Mechanical Properties of Fluids

1. Fluids – Density & Relative Density

• Fluid : material which has the property to flow → liquids + gases.
• Liquid : no fixed shape, but fixed volume. Gas : neither fixed shape nor fixed volume.

• Density = mass per unit volume ; unit kg/m³. Water ≈ 1000 kg/m³ = 1 g/cm³ (1 litre ≈ 1 kg).
• 1 cm³ = 10⁻⁶ m³ ;  1 litre = 10⁻³ m³.

• A ratio ⇒ no unit. e.g. specific gravity 7.2 ⇒ ρ = 7200 kg/m³ = 7.2 g/cm³.

2. Density of a Mixture

• Liquids mixed together (total volume = sum of volumes) ; use m = ρV :

• Equal volumes : ρ_eq = ρ₁ + ρ₂ + …n (arithmetic mean).
• Equal masses : ρ_eq = n1/ρ₁ + 1/ρ₂ + … (harmonic mean).

e.g. 1 : 2 by volume ⇒ (ρ₁ + 2ρ₂)/3 ; 1 : 2 by mass ⇒ 3ρ₁ρ₂/(ρ₂ + 2ρ₁).

3. Pressure & Force due to Pressure

• Pressure : force per unit area on a surface, due to collisions of the fluid molecules with it. Unit N/m² = Pascal (Pa) ; a scalar.

(F = PA when P is uniform over the area ; F = ∫P dA when P varies)

• Force due to pressure is always perpendicular to the surface and is a push (compressive) force.

Atmospheric pressure :

• Due to the weight of the air column above a point ; measured by a barometer. Decreases with altitude.

• At the interface of air and liquid, pressure is continuous ⇒ at the free surface P = P₀.

4. Variation of Pressure with Depth

• Fluid at rest ; liquid incompressible (ρ = constant). Pressure varies due to gravity, acceleration and rotation of the vessel.

• Thin layer (area A, thickness dx, mass dm = ρA dx) in equilibrium :

P₂A = P₁A + (ρA dx)g ⇒ dP = ρg dx ⇒ ∫_A^B dP = ∫₀^h ρg dx

• Pressure increases linearly as we move down.
• Holds between any two points of the same liquid at rest, along any path (h = vertical depth of B below A).
• All points at the same horizontal level in the same continuous liquid at rest have the same pressure — independent of the shape of the vessel.

5. Pressure in a Vertically Accelerating Vessel

(B is at depth h below A ; in free fall the pressure is the same everywhere in the liquid.)

6. Layered Liquids & U-tubes

• Immiscible layers in an open vessel : add ρgh layer by layer.

• U-tube rule : equate pressures at the level of the lower interface, where both points lie in the same liquid ⇒ ρ₁h₁ = ρ₂h₂ (heights measured above that level).

7. Force on the Bottom & Walls of a Vessel

• Ignore P₀. At the bottom (depth h) pressure is uniform = ρgh ⇒ F₁ = ρgh × (base area).
• On slant walls pressure is variable ⇒ F₂ = ∫P dA (always ⊥ wall).
• FBD of the liquid : weight W (down), F₁ from bottom (up), net vertical force F₂ from the walls ⇒ F₁ ± F₂ = W.

e.g. frustum, bottom radius R, top 2R : W = 73πR²ρgh, F₁ = πR²ρgh ⇒ walls pushed down by 43πR²ρgh. Top R/2, bottom R : W = 712πR²ρgh ⇒ walls pushed up by 512πR²ρgh.

8. Hydrostatic Paradox

• Vessels of different shapes, same base area A, filled to the same height h with the same liquid :

but the weights differ : W₂ > W₁ > W₃

• Force on the base depends only on h and A, not on the amount (weight) of liquid.
• Weighing machine reads the total weight (vessel + liquid), not the force on the base — the liquid also pushes on the walls.

