Mechanical Properties of Fluids

The One Thing

The mechanical behaviour of fluids is governed by pressure, conservation of mass and energy, viscous resistance, and molecular forces at liquid surfaces. These principles explain fluid pressure, flow speed, hydraulic machines, terminal motion, capillarity, droplets, and bubbles.

Reactions, Processes and Experiments

Key Terms

• Fluid: A substance that can flow and take the shape of its container, including liquids and gases.
• Pressure: Normal force acting per unit area: P = F/A. Its SI unit is pascal (Pa).
• Density: Mass per unit volume of a substance: rho = m/V.
• Pressure in a liquid: Pressure at depth h below the surface is P = P0 + rho gh, where P0 is the pressure at the surface.
• Gauge pressure: Pressure measured relative to atmospheric pressure: Pgauge = Pabsolute - Patmospheric.
• Pascal's law: A pressure applied to an enclosed fluid is transmitted equally and undiminished in all directions.
• Hydraulic lift: A device based on Pascal's law in which a small force on a small piston produces a larger force on a larger piston.
• Streamline flow: Smooth fluid flow in which each fluid particle follows a definite path and streamlines do not cross.
• Steady flow: Flow in which the velocity of fluid at a fixed point remains constant with time.
• Equation of continuity: For an incompressible fluid in steady flow, A1v1 = A2v2; fluid moves faster through a narrower region.
• Volume flow rate: Volume of fluid passing through an area per unit time: Q = Av.
• Bernoulli's principle: For steady, incompressible, non-viscous flow along a streamline, the total mechanical energy per unit volume remains constant.
• Bernoulli equation: P + 1/2 rho v^2 + rho gh = constant, where the three terms represent pressure, kinetic, and gravitational potential energy per unit volume.
• Viscosity: The internal resistance offered by a fluid to the relative motion of its layers.
• Coefficient of viscosity: The constant eta in the relation F/A = eta(dv/dy), where F/A is tangential stress and dv/dy is the velocity gradient.
• Newtonian fluid: A fluid whose shear stress is directly proportional to its velocity gradient, such as water and air under ordinary conditions.
• Terminal velocity: The constant maximum speed reached by a falling object in a viscous fluid when the net force becomes zero.
• Stokes' law: For a small sphere moving slowly through a viscous fluid, viscous drag is F = 6 pi eta r v.
• Reynolds number: A dimensionless quantity Re = rho vd/eta that helps predict whether flow is streamlined or turbulent.
• Surface tension: Tangential force per unit length acting along the surface of a liquid: T = F/l.
• Surface energy: Energy required to increase the surface area of a liquid; surface tension is also surface energy per unit area.
• Angle of contact: The angle between the tangent to the liquid surface and the solid surface at the point of contact.
• Capillarity: The rise or fall of a liquid in a narrow tube due to surface tension and adhesive or cohesive forces.

Easily Confused

• Pressure and force: Pressure is a scalar quantity, whereas force is a vector quantity; pressure is force per unit area.
• Absolute pressure and gauge pressure: Absolute pressure is measured from a vacuum reference, whereas gauge pressure is measured relative to atmospheric pressure.
• Streamline flow and steady flow: Streamline flow concerns the definite paths followed by particles and non-crossing streamlines; steady flow concerns velocity remaining constant at a fixed point with time.
• Continuity and Bernoulli's principle: The continuity equation expresses conservation of mass, whereas Bernoulli's equation expresses conservation of mechanical energy in ideal flow.
• Liquid and gas viscosity: Viscosity generally decreases with temperature for liquids but increases with temperature for gases.
• Liquid drop and soap bubble: Excess pressure is Delta P = 2T/R for a liquid drop and Delta P = 4T/R for a soap bubble because the bubble has two surfaces.
• Water rise and mercury depression: Water rises because adhesion exceeds cohesion; mercury is depressed because cohesion exceeds adhesion.
• Surface tension and surface energy: Surface tension is force per unit length, whereas surface energy is energy per unit area and is also numerically equal to surface tension.

What Gets Asked

• Pressure calculations: Determine pressure at depth using P = P0 + rho gh or pressure difference using Delta P = rho g Delta h. Marks are lost by confusing absolute pressure with gauge pressure or by treating pressure as a vector.
• Hydraulic-machine problems: Use F1/A1 = F2/A2 and F2 = F1(A2/A1) to calculate the output force. The common error is reversing the piston-area ratio.
• Continuity and flow-rate questions: Apply A1v1 = A2v2 for incompressible flow, rho1A1v1 = rho2A2v2 for compressible flow, or Q = Av. The key slip is using the incompressible form when density changes.
• Bernoulli questions: State or apply P + 1/2 rho v^2 + rho gh = constant under steady, incompressible, non-viscous flow along a streamline. Marks are lost by omitting the assumption of negligible viscosity or by failing to recognise that greater speed at the same height generally means lower pressure.
• Viscosity and terminal-velocity problems: Use F = 6 pi eta r v or vt = 2r^2(rho_s - rho_f)g/(9eta). The common error is confusing viscous drag with terminal velocity or omitting the density difference.
• Surface-tension and capillarity problems: Use Delta P = 2T/R for a liquid drop, Delta P = 4T/R for a soap bubble, and h = 2T cos theta/(rho gr) for capillary rise or depression. The main distinction is whether the system has one surface or two, and whether adhesion or cohesion dominates.