Chapter 10 — Mechanical Properties Of Fluids

Fig 1 — Educational Diagram: Fluids and Pressure

What is a fluid?
A fluid is a substance that deforms continuously under any applied shear stress. Liquids and gases are fluids. In fluids, molecules can move relative to each other so fluids flow and transmit pressure.

Pressure — definition and unit
Pressure at a point is defined as force applied per unit area on a small surface around that point: P = F/A. The SI unit is pascal (Pa): 1 Pa = 1 N/m². Often atmospheric pressure (≈ 1.01×10⁵ Pa) and its kilopascal or mm of Hg equivalents are used.

Characteristics of pressure in a fluid

• Pressure at a point in a static fluid acts equally in all directions (isotropic).
• Pressure increases with depth in a liquid because of the weight of the liquid above.

Hydrostatic pressure (pressure with depth)
For a fluid of density ρ under gravity g, pressure at depth h below the free surface (where pressure is p₀) is:

p = p₀ + ρ g h

Pascal's principle
A change of pressure applied to an enclosed incompressible fluid is transmitted undiminished to every part of the fluid and walls of the container. This is the basis of hydraulic machines. If two pistons of areas A₁ and A₂ are connected, then p = F₁/A₁ = F₂/A₂, so F₂ = F₁(A₂/A₁).

Atmospheric pressure and barometer
Atmospheric pressure is the pressure exerted by the weight of air. A mercury barometer measures it by balancing the column of mercury: p_atm = ρ_Hg g h, where h is the column height.

Hydrostatic paradox and communicating vessels
Hydrostatic paradox: pressure at a given depth depends only on vertical depth and fluid density, not on the total volume or shape of the container. In communicating vessels with the same fluid, free surfaces settle at the same vertical height regardless of the shapes or cross-sections of the connected containers.

Buoyancy (brief overview)
A body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced (Archimedes' principle): F_b = ρ_fluid V_displaced g. This force results from pressure differences between top and bottom surfaces.

Practical notes
- Gauge pressure = p - p_atm; absolute pressure = p (measured relative to vacuum).
- Pressure in gases also follows p = p₀ + ρ g h for small height ranges where ρ is approximately constant; for large vertical extents use hydrostatic equilibrium with variable ρ (ideal gas law).

Fig 2 — Educational Diagram: Variation of Pressure with Depth

Definition: Pressure in a fluid at rest increases with depth. The change of pressure with depth is called the hydrostatic variation of pressure.

Physical idea: Consider a fluid at rest with a free surface exposed to atmosphere (pressure p0). A column of fluid of vertical height h above a point produces an extra pressure at that point because of the weight of the fluid column. This extra pressure depends only on the vertical depth and the fluid density, not on the shape or total amount of fluid.

Derivation (simple): Take a small cylindrical element of fluid of cross-sectional area A and height dh located at depth h. Let p(h) be pressure at its top and p(h+dh) at its bottom. Equilibrium of forces in vertical direction gives:

p(h+dh)A - p(h)A - rho*g*A*dh = 0

Divide by A*dh and take limit dh -> 0:

d p/dh = rho g

Integrating from free surface (h = 0, pressure = p0) to depth h gives the hydrostatic formula:

p(h) = p0 + rho g h

Here p(h) is the absolute pressure at depth h, rho is fluid density, g is acceleration due to gravity. If p0 is atmospheric pressure, the gauge pressure (excess over atmosphere) is p_gauge = rho g h.

Important consequences:

• Pressure increases linearly with depth (for incompressible fluids of constant rho).
• At same horizontal level in a connected fluid, pressure is the same (principle used in communicating vessels).
• Shape independence: pressure at a depth depends only on vertical depth and density, not on total volume or container shape.
• Force on a surface: Local pressure produces force normal to surface. For a flat horizontal area A at depth h, F = p(h)A.
• On sloping or curved surfaces, pressure varies with depth; resultant force found by integrating p dA and its line of action may not pass through centroid.

Limitations / Notes:

• Formula assumes incompressible fluid (constant density). For gases, density changes with height and pressure decreases approximately exponentially (barometric formula) under ideal-gas assumptions.
• Temperature and salinity affect density in liquids (e.g., ocean).

Fig 3.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Statement. Pascal's law: When an external pressure is applied to a confined incompressible fluid, the increase in pressure is transmitted undiminished to every part of the fluid and to the walls of its container.

Physical meaning. If a pressure increment Δp is produced at one point in a closed fluid, the same Δp appears at every other point in that fluid (provided the container is rigid and there are no significant losses).