9. Force on a Vertical Wall

• Strip of area l dx at depth x (cube of side l, filled) : P = ρgx

F = ∫₀^l (ρgx)(l dx) = ρgl³2 = mg2  (bottom : ρgl · l² = mg)

P_C = pressure at the centroid C of the (projected) wall area — valid because P varies linearly with depth.

• Centre of pressure (line of action of F) lies below the centroid : at depth 2h3 for a rectangular wall (torque about the bottom edge = F·h/3 = ρgbh³/6).
• Layered liquids : apply the rule separately to the part of the wall in contact with each layer.
• Common wall between liquids ρ₂ > ρ₁ (same h, width a) : net force = (ρ₂ − ρ₁)gah²/2 towards the lighter side.
• Curved wall : liquid in equilibrium ⇒ force on the curved wall = force on the flat face it closes, e.g. half-cylinder (radius R, height h) : F = (ρgh/2)(2Rh) = ρgRh².

10. Pascal's Law & Hydraulic Lift

• Pascal's law : a pressure change applied at any point of a confined incompressible fluid is transmitted undiminished to every point of the fluid and to the walls of the container.

P₁ = P₂ ⇒ F₁A₁ = F₂A₂

• Small force on the small piston lifts a large load on the big piston — force multiplier.
• Force is multiplied, work is not : A₁d₁ = A₂d₂ ⇒ F₁d₁ = F₂d₂.
• Pistons at different levels (lower one at depth h below the other) : F_lowA_low = F_highA_high + ρgh.

11. Gauge Pressure & Manometer

• Gauge pressure = pressure measured with respect to atmospheric pressure (P itself is the absolute pressure).

• May be positive, negative or zero : P = 1.5 × 10⁵ Pa ⇒ +0.5 × 10⁵ Pa ; P = 0.4 × 10⁵ Pa ⇒ −0.6 × 10⁵ Pa.
• At depth h in an open liquid : P_gauge = ρgh.

Manometer (U-tube measuring gas pressure) :

• Several liquids : start from one end, add ρgh going down and subtract going up, until the other end ; e.g. specific gravity 4, h = 6 cm ⇒ P_gauge = 2400 Pa.

12. Barometer & Faulty Barometer

• Tube closed at the top, filled with liquid and inverted in a trough ; space above the column is (nearly) vacuum, P = 0.

• Mercury (13.6 g/cm³) : h ≈ 76 cm ; water : h ≈ 10.3 m (too tall ⇒ mercury is used).
• P₀ ≈ 10⁵ Pa ≈ 76 cm of Hg. In mercury 5 cm below the surface ⇒ 81 cm of Hg ; 2 cm above the outside level inside a column ⇒ 74 cm of Hg.

Faulty barometer (air trapped at the top) :

• Reading h is less than the true value P₀/ρg.
• Trapped gas at constant temperature : P₁V₁ = P₂V₂ ; uniform tube ⇒ V = A × (length of air column).
• Air column closed by a mercury pellet of length l : open end down ⇒ P = P₀ − l ; open end up ⇒ P = P₀ + l ; horizontal ⇒ P = P₀ (in cm of Hg).

13. Pressure in a Horizontally Accelerated Liquid

• Horizontal column (length l, area A) between B (rear) and A (front), acceleration a :

P_BA − P_AA = (ρAl)a

• Gravity : pressure increases along mg ⇒ ΔP = ρgh (vertical distance h).
• Acceleration : pressure increases along the pseudo force, i.e. opposite to a ⇒ ΔP = ρal (horizontal distance l).

14. Free Surface of an Accelerated Liquid

• Open vessel with horizontal acceleration a : at steady state the free surface is inclined.

A, B on the free surface (P₀) ; C below A and behind B : P₀ + ρgy = P₀ + ρax

• Surface rises towards the rear ; it is perpendicular to g_eff.
• U-tube (horizontal arm l) accelerated along the arm : h = alg, rear arm higher.

Vessel moving up a smooth incline (θ) with acceleration a :

(α = angle of the surface with the incline ; per unit mass : a + g sinθ along the incline, g cosθ normal to it.)

