Two-dimensional Stokes flow in a semicircle

Two-dimensional Stokes flow in a semicircle

Постpоено точное pешение задачи о двумеpном течении Стокса в полукpуге, вызванное pавномеpным движением кpуговой или пpямолинейной гpаницы. Пpиведены контуpные линии тока и типичное pаспpеделение скоpости....

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Date:1999
Main Authors: Meleshko, V.V., Gomilko, A.M.
Format: Article
Language:English
Published: Інститут гідромеханіки НАН України 1999
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/5078
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Cite this:Two-dimensional Stokes flow in a semicircle / V.V. Meleshko, A.M. Gomilko // Прикладна гідромеханіка. — 1999. — Т. 1, № 1. — С. 35-37. — Бібліогр.: 10 назв. — англ.

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spelling irk-123456789-50782010-01-11T12:01:01Z Two-dimensional Stokes flow in a semicircle Meleshko, V.V. Gomilko, A.M. Постpоено точное pешение задачи о двумеpном течении Стокса в полукpуге, вызванное pавномеpным движением кpуговой или пpямолинейной гpаницы. Пpиведены контуpные линии тока и типичное pаспpеделение скоpости. Побудовано точний pозв'язок задачi пpо двовимipну течiю Стокса у напiвкpузi, яка зумовлена piвномipним pухом кpугової або пpямолiнiйної гpаницi. Наведенi контуpнi лiнiї тока та типовий pозподiл швидкостi. The exact analytical solution for the two-dimensional Stokes flow in a semicircle due to uniformly moving circular or straight boundary is obtained. The contour streamline pattern and a typical velocity distribution are shown. 1999 Article Two-dimensional Stokes flow in a semicircle / V.V. Meleshko, A.M. Gomilko // Прикладна гідромеханіка. — 1999. — Т. 1, № 1. — С. 35-37. — Бібліогр.: 10 назв. — англ. 1561-9087 http://dspace.nbuv.gov.ua/handle/123456789/5078 532.5 en Інститут гідромеханіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description Постpоено точное pешение задачи о двумеpном течении Стокса в полукpуге, вызванное pавномеpным движением кpуговой или пpямолинейной гpаницы. Пpиведены контуpные линии тока и типичное pаспpеделение скоpости.
format Article
author Meleshko, V.V.
Gomilko, A.M.
spellingShingle Meleshko, V.V.
Gomilko, A.M.
Two-dimensional Stokes flow in a semicircle
author_facet Meleshko, V.V.
Gomilko, A.M.
author_sort Meleshko, V.V.
title Two-dimensional Stokes flow in a semicircle
title_short Two-dimensional Stokes flow in a semicircle
title_full Two-dimensional Stokes flow in a semicircle
title_fullStr Two-dimensional Stokes flow in a semicircle
title_full_unstemmed Two-dimensional Stokes flow in a semicircle
title_sort two-dimensional stokes flow in a semicircle
publisher Інститут гідромеханіки НАН України
publishDate 1999
url http://dspace.nbuv.gov.ua/handle/123456789/5078
citation_txt Two-dimensional Stokes flow in a semicircle / V.V. Meleshko, A.M. Gomilko // Прикладна гідромеханіка. — 1999. — Т. 1, № 1. — С. 35-37. — Бібліогр.: 10 назв. — англ.
