ma3102soln2-11

ma3102soln2-11 - MATH3102 — Assignment 2, 2011 Asterisked...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH3102 — Assignment 2, 2011 Asterisked questions form this assignment. Total marks: 25 Due Date: Hand in during tutorial on 8 September, 07‘ deposit in the MATH3102 assignment collection box on Level 4 of Building 67 before that date 1. The velocity field in a 2—dimensional unsteady flow is given by q(a:, y, t) = yeti + 2xe'tj. Show that (a) the streamline at t = O which passes through the point (1, 2) is (one sheet of) the hyperbola y2 — 232:2 = 2. (b) the pathline for the fluid particle which is at the point (1, 2) at t = 0 is the parabola y2 = 4m. ' Sketch the two curves, indicating direction of motion on each, and note that they have that same slope. where they meet at (1, 2). 2. Show that (I) = K (x — t) (y —t) represents the velocity potential of an incompressible two-dimensional flow. Show that the streamlines at time t are the curves (as — t)2 — (y — t)2 = constant, and that the paths of the fluid particles have the equations . K -1nlw-yl=3{(x+y)— m y} + B, where A, B are consts. 3. Compressible inviscid fluid flows along a very narrow tube of cross—sectional area {3(5) where s is the distance measured along the tube. Show that the equation; of continuity is ‘ 8p (9 . _-_ __ = 0 [3’ at + a8(pflv) . Where p(s, t) and 11(5, t) are the density and the speed of the fluid. Show that under the steady conditions the density at any. section is inversely pro- portional to the volume flux at the section. ' ‘ 4. * (6 marks.) A two—dimensional point source of homogeneous, incompressible, irro— tational fluid is located at the point (a, b) with a, 6 positive constants. The source has a constant strength m, and the fluid density is p. There is a solid wall along y = 0. Use the method of images to find the velocity potential and hence velocity field q = ui + 'uj in the region y 2 0. Verify that o = 0 on the wall and show that it is the only stagnation point. The pressure P at infinity is Poo; find the maximal AND minimal values of P on the wall. 5. A point source of homogeneous, incompressible, irrotational fluid is situated at the point with Cartesian coordinates (a, b, c), where a, b and c are positive. A solid wall occupies the region :1: g 0. The source has a constant strength m, the fluid has density p, and there are no external forces acting. (a) Use the method of image to find the velocity potential and hence the velocity field q = ui + vj + wk in the region :r 2 O. (b) Show that the pressure on the wall is a minimum on the circle (y—b)2+(z—c)2 = éa2, and that it has there the value p m2 54m2 a4 where PS is the pressure on the wall at the stagnation point. 6. * (7 marks.) P=P5— (a) Find the velocity potential (say gbl) of a uniform velocity field q = U i (U const) such that q = ngl. - (b) Consider a2D doublet of strength ,u at O with doublet axis along 03:, obtained from the limiting configuration of a source—sink pair as described in lectures. Given that the velocity potential of this 2D doublet is (fig = —/.b cos 6/27rr. Show that qb = ¢i+¢2 is the velocity potential for “uniform flow past a cylinder” with axis along Oz and radius 0., provided [.11 = —27rUa2. (c) Find the pressure distribution on the cylinder. 7. Two infinitely long concentric circular cylinders have their common axis along the z— axis. The inner and outer cylinders have radii r0 and r1 respectively. A non—viscous, incompressible fluid flows between the cylinders with velocity field q = q(7‘, (9)89, where (r, 6, z) are cylindrical polar coordinates. Show that if the flow is irrotational, then q = %, where a is constant. Find a corresponding velocity potential (I) and stream function W, noting that q /= curl(l\I/k). Working from Euler’s equation of motion, show that if there are no external forces acting, and the fluid is homogeneous, then the fluid. pressure at the outer cylinder is greater than that at the inner cylinder by an amount 1 2 1 1 —a ———. 2p 7% if (a) Show that the result div(D) =1 p implies f f S n - D d8 is the total charge within the closed surface S, where D is the electric displacement and p the charge density. 8. * (6 marks.) (b) The density of electric charge in a spherical region of radius 1) is given by p = b — 7" where 7" is the distance from the center of the sphere. There is no charge outside the sphere. Find the electric field E and the electric potential gb at points inside and outside the sphere. 