Closed vessel completely filled :

• Closed vessel (length l, height h), completely filled, acceleration a towards the front : P_{rear top} − P_{front top} = ρal ; P_{front bottom} − P_{front top} = ρgh ; P_{rear bottom} − P_{front top} = ρgh + ρal (rear bottom maximum, front top minimum).

• General acceleration : pressure is maximum at the corner farthest along g_eff = g − a and minimum at the opposite corner. Cube of side l, a = a₁î + a₂ĵ + a₃k̂ (ĵ up) : P_max − P_min = ρ(g + a₂)l + ρa₁l + ρa₃l.

15. Pressure in a Rotating Liquid

• Horizontal tube of liquid rotating with angular velocity ω about a vertical axis through O.

Element (dm = ρA dx) needs centripetal force dm ω²x from the pressure difference :

(P + dP)A − PA = (ρA dx)ω²x ⇒ ∫_P₀^P_A dP = ∫₀^l ρω²x dx

(P₀ = pressure on the axis ; A at distance l from the axis.)

• Pressure increases away from the axis (l₂ > l₁ measured from the axis).
• Rotating U-tube, arms at distances r₁ and r₂ from the axis : h = ω²(r₂² − r₁²)2g, the farther arm higher (e.g. 2l and l ⇒ 3ω²l²/2g).
• Tube closed at both ends, full of liquid of mass m, rotated about one end : force on the far end = ½ρω²L²A = mω²L/2 (= net centripetal force on the liquid).

16. Rotating Vessel – Paraboloid Surface

e.g. R = 5 cm, 2 rev/s (ω = 4π rad/s) ⇒ h ≈ 2 cm.

17. Archimedes' Principle & Buoyant Force

• A body immersed in a fluid, fully or partially, experiences an upward buoyant force equal to the weight of the fluid displaced by it.

Spring balance : 5 kg in air, 3 kg immersed ; the overflow water weighs 2 kg = loss in weight.

• ρ → density of the liquid ; V → immersed volume of the body.
• Accelerating frame (lift / vessel) : F_b = ρ V g_eff.
• Mass of the body M = σV₀ (σ = density of body, V₀ = total volume).

18. Buoyancy from Pressure

• Buoyant force = net force exerted by the liquid on the body due to pressure.

Side forces cancel (F₂ = F₄). F_net = F₃ − F₁ = [P₀ + ρg(h + h₁)]A − [P₀ + ρgh₁]A = ρ(hA)g

• Independent of P₀ and of the depth h₁ of the body.
• Any shape : replace the body by liquid of the same shape ; that liquid is in equilibrium, so the net pressure force on it = its weight = ρVg.
• Trick : F_b = ρVg is a short-cut to the net pressure force ; conversely F_bottom − F_top = F_b gives an unknown pressure force on one surface.

19. Sinking, Floating & Free Float

• Free float (only F_b and mg act) : ρVg = σV₀g

20. Floating Results & Apparent Weight

• Extra load to just submerge a floating body : mg = ρ(V₀ − V)g (extra buoyancy of the part above the liquid).
• Boat (base area A) sinks further by Δh when load m is added : m = ρAΔh.
• Hollow body : displaced volume uses the outer volume, weight uses the volume of material : ρV_disp = σV_material (e.g. shell of radii R, 2R floating half-immersed ⇒ σ = 4ρ/7).
• Body across two liquids (length h₁ in ρ₁, h₂ in ρ₂, h₁ + h₂ = h) :

Apparent weight :

• The body pushes the liquid down with F_b (Newton's 3rd law) ⇒ weighing machine under the beaker reads more by F_b ; total of both readings is unchanged.

21. Vertical Oscillation of a Floating Block

• Block (height h, area A, density σ) floats in liquid ρ with immersed depth h₁ : ρh₁ = σh.

Push down by x : F_net = extra F_b = (ρAg)x (restoring) ⇒ SHM with k = ρAg

(Viscous drag and liquid motion neglected ; block stays partly immersed.)

22. Force on Curved Surfaces

• Treat the curved part as one face of a closed body : F_curved (net, vertical) and F_flat differ by the buoyant force.

