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Zorlanmış köşe akışlarının stokes rejiminde teorik ve deneysel incelenmesi

Theoretical and experimantal analysis of forced corner flows for the stokes regime

  1. Tez No: 39273
  2. Yazar: EMİN FUAD KENT
  3. Danışmanlar: PROF.DR. AHMET R. BÜYÜKTÜR
  4. Tez Türü: Doktora
  5. Konular: Makine Mühendisliği, Mechanical Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1994
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 136

Özet

ÖZET Bu tez çalışmasında zorlanmış köşe akışlarının Stokes rejiminde teorik ve deneysel incelemesi yapılmıştır. Bu amaçla V-şeklinde, üstten silindirik bir yüzey ile kapatılmış bir kanal seçilmiştir. Köşedeki akışı zorlayan mekanizma kanal içinde dönen bir silindirdir. Bu yapı için vr-o biharmonik hareket denkleminin kutupsal koordinatlarda çözümü yapılmıştır. Vortisite, basınç, hızın radyal ve teğetsel bileşenlerinin ifadeleri ve silindire gelen moment ifadesi çıkarılmıştır. Silindir üzerindeki sınır şartları kutupsal koordinatların kullanılmasından dolayı tam olarak sağlanmış, kanal sınırlarındaki sınır şartları ise, kanal sınırlarının bu sistemde birer koordinat yüzeyi olmamalarından ötürü minimizasyon işlemi ile sağlanmıştır. Çeşitli kanal uzunlukları için akım fonksiyonu, basınç ve vortisite grafikleri sunulmuştur. Elde edilen akım fonksiyonu grafiği aynı özelliklere sahip kanalda yapılan deney sonucu çekilen fotoğraf ile karşılaştırıldığında tam olarak çakıştığı, ayrılma çizgisinin yeri ve gözenin merkezinin mükemmel bir uyum içinde olduğu görülmüştür. Kanal içindeki akışın görünür hale getirilmesi magnezyum parçacıkları ilavesiyle sağlanmış ve uzun poz süreli çekimler sonucu, oluşan gözeli yapılar görüntülenmiştir. Bir gözeden diğer gözeye hızların çok hızlı düşmesi nedeniyle ikinci gözeyi görebilmek için çok uzun poz süreleri (yaklaşık 2.5 saat) vermek gerekmiştir. Bu uzun poz süreli çekimler aralıklı çekim yapılarak elde edilmiştir. Bu yolla gözeler içindeki hızların bağıl olarak düşmesi gözlenmiştir. Uzun poz süreli çekimler sonucu esas akımın ardından birinci ve ikinci gözeyi görebilmek mümkün olmuştur. Ayrıca bir torkmetre ile silindire gelen moment ölçülmüştür. Birbirleri ile 2a açısı yapan iki düzlemin kesişmesiyle oluşan keskin köşede hareket kaynağından bağımsız olarak akımlar incelenmiştir. Kanal yüzeyi üzerindeki sınır şartlarını, minimizasyon işlemi yerine orijini köşe noktasında seçerek kanal sınırlarını birer koordinat yüzeyi haline getirmekle tam olarak sağlamak mümkün olmuş ve üstten açık bir kanal için hareket kaynağından bağımsız durum incelenmiştir. Bu durumda hareket kaynağından bağımsız olarak köşe noktasına doğru birbiri ardına gözeler elde edilmiştir. Ayrıca hareketin kaynağı olan silindiri de içeren nümerik bir metod geliştirilerek gözeli yapıların grafiği verilmiştir. xiı

Özet (Çeviri)

