Class 11 Physics · Chapter 9 NotesMechanical Properties of Fluids

A fluid is a substance that can flow, so liquids and gases are both fluids. Unlike a solid, a fluid has no definite shape of its own; a liquid has a fixed volume, while a gas fills the whole container. The key mechanical difference is that fluids offer very little resistance to shear stress — their shape changes under even a tiny shearing stress. When a fluid is at rest, the force it exerts on any surface is always normal (perpendicular) to that surface, because a parallel component would set the fluid in motion. Pressure is the normal force per unit area, P = F/A, and it is a scalar quantity. Its SI unit is the pascal (Pa), that is N m⁻². Density is mass per unit volume, ρ = m/V, with SI unit kg m⁻³. Relative density compares a substance's density with that of water at 4 °C and has no unit.

• PressureP = F/A, unit pascal (Pa), 1 atm = 1.013 × 10⁵ Pa
• Densityρ = m/V, unit kg m⁻³; water at 4 °C has ρ = 1.0 × 10³ kg m⁻³
• Relative density = density of substance ÷ density of water at 4 °C (dimensionless)
• Liquids are largely incompressible; gases change density greatly with pressure

Pascal's law states that pressure in a fluid at rest is the same at all points at the same height, and that a change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container. This is why a small force on the small piston of a hydraulic lift produces a large force on the larger piston: F₂ = F₁A₂/A₁, and the factor A₂/A₁ is the mechanical advantage. Pressure also increases with depth. For a fluid column of height h, the pressure difference is P₂ − P₁ = ρgh. If the surface is open to the atmosphere, P = Pₐ + ρgh. The excess pressure P − Pₐ is called gauge pressure. The shape or base area of the container does not matter — only the height of the fluid column does, which explains the hydrostatic paradox.

• P₂ − P₁ = ρgh; P = Pₐ + ρgh for a liquid open to the atmosphere
• Gauge pressure = actual pressure − atmospheric pressure
• Hydraulic liftF₂ = F₁A₂/A₁ (Pascal's law application)
• Mercury barometerPₐ = ρgh; at sea level h ≈ 76 cm of Hg
• 1 bar = 10⁵ Pa; 1 torr = 1 mm of Hg = 133 Pa

Fluid dynamics studies fluids in motion. Flow is steady (streamline) if at any given point the velocity of each passing particle stays constant in time, though the velocity may differ from point to point. The path of a particle in steady flow is a streamline, and no two streamlines can cross, because that would give a particle two possible velocities at one point. Beyond a limiting critical speed, steady flow becomes turbulent, as seen in white-water rapids. For an incompressible fluid in steady flow, the mass entering a pipe must equal the mass leaving it. This gives the equation of continuity: A₁v₁ = A₂v₂, or Av = constant. The product Av is the volume flow rate. Where the pipe narrows, the streamlines crowd together and the speed increases.

• Steady flowvelocity at a given point does not change with time
• Streamlinecurve whose tangent gives the fluid velocity direction
• Equation of continuityA₁v₁ = A₂v₂ (conservation of mass)
• Av = volume flux or flow rate, constant along the pipe

Bernoulli's principle applies conservation of energy to a non-viscous, incompressible fluid in steady flow. Along a streamline, the sum of pressure, kinetic energy per unit volume and potential energy per unit volume stays constant: P + ½ρv² + ρgh = constant. When a fluid speeds up at a constriction, its pressure falls. Torricelli's law follows from this: the speed of efflux from a small hole at depth h in an open tank is v = √(2gh), the same as the speed of a freely falling body. Bernoulli's principle also explains dynamic lift. A spinning ball drags air with it, so air moves faster above it and slower below, lowering the pressure above and giving an upward force — the Magnus effect. An aerofoil works the same way, with crowded streamlines above the wing producing lift.

• Bernoulli's equationP + ½ρv² + ρgh = constant
• Torricelli's lawv = √(2gh) for efflux from an open tank
• Magnus effectdynamic lift on a spinning ball
• Valid only for non-viscous, incompressible fluids in steady flow

Viscosity is the internal friction that opposes relative motion between layers of a fluid. When a fluid is sandwiched between a fixed plate and a moving plate, the layer touching each plate moves with that plate, and the velocity changes uniformly in between. This layered flow is called laminar flow. The coefficient of viscosity η is defined as the ratio of shearing stress to the rate of shear strain: η = (F/A) ÷ (v/l). Its SI unit is the poiseuille (Pl), also written Pa s or N s m⁻². Thin liquids like water and alcohol are less viscous than thick ones like glycerine and honey. Viscosity of liquids decreases with rising temperature, while that of gases increases. Stokes' law gives the viscous drag on a small sphere of radius a moving with velocity v through a fluid of viscosity η: F = 6πηav.

• Coefficient of viscosityη = shearing stress ÷ strain rate, unit Pa s
• Stokes' lawF = 6πηav (drag on a sphere)
• Laminar flowlayers slide smoothly, velocity zero at the wall
• Liquidsη falls with temperature; gases: η rises with temperature

When a body falls through a fluid, it drags the surrounding layers and experiences a retarding viscous force. A raindrop accelerates at first under gravity, but as its speed grows, the viscous drag also grows. When the viscous force plus the buoyant force equals the weight of the body, the net force becomes zero and the body moves with a constant velocity called the terminal velocity. For a sphere of radius a and density ρ falling through a fluid of density σ and viscosity η, the terminal velocity is vₜ = 2a²(ρ − σ)g / (9η). This shows that terminal velocity is proportional to the square of the sphere's radius and inversely proportional to the viscosity of the medium.

• Terminal velocityconstant velocity when net force on the body is zero
• vₜ = 2a²(ρ − σ)g / (9η)
• Proportional to a² and inversely proportional to η
• At equilibrium6πηavₜ = (4π/3)a³(ρ − σ)g

Molecules at the surface of a liquid have fewer neighbours than molecules in the interior, so they have extra potential energy. This surface energy means a liquid tends to minimise its surface area, and it appears as surface tension S — a force per unit length acting in the plane of the interface, equal to the surface energy per unit area. Because a sphere has the least surface area for a given volume, small drops and bubbles are spherical. The pressure inside a curved surface is greater on the concave side. For a liquid drop, Pᵢ − Pₒ = 2S/r; for a bubble in a liquid, which has only one surface, the same formula applies, while a soap bubble in air has two surfaces and Pᵢ − Pₒ = 4S/r.

• Surface tension S = force per unit length = surface energy per unit area, unit N m⁻¹
• Drop or cavityPᵢ − Pₒ = 2S/r
• Soap bubble (two surfaces)Pᵢ − Pₒ = 4S/r
• Surface tension of a liquid usually decreases as temperature rises

The angle between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid is called the angle of contact, θ. If the liquid molecules attract the solid strongly, θ is acute and the liquid wets the solid — water on clean glass or plastic. If the liquid molecules attract each other more strongly than the solid, θ is obtuse and the liquid does not wet the solid — mercury on glass, water on a waxy leaf. At the line of contact, S_la cos θ + S_sl = S_sa. Because of the pressure difference across the curved meniscus, a liquid rises in a narrow capillary tube. The capillary rise is h = 2S cos θ / (ρga), so the rise is larger for a narrower tube. If θ is obtuse, as for mercury, the liquid is depressed instead of rising.

• Angle of contact θacute for wetting liquids, obtuse for non-wetting liquids
• S_la cos θ + S_sl = S_sa at the line of contact
• Capillary riseh = 2S cos θ / (ρga)
• Rise is larger for smaller tube radius a; mercury is depressed in glass