3-7 FLUIDS IN RIGID-BODY MOTION

Transcription

1 3-7 FLUIDS IN IGID-BODY MOTION S FLUIDS IN IGID-BODY MOTION We ae almost eady to bein studyin fluids in motion (statin in Chapte 4), but fist thee is one cateoy of fluid motion that can be studied usin fluid statics ideas: iid-body motion. As the name implies, this is motion in which the entie fluid moves as if it wee a iid body individual fluid paticles, althouh they may be in motion, ae not defomin. This means that thee ae no shea stesses, as in the case of a static fluid. What kind of fluid flow has iid-body motion? You ecall fom kinematics that iid-body motion can be boken down into pue tanslation and pue otation. Fo tanslation the simplest motion is constant velocity, which can always be conveted to a fluid statics poblem by a shift of coodinates. The othe simple tanslational motion we can have is constant acceleation, which we will conside hee (Eample Poblem 3.9). In addition, we will conside motion consistin of pue constant otation (Eample Poblem 3.10). As in the case of the static fluid, we may apply Newton s second law of motion to detemine the pessue field that esults fom a specified iid-body motion. In Section 3-1 we deived an epession fo the foces due to pessue and avity actin on a fluid paticle of volume dv. We obtained df = ( p + ) dv o df dv = p + (3.) Newton s second law was witten df df = a dm = a dv o dv = a Substitutin fom Eq. 3., we obtain p + = a (3.16) If the acceleation a is constant, we can combine it with and obtain an effective acceleation of avity, eff = a, so that Eq has the same fom as ou basic equation fo pessue distibution in a static fluid, Eq This means that we can use the esults of pevious sections of this chapte as lon as we use in eff place of. Fo eample, fo a liquid undeoin constant acceleation the pessue inceases with depth in the diection of eff, and the ate of incease of pessue will be iven by eff,whee eff is the manitude of eff. Lines of constant pessue will be pependicula to the diection of eff. The physical sinificance of each tem in this equation is as follows: p + = a net pessue foce body foce pe pe unit volume + unit volume = at a point at a point mass pe acceleation unit of fluid volume paticle

2 S- CHAPTE 3 / FLUID STATICS This vecto equation consists of thee component equations that must be satisfied individually. In ectanula coodinates the component equations ae p + = a diection p + = y ay ydiection y p + = a diection Component equations fo othe coodinate systems can be witten usin the appopiate epession fo p. In cylindical coodinates the vecto opeato,, is iven by = eˆ ˆ ˆ e k (3.17) (3.18) ê whee and ae unit vectos in the and diections, espectively. Thus ê = p eˆ e k p ˆ ˆ (3.19) EXAMPLE 3.9 Liquid in iid-body Motion with Linea Acceleation As a esult of a pomotion, you ae tansfeed fom you pesent location. You must tanspot a fish tank in the back of you minivan. The tank is 1 in. 4 in. 1 in. How much wate can you leave in the tank and still be easonably sue that it will not spill ove duin the tip? EXAMPLE POBLEM 3.9 GIVEN: FIND: Fish tank 1 in. 4 in. 1 in. patially filled with wate to be tanspoted in an automobile. Allowable depth of wate fo easonable assuance that it will not spill duin the tip. SOLUTION: The fist step in the solution is to fomulate the poblem by tanslatin the eneal poblem into a moe specific one. We econie that thee will be motion of the wate suface as a esult of the ca s tavelin ove bumps in the oad, oin aound cones, etc. Howeve, we shall assume that the main effect on the wate suface is due to linea acceleations (and deceleations) of the ca; we shall nelect sloshin. Thus we have educed the poblem to one of deteminin the effect of a linea acceleation on the fee suface. We have not yet decided on the oientation of the tank elative to the diection of motion. Choosin the coodinate in the diection of motion, should we alin the tank with the lon side paallel, o pependicula, to the diection of motion? If thee will be no elative motion in the wate, we must assume we ae dealin with a constant acceleation, a. What is the shape of the fee suface unde these conditions? Let us estate the poblem to answe the oiinal questions by idealiin the physical situation to obtain an appoimate solution. GIVEN: Tank patially filled with wate (to depth d) subject to constant linea acceleation, a. Tank heiht is 1 in.; lenth paallel to diection of motion is b. Width pependicula to diection of motion is c.