SUMMARY THEORETICAL AND EXPERIMENTAL ANALYSIS OF FORCED CORNER FLOWS FOR THE STOKES REGIME In this work, the theoretical and experimental analysis of forced corner flows for the Stokes regime is made. The subject of Stokes flow, defined to be that in which the inertia forces are negligible, has its foundations in the last century. Mathematically it is attractive since the simplification obtained by dropping the non- linear inertia terms makes the resulting equations much more tractable. The literature shows that during the last decade numerous works on the Stokes flows have been devoted to separation phenomena and to the viscous cells which generally result. This cellular motion interests scientists not only from a fundamental point of view, but also because it is encountered in numerous applications. There are a number of areas in the industrial environment and in natural systems where low-Reynolds-number flows are important. The industrial applications include the effect of surface roughness on lubrication, the design of polymer dies for favorable extension rates and the development of peristaltic pumps for sensitive viscous materials. In natural systems, low-Reynolds-number are important in biomedical applications and studies of animal locomotion. In Chapter 1, a review of the previous studies in this field is presented and basic contents of the thesis are also outlined. In Chapter 2, the formulation of the problem is given. In this study a special wedge-shape channel bounded by a cylindrical surface is chosen. The horizontal cross-section of this channel is given in Fig. 2.1. The purpose of this choice is the geometry of the channel which has only one singular point at the vertex of the wedge. A vertical circular cylinder of radius Rx is placed at the center of the cylindrical part the channel. This channel is filled with a highly viscous fluid of constant physical properties; its kinematic viscosity is denoted by y. The motion is XlllIn Chapter 4, the flow of a viscous fluid near a sharp corner between two fixed rigid walls intersecting at an angle 2a is considered. It is possible to induce a flow in the corner region so formed by stirring the fluid at a distance from the corner. In the region close to the corner the flow will be slow enough to be described by the Stokes equations. It might be anticipated that, sufficiently near the corner, the flow pattern may be independent of the stirring forces far from the corner that agitate the fluid. In terms of the stream function nr.e) in the plane polar coordinate, it is simply the biharmonic equation, Eqn.(2). By solving this equation, we obtain streamline patterns of the flow for different corner angles 2a and the first, second and the third families of the streamlines for corner angle 2a = 30*, Some results are given and compared with the visualization photographs concerning the angle at which the separating streamline leaves the rigid boundaries which form the corner. Finally, in the last section the obtained results are compared with each other and possible extensions are proposed. XVlllP(r,8) = 2b1r ~l sinS- SdjrsinS -2b ^'r _1 cos0-*-8d1'rcos0 + Y, [(-4n+4)b“r ”n - (4n+ 4)d“rn]sin(a9) (1-2 - £ [(-4n+4)b”'r“n -(4rt + 4)dn'rn]cos(nÖ) (13) B-2 for the pressure equation. As opposed to the no-slip condition on the cylinder, which is satisfied exactly by the polar coordinate series, no-slip condition on the channel walls can not foe satisfied exactly by these series because these walls are not coordinate surfaces of the frame that we have used to define the stream function. Thus, this latter condition has been satisfied optimally by using a numerical method. For this purpose, the quadratic minimization method to resolve the satisfaction of the boundary conditions has been used. This method consists of minimizing the quadratic difference between the imposed velocity on the boundary and the velocity deduced from the series, In this way, the rest of the coefficients of the series have been found. Then the graphic of the streamline pattern, the equivorticity pattern and pressure curves are given. The sketch of the apparatus constructed to visualize the flow is shown in Fig. 3.1. The channel is filled with a highly viscous silicon Rhodorsil oil. The visualization is carried out using very small cuttings of magnesium of about 40 iim in length and 4 urn in thickness. They are illuminated by a horizontal thin sheet of light coming from a laser device. Given that the velocities decrease exponentially along the channel axis, very long times of exposure are required to visualize the flow in the whole domain, particularly for observing the small particle displacements. Intermittent lighting is used and two oppositely rotating successive cells which occur beyond the main current driven by the cylinder have been visualized experimentally. The photographs obtained in these experiments are given in Chapter 3. An excellent agreement is found between the visualization photograph in Fig. 3, 6 and the corresponding result in Fig. 2. 10. xvnIn Chapter 4, the flow of a viscous fluid near a sharp corner between two fixed rigid walls intersecting at an angle 2a is considered. It is possible to induce a flow in the corner region so formed by stirring the fluid at a distance from the corner. In the region close to the corner the flow will be slow enough to be described by the Stokes equations. It might be anticipated that, sufficiently near the corner, the flow pattern may be independent of the stirring forces far from the corner that agitate the fluid. In terms of the stream function nr.e) in the plane polar coordinate, it is simply the biharmonic equation, Eqn.(2). By solving this equation, we obtain streamline patterns of the flow for different corner angles 2a and the first, second and the third families of the streamlines for corner angle 2a = 30*, Some results are given and compared with the visualization photographs concerning the angle at which the separating streamline leaves the rigid boundaries which form the corner. Finally, in the last section the obtained results are compared with each other and possible extensions are proposed. XVlllP(r,8) = 2b1r ~l sinS- SdjrsinS -2b ^'r _1 cos0-*-8d1'rcos0 + Y, [(-4n+4)b”r“n - (4n+ 4)d”rn]sin(a9) (1-2 - £ [(-4n+4)b“'r ”n -(4rt + 4)dn'rn]cos(nÖ) (13) B-2 for the pressure equation. As opposed to the no-slip condition on the cylinder, which is satisfied exactly by the polar coordinate series, no-slip condition on the channel walls can not foe satisfied exactly by these series because these walls are not coordinate surfaces of the frame that we have used to define the stream function. Thus, this latter condition has been satisfied optimally by using a numerical method. For this purpose, the quadratic minimization method to resolve the satisfaction of the boundary conditions has been used. This method consists of minimizing the quadratic difference between the imposed velocity on the boundary and the velocity deduced from the series, In this way, the rest of the coefficients of the series have been found. Then the graphic of the streamline pattern, the equivorticity pattern and pressure curves are given. The sketch of the apparatus constructed to visualize the flow is shown in Fig. 3.1. The channel is filled with a highly viscous silicon Rhodorsil oil. The visualization is carried out using very small cuttings of magnesium of about 40 iim in length and 4 urn in thickness. They are illuminated by a horizontal thin sheet of light coming from a laser device. Given that the velocities decrease exponentially along the channel axis, very long times of exposure are required to visualize the flow in the whole domain, particularly for observing the small particle displacements. Intermittent lighting is used and two oppositely rotating successive cells which occur beyond the main current driven by the cylinder have been visualized experimentally. The photographs obtained in these experiments are given in Chapter 3. An excellent agreement is found between the visualization photograph in Fig. 3, 6 and the corresponding result in Fig. 2. 10. xvnIn Chapter 4, the flow of a viscous fluid near a sharp corner between two fixed rigid walls intersecting at an angle 2a is considered. It is possible to induce a flow in the corner region so formed by stirring the fluid at a distance from the corner. In the region close to the corner the flow will be slow enough to be described by the Stokes equations. It might be anticipated that, sufficiently near the corner, the flow pattern may be independent of the stirring forces far from the corner that agitate the fluid. In terms of the stream function nr.e) in the plane polar coordinate, it is simply the biharmonic equation, Eqn.(2). By solving this equation, we obtain streamline patterns of the flow for different corner angles 2a and the first, second and the third families of the streamlines for corner angle 2a = 30*, Some results are given and compared with the visualization photographs concerning the angle at which the separating streamline leaves the rigid boundaries which form the corner. Finally, in the last section the obtained results are compared with each other and possible extensions are proposed. XVlll

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