Simple derivation and hydraulic device model. Consider two connected pistons of areas A1 and A2 in the same confined fluid. A small force F1 applied to piston 1 produces a pressure increase Δp = F1 / A1. By Pascal's law the same pressure increase acts on piston 2, so the upward force produced there is F2 = Δp · A2. Hence

• Δp = F1 / A1 = F2 / A2
• F2 = F1 · (A2 / A1). (Force multiplication)

Displacement and work (energy) consideration. If piston 1 moves down by distance s1, it displaces a volume V = A1·s1 which causes piston 2 to move up by s2 where A1·s1 = A2·s2. Thus s2 = (A1/A2)·s1. Neglecting losses, input work equals output work:

• W_in = F1·s1 = W_out = F2·s2
• Therefore F2/F1 = A2/A1 and s2/s1 = A1/A2.

Conditions and limitations. Pascal's law assumes the fluid is essentially incompressible, the container is closed and rigid (or deformations negligible), flow is quasi-static, and losses from viscosity, leakage or turbulence are negligible. In real devices friction and seal leakage reduce theoretical performance.

Relation to other fluid concepts. Pascal's law describes transmission of added pressure in confined fluids. It is distinct from hydrostatic pressure variation with depth (p = p0 + ρgh), though both use the concept of pressure in fluids.

Common applications. Hydraulic press, hydraulic jack (car jack), hydraulic lifts/elevators, automobile braking systems (master cylinder → wheel cylinders), hydraulic steering, hydraulic actuators in heavy machinery, dental chairs and hospital beds, hydraulic presses in industry.

Fig 4 — Educational Diagram: Atmospheric Pressure and Barometer

Atmospheric pressure — definition: Atmospheric pressure is the force per unit area exerted on a surface by the weight of the column of air above that surface. At sea level the standard atmospheric pressure is p0 = 1 atm = 1.013 × 10⁵ Pa ≈ 760 mmHg.

Why there is atmospheric pressure: Air has mass. Gravity acts on the air column above a given area, producing a downward force. Pressure at any level is the weight of the air above per unit area.

Hydrostatic equilibrium (general): For a fluid column in equilibrium, dp/dh = −ρ g, where p is pressure, h vertical coordinate (positive upward) and ρ is density. For incompressible liquids ρ is constant and integrating gives p = p(surface) + ρ g h (with h measured downward from the surface).

Compressible atmosphere: Air is compressible and its density varies with pressure and temperature. Using the ideal gas law (ρ = pM/RT or ρ = p/RT' with appropriate constants) and dp/dh = −ρ g, we get the barometric (isothermal) formula:

p(h) = p(0) exp(−M g h / R T) = p(0) exp(−h / H)

where M is molar mass of air, R universal gas constant, T absolute temperature and H = RT/(M g) is the scale height (≈ 8–8.5 km at standard conditions). For small height ranges or incompressible fluids, pressure change approximates linearly: Δp ≈ −ρ g Δh.

Torricelli's experiment and the mercury barometer: Evangelista Torricelli (1643) showed that atmospheric pressure can support a mercury column. A glass tube, closed at one end, is filled with mercury and inverted into a mercury reservoir. Some mercury falls leaving a near vacuum at the top (Torricellian vacuum). The height h of the mercury column satisfies

p_atm = ρ_Hg g h

Thus measuring h gives p_atm. For mercury (ρ ≈ 13,600 kg m⁻³) at sea level h ≈ 760 mm.

Why mercury? Mercury is used because: (1) high density yields a practical column height (~760 mm) instead of ~10.3 m for water; (2) low vapor pressure so the top of the tube remains nearly vacuum; (3) non-wetting, good visibility. Water would require a ~10.3 m column and its vapour pressure would spoil the vacuum at the top.

Practical barometers: - Mercury barometer (column of mercury) — classic standard. - Aneroid barometer — an evacuated flexible metal box (no liquid); pressure changes deform the box and are converted to dial readings. Both are used for weather and altitude measurements.

Corrections and limitations: A simple p = ρ g h assumes constant ρ and a perfect vacuum. Real barometer readings need corrections for (i) temperature (density of mercury changes), (ii) vapor pressure of the liquid (nonzero), (iii) capillarity if the tube is narrow, and (iv) local gravity variations. Meteorological pressure is often reported as reduced to sea level to compare stations at different heights.

Applications / significance: Atmospheric pressure is crucial in weather forecasting (low pressure indicates storms), determining altitude (altimeters), understanding breathing and aviation, design of vacuum systems, and in everyday phenomena such as why ears pop on ascent/descent and why liquids boil at different temperatures with pressure.

Fig 5 — Educational Diagram: Buoyancy and Archimedes' Principle

What is buoyancy?
Buoyancy (upthrust) is the upward force exerted by a fluid on an object immersed in it. It arises because fluid pressure increases with depth, so pressure on the lower surface of an immersed object is greater than on its upper surface. The resultant of these pressure forces is an upward net force called the buoyant force.

Pressure in a fluid
Pressure at depth h in a fluid of density rho is p = p0 + rho g h, where p0 is surface pressure (often atmospheric). The difference of pressure on opposite faces produces the buoyant force.

Archimedes' principle (statement)
An object wholly or partially immersed in a fluid experiences a buoyant force equal in magnitude to the weight of the fluid displaced by the object.

