Mechanical Properties of Fluids
Socho tum kitchen mein paani se bhara balti utha rahe ho — paani ka shape balti jaisa ho jata hai. Ab lohe ka dibba uthao toh shape same rehta hai. Farak kya? Paani behta hai — fluid hai. Fluid ka matlab jo beh sake — liquid aur gas dono.
Ab wahi paani haath se dabao toh haath geela hota hai par shape badalta hai. Aur honey shahad ko ulatne mein time lagta hai paani se zyada. Koi cheez patla behti hai koi gaadha, koi upar udti hai koi neeche girti hai. Is chapter mein yehi khelenge: pressure, upar neeche paani ka dabav, behta hua paani ka speed, chipchipahat viscosity, aur upar ki chamdi surface tension.
Jaise cycle ke tyre mein hawa ka pressure, syringe mein paani ka pressure, hawai jahaj ka udna, baarish ke paani ka nal se nikalna — sab isi chapter ke formulas se samjhenge.
1. Pressure & Pascal's Law - Hydraulic Lift Ka Jadoo
Socho tum chapati pe belan se pressure lagate ho — force ek area pe lagta hai. Thrust = total normal force, Pressure = thrust per unit area = F/A. Unit N/m² = Pascal Pa. Pressure scalar hai, vector nahi, kyunki direction area ka normal hota hai.
Ab ek dabbe mein paani bharo aur charo taraf chhed kar do — paani har chhed se same speed se niklega. Kyun? Pascal's law: Ek enclosed fluid mein agar kahin pressure badhao toh wo bina kam hue har point aur walls pe same transmit hota hai. Jaise ek balloon ko kahin se dabao toh pura balloon tight hota hai.
Iska sabse mast use hydraulic lift. Do pistons jode hue — chhota area A1 pe chhota force F1 lagao toh pressure P=F1/A1. Wahi pressure bade piston A2 pe P=F2/A2, toh F2 = F1 * A2/A1. Area ratio 100 hai toh 100 kg ka bacha 10,000 kg ka truck utha dega! Car service station pe gaadi upar jaati hai isi se. Hydraulic brakes mein bhi same — pair se chhota piston dabao toh charo pahiyon pe bada force.
1 atm = 1.013×10⁵ Pa = 76 cm mercury ka pressure. Matlab samundar ke level pe hawa ka itna dabav hum pe hai par feel nahi hota kyunki andar se bhi same pressure.
• Thrust = total normal force, Pressure = F/A, Pa, scalar.
• Pascal's law: enclosed fluid mein extra pressure har jagah same transmit hota hai, no loss.
• Hydraulic lift: P same → F2 = F1 × (A2/A1) → small force lifts big load.
• 1 atm = 1.013×10⁵ Pa = 76 cm Hg column, barometer mein use.
2. Pressure With Depth & Barometer - Neeche Jao Toh Pressure Badhe
Swimming pool mein neeche jaate hi kaan mein dabav mehsoos hota hai? Wahi formula hai. Socho paani ka ek cylinder height h, area A, density ρ. Upar ka paani neeche wale paani pe weight dalta hai. p = pa + ρgh. pa upar atmosphere ka, ρgh paani ke column ka. g depth ke saath almost same.
Derivation simple: Neeche wale face pe upar ka weight + atmosphere = pressure × area. W = ρ × volume × g = ρ A h g, toh pA = paA + ρAhg → p = pa+ρgh. Matlab har 10 m paani mein ~1 atm badhta hai (ρ=1000, g=10, h=10 → 1e5 Pa).
Isi se barometer banta hai. Torricelli ne mercury ka tube ulta kiya paare ke bartan mein — upar vacuum, neeche pa=ρgh. Height 76 cm aayi mercury ki, kyunki mercury dense 13600 kg/m³. Agar paani use karte toh 10 m chahiye! Aneroid barometer mein bina liquid ke diaphragm se measure karte hain.
