Fluid Mechanics – Part 1 (9a)

What are fluids?

• include both liquids and gases.

• They are common in everyday life, such as the air we breathe, water in lakes/rivers/sea, and the liquids we drink.

• Their behaviour is widely understood through everyday experiences.

What is the difference between liquids and gases?

• Liquids have a definite volume but no fixed shape, taking the shape of their container.

• Gases have no fixed volume or shape — they expand to fill any container.

• Both flow, so both are fluids.

Why do liquids have a definite volume?

• In a liquid, the forces of attraction between molecules (called cohesion) prevent the molecules from moving far apart.

• These cohesive forces keep the liquid’s volume fixed, even though it can still change shape.

What happens when a shear stress is applied to a fluid?

• A fluid at rest cannot support shear stress.

• Any applied shear causes continuous deformation (flow).

• When the shear is removed, the fluid has no residual stress, regardless of molecular displacement.

Why do liquids have a definite volume?

• Liquids have molecular cohesion — attractive forces between molecules.

• These cohesive forces keep the molecules close, preventing them from spreading out too far.

• Therefore, liquids maintain a definite volume.

Why do gases expand to fill their container?

• Gas molecules move rapidly and freely.

• Their large molecular motion overcomes cohesive forces between molecules.

• As a result, gas molecules spread out and fill the entire container, meaning gases have no free surface.

Why are both liquids and gases considered fluids?

• Both flow.

• Liquids have definite volume but no fixed shape.

• Gases have no fixed volume or shape and expand to fill a container.

What is the molecular difference between a liquid and a gas?

• Liquids: Molecules experience cohesive forces, keeping them close and giving liquids a definite volume.

• Gases: Molecules move rapidly; cohesive forces are overcome, so they spread out and fill the entire container.

Why does a gas expand to fill its container?

• Because the molecules move freely and spread out until they occupy all available space, creating no free surface.

When does a liquid turn into a gas?

What happens when a shear stress is applied to a fluid at rest?

• A stationary fluid cannot resist shear stress.

• Any applied shear causes continuous deformation (flow).

• Once the shear is removed, the fluid has no residual stress regardless of molecular displacement.

What defines a fluid in terms of shear stress?

• A fluid at rest cannot support shear stress.

• Any shear force causes flow, not elastic deformation.

What is the continuum assumption in fluid mechanics?

• Matter is made of atoms/molecules, but in engineering, fluid volumes are large enough that properties can be treated as continuous and homogeneous, not particle-based.

• Example: A tiny 1-mm water droplet contains ~2.9 × 10¹⁸ molecules, so molecular detail is unnecessary for calculations.

• Therefore, fluids are treated as a continuum, allowing use of smooth fields (density, pressure, velocity).

Why can’t very tiny fluid samples be treated as homogeneous?

• In a very, very small fluid sample, only a few molecules may be inside the volume at any moment.

• Due to random molecular motion, the number of molecules, their speeds, and their directions constantly change.

• Because of this randomness, the fluid cannot be considered homogeneous at that microscopic scale.

Why can fluids be treated as a continuum despite random molecular motion?

• Even though atoms and molecules move randomly, the timescales of this motion are extremely short.

• For any engineering-sized volume and any practical time interval, the properties of the fluid remain effectively constant.

• Therefore, we assume the fluid behaves as a continuous (homogeneous) medium, unaffected by random molecular fluctuations.

What causes pressure in a fluid?

Pressure is caused by atoms and molecules in a fluid colliding with each other and with the surfaces of a container.

What causes pressure inside a gas?

• Gas molecules are in constant random motion.

• When they collide with each other, they exert equal and opposite forces, producing no net force on the container.

• However, these collisions create a continuous internal pressure within the fluid.

What happens when gas molecules collide with a wall, and how does this relate to pressure?

• Gas molecules are in constant random motion and collide with the walls of their container.

• In a collision with a wall (assumed perfectly elastic):

  • The parallel component of velocity stays the same.

  • The perpendicular component keeps the same magnitude but reverses direction.

• These repeated molecular impacts produce a force on the walls, which is experienced as pressure.

How do many molecular collisions create pressure on a surface?

• Each molecule collides with the surface, reversing its velocity perpendicular to the wall.

• A single collision produces only a tiny force.

• But because many molecules collide repeatedly, their combined impacts create a uniform distribution of force, which we observe as pressure.

Is pressure a scalar or vector, and how does it act in a fluid?

• Pressure is a scalar because it has no specific direction — it acts equally in all directions within a fluid due to molecular collisions.

• But when a fluid is in contact with a surface, the force exerted is perpendicular to that surface, making the force a vector quantity.