Derivation (simple)
Consider a body of submerged volume V_sub in a fluid of density rho_f. The net upward force equals the integral of pressure over the body surface and equals rho_f V_sub g. Therefore buoyant force F_B = rho_f g V_sub. If the body is fully submerged, V_sub is the object's volume. If partially submerged, V_sub is the displaced fluid volume.

Apparent weight
If W = m g is the object's real weight and F_B is buoyant force, the apparent weight W_app measured by a scale is W_app = W - F_B. If F_B >= W, the object will accelerate upward (float or rise).

Floating and sinking conditions
- If F_B < W: object sinks.
- If F_B = W: object floats in equilibrium.
- If F_B > W: object accelerates upward until it reaches a position where equilibrium is possible (partially immersed) or escapes the fluid.

For a floating object (mass m, volume V): rho_f V_sub g = m g, so V_sub / V = rho_object / rho_f. Thus the fraction submerged equals the ratio of object density to fluid density.

Stability (brief)
Two points matter: centre of gravity (G) of the body and centre of buoyancy (B), the centroid of displaced fluid. For stable equilibrium when floating, small tilts should produce a righting moment; this often involves the metacentre (M). If M lies above G, equilibrium is stable.

Applications and consequences
Archimedes' principle allows density determination by weighing in air and in fluid, design of ships and submarines, understanding hot-air and gas balloons, and predicting icebergs buoyancy (large fraction submerged if densities close).

Fig 6 — Educational Diagram: Floatation and Stability of Floating Bodies

Basic idea (Archimedes' principle)
A body immersed (fully or partly) in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. For a floating body the buoyant force equals the weight of the body.

Condition of floatation
If a body of volume V and density ρ_b floats in a fluid of density ρ_f, the volume V_sub submerged satisfies: ρ_f V_sub g = ρ_b V g. Hence the fraction submerged is V_sub / V = ρ_b / ρ_f. A body floats if its average density is less than or equal to the fluid density.

Centers: gravity and buoyancy
The weight acts through the centre of gravity (G) of the body. The buoyant force acts through the centre of buoyancy (B) — the centroid of the displaced fluid volume. For a symmetric body at rest B is vertically below G.

Stability on tilting — metacentre
When a floating body is given a small angular tilt, the submerged shape changes and B shifts. The lines of action of buoyant force for the tilted positions intersect at a point called the metacentre (M), which for small angles is fixed relative to the body. The metacentric height GM = distance between G and M (positive if M is above G) determines initial stability:

• GM > 0 (M above G): stable equilibrium — a small tilt produces a restoring couple bringing the body back upright.
• GM = 0: neutral equilibrium — body stays in the new position.
• GM < 0 (M below G): unstable — tilt produces a overturning couple.

Metacentric relations
For small angles: BM = I / V_sub, where I is the second moment of area (moment of inertia) of the waterplane area about the tilt axis and V_sub is the submerged volume. Then GM = BM - BG, where BG is the distance between B and G.

Restoring moment (righting couple)
If W is the weight of the body and θ the small heel angle, the restoring (righting) moment M_r ≈ W · GM · sinθ (≈ W · GM · θ for small θ). A positive M_r tends to restore the upright position.

Design consequences and qualitative points

• Wider waterplane increases I and thus BM -> increases GM (more initial stability).
• Raising G (top-heavy load) reduces GM and can make the vessel unstable.
• Submarines control floatation by changing displaced volume (ballast tanks) to alter V_sub and hence buoyancy.
• Very large GM gives a stiff ship (quick, uncomfortable roll); too small GM gives a tender ship (slow but large roll).

Important caveat
Metacentric analysis is for small angles. For large angles the position of M changes and full hydrostatic stability curves (righting arm vs heel angle) are used.

Fig 7.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Definition: Viscosity is a measure of a fluid's internal friction or resistance to flow. It quantifies how strongly layers of fluid resist relative motion.

Physical origin: When adjacent layers of a fluid move at different speeds, momentum is transferred by molecular interactions between layers. This transfer produces a tangential (shear) force that opposes the relative motion — the viscous force.

Shear stress and velocity gradient: For a fluid flowing between parallel plates (or in general locally), the shear stress τ (force per unit area) is proportional to the velocity gradient (rate of shear) dv/dy:

• τ = η (dv/dy)

Here η is the dynamic (shear) viscosity. This relation is the defining law for Newtonian fluids (linear relation).

Newtonian vs Non-Newtonian fluids:

• Newtonian fluids: τ ∝ dv/dy with constant η (examples: water, air, most simple liquids).
• Non-Newtonian fluids: viscosity depends on shear rate (examples: ketchup — shear thinning; corn‑starch suspension — shear thickening; toothpaste — yield stress fluid).

Dynamic and kinematic viscosity:

• Dynamic viscosity η (SI unit: Pa·s = N·s/m²).
• Kinematic viscosity ν = η/ρ (SI unit: m²/s), where ρ is fluid density.