Manometer se kisi bartan ka pressure nikalte hain — U-tube mein ek side pa, dusri side p, height difference se p-pa=ρgh. Blood pressure meter bhi manometer hi hai.
• p = pa + ρ g h, h depth, ρ density, same level same pressure.
• Torricelli: mercury barometer 76 cm Hg = 1 atm, water barometer 10.3 m.
• Gauge pressure = p - pa = ρgh, absolute = gauge + pa.
• Hydraulic machines work because p same everywhere at same horizontal level.
3. Streamline Flow & Bernoulli - Behta Paani Ka Energy
Socho tum ek pipe se paani daal rahe ho, kabhi seedha patli dhara, kabhi gol gol ghumta hua turbulent. Streamline flow: Har point pe har particle jo aaye uska velocity same rahe, path ek dusre ko cross na kare, smooth lines. Jaise train ke tracks. Turbulent mein eddies, swirls — jaise dhuaan upar jaake gol gol ghumta hai.
Ab continuity: Pipe patla mota ho toh? Paani compress nahi hota, jitna andar utna bahar. Av = constant. A1v1 = A2v2. Patli jagah pe speed zyada, moti jagah pe kam. Garden pipe ka muh dabao toh patla hota hai aur paani tez jaata hai — wahi.
Bernoulli ka principle toh mast hai — ye energy conservation hai flowing fluid ke liye. p + ½ ρv² + ρgh = constant along a streamline. p = pressure energy per volume, ½ρv² = kinetic per volume, ρgh = potential per volume. Matlab speed badhe toh pressure ghathe.
Isse hawai jahaj udta hai! Wing ka upar ka surface curve zyada, hawa ko zyada distance tay karna padta hai, speed upar zyada → pressure upar kam → neeche se upar lift. Chimney se dhuaan tez hawa mein jaldi nikalta hai kyunki bahar pressure kam. Spray bottle mein tez hawa se paani khinch aata hai.
• Streamline smooth, no crossing, steady at point. Turbulent irregular eddies.
• Continuity: Av = const → narrow → v more. Volume flow rate Q=Av m³/s.
• Bernoulli: p + ½ρv² + ρgh = const → energy per volume conserved. Valid for ideal fluid: incompressible, non-viscous, steady, irrotational.
• Application: Wing lift, chimney draft, atomizer spray — faster fluid lower pressure.
4. Torricelli, Venturi & Magnus - Tank Se Paani Kitna Tez?
Ek bade tank mein neeche chhed kar do, paani kitni speed se niklega? Bernoulli lagao: upar surface pe v≈0 (bada tank), pressure pa, height h. Chhed pe pressure pa, height 0, speed v. Toh pa+ρgh = pa+½ρv² → v=√(2gh). Ye Torricelli's law hai — same jaise upar se girao toh speed √(2gh) hoti hai free fall mein! Matlab paani utni hi speed se niklega jitna upar se girke aaye.
Venturi-meter se flow rate nikalte hain. Pipe mein beech mein patla hissa. Continuity se A1v1=A2v2, patle mein v2 zyada. Bernoulli se p1+½ρv1² = p2+½ρv2² → p1-p2=½ρ(v2²-v1²). Manometer se p1-p2 measure karo toh v nikal jayega, phir Q=A1v1. Factory mein paani, tel ka flow isi se measure hota hai.
Magnus effect — cricket mein spinning ball swing kyun hoti hai? Ball ghume toh ek side pe hawa aur spin same direction → speed zyada → pressure kam, dusri side opposite → pressure zyada → ball curve ho jaati hai pressure difference se lift. Football ka banana kick bhi same. Aeroplane ka spinning cylinder wala concept bhi.
Blood flow mein bhi Bernoulli — artery mein blockage (atherosclerosis) pe area kam → speed zyada → pressure kam → artery collapse ka risk. Isliye doctor Venturi jaise sochte hain.
• Torricelli efflux: v = √2gh — free-fall equivalent, independent of density.