3 3-7 FLUIDS IN IGID-BODY MOTION S-3 FIND: (a) Shape of fee suface unde constant a. (b) Allowable wate depth, d, to avoid spillin as a function of a and tank oientation. (c) Optimum tank oientation and ecommended wate depth. y d a SOLUTION: Govenin equation: p + = a O b ˆ p + ˆ p + i j kˆ p + (ˆ i + ˆ + ˆ ) = (ˆ + ˆ + ˆ jy k ia jay ka) y Since p is not a function of, / = 0. Also, = 0, =, = 0, and a = a = 0. y y ˆ p ˆ p i j jˆ = iˆ a y The component equations ae: = a y = ecall that a patial deivative means that all othe independent vaiables ae held constant in the diffeentiation. The poblem now is to find an epession fo p p(, y). This would enable us to find the equation of the fee suface. But pehaps we do not have to do that. Since the pessue is p p(, y), the diffeence in pessue between two points (, y) and ( d, y dy) is Since the fee suface is a line of constant pessue, p constant alon the fee suface, so dp 0 and 0 = p + = d p dy a d dy y Theefoe, dy a = {The fee suface is a plane.} d fee suface Note that we could have deived this esult moe diectly by convetin Eq into an equivalent acceleation of avity poblem, whee = ia ˆ = ia ˆ j ˆ. Lines of constant pessue (includin the fee suface) will then be eff p dp = d p + y dy p + = 0 1 a pependicula to the diection of eff, so that the slope of these lines will be =. a eff

4 S-4 CHAPTE 3 / FLUID STATICS In the diaam, d oiinal depth e heiht above oiinal depth b tank lenth paallel to diection of motion e 1 in. d θ a b b b dy b a e = = tan = d fee suface Only valid when the fee suface intesects the font wall at o above the floo Since we want e to be smallest fo a iven a, the tank should be alined so that b is as small as possible. We should alin the tank with the lon side pependicula to the diection of motion. That is, we should choose b 1 in. b With b 1 in., The maimum allowable value of e 1 d in. Thus a a 1 d = 6 and dma = a e = 6 in. If the maimum a is assumed to be, then allowable d equals 8 in. To allow a main of safety, pehaps we should select d 6 in. d ecall that a steady acceleation was assumed in this poblem. The ca would have to be diven vey caefully and smoothly. This Eample Poblem shows that: Not all enineein poblems ae clealy defined, no do they have unique answes. Fo constant linea acceleation, we effectively have a hydostatics poblem, with avity edefined as the vecto esult of the acceleation and the actual avity. EXAMPLE 3.10 Liquid in iid-body Motion with Constant Anula Speed A cylindical containe, patially filled with liquid, is otated at a constant anula speed,, about its ais as shown in the diaam. Afte a shot time thee is no elative motion; the liquid otates with the cylinde as if the system wee a iid body. Detemine the shape of the fee suface. ω

5 3-7 FLUIDS IN IGID-BODY MOTION S-5 EXAMPLE POBLEM 3.10 GIVEN: A cylinde of liquid in iid-body otation with anula speed about its ais. FIND: Shape of the fee suface. SOLUTION: Govenin equation: p + = a h 1 h 0 ω It is convenient to use a cylindical coodinate system,,,. Since 0 and, then Also, a a 0 and a. The component equations ae: Fom the component equations we see that the pessue is not a function of ; it is a function of and only. Since p p(, ), the diffeential chane, dp, in pessue between two points with coodinates (,, ) and ( d,, d) is iven by Then To obtain the pessue diffeence between a efeence point ( 1, 1 ), whee the pessue is p 1, and the abitay point (, ), whee the pessue is p, we must inteate Takin the efeence point on the cylinde ais at the fee suface ives Then eˆ eˆ kˆ p ˆ = ( ˆ + ˆ + ˆ k ea e a ka) + + eˆ eˆ 1 p kˆ p = ˆ + ˆ e k = = 0 = p p p dp = d + d dp = d d dp = d d p1 1 1 p p1 = ( 1 ) ( 1) p = p = 0 = h 1 atm p patm = ( h1 )

6 S-6 CHAPTE 3 / FLUID STATICS Since the fee suface is a suface of constant pessue (p p atm ), the equation of the fee suface is iven by o 0 = ( h1 ) ( ) = h1 + The equation of the fee suface is a paabaloid of evolution with vete on the ais at h 1. We can solve fo the heiht h 1 unde conditions of otation in tems of the oiinal suface heiht, h 0, in the absence of otation. To do this, we use the fact that the volume of liquid must emain constant. With no otation With otation Then V = d d = d = h V h = 1 + = h V = h0 1 4 d Finally, h ( ) = h + and h1 = h ( ) ( ) ( ) 1 = h + = h () Note that the epession fo is valid only fo h 1 0. Hence the maimum value of is iven by ma h0. This Eample Poblem shows: The effect of centipetal acceleation on the shape of constant pessue lines (isobas). Because the hydostatic pessue vaiation and vaiation due to otation each depend on fluid density, the final fee suface shape is independent of fluid density.