A 1. EN2210: Continuum Mechanics. Homework 7: Fluid Mechanics Solutions

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1 EN10: Continuum Mechanics Homewok 7: Fluid Mechanics Solutions School of Engineeing Bown Univesity 1. An ideal fluid with mass density ρ flows with velocity v 0 though a cylindical tube with cosssectional aea A 0. A body extending downsteam to infinity with coss-sectional aea A 1. Find an expession fo the foce acting on the body in the tube Conside the contol volume shown in the figue. Let F denote the foce acting on the aea A0 A1 (positive foce acts to the ight Mass consevation gives v0a0 = v1( A0 A1 v 0 A 0 A 1 Linea momentum gives F + p0a0 p1( A0 A1 = ρv0v0a0 + ρv1v1( A0 A1 (the left hand side is the integal of the taction on the contol vol; the ight hand side is the ate of change of linea momentum. Benoulli gives 1 1 p + ρv = p + ρv Solving these equations gives The foce on the body is equal and opposite. A A0 A1 F= Ap +ρv ( [3 POINTS]. An incompessible Newtonian viscous fluid with viscosity η and density ρ occupies the egion y < 0. At time t=0 the fluid has velocity distibution v1 = u0exp( y / b with all othe velocity components zeo. The goal of this poblem is to calculate the velocity in the fluid fo t=0. Gavity and pessue vaiations in the fluid may be neglected. By consideing a velocity field of the fom v1 = u0f( texp( y f( t / b, calculate the vaiation of velocity in the fluid with position and time.

2 The Navie-Stokes equation educes to η v 1 v + 1 ρ = y t η 3 y y f( t u0 f( t exp( y f( t / b = u0f '( t 1 exp( y f( t / b ρ y b b η 3 y f( t u 0 exp( y f( t / b + 4 f( t exp( y f( t / b ρb b y u f( t f '( t 1 exp( y f( t / b b η 3 1 4ηt f( t = f '( t = + C ρb f( t ρb The initial condition gives C=1, which gives u0 v1 = exp( y / ( b + 4 ηt / ρ 1+ 4 ηt / ρb [3 POINTS] 3. Two paallel cicula plates with adius R ae sepaated by an incompessible Newtonian fluid with viscosity η. The plates ae a distance h apat and appoach one anothe slowly with elative speed U. The goal of this poblem is to find an appoximate solution fo the velocity distibution in the fluid between the plates. Assume Stokes flow and neglect the acceleation, and assume a velocity field of the fom v = f ( z v = g( z z R z h 3.1 Use the incompessibility condition to detemine f( z in tems of gz ( Incompessibility equies v, which gives v v vz dg 1 dg + + = f( z + f( z = z dz dz [ POINTS] 3. Hence, find an expession fo the velocity gadient in tems of gz ( and (use cylindical pola coodinates

3 1 dg v= e + gez dz 1 dg 1 dg 1 dg dg v= e e eθ eθ + e e + e e dz dz z dz dz z z z [ POINTS] 3.3 Wite down two bounday conditions fo gz ( at z = ± h The conditions v = vz give g( ± h = U dg dz [ POINTS] 3.4 Show that fo this poblem the pinciple of vitual wok educes to hd: dv dz πd dv : dv σ: dd dz πd = hd: dd dz πd ( hd: dl dw dz πd hd: dldz πd [ POINTS] 3.5 Hence, show that R 0 h 3 dg ddg d g d dg d π + dz dz dz 8 dz dz h