work_keys_str_mv AT meleshkovv twodimensionalstokesflowinasemicircle
AT gomilkoam twodimensionalstokesflowinasemicircle
first_indexed 2025-07-02T08:15:46Z
last_indexed 2025-07-02T08:15:46Z
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fulltext �ਪ« ¤­  £÷¤à®¬¥å ­÷ª . 1999. �®¬ 1(73), N 1. �. 35 { 37��� 532.5TWO-DIMENSIONAL STOKES FLOW IN A SEMICIRCLEV. V. MELESKO, A. M. GOMILK�Institute of Hydromechanics National Academy of Sciences, Kiev, UkraineReceived 11.11.98�®áâp®¥­® â®ç­®¥ p¥è¥­¨¥ § ¤ ç¨ ® ¤¢ã¬¥p­®¬ â¥ç¥­¨¨ �â®ªá  ¢ ¯®«ãªp㣥, ¢ë§¢ ­­®¥ p ¢­®¬¥p­ë¬ ¤¢¨¦¥­¨¥¬ªp㣮¢®© ¨«¨ ¯pאַ«¨­¥©­®© £p ­¨æë. �p¨¢¥¤¥­ë ª®­âãp­ë¥ «¨­¨¨ ⮪  ¨ ⨯¨ç­®¥ p á¯p¥¤¥«¥­¨¥ ᪮p®áâ¨.�®¡ã¤®¢ ­® â®ç­¨© p®§¢'燐ª § ¤ ç÷ ¯p® ¤¢®¢¨¬÷p­ã â¥ç÷î �â®ªá  ã ­ ¯÷¢ªpã§÷, 猪 §ã¬®¢«¥­  p÷¢­®¬÷p­¨¬ pã宬ªp㣮¢®ù  ¡® ¯pאַ«÷­÷©­®ù £p ­¨æ÷. � ¢¥¤¥­÷ ª®­âãp­÷ «÷­÷ù ⮪  â  â¨¯®¢¨© p®§¯®¤÷« 袨¤ª®áâ÷.The exact analytical solution for the two-dimensional Stokes ow in a semicircle due to uniformly moving circular orstraight boundary is obtained. The contour streamline pattern and a typical velocity distribution are shown.INTRODUCTIONTwo-dimensional uid motions in which inertiaforces are negligible compared to viscous forces(creeping or Stokes ows) have been widely studied.The linear form of the governing biharmonic equa-tion yield, in many cases, closed-form solutions for ows in some canonical domains. On the other hand,there exists only an approximate solution [5, 6] for aproblem of motion in a semicircle induced by a uni-form tangential velocity at a circular boundary. Inthe present note we provide the exact analytical so-lution for this problem.The method of solution is based on a usage of thebipolar coordinate system. This system for the two-dimensional biharmonic equation was employed earlyfor exact solutions of problems for both Stokes ow[3, 4, 8] and elastic stresses [2, 7] in domains enclosedby eccentric cylinders.1. STATEMENT OF THE PROBLEMConsider Stokes ow inside a semicircle 0 � r � a,0 � � � � generated by either a constant tangentialvelocity V applied at the curved part of the boundary,or a constant tangential velocity U applied at theplane part of the boundary. The rest of the boundaryremains unmovable.The velocity �eld is given in terms of the streamfunction (r; �) byur = 1r @ @� ; u� = �@ @r ; (1)where satis�es the biharmonic equation�� = 0; (2)with � = @2@r2 + 1r @@r + 1r2 @2@�2 is the Laplace operatorin the polar coordinates (r; �). The boundary conditions for equation (2) are either = 0; @ @� = 0 at � = 0; �; 0 � r � a; = 0; @ @r = �V at r = a; 0 � � � �: (3)or = 0; @ @� = U r at � = 0; 0 � r � a; = 0; @ @� = �U r at � = �; 0 � r � a; = 0; @ @r = 0 at r = a; 0 � � � �: (4)2. METHOD OF SOLUTIONThe biharmonic equation (2) for both boundaryconditions (3) and (4) admits exact solution whichcan be obtained by means of bipolar coordinates. Thebipolar coordinate system (�; �) is introduced accord-ing the relationx+ iy = a i ctg � + i�2 ; (5)so that the two poles of the coordinates are locatedon the x-axes at the points (� a; 0). Thenx = r cos � = Jsh �; y = r sin � = J sin �; (6)with J = a=(ch � � cos �), and the semicircle 0 � r �a, 0 � � � � in polar coordinates transforms intothe strip �1 � � � 1, �=2 � � � � in the bipolarcoordinates.The biharmonic equation (2) in the bipolar coor-dinates can be rewritten as@4 @�4 +2 @4 @�2 @ �2+@4 @�4 +2 @2 @�2 �2 @2 @�2 + = 0 (7)c �. �. �¥«¥èª®, �. �. �®¬¨«ª®, 1999 35 �ਪ« ¤­  £÷¤à®¬¥å ­÷ª . 