• Cone (base area πR², height H) hung vertex down, base at depth h : F_curved = ρgπR²(h + H3) upward — independent of the cone's density.
• Sphere (radius R) with top at depth h, force on the lower hemisphere (flat face at depth h + R) :

23. Two Balls & Two-Liquid Floating

• Two balls (same V) joined by a taut string, in equilibrium inside liquid ρ : upper σ₂ < ρ < σ₁ lower.

• Block floating across two liquids : the upper liquid presses only on the top face, the lower only on the bottom face ; F_bottom − F_top = F_b1 + F_b2 = mg.

24. Buoyancy in an Accelerating Vessel

• Vessel with horizontal acceleration a : in its frame a pseudo force σVa acts backward ; the liquid gives an extra buoyant force ρVa along a (replace g by a).
• W.r.t. the vessel : heavier body (iron) moves backward, lighter body (wood) moves forward.

Ball (σ > ρ) on a string in a closed, filled vessel : T sinθ = (σ − ρ)Va , T cosθ = (σ − ρ)Vg

(String lies along g_eff ; for a light ball σ < ρ it tilts forward.)

25. Ball Dropped into a Liquid

• Ball (density σ < ρ) released from height h above the surface ; no loss of speed on entry, no viscosity.

In air : v₀² = 2gh. In liquid (upward retardation) : ρVg − σVg = σVa ⇒ a = (ρσ − 1)g

Work–energy : σVg(h + s) − ρVgs = 0 ⇒

e.g. σ = 800, ρ = 1000 kg/m³ ⇒ a = g/4, s = 4h.

26. Melting of Floating Ice

• Similarly : a steel block on a floating wooden block thrown into the water ⇒ level decreases.

27. Centre of Buoyancy (COB)

• F_b acts at the centre of buoyancy = centroid of the displaced volume.
• Weight acts at the COM. If COB and COM are not on one vertical line, F_b and mg form a couple ⇒ the body rotates.

Wood–iron rod (equal volumes) fully immersed : COB at the geometric centre, COM towards iron ⇒ iron end goes down.

• Rod hinged at the bottom (length l, density σ < ρ, liquid depth h) : immersed length L = h secθ ; torque about the hinge ⇒ ρL² = σl² :

(Valid when h < l√(σ/ρ) ; otherwise the rod stays vertical.)

28. Fluid Dynamics – Streamlines & Types of Flow

• Streamlines : curves that are, at every instant, tangent to the velocity of the flow at each point.

Laminar (streamline) flow :

• Streamlines do not change with time (steady flow) ; every particle through a given point follows the same path.

Turbulent flow :

• Streamlines change with time ; particles through a point do not follow the same path (eddies form).

29. Reynolds Number

• Dimensionless ratio of inertial force to viscous force ; decides whether pipe flow is laminar or turbulent.

ρ → density of liquid, v → speed, d → diameter of tube, η → coefficient of viscosity (Pa·s), Q → volume flow rate (v = 4Q/πd²)

• R_e < ≈ 2000 : laminar ; R_e > ≈ 3000 : turbulent ; in between : unstable (transition). Exact limits vary slightly from book to book.
• Large ρ, v, d ⇒ turbulence more likely ; large η ⇒ laminar.
• For the same Q : R_e ∝ 1/d (wider tube ⇒ smaller R_e).

30. Flow Rate & Equation of Continuity

• Pipe of area A, speed v : in time dt the liquid crossing a section fills length v dt.

Units : m³/s (or L/s) and kg/s. Water : 1 L/s ↔ 1 kg/s ; 1 cm³/s ↔ 1 g/s.

• Mass is conserved (steady flow, no leak) : mass in per second = mass out per second, at any section or junction.

• Incompressible liquid (ρ constant) :

• Narrower section ⇒ faster flow (v ∝ 1/A ∝ 1/r²) ; streamlines crowd where speed is large.
• Junction : Av = A₁v₁ + A₂v₂ + … (volume in = volume out).