Is pressure a scalar or vector, and how does it act on surfaces?

• Pressure is a scalar quantity because it has no specific direction — it acts equally in all directions within a fluid as molecules collide.

• When a fluid contacts a surface, the force exerted on that surface is always perpendicular (normal) to the surface, making that force a vector quantity.

How should pressure and force be represented in diagrams?

• Pressure is a scalar and acts in all directions, so it must NOT be shown as an arrow on a diagram.

• The force caused by pressure acting on a surface is a vector and can be shown with an arrow, because it acts perpendicular to the surface.

Why is pressure a scalar, but the force it creates on a surface a vector?

• Pressure is a scalar because it has no direction—it acts equally in all directions within a fluid as molecules collide randomly.

• When pressure acts on a surface, it produces a force that is always perpendicular to that surface.

• This force has direction, so it is a vector.

What is the difference between absolute pressure and gauge pressure?

• Absolute pressure is measured relative to a perfect vacuum (0 pressure).

• Gauge pressure is the pressure above atmospheric pressure.

• Atmospheric pressure is all around us, so in engineering we often use gauge pressure because it tells us how much pressure is in addition to the surrounding air pressure.

Why is pressure a scalar, but the force it creates is a vector?

• Pressure is a scalar because it acts equally in all directions within a fluid and has no specific direction.

• When a fluid touches a surface, the force caused by the pressure is always perpendicular to the surface, giving it a specific direction, so the force is a vector.

How do pressure and force behave in a fluid?

• Pressure is a scalar because it has no direction — it acts equally in all directions within a fluid.

• The force created by pressure on a surface does have a direction: it always acts perpendicular (normal) to the surface.

What is the perfect gas equation of state?

• A perfect gas is idealised as rigid, perfectly elastic spheres in constant motion.

• This model leads to the perfect gas equation of state, which relates pressure, volume, and temperature of a gas.

• It accurately describes gas behaviour over a wide range of typical engineering conditions.

What is the perfect gas equation of state?

• A gas can be modelled as rigid, perfectly elastic spheres in constant random motion that collide with each other and the container walls.

• This idealised model leads to the perfect (ideal) gas equation of state, which describes the relationship between pressure, volume, and temperature in gases over a wide range of engineering conditions.

Why is pressure considered a scalar, but the force it creates is a vector?

• Pressure is a scalar because it has no direction—it acts equally in all directions inside a fluid.

• But when pressure acts on a surface, it produces a force that is perpendicular to that surface, and force has direction, so it is a vector.

What is the perfect gas equation of state and what concept is it based on?

• A gas can be modelled as rigid, perfectly elastic spheres in constant motion that collide with each other and the container walls.

• This idealized model leads to the perfect (ideal) gas equation of state, which accurately describes the relationship between pressure, volume, and temperature for many engineering conditions.

What is the Perfect Gas (Ideal Gas) Equation of State based on?

• A gas is modeled as rigid, perfectly elastic spheres moving randomly and colliding with each other and container walls.

• This idealized model leads to the Perfect Gas (Ideal Gas) Equation of State, which describes the relationship between pressure, volume, and temperature.

• It is valid across a wide range of engineering conditions.

What is the perfect gas (ideal gas) equation of state based on?

• A gas is modelled as rigid, elastic spheres in constant motion.

• This ideal behaviour leads to the perfect gas equation of state:

  pV=nR~TpV = n \tilde{R} TpV=nR~T

• It is derived from Boyle’s law ( V∝1/pV \propto 1/pV∝1/p at constant T ) and Charles’s law ( V∝TV \propto TV∝T at constant p ).

• R~=8314 J kmol−1K−1\tilde{R} = 8314\ \text{J kmol}^{-1}\text{K}^{-1}R~=8314 J kmol−1K−1 is the universal gas constant.

What is compressibility in fluids?

• Increasing pressure reduces the volume of a substance and increases its density.

• Liquids and solids: Volume change is very small (almost incompressible).

• Gases: Even small pressure changes cause large changes in volume and density (highly compressible).

When can gases be treated as incompressible?

• Gases compress easily: small pressure changes cause large volume changes.

• For gases, the bulk modulus KKK is roughly equal to their pressure.

• A very small volumetric strain (e.g., 0.1%) needs only a very small pressure change.

• Therefore, gases can only be treated as incompressible when pressure changes are extremely small.

What is surface tension and why does it occur?

• Surface tension comes from cohesive forces between liquid molecules.

• At the liquid’s surface, these forces create a “skin” that holds the liquid together.

• This is why small drops form spherical shapes.

