Introduction to Fluid Mechanics

Transcription

1 Chapte Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = ft/s. (a) What is its mass in kg? (b) What will the weight of this body be in N if it is exposed to the moon s standad acceleation g moon = 1.6 m/s? (c) How fast will the body acceleate if a net foce of 4 lbf is applied to it on the moon o on the eath? F = weight and a = g eath : F=W=mg=1 lbf = (m slugs) (3.174 ft/s ) o m = 1/3.174 = (31.8 slugs)( kg/slug) = kg Ans. (a) The change fom 31.8 slugs to kg illustates the pope use of the convesion facto kg/slug. The mass of the body emains kg egadless of its location. F = W moon = m.g moon = (453.6 kg)(1.6 m/s ) = 735 N Ans. (b) This poblem does not involve weight o gavity o position and is simply a diect application of Newton s law with an unbalanced foce: F = 4 lbf = m.a = (31.8 slugs)(a ft/s ) o a =4/31.8 = 1.43 ft/s = 3.79 m/s Ans. (c) This acceleation would be the same on the moon o eath o anywhee. Example 1. Dimensions and Units An ealy viscosity unit in the cgs system is the poise (abbeviated P), o g/(cm.s), named afte J. L. M. Poiseuille, a Fench physician. The viscosity of wate (fesh o salt) at K = C is appoximately μ =.1 P. Expess this value in (a) SI and (b) BG units.

2 Chapte 1 μ = [.1 g/(cm. s)] (1 kg/1 g ) (1cm/m) =.1 kg/(m.s) Ans. (a) μ = [.1 kg/(m. s)] (1 slug/14.59 kg ) (.348 m/ft) = slug/(ft.s) Ans. (b) Note: Result (b) could have been found diectly fom (a) by dividing (a) by the viscosity convesion facto listed in Table (1.). Example 1.3 Popeties of a Fluid Suppose that the fluid being sheaed in Figue (1.5) is SAE 3 oil at C. Compute the shea stess in the oil if u = 3 m/s and h = cm. The shea stess is found fom Eq. (1.13) by diffeentiating Eq. (1.14): du u (E1.1) d y h Fom Table (1.5) fo SAE 3 oil, μ =.9 kg/(m. s). Then, fo the given values of u and h, Eq. (E1.1) pedicts.9kg/( m. s) (3m / s) 43kg/( m. s ) 43N / m 43Pa Ans..m Although oil is vey viscous, this is a modest shea stess, about 4 times less than atmospheic pessue. Viscous stesses in gases and thin liquids ae even smalle. Example 1.4 (14 final Exam) Popeties of a Fluid The velocity pofile is a lamina flow though a ound pipe is expessed as, u U1 whee U = aveage velocity, (a) Daw dimensionless shea stess pofile = adius of pipe. against whee

3 Chapte 1 is wall shea stess. (b) Find, when oil flows with absolute viscosity 4 1 N.s/m and velocity of 4 m/s in a pipe of diamete 15 mm. Given u U1 du 4U du 4 U Then and d d And du d 4 U.. (E1.) So, and theplotis shownin Figue (E4.1) Ans. (a) Fom Eq. (E1.), N m Ans. (b) Fig. E1.4: dimensionless shea stess pofile Example 1.5 Popeties of a Fluid against. Deive an expession fo the change in height h in a cicula tube of a liquid with suface tension σ and contact angle θ, as in Figue (E1.5). The vetical component of the ing suface-tension foce at the inteface in the tube must balance the weight of the column of fluid of height h R cos R gh

4 Chapte 1 3 Solving fo h, we have the desied esult cos h Ans. gr Fig. E1.5 Thus the capillay height inceases invesely with tube adius R and is positive if θ < 9 (wetting liquid) and negative (capillay depession) if θ > 9. Suppose that R = 1 mm. Then the capillay ise fo a wate-ai-glass inteface, θ, σ =.73 N/m, and ρ = 1 kg/m 3 is (.73 N / m)(cos ) h.15( N. s ) / kg.15m 1. 5cm 3 (1 kg/ m )(9.81m / s )(.1m) Fo a mecuy-ai-glass inteface, with θ = 13,, σ =.48 N/m, and ρ = 1136, the capillay ise is h = -.46 cm When a small-diamete tube is used to make pessue measuements, these capillay effects must be coected fo.