Laminar flow in a pipe (Poiseuille flow): For steady, incompressible, laminar flow of a Newtonian fluid in a circular pipe, viscous forces produce a parabolic velocity profile:

• v(r) = (ΔP/(4 η L)) (R² − r²), maximum at centre r = 0.
• Volume flow rate Q = (π R^4 ΔP) / (8 η L).

Viscous drag on a sphere (Stokes' law): For a small sphere of radius r moving slowly (low Reynolds number) through a viscous fluid, the viscous drag is

• F_drag = 6 π η r v.

Equating Stokes drag with net weight (gravitational minus buoyant) gives terminal velocity for a small sphere:

• v_t = (2 r^2 (ρ_s − ρ_f) g) / (9 η) , valid for small Re.

Role of Reynolds number: The Reynolds number Re = ρ v D / η compares inertial to viscous forces. Low Re indicates viscous-dominated (laminar) flow; high Re leads to turbulence. In pipes, transition typically near Re ≈ 2000–2300.

Temperature dependence: For liquids, viscosity typically decreases with increasing temperature (molecules move more easily). For gases, viscosity increases with temperature.

Practical importance: Viscosity determines how fast fluids flow through pipes, affects lubrication, droplet formation, sedimentation rates, blood flow in vessels, and design of hydraulic systems.

Fig 8 — Educational Diagram: Flow Types and Reynolds Number

Overview
Flow of a fluid is described by how fluid particles move. Important distinctions are steady vs unsteady, uniform vs non-uniform, compressible vs incompressible, rotational vs irrotational, and — most important for practical engineering — laminar vs turbulent flow.

Steady and Unsteady
Steady flow: fluid properties at a point do not change with time. Unsteady flow: properties vary with time.

Uniform and Non-uniform
Uniform flow: velocity is same at all points along a streamline direction. Non-uniform: velocity changes from point to point.

Laminar vs Turbulent
Laminar flow: fluid moves in smooth layers (streamlines do not cross). Particle paths are ordered and momentum transfer is mainly by molecular viscosity. Velocity profile in a circular pipe is parabolic.
Turbulent flow: flow contains random fluctuations, eddies and mixing. Momentum transfer is dominated by turbulent mixing (eddy viscosity). Velocity profiles are fuller (flatter) near the center and steep gradients near the wall.

Reynolds Number (Re)
Reynolds number is a dimensionless quantity that predicts whether flow will be laminar or turbulent. It represents the ratio of inertial forces to viscous forces in the flow:

Re = (inertial forces) / (viscous forces) = (rho v L) / mu = v L / nu

where rho is fluid density, v is characteristic velocity, L (or D) is characteristic length (e.g., pipe diameter), mu is dynamic viscosity, and nu = mu/rho is kinematic viscosity.

Physical meaning
Low Re means viscous forces dominate and fluctuations are damped → laminar. High Re means inertial forces dominate and small perturbations grow → turbulent.

Critical and Transitional Values
For internal flow in a circular pipe: Re < ~2000 usually laminar; Re > ~4000 usually turbulent; 2000–4000 transitional (possible intermittent turbulence). For external flows and different geometries critical Re differs (e.g., flat plate boundary layer transition often around Re_x ~ 5×10^5 based on distance from leading edge).

Reynolds Experiment (qualitative)
Osborne Reynolds injected a colored dye into water flow in a pipe. At low velocities the dye formed a steady straight line (laminar). At high velocities the dye mixed quickly with water showing turbulent flow. He used this to identify the critical Re value for pipes.

Why it matters
Knowing the flow regime helps predict pressure drop, heat/mass transfer, mixing, noise and design features (pump power, pipe sizing, aerodynamic drag). For example, turbulent flow increases mixing and heat transfer but also frictional losses.

Simple example calculation
Water at 20°C (nu ≈ 1.0×10^-6 m2/s) in a pipe D = 0.05 m with mean speed v = 1.0 m/s: Re = v D / nu = (1.0 × 0.05) / (1.0×10^-6) = 5×10^4 → turbulent.

Useful empirical relations
For laminar flow in a circular pipe the friction factor f = 64 / Re. For turbulent flow f depends on Re and relative roughness (Moody chart).

Summary
Reynolds number gives a simple rule-of-thumb to classify flow: low Re → laminar and ordered; high Re → turbulent and mixing. The exact transition depends on geometry, surface roughness and disturbances, so critical values are approximate.

Fig 9 — Educational Diagram: Equation of Continuity

Definition: The equation of continuity expresses conservation of mass for a fluid in motion. For a steady flow it states that the mass of fluid passing any cross-section of a stream per unit time is constant.

Simple derivation (one-dimensional, steady flow): Consider two cross-sections 1 and 2 of a pipe with areas A1 and A2 and fluid speeds v1 and v2. In time Δt, volumes passing the sections are A1 v1 Δt and A2 v2 Δt. Masses are ρ A1 v1 Δt and ρ A2 v2 Δt. Conservation of mass ⇒ ρ A1 v1 Δt = ρ A2 v2 Δt. For incompressible fluid (ρ constant) this reduces to A1 v1 = A2 v2.