• Definition: Surface tension is the tensile force per unit length acting at a liquid’s surface.

What determines whether a liquid wets a solid surface?

• Adhesion > Cohesion: The liquid spreads out and wets the surface.

• Cohesion > Adhesion: The liquid does not wet the surface and forms droplets.

• Examples:

  • Water wets clean glass.

  • Mercury does NOT wet glass.

  • Water does NOT wet a greasy surface.

How do adhesion and cohesion affect whether a liquid wets a solid surface?

• Adhesion > Cohesion: The liquid spreads out and wets the solid surface.

• Cohesion > Adhesion (high surface tension): The liquid does not wet the surface and forms droplets.

• Examples:

  • Water wets clean glass.

  • Mercury does not wet glass.

  • Water does not wet a greasy surface.

What is hydrostatics?

• Hydrostatics is the study of fluids at rest or fluids in uniform motion where all parts move with the same velocity.

• This means there is no relative motion within the fluid.

• Example: Liquid inside a moving road tanker can be treated as a hydrostatics problem if the liquid is not sloshing around.

How does pressure vary with elevation in a stationary fluid?

• In a stationary fluid, gravity acts downward, creating a vertical pressure gradient.

• Pressure increases with depth and decreases with elevation.

When does a liquid wet or not wet a solid surface?

• If adhesion (liquid–solid attraction) is greater than cohesion (liquid–liquid attraction), the liquid spreads out and wets the surface.

• If cohesion is greater than adhesion, the liquid does not wet the surface and forms droplets.

• Examples:

  • Water wets clean glass.

  • Mercury does not wet glass.

  • Water does not wet a greasy surface.

What is a piezometer used for?

• A piezometer is the simplest type of manometer.

• It is used for measuring the pressure in a liquid.

How do surface tension and adhesion affect manometer readings?

• Surface tension and adhesion can create a meniscus, meaning the liquid surface in the tube is not flat.

• This makes it hard to read the exact liquid level.

• The contact angle between the liquid and the tube depends on both the type of liquid and the tube material.

What causes capillary rise in narrow tubes, and how is it estimated?

• Capillary rise occurs when liquid climbs up a narrow tube because of adhesion between the fluid molecules and the tube wall.

• If the meniscus is approximated as a hemisphere, the rise can be estimated by equating the surface tension force (γ) with the weight of the lifted fluid.

• Surface tension (γ) is the force per unit length acting at the liquid surface.

What causes buoyancy and when does a body float?

• In a fluid at rest, pressure increases with depth, so the lower surfaces of an immersed body experience higher pressure than the upper surfaces.

• This creates a net upward force called buoyancy.

• A body floats when the buoyancy force is equal to its weight.

Why does a buoyant force act upward on an immersed body?

• The buoyancy force has no horizontal component because horizontal pressures balance on opposite sides.

• The upward thrust on the lower surface of a fully immersed body equals the weight of the fluid directly above that surface.

• This volume of fluid (real or imaginary) represents the buoyant force that makes the body float.

How can buoyancy be explained using an equilibrium argument?

• Imagine removing the submerged body and replacing it with an identical volume of fluid.

• This imagined fluid volume is in equilibrium under its own weight and surrounding pressure forces.

• Therefore, the upward thrust (buoyant force) on the original body must equal the weight of the displaced fluid.

What is the centre of buoyancy?

• When a body is immersed in a fluid, the buoyancy force equals the weight of the displaced fluid.

• This force acts through the centre of buoyancy — the centroid of the displaced fluid volume.

• It is not usually the same as the body’s own centre of gravity.

What determines whether a floating body is stable in a given orientation?

• Buoyancy alone does not determine stability.

• Stability depends on the relative positions of the centre of buoyancy and the centre of gravity.

• A body floats in a stable orientation when the centre of buoyancy rises to support the centre of gravity after small disturbances.

• Example: A pencil floats horizontally but not vertically because the centre of gravity and centre of buoyancy alignment is unstable in the vertical orientation.

What are the key properties and pressure concepts in fluid mechanics?

• Key properties: mass, weight, density, pressure.

• Pressure comes from molecular collisions and acts equally in all directions, but the force on a surface is always perpendicular.

• Distinguish between absolute pressure (relative to vacuum) and gauge pressure (above atmospheric).

• The ideal gas law relates pressure, volume, and temperature in gases.

• Liquids are nearly incompressible, while gases change density significantly with pressure.

How do pressure and force act in a fluid?

• Pressure is a scalar and acts equally in all directions within a fluid.

• The force exerted by a fluid on a surface is always perpendicular to that surface.