5 ϕ18cm ϕ15cm Chapte 1 4 Example 1.6 Popeties of a Fluid A cylinde 7.5 cm adius and 6 cm in length otate coaxially inside a fixed cylinde of the same length and 9 cm inne adius as shown in Figue (E1.6). Glycein μ = 8 Poise fills the space between to cylindes. A Toque.4 N.m is applied to the inne cylinde. Afte a constant velocity is attended, calculate the following: (a) velocity gadient at the cylinde walls, (b) the velocity ustling and (c) the powe dissipated by the fluid esistance. 6cm Fig. E1.6 The shea stess is found fom Eq. (1.13) du (E1.3) d y L L Toque F A.... (E1.4) whee L is the cylinde length then fom Eq. (E1.4).4( N. m) du d y ( m)

6 Chapte 1 5 du dy 8( Poise) 1 du d y innewall (7.51 ).1375 N / m..... (E1.5) 3.6 Ans.(a) du d y outewall (91 ) Ans.(a) Fom Eq. (E1.4) and whee dy =-d du du.1375 d y d.1375 du d (E1.6) by integating Eq. (E1.6): u.75 1 du.1375 d.9 Whee u() = at =.9 m Then u 9.48 m / s Ans.(b).9 N Whee u 6 N N Powe Toque. 1HP (N: evolution pe minute) N 37.5pm Ans.(c)

7 Chapte Poblems 1. Deive the SI unit of foce fom base units.. Explain dynamic viscosity and kinematic viscosity. Give thei dimensions. 3. Explain the phenomenon of capillaity. Obtain an expession fo capillay ise of a fluid. 4. Expess the viscosity and the kinematics' viscosity in SI units. 5. Fo low-speed (lamina) steady flow though a cicula pipe, the velocity u vaies with adius and takes the fom p u B whee μ is the fluid viscosity and Δp is the pessue dop fom entance to exit. What ae the dimensions of the constant B? 6. The density of wate at 4 C and 1 atm is 1 kg/m 3. Obtain the specific volume. 7. The specific weight of a cetain liquid is 1 KN/m 3. Detemine its density and specific gavity. 8. A liquid when poued into a gaduated cylinde is found to weigh 8 N when occupying a volume of 5 ml (millilites). Detemine its specific weight, density, and specific gavity. 9. Obtain the pessue in SI (Pa) necessay fo shinking the volume of wate by 1% at nomal tempeatue and pessue. Assume the compessibility of wate β= Pa A block of weight W slides down an inclined plane while lubicated by a thin film of oil, as in Figue (1.P1). The film contact aea is A and its thickness is h. Assuming a linea velocity distibution in the film, deive an expession fo the teminal (zeo-acceleation) velocity V of the block.

8 Chapte Deive an expession fo the capillay height change h fo a fluid of suface tension σ and contact angle θ between two vetical paallel plates a distance W apat, as in Figue (P1.11). What will h be fo wate at C if W =.5 mm? Fig. 1.P1. Fig. 1.P Find suface tension of a soap bubble of 48 mm diamete while pessue inside is 3.1 Pa highe than atmospheic one. 13. A Newtonian fluid having a specific gavity of.9 and a kinematics viscosity of m /s flows past a fixed suface. Due to the no-slip condition, the velocity at the fixed suface is zeo (as shown), and the velocity pofile nea the suface is shown in Figue (1.P13). Detemine the magnitude and diection of the sheaing stess developed on the plate. Expess you answe in tems of U and δ, with U and δ expessed in units of metes pe second and metes, espectively. Fig. 1.P13.

9 Chapte As shown in Figue (1.P14), a cylinde of diamete 1mm and length mm is placed inside a concentic long pipe of diamete 15 mm. An oil film is intoduced in the gap between the pipe and the cylinde. What foce is necessay to move the cylinde at a velocity of l m/s? Assume that the kinematic viscosity of oil is 3 cst and the specific gavity is.9. Fig. 1.P14