• Venturi: continuity + Bernoulli → measure Δp via manometer → get flow rate Q = A1v1 = A2v2.
• Magnus dynamic lift: spinning ball → pressure difference → sideways lift, curve ball.
• Conditions: ideal fluid, tank large, hole small, steady flow.
5. Viscosity & Stokes & Reynolds - Chipchipahat Ka Science
Shahad aur paani — dono behta hai par shahad dheere. Kyun? Chipchipahat viscosity. Socho fluid ki layers ek dusre pe slide karti hain, beech mein friction jaise. Viscosity η se measure: Viscous force F = η A dv/dx. dv/dx velocity gradient — ek layer tez dusri dheere toh gradient. Unit Pa·s = N·s/m². Paani ka η≈1e-3, shahad ~10, glycerine ~1 Pa·s.
Ek chhota gola viscous liquid mein gir raha hai toh drag lagta hai — Stokes' law: F = 6π η r v. r radius, v speed. Ye small sphere, low speed, laminar ke liye.
Ab gola girte girte ek speed pe weight = buoyant + Stokes drag → net zero → constant speed se girta hai, isko terminal velocity vt kehte hain. vt = 2 r² (ρ-σ) g / 9 η. ρ sphere density, σ fluid density. r² se proportional — mota gola tez girta hai. Isi se raindrop slow aata hai, fog mein chhote drops terminal chhota isliye hava mein latakte hain.
Flow kab smooth rahega kab turbulent? Reynolds ne number diya: Re = ρ v d / η. d characteristic length diameter. Re <1000 laminar smooth, >2000 turbulent chaotic, beech mein transitional. Re dimensionless. Critical velocity vc = Re_c η / ρ d. Paani ka pipe mein Re zyada → turbulent, shahad mein kam → laminar.
• Viscosity η = resistance to flow, F = η A dv/dx, unit Pa·s.
• Stokes: F_drag = 6π η r v, valid small sphere, slow, laminar, Re≈1.
• Terminal: mg = Fb + Fd → vt = 2r²(ρ-σ)g/9η, r² → bigger sphere falls faster.
• Re = ρ v d / η: <1000 laminar orderly, >2000 turbulent eddies, critical velocity vc = Rec η/ρd.
6. Surface Tension - Paani Ki Chamdi
Paani ke upar sui halki rakh do toh doobti nahi, kyun? Paani ki upar ek khaal si hai — surface tension. Molecules andar wale charo taraf khinche jaate hain, surface wale sirf neeche aur side se khinche jaate hain → net andar → surface ko tight karne ki koshish → minimum area lena chahta hai sphere.
Surface tension S = Force per unit length = F/l = Energy per unit area = E/A. Unit N/m = J/m². Paani ka 0.0725 N/m at 20°C, mercury 0.4355.
Angle of contact θ: jahan liquid, solid, air milte hain wahan liquid surface ka solid se angle. Paani glass pe 0° acute → wetting, mercury glass 140° obtuse → non-wetting. Isi se meniscus concave convex banta hai.
Excess pressure: Drop ke andar bahar se zyada pressure. Drop (1 surface): Δp = 2S/r, Bubble (2 surfaces film): Δp = 4S/r. Chhota radius → pressure zyada. Isliye chhote bubbles milke bada banta hai.
Capillary rise: Patli tube mein paani upar chadh jata hai. Kyun? Surface tension ka upward component = weight of column. h = 2S cosθ / r ρ g. r tube radius, θ contact angle. Patli tube r kam → h zyada. Pedon mein paani jad se upar isi se jaata hai, blotting paper ink isi se khinchta hai. Detergent S kam karta hai → dhुलाई acchi, paani zyada failta hai.
• S = F/l = energy/area, N/m, water 0.0725 N/m. Surface tends to minimize area → spherical drops.
• Angle of contact θ: water-glass ~0° wetting, mercury-glass ~140° non-wetting.
