{1000\times0.5\times0.02}{0.001}
]
[
Re=10,000
]
Answer: 10,000. The flow is generally turbulent in a straight circular pipe under ordinary conditions.
Example 4: Effect of Pipe Radius
If the radius of a pipe doubles, what happens to the ideal volume flow rate according to Poiseuille’s law, assuming all other variables remain constant?
[
Q\propto r^4
]
[
Q_{new}=2^4Q_{old}=16Q_{old}
]
Answer: The flow rate increases sixteen times.
8. Important Examination Distinctions
| Concept | Meaning |
|---|---|
| Surface tension | Force per unit length along a liquid surface |
| Cohesion | Attraction between similar molecules |
| Adhesion | Attraction between different substances |
| Capillary action | Liquid movement through narrow spaces |
| Viscosity | Resistance of a fluid to flow |
| Dynamic viscosity | Resistance to shear deformation |
| Kinematic viscosity | Dynamic viscosity divided by density |
| Laminar flow | Smooth and orderly fluid movement |
| Turbulent flow | Irregular fluid movement |
| Reynolds number | Dimensionless indicator of flow regime |
| Terminal velocity | Constant speed when net force becomes zero |
9. 36 Key Takeaways for Exams
- Surface tension is a property of liquid surfaces arising from cohesive molecular forces.
- Surface tension tends to minimise the surface area of a liquid.
- Cohesion is attraction between molecules of the same substance.
- Adhesion is attraction between molecules of different substances.
- Surface tension is measured in N/m.
- Surface tension generally decreases as temperature increases.
- Soaps and detergents generally reduce the surface tension of water.
- Small liquid droplets tend to become spherical because of surface tension.
- Surface tension helps explain the movement of water striders on water.
- Capillary action is the movement of liquid through narrow tubes or porous materials.
- Capillary action depends on surface tension, adhesion and cohesion.
- Water generally rises in a clean glass capillary tube.
- Mercury generally shows capillary depression in glass.
- A concave meniscus is commonly formed by water in clean glass.
- A convex meniscus is commonly formed by mercury in glass.
- Viscosity is a fluid’s resistance to flow.
- Honey is more viscous than water under comparable conditions.
- Dynamic viscosity is measured in Pa·s.
- Kinematic viscosity is dynamic viscosity divided by density.
- The SI unit of kinematic viscosity is m²/s.
- The viscosity of most liquids decreases as temperature increases.
- The viscosity of most gases increases as temperature increases.
- Newtonian fluids have viscosity independent of shear rate at fixed conditions.
- Non-Newtonian fluids do not follow a simple linear shear-stress relationship.
- Laminar flow is smooth and orderly.
- Turbulent flow involves irregular movement and eddies.
- Reynolds number helps predict the flow regime.
- Reynolds number is dimensionless.
- The continuity equation expresses conservation of mass.
- Bernoulli’s principle expresses conservation of mechanical energy in ideal fluid flow.
- A Venturi meter uses changes in fluid pressure and velocity for flow measurement.
- Poiseuille’s law applies to idealised laminar flow through a cylindrical tube.
- Poiseuille’s flow rate is proportional to the fourth power of tube radius.
- Terminal velocity occurs when the net force on a falling object becomes zero.
- At terminal velocity, weight is balanced by buoyancy and drag.
- Surface tension, viscosity and fluid flow are important in engineering, biology, medicine and industrial processes.