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(1)BEM simulation for glass parisons Citation for published version (APA): Wang, K. (2002). BEM simulation for glass parisons. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR554076. DOI: 10.6100/IR554076 Document status and date: Published: 01/01/2002 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne. Take down policy If you believe that this document breaches copyright please contact us at: openaccess@tue.nl providing details and we will investigate your claim.. Download date: 13. Sep. 2021.

(2) BEM simulation for glass parisons. PROEFSCHRIFT. ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van the Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 21 januari 2002 om 16.00 uur. door. Kaichun Wang geboren te Honghu, China.

(3) Dit proefschrift is goedgekeurd door de promotoren:. prof.dr. R.M.M. Mattheij en prof. C.S. Chen. Copromotor: dr. H.G. ter Morsche.

(4) Contents 1. Introduction 1.1 Formation of parisons . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .. 2. Mathematical modelling 2.1 Modelling the flow . . . . 2.2 Modelling the temperature 2.2.1 The pressing . . . 2.2.2 The dwell . . . . . 2.3 Choosing solution methods. 3. 4. 5 6 6. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 9 9 14 14 16 17. BIEs and their uniqueness 3.1 The BIE for the Laplace equation . . . . . 3.2 An instructive example . . . . . . . . . . 3.3 Some results from classic potential theory 3.4 Uniqueness of BIEs . . . . . . . . . . . . 3.4.1 Dirichlet boundary condition . . . 3.4.2 Mixed boundary condition . . . . 3.4.3 Robin boundary condition . . . . 3.4.4 Neumann boundary condition . . 3.5 The BIE for the Stokes equation . . . . . 3.6 Uniqueness of the Stokes equation . . . . 3.7 Axisymmetric potential problems . . . . . 3.8 Axisymmetric Stokes problems . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. 19 20 23 26 28 28 29 31 32 33 35 37 40. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. Computational aspects 43 4.1 Discretisation of BIEs . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Singular and nearly singular integrals . . . . . . . . . . . . . . . 47.

(5) CONTENTS. 4 4.2.1 Singular integrals . . . . . . . . . . 4.2.2 Nearly Singular Integrals . . . . . . Boundary conditions and corners . . . . . . Discretisation of the extra condition . . . . Adaptive discretisation . . . . . . . . . . . 4.5.1 Local and global errors . . . . . . . 4.5.2 Equidistribution . . . . . . . . . . . 4.5.3 Equidistributing the local error . . . 4.5.4 Equidistribution of the global error . 4.5.5 An algorithm for equidistribution . 4.5.6 An algorithm for error control . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 47 50 52 57 63 63 64 65 68 68 70. 5 Dual reciprocity method 5.1 Basic formulations . . . . . . . . . . . . . 5.2 Particular solutions . . . . . . . . . . . . . 5.2.1 Dual reciprocity method . . . . . . 5.2.2 Kansa’s method . . . . . . . . . . . 5.3 Computational schemes . . . . . . . . . . . 5.4 The DRM for the pseudo–Poisson equation 5.5 A more efficient method in linear case . . . 5.6 Axisymmetric DRM . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 75 76 77 77 79 80 81 84 87. . . . .. 93 93 94 98 100. . . . .. 105 105 107 110 112. 4.3 4.4 4.5. 6 Stokes flow during the pressing 6.1 Revisiting the problem . . 6.2 Steady Stokes flow . . . . 6.3 Time Integration . . . . . . 6.4 Numerical Results . . . . .. . . . .. . . . .. 7 Heat conduction during the dwell 7.1 Revisiting the problem . . . . 7.2 Discretisation . . . . . . . . . 7.3 A Model problem . . . . . . . 7.4 Numerical results . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. Bibliography. 117. Index. 123.