1999. �®¬ 1(73), N 1. �. 35 { 37for the auxiliary function = =J .By means of equality@ @� = 1J @ @� + @(1=J)@� = @ @n� + sin �a ;(where n� denotes the outer normal to the line � =const) we can reformulate the boundary conditions(3) in terms of as = 0; @ @� = 0 at � = 12�; j � j � 1; = 0; @ @� = V at � = �; j � j � 1: (8)By choosing a solution of equation (7) = A sin � +B cos � + C� sin � +D � cos � ; (9)we can satisfy all boundary conditions (8), providedthat the values of the constants A;B;C;D are:A = 2��2 � 4 V; B = � 2�2�2 � 4 V;C = � 4�2 � 4 V; D = 2��2 � 4 V: (10)Returning from (�) in (9) to the stream function (r; �) by means of equalitiesJ sin � = r sin �; J cos � = r2 � a22a ;� = � � arctg 2ar sin �a2 � r2 ;after some reductions, we come to the expression = 2V�2 � 4h�(a2 � r2) + 4ar sin �2a � (11)�arctg 2ar sin �a2 � r2 � �r sin �i:The behaviour of the stream function near thecorner point rc = a; �c = 0 can be obtained fromexpansion into Taylor series the expression (12) inthe local polar coordinates (�; �) with r cos � = a �� sin�; r sin � = � cos �. The �rst linear on � term is: loc = 4V �4� �2 �� cos �+ 12�� sin�� �24 sin�� ;which corresponds to the Goodier [1] - Taylor [10]solution for a quarter plane � � 0; 0 � � � �=2 withthe constant tangential velocity �V applied at theplane � = 0.The exact solution of (2) for the boundary condi-tions (4) can be derived in a similar way, = � 2U�2 � 4h�ar sin � + a2 � r2a � (12)�arctg 2ar sin �a2 � r2 � 12�2r sin �i: Fig. 1. Streamline pattern of the Stokes ow in a semi-circle according to exact solution (12). The streamlinesare plotted for the contour lavels n = 0:015V a n withn = 1; :::;10. The maximum value of the stream functionat the stagnation point is 0:1713V a. Fig. 2. Streamline pattern of the Stokes ow in a semicir-cle according to approximate solution (14). The stream-lines are plotted for the contour lavels ~ n = 0:015V a nwith n = 1; :::;10.3. RESULTS AND DISCUSSIONFig. 1 shows a streamline pattern for the ow de-scribed by equation (12). The ow forms a singlevortex cell which has one stagnation elliptic pointre = 0:636a; �e = �=2.It was reported [5, 6] that the approximate solutionof the problem (2), (3) is~ = 4V a3 � ra�2 h1� � ra� i sin2 �; (13)with = (11=3) 12 � 1 = 0:915. It is worth notingthat this solution does not satisfy both the governingbiharmonic equation (2) for the stream function andthe non-slip conditions (3) at the moving boundary.However, the streamline pattern presented in Fig. 2(see also Fig. 8.2.3 in [9]) for the approximate solution(14) has qualitatively similar appearence with Fig. 1.Fig. 3 represents a distribution of the tangentialvelocity u� along the middle line � = �=2 of the cav-ity. Again, we see a reasonable quantitative corre-36 �. �. �¥«¥èª®, �. �. �®¬¨«ª® �ਪ« ¤­  £÷¤à®¬¥å ­÷ª . 1999. �®¬ 1(73), N 1. �. 