• Excess pressure: drop 2S/r, soap bubble 4S/r (two films), smaller r more pressure.
• Capillary: h = 2S cosθ / rρg, narrow tube higher rise, detergent reduces S → better cleaning, plants rise via capillarity.
| Concept | Formula (Correct) | Daily Example |
|---|---|---|
| Pressure | P=F/A Pa scalar, 1 atm=1.013e5 Pa=76 cm Hg | Syringe thrust, tyre air |
| Pascal | Δp same everywhere enclosed, Lift F2=F1 A2/A1 | Car lift, hydraulic brake |
| Depth | p=pa+ρgh linear, gauge=ρgh, barometer pa=ρgh | Swim ear pressure 10m~1atm |
| Continuity | Av=const Q=Av m³/s incompressible mass conservation | Pipe mouth press speed↑ |
| Bernoulli | p+½ρv²+ρgh=const ideal steady incompressible non-viscous irrotational | Wing lift speed↑ pressure↓ |
| Torricelli | Efflux v=√2gh free-fall same | Tank hole jet |
| Venturi/Magnus | Δp=½ρ(v2²-v1²) Q=A1v1, Magnus pressure diff due to spin lift | Flow meter, banana kick |
| Viscosity | F=ηA dv/dx Pa·s, Stokes F=6πηrv, vt=2r²(ρ-σ)g/9η | Honey slow, rain terminal |
| Reynolds | Re=ρvd/η <1000 laminar >2000 turbulent | Tap slow smooth fast turbulent |
| Surface tension | S=F/l=E/A N/m, water 0.0725, contact θ wet acute non-wet obtuse, Δp drop 2S/r bubble 4S/r, capillary h=2S cosθ/rρg | Needle on water, plants rise, detergent |
1. Pressure Basics: P=F/A Pa scalar. Pascal's law: enclosed fluid Δp transmits equally → hydraulic lift F2=F1(A2/A1) force multiplier. p depth: p=pa+ρgh same horizontal level same p, gauge=ρgh, barometer pa=ρgh Hg 76 cm.
2. Flow Continuity & Bernoulli: Continuity Av=const Q=Av (mass conservation incompressible). Bernoulli p+½ρv²+ρgh=const along streamline for ideal fluid (steady, incompressible, non-viscous, irrotational) → energy per volume conserved. Consequence speed↑ pressure↓.
3. Torricelli & Venturi & Magnus: Torricelli efflux v=√2gh (Bernoulli with large tank). Venturi: narrow v↑ p↓ Δp=½ρ(v2²-v1²) measure via manometer → Q. Magnus dynamic lift spinning ball pressure difference one side fast air low p → curve. Blood artery blockage same.
4. Viscosity & Re: Viscous force F=ηA dv/dx η Pa·s. Stokes drag F=6πηrv small sphere low Re. Terminal when mg=Fb+Fd → vt=2r²(ρ-σ)g/9η ∝r² bigger falls faster. Reynolds Re=ρvd/η dimensionless: <1000 laminar orderly, >2000 turbulent eddies, critical vc=Rec η/ρd.
5. Surface Tension & Capillary: S=F/l=E/A N/m J/m² water 0.0725. Contact angle θ: water-glass ~0° wet concave, mercury-glass ~140° non-wet convex. Excess Δp: drop 2S/r (1 surface), bubble 4S/r (2 surfaces) small r large p. Capillary rise h=2S cosθ/rρg ∝1/r narrow higher, detergent lowers S → better spreading cleaning, plant rise.
Numerical Checklist: P=F/A, p=pa+ρgh, Q=Av, Bernoulli p+½ρv²+ρgh=const, v=√2gh, Venturi Δp=½ρ(v2²-v1²), Stokes 6πηrv, vt=2r²Δρg/9η, Re=ρvd/η, S=F/l, Δp=2S/r /4S/r, h=2S cosθ/rρg. Units Pa, m³/s, Pa·s, N/m.