(6) Chapter 1 Introduction Glass is an interesting material, having multiple applications on the one hand and being available in unlimited quantities on the other hand. Indeed, it can be transparent, making it indispensable for applications such as window–panes, but at the same time, its constitutive properties are such that it appears to be flexible when used as fibre, making it a favourite material for data transport by light. The raw material is mainly silicon–dioxide, i.e. ordinary sand. Production of glass goes more or less along the following lines: First, grains and additives, such as soda, are heated in a tank. This can be a device several tens of metres long and a few metres high and wide (the width being larger than the height). Here gas burners  . or electric heaters provide the heat necessary to warm the material up to some The liquid glass comes out at one end and is, for example, led to a pressing or blowing machine, or ends up on a bed of liquid tin, where it spreads out to become float glass (for window–panes, wind–screens etc.). In the latter case the major problems are to ensure a smooth flow from the oven to the bed and to control the spreading and flattening. An essential process is then the cooling of the product. Hence the production involves a high energy cost factor, inducing a continuous search for more efficient techniques. For one part these are to be found in a better control of the combustion (or more general heating) process and the design of the oven. For another they can be formed by a better control of the end product. Indeed, better monitoring of the cooling, for example, may reduce residual stresses in the material to allow the production of thinner glass (thus reducing the material costs), and another example is the actual morphology phase, where a piece of hot glass is formed into the desired shape; in fact, this may be even more important for obtaining thinner glass products..

(7) 6. CHAPTER 1. INTRODUCTION. Although glass technology has long been based on expertise and experimental knowledge, it turns out that this is no longer sufficient for design improvement. The need for such improvement derives not only from governmental requirements but also from fierce competition from materials such as polymers. Hence mathematical modelling and numerical simulation are needed. We shall consider a typical example of the morphology problem, i.e., the production of a parison. This is a preform which occurs in the production of packing glass, such as bottles and jars.. 1.1 Formation of parisons The complete forming process goes as follows: First, a piece of hot glass, the gob, coming directly from the oven, drops into a mould. After the mould has been closed (see Figure 1.1), a plunger moves up quickly (in say 0.5 seconds) and forces the glass to fill the free space in between. When the plunger stops the glass is left in the mould for a second for cooling, this stage is called the dwell. By then, a bell–shaped parison is formed. Afterwards the parison is taken out of the mould by grabbing it at the lower part; then it stays outside the mould for a short period, known as the reheating phase. Finally the parison is put in a second mould and blown into a final shape (see Figure 1.2), this is known as the blowing phase. In this thesis we only concentrate on the formation of the parison, i.e. the pressing phase, including the dwell (see Figure 1.1).. 1.2 Outline of the thesis We formulate the mathematical model for the flow and the heat transfer during the pressing    ), the glass and the dwell in Chapter 2. At sufficiently high temperatures (say above  behaves like an incompressible Newtonian fluid. After a dimension analysis we arrive at a Stokes equation. We also show that the heat transfer during the pressing is negligible. During the dwell, the plunger stands still and there is no flow any more; the heat transfer in this stage is described by a diffusion equation. The glass domain is considered to be axisymmetric. From Chapter 3 to 5, we discuss some boundary element techniques. Topics on boundary integral equations (BIE) are collected in Chapter 3. We show how to derive the axisymmet-.

(8) 1.2. OUTLINE OF THE THESIS. 7. Figure 1.1: The pressing phase.. ric BIE from a three–dimensional BIE. Since the solution of a partial differential equation (PDE) is also a solution of the corresponding BIE, the existence of a solution of the BIE is always guaranteed. However, in general the PDE and the corresponding BIE are not exactly equivalent, in the sense that the BIE may have solution(s) not satisfying the original PDE. In some cases, although the solution of the original PDE is unique, the corresponding BIE may have more than one solution. In these cases, we need extra condition(s) to single out the desired solution. We discuss the uniqueness of BIEs for two–dimensional problems. Chapter 4 is about computational aspects, such as (nearly) singular integrals, corner treatment and the discretisation of the extra condition mentioned above. Sometimes, for the solution of a boundary integral equation to be unique, an extra condition is needed; but numerically, we can show that the extra condition is less important. At the end of this chapter we develop an adaptive BEM based on the equidistribution principle, which are applicable for both two–dimensional problems and axisymmetric problems. We deal with inhomogeneous terms in Chapter 5, i.e. the dual reciprocity method (DRM). A more efficient DRM formulation is derived for a type of pseudo–Poisson equations. We also develop an axisymmetric DRM formulation, which can be applied to inhomogeneous problems on an axisymmetric domain. Chapter 6 and 7 are devoted to the application of these BEM–related techniques to the problems formulated in Chapter 2. In chapter 6, we apply the adaptive BEM developed.