35 { 37 Fig. 3. Cross-sectional distribution of the tangential ve-locity u� along the line � = �=2 for exact solution (12),solid line, and approximate solution (14), dashed line.spondence between exact and approximate solutionsin the centre of the cavity. One should note theessential di�erence between velocities at the curvedboundary: the approximate solution provides the val-ue 1:333V instead of the prescribed value V . Besides,the approximate solution underestimates signi�cantlythe velocity near the corner points: at the boundaryr = a the tangential velocity u� varies as 43 V sin2 �instead of being constant. These discrepancies mayhave a crucial importance for a quantitative study ofmixing process in such a cavity.4. CONCLUSIONThus, usage of the bipolar coordinates providesexact analytical solution for Stokes ow in a semi-circle induced by uniform tangential velocities atthe boundary. Such solution seems to be impor-tant for accurate analysis of mixing process in thepartitioned-pipe mixer [5, 6, 9].For a nonuniform velocity distribution along theboundaries the exact solution can be constructed ina form of some integrals. In particular, the case whennonzero constant tangential velocity is applied at theboundary r = a with �0 � � � � � �0 is particularlyinteresting because it can provide both an analyticalexpression of the amplitudes of the Mo�att eddies in a �nite cavity and a clear picture of the competitionbetween local and far-�eld e�ects for dominance inthe neighbourhood of the corner points.[1. Goodier J. N. An analogy between the slow motionof a viscous uid in two dimensions, and systemsof plane stress // Phil. Mag. (Ser. 7).{ 1934.{ 17.{P. 554{564.2. Je�ery G. B. Plane stress and plane strain in bipolarco-ordinates // Phil. Trans. R. Soc. London.{ 1920.{ser A 221.{ P. 265{293.3. �㪮¢áª¨© �. �. � ¤¢¨¦¥­¨¨ ¢ï§ª®© ¦¨¤ª®áâ¨,§ ª«î祭­®© ¬¥¦¤ã ¤¢ã¬ï ¢p é î騬¨áï íªá-業âp¨ç­ë¬¨ 樫¨­¤p¨ç¥áª¨¬¨ ¯®¢¥på­®áâﬨ //�®®¡é. � p쪮¢-£® ¬ â¥¬. ®¡é-¢  ¯à¨ � à쪮¢.�­-â¥.{ 1887.{ �ë¯ 1.{ �. 31{46. � ª¦¥: �ã-ª®¢áª¨© �. �. �®«­®¥ ᮡp ­¨¥ á®ç¨­¥­¨©, ⮬IV, 250{278, �.-�.: ����, 1937. � ª¦¥: �㪮¢-᪨© �. �. �®¡p ­¨¥ á®ç¨­¥­¨©, ⮬ III, 121{132,�.-�.: �����, 1949.4. �㪮¢áª¨© �. �., � ¯«ë£¨­ �. �. � âp¥­¨¨ ᬠ-§®ç­®£® á«®ï ¬¥¦¤ã 訯®¬ ¨ ¯®¤è¨¯­¨ª®¬ // Tpã-¤ë ®â¤. 䨧. ­ ãª �¡é-¢  «î¡¨â. ¥áâ¥á⢮§­ ­¨ï.{1906.{ 13, No 1.{ �. 24{33. � ª¦¥: �㪮¢-᪨© �. �. �®«­®¥ ᮡp ­¨¥ á®ç¨­¥­¨©, ⮬ IV,279{319, �.-�.: ����, 1937. � ª¦¥: �㪮¢-᪨© �. �. �®¡p ­¨¥ á®ç¨­¥­¨©, ⮬ III, 133{151,�.-�.: �����, 1949. � ª¦¥: � ¯«ë£¨­ �. �. �®«-­®¥ ᮡp ­¨¥ á®ç¨­¥­¨©, ⮬ II, 91{106, �.: ������, 1933. � ª¦¥: � ¯«ë£¨­ �. �. �®¡p ­¨¥ á®-稭¥­¨©, ⮬ III, 7{26, �.-�.: �����, 1950.5. Khakhar D. V., Franjione J. G., Ottino J. M. A casestudy of chaotic mixing in deterministic ows, thepartitioned-pipe mixer // Chem. Engng. Sci.{ 1987.{42.{ P. 2909{2919.6. Kusch H. A., Ottino J. M. Experiments on mixing incontinuous chaotic ows // J. Fluid Mech.{ 1992.{236.{ P. 319{357.7. M�uller W. Ebene Spannungs- und Str�omungsfeldermit zwei kreiszylindrischen Grenzen // Ing.Archiv.{1942.{ 13.{ S. 37{58.8. M�uller, W. Beitrag zur Theorie der langsamenDrehung zweier exzentrischer Kreiszylinder in derz�ahen Fl�ussigkeit // Z. angew. Math. Mech.{ 1942.{22.{ S. 177{189.9. Ottino J. M. The Kinematics of Mixing: Stretching,Chaos and Transport.{ Cambridge: Cambridge Uni-versity Press, 1989.{ 364 p.10. Taylor G. I. On scraping viscous uid from a planesurface // Miszellangen der Angewandten Mecha-nik (Festschrift Walter Tollmien).{ Berlin, Akademie-Verlag, 1962.{ P. 313{315. �. �. �¥«¥èª®, �. �. �®¬¨«ª® 37

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