9. (a) Show that the electric potential for a dipole in two dimensions is . _ pcos0 9b 27mm" Show the equipotentials are circles. (962 + y2 = Am.) 2 (b) A conducting cylinder of infinite length and radius a is placed with its axis normal to a uniform electric field of strength E. Use the result in part (a) (or any other method) to find the potential at any point outside the cylinder. 10. (a) A point source emits Q units of heat at the origin 0 at t = O in the plane Ozzy. For circular symmetry the heat diffusion equation has the form 8T 82T 18T a—k<w+;w>’ t>0 The solution is known to have the form A 2 T(7",t) = — exp (—1—) . Find A by considering the total amount of heat in the plane at any instant t > 0. (b) A point source of pollutant in three dimensions constantly emits Q units of mass per unit time in a uniform fluid velocity field q = U k, where U is a const. Write the approximate form of the steady convection—diffusion equation that is appropriate for the downstream wake region. By referring to part (a) find the concentration C(r, z), where (r, 0, z) are cylindrical polar coordinates. ll. * (6 marks.) Find the steady concentration field for a point source of pollutant of strength Q at 0 in three dimensions in a uniform field q = U k. Hint: Write the exact convection—diffusion equation in terms of dimensionless length coordinates, for example Z = Uz/k. Write O = 62/ng and show 1 V2 _ Since spherically symmetric solutions (MR) are expected (where R = U r / k; is a di— mensionless spherical polar coordinate) write the above equation in the appropriate form with R as independent variable. Next let f = Rqfi and solve f. _' @Lh MATH’sm’L gBVL ’L. I ' “LAX. Ugt‘R/xi Mo:\-\Ao<\ of ?Mjo,s - ‘An’wd. H40; Qciguix'mvhmk PNfiAQ/M ‘ ' “YMO-A :— e (a ' '1 I l" (EC A, Sofia ULS 3, “yo §3 \ I. ’ G) I waif rm C21” “BACON «QSSAMQ ffvcokq’c‘ocf \ V 1 V\ J Cm . WNW fr z. .3 q k - u g» + 1.“ 1D ‘ ‘Afl ‘ 'JC'Q\ ¥ - 5 2:1 fl + "1 \ ‘1 " W: a<\> M ‘Ho - 23*“? “M Z ‘1 “T1 "' % Oh Wt WMK (\l ~31 ( .1 z0\ :0 $20 2c Vdo‘zkfl ‘aaolten‘k'7qk-fur ’“I’Ammiawss'i‘m %\ 'Kppg¥é\x:fg/\m¥ flux”; flows Us) :A ajxéngrtm mugs (a .e I) 3) «WW ,t ‘5sz Losg $9 Ck" 1. U f CQSB ' I __ MC.ch . £3 I 1‘7» “A :D ‘ V '1 use nLk'COSQ ‘ > _ ~ . ‘ ' , «Jar lib ‘Tx-Ws (Q, €0.23an5 ‘KI-Q‘Lodix {Q‘ng/‘Amak QQ( ‘1.“ fixflé :i’fGJC‘AAch‘OOSK A?!” n $«QIHSINQ—Sknk \‘.UA‘\CD(M ,C’Kw5_ 4&M: m C \anqwfl ‘ A \ : A - ‘ ‘ ‘ A ' CQOAG (05‘s 93’- CmeA‘ius a 9:93;“; gt 910K . I], _ , ) ./\ o ' '~ ' I ‘ t F «UN to we, RNUVL ts “0 POVA‘MK tOMGOWWN' ‘N w ' f ’ ac deicstfiij CVQQy 59C? (‘Tfox‘e _ W‘Lomgmw meiak. cksqqufi‘. 15% s ‘ ' T1 3. "NOT Ifmqmgkic. . \AJUA A: as 1 , v EDI“): W E gbkw fcéx Q/AnguxQ; b~r b. 'SZIIw/fngiricl {MM '0 ’ a<fi> ‘ S . ’A S 0 3r 1 V ..._C ED “W J: .33. I.) ‘ m? ‘ as 3 ~ H— .- sdwfl ~=4> q? ‘ Rf~1b§+ C, . ‘ his 7 rib C ZCMS‘: - ‘ ) ref f‘ >\D 1 3V\/\L C.\/\°\fjiz K‘AS’TQX‘P— TMQ SLUCQCQ S $3 Q13 H53 Ln—JIAH Jr g ffir‘B Lmr'LA H O . ' Sb- \3 I ' I . ‘ .' (FL [ \QLI‘ “‘1 SLV‘ova'UCRF Ar +Q ‘~ U—‘L4—~ o ,. I . I «1 ww._\_‘° .. {p-b +b wk "Dr ‘ V1.26? «K ' ' ‘ ’VL-io‘r ) ‘ ..__ ._ \oz' ‘ » I “‘9 C ~ Load 30- -' I p < \8 . q>LF\1 «1L5 Cfilb»+ (590 0 it: ‘5 ' I b4 ‘). . «mEOF ) V>b *2 n A .50 EZ‘VL ""¢Cr3§:’ K” h 1‘ A _ DEVEB 9—— . D ' '\D\~ A. - h , {\ 1319.95“ "V 2 >3 L.Ov\lUQ;cEQ_~AQC£/RQ?OA sscwéj mm: cx/ :U h fig 1‘ " U]? : REE/C —. hag-L0“ Er- ‘ U "h": V Ut M Cofigw“? 0 I A—j Igg‘SCMfQS ) \A‘SYEDAKQQ $fmgnsfi'on\o_g$ 6.00633 '~. 2 '9”? I I > ' K Z Z ‘QAC'Q / 18.0 Ok'kahiq (ac “4 .- [‘L A $1 ~ Q/C "Z :. Laqflmc‘lén (a . &3M§,A$TQA\Q§S 16013.3 3 “TV‘NS LOAv-La-X; .7”<’:Q{MS ' N ” I. I . I 39 Q'me/Ck‘ SfK/LOJ:COV\\VX g/J‘NMQADHC. <§v~41><m ,, joxgygy M 4 0 ~ _ I alMQnngwA/QSS LQO‘FJ \— ‘ $930 . Ci \1 E§> i‘nté A« Q a\\oowz xii? H ., _ I I ' ,. I+ -_ Col 30V: 350 \ch \DOUMAQA QB Rgak ) "Cka . . ' I “R ‘7. I " 1 A 2’ A i. COASE x I ‘ _ es~-R>,/ CZQ,"AEZQ\Q, XII—{2:9 1 ._ UP 52/ - ’ in, qum 0 C ugymx («DIVA cg—fimkks “3)? m; 'vIFQ (AQ‘XGQIfMfi’AQ' /. “icky, \p‘aVQ_ ' ~— 2 r \ «o C AL V” mmgu sBfiS’sGo/o $3,620 / an'giAflzJ 5%{ngg' S "0C4 $M0.&'k $I3\’\O_JL“\QAO\4~L§ (<41 C¢43§$QA (RE D ...
View Full Document

This note was uploaded on 11/26/2011 for the course MATH 3102 taught by Professor Yeo during the Three '11 term at Queensland.

Page1 / 13

ma3102soln2-11 - MATH3102 — Assignment 2, 2011 Asterisked...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online