Physical meaning: If the cross-sectional area decreases, the flow speed increases so that the same volume (or mass) passes per second. For compressible fluids the density changes and must be included.

Differential (general) form: The continuity equation for a general (possibly unsteady, compressible) flow is ∂ρ/∂t + ∇·(ρ v) = 0. For incompressible flow (constant ρ) this simplifies to ∇·v = 0 (zero divergence of velocity field).

Assumptions and applicability: The simple A v = constant form assumes steady, one-dimensional flow along a conduit and constant fluid density (good for liquids and low-speed gases, typically Mach < 0.3). The full differential form applies to varying density, unsteady and three-dimensional flows.

Connection with other concepts: The continuity equation is independent of energy considerations; it complements Bernoulli's principle. In a narrowing pipe, continuity predicts increased velocity; Bernoulli then relates that increased velocity to pressure change.

Units: Area A in m2, speed v in m/s, volumetric flow rate Q = A v in m3/s, density ρ in kg/m3, mass flow rate \u200b\u200bm_dot = ρ A v in kg/s.

Fig 10 — Educational Diagram: Bernoulli's Theorem

Statement: For a steady, incompressible, non-viscous flow of a fluid, the sum of the pressure energy, kinetic energy per unit volume and potential energy per unit volume remains constant along a streamline. Mathematically: p + (1/2)ρv² + ρgh = constant along a streamline.

Physical meaning: Bernoulli's theorem is an energy conservation statement for flowing fluids. The term p is the static pressure (energy per unit volume due to pressure), (1/2)ρv² is the kinetic energy per unit volume (dynamic pressure), and ρgh is the potential energy per unit volume due to height in a gravitational field. If the fluid speeds up (v increases), its pressure p must decrease or its height h must change so the total remains constant.

Derivation outline (work–energy approach): Consider a streamline and two close cross-sections 1 and 2 in steady flow. Work done on the fluid by pressure forces moving a fluid element from 1 to 2 is p1A1Δx1 - p2A2Δx2. That work changes the kinetic energy and potential energy of the element. Dividing by the element volume and simplifying yields p1 + (1/2)ρv1² + ρgh1 = p2 + (1/2)ρv2² + ρgh2. Alternatively, start from Euler's equation and integrate along a streamline to obtain the same relation.

Common alternate forms: Per unit mass: p/ρ + v²/2 + gh = constant. Head form (divide by ρg): p/(ρg) + v²/(2g) + h = constant. Dynamic pressure: q = (1/2)ρv². Pitot relation: v = sqrt(2(p_total - p_static)/ρ).

Assumptions and limitations: The theorem applies only for steady flow of an incompressible and non-viscous fluid, along a single streamline, and when no external work (pumps, turbines) or heat addition removes or adds mechanical energy. In real flows viscous losses, turbulence, and compressibility (high-speed gas flows) limit direct applicability; corrections (head loss terms) are then needed.

Energy interpretation and use: Bernoulli's theorem lets you trade pressure, speed and elevation. It is widely used with the continuity equation (A1v1 = A2v2 for incompressible flow) to relate velocities and pressures in pipes, nozzles, and constrictions.

Fig 11 — Educational Diagram: Applications of Bernoulli's Principle

Statement: Bernoulli's principle (for an incompressible, non-viscous, steady flow along a streamline) says that the sum of pressure energy, kinetic energy per unit volume and potential energy per unit volume is constant:

p + 1/2 ρ v² + ρ g h = constant (along a streamline).

Assumptions: incompressible fluid, no viscosity (ideal fluid), steady flow, along the same streamline and no work done by non‑conservative forces between the two points.

Physical idea / quick derivation: From work–energy: work done by pressure forces + change in potential energy = change in kinetic energy. Rearranging gives the Bernoulli equation above. The continuity equation A₁v₁ = A₂v₂ links cross‑sectional area and speed.

How this leads to applications: Bernoulli shows that where a fluid speeds up, static pressure drops (and vice versa). Many devices exploit this pressure–speed tradeoff to measure flow speed, create suction, or generate lift.

Common applications (brief):

• Aircraft wing (lift): Air moves faster over the curved top surface producing lower pressure than underneath → net upward lift (Bernoulli effect contributes; circulation and Newtonian action also important).
• Venturi meter: A constriction increases speed and lowers pressure. Measuring the pressure difference gives the flow rate.
• Pitot tube: Measures stagnation (total) pressure; difference between total and static pressure gives dynamic pressure and hence velocity: v = sqrt(2(p_total − p_static)/ρ).
• Atomizers, perfume sprayers, and carburettors: Fast air flow in a narrow region lowers pressure and draws liquid into the flow (suction) to produce a spray or mix fuel with air.
• Blood flow and stenosis: Narrowing of arteries raises flow speed and lowers pressure locally; important in medical diagnostics (Doppler + pressure relations).
• Roof uplift in storms / chimney draft: High wind speed over a roof lowers pressure above it (can lift roof); wind over a chimney produces lower pressure and increases draft.