31. Bernoulli's Equation

• For steady, streamline flow of an incompressible, non-viscous liquid, the total energy per unit volume is the same at all points of a streamline.

Energy of a liquid element (mass dm, volume dV) :

(Pressure energy per unit volume equals the pressure P itself.)

together with continuity : A₁v₁ = A₂v₂

• Horizontal tube : where speed is more, pressure is less.
• Uniform tube ⇒ same v everywhere ⇒ P_lower − P_upper = ρgh (as in statics).
• Note : P = P₀ + ρgh holds only for a liquid at rest ; for moving liquid use Bernoulli.
• Point open to atmosphere (free jet / open end) ⇒ P = P₀.
• Valve in a horizontal pipe : closed reading P₁, open P₂ ⇒ v = √(2(P₁ − P₂)/ρ).

32. Venturi Meter

• Device to measure the volume flow rate in a pipe.

Continuity : v₂ = A₁v₁/A₂ ; Bernoulli (same height) : P₁ − P₂ = ½ρ(v₂² − v₁²) = ½ρv₁²A₁² − A₂²A₂²

Manometer tubes : P₁ − P₂ = ρgh ⇒ v₁ = A₂ √(2gh)√(A₁² − A₂²)

• Wide part : v less, P more ; throat : v more, P less.

33. Siphon

• Uniform tube draining a vessel : A = inlet (depth h₁), B = highest point (h₂ above the surface), C = open outlet (h₃ below A).

• Uniform tube ⇒ v_A = v_B = v_C = v ; C open to air ⇒ P_C = P₀.
• Liquid at A is moving, so P_A ≠ P₀ + ρgh₁ ; Bernoulli with C :

• Flow condition : P_B > 0 ⇒ P₀ > ρg(h₁ + h₂ + h₃) ; C must be below the free surface.
• Free surface → C (tube area ≪ vessel area) : v = √(2g(h₁ + h₃)).

34. Speed of Efflux (Torricelli) & Range

• Small hole (area a) in the wall of a vessel (area A), depth h below the free surface.

Continuity : surface speed v_s = av/A ; Bernoulli : P₀ + ½ρv_s² + ρgh = P₀ + ½ρv²

• Same speed as a body falling freely through h.

Range on the ground (hole at height h₁ = H − h) :

time of fall t = √(2h₁/g) ⇒ R = vt = √(2gh)·√(2h₁/g)

• R is maximum at h = h₁ = H/2 ; R_max = H.
• Same R if h and h₁ are interchanged (hole at depth d from the top and at height d from the bottom).

Two liquid layers (ρ₁, h₁ over ρ₂, h₂), hole at the bottom :

• Piston pushing liquid out of a nozzle : P_piston − P₀ = ½ρ(v₂² − v₁²) ⇒ force F = (P_piston − P₀)A₁.
• A falling stream narrows (R_C = R(H/(H + h))^1/4 at distance h below the tap) ; a rising jet slows and widens.

35. Thrust Force

• Liquid flowing out with speed v (relative to the vessel) pushes the vessel the opposite way.

• v = √(2gh) (a ≪ A). Smooth ground : acceleration = F_th/M ; rough ground : on the verge of sliding when F_th = μMg (M = vessel + liquid).
• Light vessel, hole at the bottom : μ_min = 2ρaghρAhg = 2aA.

36. Time to Empty a Tank

A(−dhdt) = a√(2gh) ⇒ ∫_H^h dh√h = −aA√(2g) ∫₀^t dt

• T ∝ √H : the lower half takes longer ; t_{2nd half}/t_{1st half} = 1/(√2 − 1) = √2 + 1 (e.g. 200 s → ≈ 483 s).
• Steady level with inflow Q (volume per second) : Q = a√(2gh) ⇒ h = Q²2ga².

37. Dynamic Lift (Bernoulli in Air)

• Air above the wing moves faster (v₂ > v₁) ⇒ pressure above is less ⇒ net upward force.

