Fluids - liquids and gases - are substances that flow. Unlike solids, they cannot sustain a shearing stress at rest; any shear stress makes them flow. The study of fluids at rest is called hydrostatics, and the study of fluids in motion is called hydrodynamics. This chapter introduces pressure in fluids, buoyancy, fluid flow, viscosity, and surface tension.
Fluids surround us everywhere - air is a fluid, water is a fluid, and even blood flowing in our veins is a fluid. Understanding the behaviour of fluids helps us explain how a ship floats, why a plane's wing provides lift, how a syringe works, why raindrops are spherical, and why oil reduces friction.
This chapter builds on the concept of pressure, extends it to depth in a fluid through Pascal's law and Archimedes' principle, and then studies the dynamics of ideal fluids through the equation of continuity and Bernoulli's theorem. Real fluids are studied through viscosity and surface tension.
Pressure is defined as the normal force per unit area:
P = F / A
The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m^2. Other units include bar (1 bar = 10^5 Pa) and the atmosphere (1 atm = 1.013 x 10^5 Pa). Pressure is a scalar quantity; at any point in a fluid at rest, it acts equally in all directions.
The pressure at a depth h in a fluid of density rho, with the free surface open to the atmosphere, is:
P = P_atm + rho g h
where g is the acceleration due to gravity. The pressure increases with depth because of the weight of the fluid above. The gauge pressure is the excess pressure above atmospheric pressure, rho g h. This formula explains why dams are made thicker at the bottom - the pressure is greatest at the greatest depth.
Pascal's law states that a pressure change applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of its container. This principle is used in hydraulic machines such as the hydraulic lift, hydraulic press, and hydraulic brakes.
In a hydraulic lift, a small force F1 is applied on a piston of small area A1, creating a pressure P = F1/A1. This pressure is transmitted undiminished to a piston of larger area A2, producing a larger force:
F2 = P * A2 = F1 * A2 / A1
Since A2 is much larger than A1, a small force can lift a heavy load. The hydraulic lift is a force multiplier that demonstrates the practical power of Pascal's law.
When a body is immersed in a fluid, the fluid exerts an upward force on the body called the buoyant force. The buoyant force arises because the pressure at the bottom of the body is greater than the pressure at the top, since pressure increases with depth. The magnitude of the buoyant force is given by Archimedes' principle.
Archimedes' principle states that the buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by the body. If the body is completely immersed in a fluid of density rho, the volume of fluid displaced equals the volume of the body, and the buoyant force is:
F_b = rho * V * g
where V is the volume of the body. A body floats when its weight is balanced by the buoyant force, which requires its average density to be less than or equal to the density of the fluid. A body sinks when its density is greater than that of the fluid. This is why a ship of iron floats - its average density (mass divided by total volume including the empty hull) is less than that of water.
Fluid flow can be streamline (laminar) or turbulent. In streamline flow, every particle passing through a given point follows the same path, and the fluid layers slide over each other without mixing. In turbulent flow, the flow is irregular and the particles follow chaotic, fluctuating paths. The speed at which flow changes from streamline to turbulent is called the critical velocity.
For an incompressible, non-viscous fluid in steady streamline flow, the mass flowing into any section must equal the mass flowing out. If A is the cross-sectional area and v the flow speed, this gives the equation of continuity:
A1 * v1 = A2 * v2
This means that where the cross-sectional area of a tube is smaller, the fluid flows faster. The product Av is called the volume flow rate or the flux, and it has units of m^3/s. This equation explains why a river flows faster where it is narrow.
Bernoulli's theorem states that for an incompressible, non-viscous fluid in steady streamline flow, the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline:
P + (1/2) rho v^2 + rho g h = constant
Here P is the static pressure, (1/2)rho v^2 is the dynamic pressure, and rho g h is the pressure due to the height of the fluid. The theorem is essentially the law of conservation of energy applied to fluid flow.
Bernoulli's theorem explains many phenomena: the lift on an airplane wing (air moves faster over the top, so the pressure there is lower), the swing of a cricket ball, the spraying of perfume from a bottle, and the working of a venturi meter. When the speed of a fluid increases, its pressure decreases.
Viscosity is the property of a fluid that opposes the relative motion between its layers, analogous to friction between solids. A moving fluid experiences an internal resistance because of the viscous forces between adjacent layers. Honey has high viscosity and flows slowly, while water has low viscosity and flows easily.
For a fluid between two parallel plates, the viscous force between layers is given by Newton's law of viscosity:
F = - eta A dv/dx
where eta is the coefficient of viscosity, A is the area of the layer, and dv/dx is the velocity gradient. The SI unit of viscosity is the pascal-second (Pa s), also called the poise (1 poise = 0.1 Pa s).
Viscosity depends on temperature. The viscosity of liquids decreases as temperature rises, but the viscosity of gases increases with temperature. When a sphere falls through a viscous fluid, it reaches a terminal velocity when the buoyant force and viscous drag balance its weight:
v_t = 2 r^2 (rho - sigma) g / (9 eta)
where r is the radius of the sphere, rho its density, sigma the density of the fluid, and eta the viscosity.
Surface tension is the property of the free surface of a liquid that makes it behave like a stretched elastic membrane. It arises because the molecules at the surface are attracted inward by the molecules below, so the surface tends to contract to the minimum area. Surface tension is defined as the force per unit length acting along the surface:
S = F / L
The SI unit of surface tension is N/m or J/m^2. Surface tension explains why a needle floats on water, why raindrops are spherical (a sphere has the minimum surface area for a given volume), and why water rises in a capillary tube.
The excess pressure inside a liquid drop of radius r is:
delta P = 2 S / r
For a soap bubble with two surfaces, the excess pressure is:
delta P = 4 S / r
Capillary rise is another consequence of surface tension; in a capillary tube of radius r, a liquid of surface tension S rises to a height h = 2 S cos theta / (rho g r), where theta is the angle of contact.
| Quantity | Formula | Unit |
|---|---|---|
| Pressure at depth | P = P_atm + rho g h | Pa |
| Buoyant force | F_b = rho V g | N |
| Continuity equation | A1 v1 = A2 v2 | m^3/s |
| Bernoulli's theorem | P + (1/2)rho v^2 + rho g h = constant | Pa |
| Viscous force | F = - eta A dv/dx | N |
| Terminal velocity | v_t = 2r^2(rho - sigma)g/(9 eta) | m/s |
| Surface tension | S = F / L | N/m |
| Situation | Excess Pressure |
|---|---|
| Liquid drop | 2S/r |
| Soap bubble | 4S/r |
| Capillary rise height | h = 2S cos theta/(rho g r) |
In this chapter we studied the behaviour of fluids at rest and in motion. Pressure increases with depth as P = P_atm + rho g h, Pascal's law allows the transmission of pressure in enclosed fluids enabling hydraulic machines, and Archimedes' principle explains why objects float or sink. For fluids in motion, the equation of continuity A1v1 = A2v2 and Bernoulli's theorem P + (1/2)rho v^2 + rho g h = constant describe ideal flow and explain phenomena from airplane lift to the venturi meter. Real fluids are characterised by viscosity, which gives rise to terminal velocity, and by surface tension, which governs drops, bubbles, and capillary rise. These ideas connect directly to the study of thermal properties of matter, where fluids expand and transport heat.