Limitations: Not applicable where viscous effects, turbulence, or compressibility (high Mach number) are significant. Energy losses (head losses) must be included in real systems.

Fig 12.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Overview: Poiseuille's law describes steady, laminar flow of an incompressible Newtonian fluid through a long, straight, rigid circular pipe. It gives the volumetric flow rate in terms of pressure difference, pipe geometry and fluid viscosity.

Key assumptions: steady flow, incompressible Newtonian fluid, laminar regime (Re < ~2000), straight circular pipe of constant radius, no slip at the wall, negligible entrance and exit effects (pipe is long compared to radius).

Velocity profile and derivation (sketch): For laminar flow the fluid moves in concentric cylindrical layers. Balance of pressure force and viscous shear leads to a differential equation whose solution is a parabolic velocity profile:

v(r') = vmax · (1 - (r'^2/R^2))

Here r' is the radial distance from the pipe center, R is the pipe radius, and vmax is the centreline (maximum) velocity. Integrating this profile over the pipe cross-section gives Poiseuille's law for the volumetric flow rate Q.

Physical meaning: Flow rate is directly proportional to the pressure difference and very strongly dependent on radius (r^4), and inversely proportional to viscosity and pipe length. A small decrease in radius drastically reduces flow. The flow dissipates mechanical energy as heat by viscous friction; power dissipated = ΔP · Q.

Limits and transition: Poiseuille's law applies only in the laminar regime. When Reynolds number Re = (ρ v_avg D)/η exceeds a critical value (≈2000 for pipe flow), flow becomes transitional/turbulent and Q no longer follows the r^4 law or linear dependence on ΔP.

Important derived results: shear stress varies linearly with radius (zero at center, maximum at wall), average velocity v_avg = Q/(πR^2), and vmax = 2 v_avg for the parabolic profile.

Fig 13.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Definition: Stokes' law gives the viscous drag force experienced by a small rigid sphere moving slowly through a viscous (Newtonian) fluid. Terminal velocity is the constant speed reached by the sphere when the net force on it becomes zero (no further acceleration).

Stokes' law (statement): For a sphere of radius r moving with speed v through a fluid of viscosity η (under conditions of laminar flow, i.e. very low Reynolds number), the viscous drag force is

F_d = 6π η r v.

Derivation outline (brief):

• Consider a small sphere of volume V = (4/3)π r³ and density ρ_s falling in a fluid of density ρ_f.
• Forces on the sphere: weight W = V ρ_s g downward, buoyant force B = V ρ_f g upward, viscous drag F_d = 6π η r v upward (opposes motion).
• Equation of motion: m dv/dt = W − B − F_d = V g (ρ_s − ρ_f) − 6π η r v.

Terminal velocity: At steady state dv/dt = 0, so net force = 0. Solve for terminal speed v_t:

v_t = (2/9) · (r² g (ρ_s − ρ_f)) / η.

Time dependence: Solving the linear first-order ODE gives exponential approach to v_t:

v(t) = v_t [1 − exp(−t/τ)], where τ = (2/9) (ρ_s r²)/η is the relaxation (time) constant.

Assumptions and applicability:

• Flow around the sphere is steady and laminar (Reynolds number Re << 1). Re is given by Re = (2 r ρ_f v)/η (or Re = ρ_f v d / η with diameter d = 2r).
• Sphere is rigid and smooth; fluid is infinite and homogeneous; no-slip boundary condition at the sphere surface.
• For larger Re (moderate to high speeds or larger spheres) drag ∝ v² and Stokes' law no longer holds.

Physical meaning: Stokes' law shows that at very low speeds viscous forces are proportional to speed (linear drag). Terminal velocity arises because weight minus buoyancy is balanced by viscous drag; for small particles v_t grows ∝ r², so larger particles settle much faster.

Fig 14 — Educational Diagram: Surface Tension and Surface Energy

Definition: Surface tension (γ) is the property of a liquid surface that makes it behave like a stretched elastic membrane. It is defined as the tangential force per unit length acting along the surface, perpendicular to an imaginary line on the surface. Surface energy (also called surface free energy) is the work required to create a unit area of new surface.

Origin: Surface tension arises from intermolecular forces (cohesion) between liquid molecules. Molecules at the surface experience an imbalance of forces (fewer neighbors on the outside), producing a net inward pull that minimizes the surface area. Adhesion (attraction between liquid and solid) competes with cohesion and determines wetting behaviour (contact angle).

Quantitative definitions and relation:

• Surface tension γ = F / l, where F is the force acting along the surface across a line of length l. SI unit: newton per metre (N m^-1).
• Surface energy E_s = W / A, where W is the work done to create area A. SI unit: joule per metre squared (J m^-2).
• For a pure liquid surface under usual conditions, surface tension equals surface energy per unit area: γ = dW / dA. Numerically γ (N m^-1) = E_s (J m^-2), because 1 N·m = 1 J.