• Take-off when lift = Mg.
• Wind of speed v over a roof (still air inside) : ΔP = ½ρv² ⇒ force ½ρv²A upward (roof blown off).

38. Viscosity & Viscous Force

• Viscosity : property of a fluid by which it resists flow — opposes relative motion between its layers and of bodies moving through it (honey : high, water : low).

Newton's law of viscosity :

• η = coefficient of viscosity ; (−) : the force opposes relative motion — faster layer dragged back, slower layer dragged forward.

Water (room temperature) : η ≈ 1 mPa·s = 10⁻³ Pa·s (= 0.01 poise) ;  dimensions [ML⁻¹T⁻¹]

Linear velocity profile (e.g. river of depth h) :

• v varies linearly from 0 at the bed to v at the top ⇒ dvdy = Δvh :

• No-slip condition : the fluid layer touching a solid surface moves with that surface (layer at the river bed is at rest).

39. Viscous Force between Two Plates

• Plates of area A separated by a liquid layer of thickness h : only the relative velocity matters.

• Constant velocity ⇒ applied force = total viscous drag, e.g. boat (area A) on a river of depth D : F = ηAv₀/D ; block sliding down an oiled incline : mg sinθ = ηAv/h.
• Plate between two others : add the drags from both liquid layers.
• Cylinder (radius R, length l, mass m) falling in a pipe with lubricant gap r ≪ R : mg = η(2πRl)v/r ⇒ v = mgr2πηRl.

40. Stokes' Law & Terminal Velocity

• Sphere of radius r moving with velocity v through a liquid of viscosity η (small speed, streamline flow) :

Terminal velocity :

F_net = mg − F_b − F_v ; F_v grows with v until F_net = 0 ⇒ constant terminal speed :

6πηrv = (σ − ρ)43πr³g

• σ > ρ (steel ball in water) : moves down ; σ < ρ (air bubble in water) : v_T negative ⇒ rises with terminal speed.
• Same r, ρ, η : v_T ∝ (σ − ρ). At terminal speed, viscous force = mg − F_b.
• Height of fall for which the speed does not change on entering the liquid : √(2gh) = v_T ⇒ h ∝ r⁴.

Graphs (released from rest) :

• Acceleration falls from g(1 − ρ/σ) to zero ; speed rises and approaches v_T asymptotically.

Rate of heat production at terminal velocity :

v_T ∝ r² ⇒ P ∝ r⁵ ;  P ∝ v_T^5/2

41. Merging or Breaking of Drops

Volume conservation : 43πR³ = N · 43πr³

42. Surface Tension & Soap Film

• Surface tension : tendency of a liquid surface at rest to shrink to the minimum possible area — it behaves like a stretched elastic membrane.

• e.g. spherical drops (minimum area), brush hair clinging together out of water, insects walking on water, capillary rise ; soap / detergents decrease it (cleaning).
• Surface tension decreases with increase in temperature and with impurities like soap.
• Force acts along (tangent to) the surface, perpendicular to the line, and is attractive :

T in N/m (= J/m², energy per unit area) — fixed for a liquid at a given temperature.

• A soap film has two free surfaces :

43. Excess Pressure inside Drops & Bubbles

Soap bubble (two surfaces) :

Hemisphere in equilibrium : PπR² = P₀πR² + T(2πR)·2

Liquid drop (one surface) :

Air bubble at depth h in a liquid (one surface) :

• Smaller radius ⇒ larger excess pressure (higher on the concave side) ⇒ two connected bubbles : air flows from the smaller to the larger.
• Bubble of radius R inside one of 2R : P_outer = P₀ + 2T/R, P_inner = P₀ + 6T/R.
• Work to expand a soap bubble R₁ → R₂ : W = 8πT(R₂² − R₁²) (R → 2R : 24πR²T).
• Drop of radius R broken into N drops : ΔE = 4πR²T(N^1/3 − 1) increase ⇒ the liquid cools ; merging drops releases energy (heats).

44. Formula Summary

ρ = liquid density, σ = body density, η = viscosity, T = surface tension (in the SHM row, T = time period).