Stokes law. Viscosity coefficient

Transcription

1 EXERCISE 7 Stokes law. Viscosity coeicient 7.1. Intoduction Real luid has a cetain amount o intenal iction, which is called viscosity. Viscosity exists in both liquids and ases, and is essentially the ictional oce between the adjacent layes o luid as the layes move past one anothe. In liquids, viscosity appeas due to the cohesive oces between the molecules. In ases, it aises om collisions between the molecules. x movin plate v luid velocity adient l static plate i. 7.1 Expeiment setup o obtainin o viscosity coeicient Dieent luids posses dieent amounts o viscosity: syup is moe viscous than wate; ease is moe viscous than the enine oil; liquids in eneal ae much moe viscous than ases. The viscosity o dieent luids can be expessed quantitatively by the coeicient o viscosity, η (the Geek lowecase lette eta), which could be deined usin the ollowin expeiment. A thin laye o luid is placed between two lat plates. One plate is static and the othe is made to move (see i. 7.1). The luid diectly in contact with each plate is held to the suace by the adhesive oce between the molecules o the liquid and those o the plate. Thus the uppe suace o the luid moves with the same speed v as the uppe plate, wheeas the luid in contact with the stationay plate emains stationay. The stationay laye o luid etads the low o the 53

2 laye just above it, which in tun etads the low o the next laye, and so on. Thus the velocity vaies continuously om 0 to v, as shown. The incease in velocity divided by the distance ove which the chane is made - equal to v/l - is called the velocity adient. To move the uppe plate equies a oce, which you can veiy by movin a lat plate acoss the puddle o syup on the table. o a iven luid, it is ound that the equied oce, is popotional to the aea o a luid in contact with each plate A, and to the speed v, but is invesely popotional to the sepaation l, o the plates, what comes down to the ollowin elation: va / l. o dieent luids, the moe viscous the luid, the eate is the equied oce. Hence the popotionality constant o this equation is deined as the coeicient o viscosity, η: Av =η (7.1) l Solvin o η, we ind 2 η = l/ va. The SI unit o η is N s / m = Pa s. In the CGS system, the unit is 2 dyne s / cm and the unit is called a poise (P). Viscosities ae oten iven in centipoise * 2 ( 1cP= 10 P ). Viscosity is a unction o the tempeatue o example a hot enine oil is less viscous than the cold one. The Table 7.1 lists the viscosity coeicients o vaious luids at the speciied tempeatues Table 7.1. The viscosity coeicients o vaious luids at the speciied tempeatues luid Tempeatue [ o C] Viscosity [Pa. s] Wate Wate Wate Ethyl alcohol Enine oil Ai Hydoen Wate vapou * 1[Pa*s] = 10[P] = 1000[cP] 54

3 In eneal the equation o iction oce o any velocity adient occuin duin a lamina low is iven below (Newton s equation): dv =η A (7.2) dx The equation is valid only o small velocities (low values o Reynolds numbe, ρv Re<1160, Re = ). luids, which obey this equation, ae called Newtonian luids. tπη It would be athe diicult to calculate viscosity o the liquids diectly om the above equation. Especially it would be diicult to measue the velocity adient and make sue that the aea o contact between the plates is kept constant. Instead, a Stokes viscosimete is used, in which small metal balls ae dopped in a lass tube illed with liquid. liquid daed by the ball metal ball v i Scheme o avitationally allin ball in viscous liquid. When an object (like a metal ball) alls avitationally in viscous liquid it das cetain 55

4 amount o the liquid with itsel due to the molecula inteactions between suace o the object and the molecules o the liquid. These layes situated close to the movin object da athe layes (shown on i. 7.2). Thus viscosity o the luid slows down the allin object and ceates a velocity adient in the luid pependicula to the diection o motion o the object and the layes. The velocity o the object v, is small enouh that we can assume a lamina low and use the Stokes law to calculate the iction oce actin on the metal ball: = 6πηv (7.3) whee stands o the adius o the ball, v velocity, η - viscosity. Thee ae two moe oces that act on the metal ball. The ist one is obviously the avity. 4 = m = π 3 ρ m (7.4) 3 whee ρ m is a density o the metal (steel). The second oce is a buoyant oce. It occus because the pessue in the luid inceases with depth. Thus the upwad pessue on the bottom suace o the submeed object is eate than the downwad pessue on its top suace (see i. 7.3) h 1 1 A h 2 h=h 2 -h 1 2 i Detemination o buoyant oce 56