Important consequences & formulas:

• Work and energy: To increase a surface area by dA, the work required is dW = γ dA. For a film with two surfaces, dW = 2γ dA.
• Laplace pressure (curved surface): The pressure difference across a curved liquid surface is ΔP = γ(1/R₁ + 1/R₂). For a spherical drop (R₁=R₂=R): ΔP = 2γ / R. For a thin soap bubble (two interfaces): ΔP = 4γ / R.
• Capillary rise: Height h to which liquid rises (or falls) in a capillary tube of radius r: h = (2γ cosθ) / (ρ g r), where θ is the contact angle, ρ the liquid density, g acceleration due to gravity.
• Force on a frame/rod: If a wire of length l borders a liquid surface, the upward/tangential force due to surface tension is F = γ l (for single surface) or F = 2γ l (if film has two surfaces).
• Temperature dependence: γ decreases with increasing temperature and tends to zero at the critical temperature. Mathematically dγ/dT < 0.

Measurement methods (brief): Capillary-rise method (use h formula), drop-weight or pendant-drop method, and du Noüy ring (ring tensiometer) where the maximum force to detach a ring gives γ.

Wetting and contact angle: Whether a liquid spreads on a solid depends on surface tensions of solid-vapor, solid-liquid and liquid-vapor interfaces (Young's equation): γ_SV - γ_SL = γ cosθ. A small contact angle (θ < 90°) indicates good wetting; large θ indicates poor wetting.

Visual intuition: Surface tension tends to minimize surface area for a given volume — droplets tend to be spherical. Surfactants (soaps, detergents) lower γ, allowing spreading and easier formation of emulsions and foams.

Fig 15 — Educational Diagram: Capillarity

Definition: Capillarity (capillary action) is the phenomenon in which a liquid rises or falls in a narrow tube (capillary) or in porous material due to surface tension and adhesive/cohesive forces.

Cause: At a liquid–solid–gas contact, surface tension and the relative strength of adhesive forces (between liquid and solid) and cohesive forces (within liquid) determine the shape of the meniscus. The contact angle θ (measured inside the liquid at the solid surface) characterizes wetting: θ < 90° (wetting, concave meniscus, rise), θ > 90° (non‑wetting, convex meniscus, depression).

Qualitative description: In a thin vertical tube of radius r dipped into a liquid that wets the tube (e.g., water in glass), the liquid climbs the walls forming a concave meniscus. Surface tension along the contact line pulls the liquid upward. The rise continues until the upward force from surface tension is balanced by the weight of the liquid column.

Derivation (static equilibrium): Consider a vertical cylindrical tube of radius r with a meniscus of contact angle θ. The vertical component of the surface tension acting along the circumference is F_up = 2πrT cosθ, where T is the surface tension. The weight of the liquid column of height h is W = (πr^2 h)ρg. At equilibrium F_up = W, hence

h = (2 T cosθ) / (ρ g r)

This is Jurin's law: capillary rise h is inversely proportional to the tube radius r and proportional to cosθ. If cosθ is negative (θ > 90°), h is negative and the liquid is depressed (example: mercury in glass).

Other related relations: The excess pressure ΔP across a curved liquid surface (Laplace pressure) is ΔP = T(1/R1 + 1/R2). For a spherical surface of radius R, ΔP = 2T/R. Capillary pressure in a cylindrical capillary with meniscus of radius of curvature R also determines flow in porous media.

Assumptions & limits: The derivation assumes: a circular tube with radius much smaller than the capillary length so gravity and curvature approximations hold; static equilibrium (no flow); negligible tube thickness. For very small radii or very large heights other effects (evaporation, viscosity during dynamic rise, contact-angle hysteresis) may matter.

Typical values: Surface tension of water at 20°C: T ≈ 0.0728 N/m. For water (ρ ≈ 1000 kg/m³) in a 1 mm radius tube (r = 10⁻³ m) with θ ≈ 0°, approximate rise h ≈ 2×0.0728/(1000×9.8×10⁻³) ≈ 1.49 × 10⁻1 m ≈ 15 cm.

Practical notes: Real systems often behave like many small capillaries in parallel (porous materials). Dynamic capillary rise follows the Washburn equation (viscous flow); static result above is the final equilibrium height.

Fig 16 — Educational Diagram: Drops, Bubbles and Excess Pressure

Overview
Surface tension is the property of a liquid surface that makes it behave like a stretched elastic membrane. Curvature of a liquid surface produces an excess (or Laplace) pressure across the surface: the pressure on the concave side is greater than on the convex side. This is why small drops and bubbles have higher internal pressure than the surrounding fluid.