4 hd: dd dz πd L 0 0 d L 0 0 h 0 L L / : θθ z 0 dlθθ dl / z dz πd 0 Lz / Lzz 0 dlz / dlzz R h 1 dg 1 ddg 1 dg 1 ddg 1 dg 1 ddg dgddg h d dz dz dz dz 4 z dz z dz dz dz π R h 3 dg ddg 1 1 dg 1 ddg h + πd dz dz z dz z dz [ POINTS] 3.6 Deduce that and hence detemine g(z. 4 d g R d g 4 4 dz dz Integate the fist tem in the integal by pats once, the second tem by pats twice, and use the condition dd g that d g = dz h 4 d 3 dg 1 d g 8 4 dg πd dz dz dz h h 4 4 R d 3 dg 1 R d g dgdz dz dz dz h 4 3 d g R d g dz dz π This equation can be solved to give ( 6 hr / 6 hr / ( 6z/ R 6 zr / 6Uz e + e + RU e e gz ( = 6 hr / 6 hr / 6 hr / 6 hr / 6he ( + e + Re ( e [3 POINTS] 3.7 Show that fo R>>h the solution can be appoximated futhe by 3 3 gz ( = Uz ( / h 3 z/ h/ The Taylo expansion of g(z gives the equied esult. [ POINTS]

5 3.8 Fo the appoximation in 3.6, calculate the pessue distibution in the fluid, and deduce an expession fo the foce equied to squeeze the plates togethe (assume the integal of the pessue though the film thickness at =R vanishes. You can assume that the Stokes equation fo a cylindically symmetic flow is p 1 v v v + η + z p 1 vz v + η z + z z These equations can be integated to get 3Uh z 1 ( R 3 Uh ( R p = h 3 h h h h [ POINTS] The esultant foce is ( 3R 8h RU 3 RU h 8h π h π h [1 POINT] 4. The potential ω cosh( k( y3 + h φ = A sin( ky1 ωt k sinh( kh descibes small amplitude suface waves in an ideal fluid with depth h. 4.1 Calculate the velocity distibution in the fluid cosh( k( y3 + h v1 = Aω cos( ky1 ωt sinh( kh sinh( k( y3 + h v3 = Aω sin( kx1 ωt sinh( kh A y 3 y1 [ POINTS] 4. Show that the velocity at the suface of the fluid is consistent with a suface wave with a vetical displacement w = Acos( kx1 wt. What ae the tajectoies of mateial paticles in the fluid?

6 w On y 3 we have v = Awsin( ky1 wt = t Mateial paticles descibe cicula obits [ POINTS] 4.3 Find an expession fo the pessue in the fluid (neglect the tem of ode a φ ω cosh( k( y3 + h p + ρ + ρgy3 p = ρ gy3 a cos( ky1 ωt t k sinh( kh [ POINTS] 4.4 Fo small amplitude waves the bounday condition on the fluid suface can be expessed as p p( y3 + Acos( ky1 ωt 0 y 3 x 3 Find a condition elating k and ω (the dispesion elation necessay to satisfy this condition (neglect the tem of ode A Substituting into the bounday condition we get ω Acos( ky1 ωt g ω = gk tanh( kh k tanh( kh [ POINTS] Wate waves ae a ich souce of math poblems: This pape is an example 5. By consideing a flow potential of the fom Ω= Asinωt sin ky1 calculate the natual fequencies of a 1-D ai column with sound speed c s, fo the following bounday conditions: (i The end at y 1 is open (constant pessue; while the end at y 1 = L is closed (zeo velocity. (ii Both ends ae open. The govening equation is The bounday conditions give Ω v1 = y closed end Ω Ω c 0 0 s = ω k c s = t y 1 1 Ω p v o 1 Ω p= = cs t t y 1 y1 y1 open end

7 Fo the closed end the second bounday condition is satisfied by the chosen fom fo the solution; the fist ( π + n is satisfied by choosing cos kl k = n,1,... L ( π + n The fequencies ae thus ωn = c s n,1,... L Similaly if both ends ae open then nπ ω n = c s n = 1,... L [ POINTS] (iii Estimate the length of a tansvese flute whose lowest fequency is 46.9 Hz (both ends of a flute ae open. Sound speed in ai is 343. m/s π c π f = c s s L = 70cm L f [1 POINT]

Do not turn over until you are told to do so by the Invigilator.

Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.