5 To see the eect o buoyancy conside a cylinde o heiht h whose top and bottom have aeas A and which is completely submeed in the luid o density ρ, as shown on iue 7.3. The luid exets a pessue P 1 = ρ h 1 at the top suace o the cylinde. The oce due to this pessue on top o the cylinde is 1 1 ρ 1 = P A= h A, and it is diected downwad. Similaly the luid exets an upwad oce on the bottom o the cylinde equal to 2 2 ρ 2 = P A= h A. The net oce due to the luid pessue, which is a buoyant oce, B, acts upwad and has the manitude: B 2 1 = ρ A( h2 h ) = ρ Ah= ρ V (7.5) = 1 In case o a metal ball submeed in a liquid the buoyant oce is equal to: B 4 = π 3 ρ (7.6) 3 Initially, when the metal ball is dopped thouh the unnel to the liquid it steadily acceleates. As the velocity inceases, the opposin iction oce also inceases, leadin inally to the balance o oces. Hence we can assume that ate some initial time the ball moves with a constant speed and the thee oces ae in equilibium: Σ = = 0 (7.7) B B = + (7.8) Substitutin all peviously deived omulas (7.3., 7.4. and 7.5.) into the above equation, we can obseve that it links two quantities: velocity o the ball and viscosity o the liquid. Thus calculatin the velocity we can detemine the viscosity usin equation ( ρ ρ m ) η = v (7.9) 9 57

6 The Stokes equation is accuate o ininitely lae envionment and does not take into account the eect o the walls o the cylinde. A coective tem equal to 1 is R intoduced. It povides an estimate o how much the ball was additionally slowed down due to the pesence o the walls o the cylinde Measuements An expeiment is peomed in the Stokes viscosimete (see i.7.4). unnel allin metal ball lass tube lyceine ubbe cok i.7.4. The Stokes viscosimete ollow the expeimental pocedue step by step: 1. ill the cylinde with lyceine. 2. Put a unnel into the mouth o the cylinde. 3. Measue the distance between the levels, maked with blue stipes on the cylinde. 4. Dop (one by one) steel balls into the cylinde thouh the unnel and measue the time o allin between the levels. 5. Repeat the pevious point o evey ball (about 15). 58

7 6. Wite down all esults into a table in you copy-book. 7. Collect all additional data (e.. density o the steel and lyceine). 8. Take out the balls om the cylinde (pullin caeully the ubbe cok, and lettin some o the lyceine low out). Pou the lyceine back into the cylinde. The data should be collected in Table 7.2. Table 7.2 No. [m] l [m] t [s] R [m] ρ m 1. [k/m 3 ] ρ [k/m 3 ] η [Pa. s] 7.3. Results, calculation and uncetainty Calculate the viscosity o lyceine usin equation 7.9. Estimate the uncetainty o the measued values by calculatin the standad deviation. The inal esult eads: η = η ± S (7.10) η 7.4. Questions 1. What is viscosity? What kind o viscosity coeicients do you know? 2. Descibe the phenomenon o aindop allin down. 3. Deive the equation o viscosity. 4. Methods o viscosity measuement. 5. On what depends the viscosity? 6. What is Reynolds numbe? 59

8 7. What kind o conditions should be ulilled to use Stokes equation? 8. Why do we have to use coective tem? 9. Comment on Achimedes law. 10. Deive the equation o buoyant oce Reeences 1. Szydłowski H., Pacownia izyczna, PWN, Waszawa, Bobowski Cz., izyka kótki kus, WNT, Waszawa, Giancoli D.C., Physics. Pinciples with Applications, Pentice Hall, eynman R., eynmana wykłady z izyki, Tom 2.2., PWN, Waszawa,