Young–Laplace (general) relation
For any curved liquid surface the excess pressure Δp (pressure inside minus outside, sign convention: positive when interior pressure is larger) is given by the Young–Laplace equation:
Δp = γ (1/R1 + 1/R2)
where γ (or S) is surface tension and R1, R2 are the principal radii of curvature (with sign convention for orientation). For a spherical surface R1 = R2 = R this reduces to the spherical result below.

Derivation for a spherical drop
Consider a spherical drop of radius R. A virtual increase dR changes area A = 4πR² and volume V = 4/3 πR³. Equilibrium of virtual work between pressure and surface forces gives Δp dV = γ dA. Hence Δp = γ (dA/dV) = γ (8πR)/(4πR²) = 2γ/R. So for a liquid drop (or a gas bubble inside a liquid):
Δp = 2γ / R

Soap bubble (thin film) — two surfaces
A soap bubble has two interfaces (inner and outer surfaces), so surface energy change is twice that of a single surface. Repeating the virtual-work argument gives:
Δp = 4γ / R
Thus the pressure inside a soap bubble (gas) is greater than outside by 4γ/R.

Signs and examples of different cases

• Liquid drop in gas: pressure inside drop > outside gas by 2γ/R.
• Gas bubble in liquid: inside gas pressure > surrounding liquid by 2γ / R.
• Soap (double-surface) bubble in air: inside gas pressure > outside air by 4γ / R.

Physical consequences

• Smaller drops/bubbles have larger Δp (Δp ∝ 1/R). This is why very small droplets evaporate faster and why bubbles are less stable when very small.
• Two droplets of different radii in contact: the smaller (higher internal pressure) tends to feed liquid into the larger, making small droplets shrink and large ones grow (Ostwald ripening / coarsening).
• Biological example: pulmonary surfactant lowers γ in alveoli so small alveoli do not collapse due to high Δp.

Units and typical values
Surface tension γ has units N·m⁻¹. For water at 20 °C, γ ≈ 0.072 N·m⁻¹. For a water drop of radius 1 mm, Δp ≈ 2·0.072/0.001 ≈ 144 Pa.

Limitations & notes
The formulas assume a clean, isotropic interface and that gravity and viscosity effects are negligible for the shape (valid for small drops/bubbles where surface forces dominate). For non-spherical shapes use the general Young–Laplace equation with the correct curvatures.

Fig 17 — Educational Diagram: Key Formulas and Problem-Solving Tips

This section collects the essential formulas and systematic tips to solve problems in the CBSE Class 11 chapter "Mechanical Properties of Fluids." Focus on identifying the physical regime (static fluid, surface-tension-dominated, viscous flow, or ideal-flow), choose an appropriate control volume or free-body diagram, apply the relevant law (hydrostatic equilibrium, Pascal, Archimedes, Stokes/Poiseuille, Bernoulli), and check units and limits.

Core ideas:

• Hydrostatic pressure: pressure in a resting fluid increases linearly with depth. Use a consistent reference level and remember gauge vs absolute pressure.
• Pascal's principle: pressure applied to a confined fluid transmits undiminished in all directions — basis of hydraulic machines.
• Buoyancy (Archimedes' principle): a body immersed in fluid experiences an upward force equal to the weight of displaced fluid — used to determine floatation and apparent weight.
• Surface tension and capillarity: molecules at an interface produce an effective line force (surface tension) and surface energy; capillary rise is set by a balance of surface forces and weight of the column.
• Viscosity and viscous flow: internal friction resists relative motion. For slow, laminar, low-Reynolds-number flow, use Stokes' law and Poiseuille's law. For ideal (non-viscous) steady flow, use continuity and Bernoulli's equation.

Problem-solving strategy (stepwise):

1. Read and sketch the setup; mark given quantities and unknowns, indicate dimensions and directions of forces/flows.
2. Decide regime: static fluid, surface tension effect, laminar viscous, or ideal flow. This determines which formulas apply.
3. Write governing equations (force balance, pressure-depth relation, continuity, energy/Bernoulli, or viscous relations).
4. Apply boundary conditions: atmospheric pressure at open surfaces, no-slip at solid boundaries (viscous), contact angle for capillarity if given.
5. Solve algebraically, check units and limits (e.g., small-radius limit for capillarity, high/low viscosity limits), and do a quick plausibility check (signs, magnitudes).
6. When in doubt use dimensional analysis to guide the form of the result and check for missing factors of g, rho, length, or viscosity.

Keep these practical tips in mind:

• Always state whether pressure is gauge (relative) or absolute. p_absolute = p_gauge + atmospheric pressure.
• For hydrostatics, pressure difference matters more than absolute values: p2 - p1 = rho g (h2 - h1).
• For buoyancy problems, compare densities: object floats if rho_object < rho_fluid (partial submersion) and sinks if rho_object > rho_fluid.
• In viscous-flow problems, check whether flow is laminar (low Re) before using Stokes or Poiseuille; otherwise these formulas fail.
• Use approximations: thin-film, small-angle (cos θ ≈ 1), or neglect viscosity for high-Re inviscid flow when appropriate.