(9) 8. CHAPTER 1. INTRODUCTION. Figure 1.2: The blowing phase. in Chapter 4 to the Stokes equation governing the pressing. In Chapter 7, we apply the axisymmetric DRM to the heat equation during the dwell..

(10) Chapter 2 Mathematical modelling In this chapter we derive a mathematical model describing the pressing and the dwell. As mentioned in Chapter 1, the pressing phase mainly involves fluid flow; the dwell phase involves only heat transfer. We only consider heat conduction in the dwell although radiation exists as well (cf. [27]). We will model the flow during the pressing in Section 2.1, and the temperature during the pressing as well as the dwell in Section 2.2.. 2.1 Modelling the flow We first study the flow of the glass in the mould during the pressing. Let  be the density of the fluid,

(11)   the body forces (here only the gravitational force is considered), 

(12)  the stress tensor, 

(13)  the velocity vector and  the pressure. Then the flow of the glass during the pressing is governed by the following two equations. The first one deals with the conservation of mass, i.e. the continuity equation   !#"%$.

(14) &'( *)+. (2.1). The second one deals with the conservation of momentum and leads to the equation of motion  ,

(15).  . $"

(16) -. "%$ ./(. . 0 1+. (2.2).

(17) CHAPTER 2. MATHEMATICAL MODELLING. 10. We may assume the density to be constant during the process. Further we may assume the glass to be a Newtonian fluid. This yields the following constitutive equation for the stress tensor  and the rate of deformation tensor 2 #8:9. 9. 3 5467. 2<;. (2.3). where 7 is the unit matrix, is the dynamic viscosity which is supposed to be constant (see Section 2.2) and @ " 8

(18) . 2>=?. "

(19). /.ABC+. (2.4). By using the incompressibility in the continuity equation and substituting the constitutive equation and the continuity equation into the equation of motion, we have the so–called Navier–Stokes equations D. "%$. E F)G; $M" ,

(20) IHCJ

(21) L HCK. " /N 54. O9P . . 0 1+. (2.5). In order to further analyse the problem quantitatively we make these equations dimensionless. This involves some typical values of various parameters, which are listed in Table 2.1. Parameter Q/R V,R 9 R . ^`_ b. XdY X e d _ X. 9 =?.

(22) LXZY. Typical value @ )TSGU m @ )TS,W ms S,W @ 80): \ [ kgm S,W s S,W )& \ ) kgm ST] @a ) Jkg S,W K S,W @ @ +?c \ Wm S,W K S,W 80 @  ) C  c&)0) C @  )0)&) C. Meaning the typical scale of the parison the typical velocity of the plunger the dynamic viscosity of the glass the density of glass specific heat conductivity the typical temperature of the glass the typical temperature of the mould the typical temperature of the plunger. Table 2.1: Typical values. @. )&o3p&q U , the acceleration of gravity. The We replace by 9 gf with h=i kjl 6jnm R typical viscosity is defined as the characteristic viscosity. The typical average VdR velocity of the plunger can be used as a characteristic velocity. A characterQ/R istic length is taken as the average thickness of the parison . Consequently the characteristic pressure and the characteristic time can be derived by r1R. 9 RVdR. p. Q'R ;. (2.6).

(23) 2.1. MODELLING THE FLOW. 11. and X. R. Q'R. p. VdR ;. (2.7). respectively. Defining the following dimensionless quantities s. uMt v ; Kv ;. f =?. f =?. f =?. w. 3 f =?. A yx v _J ; zv ;. (2.8). we obtain from (2.5) D. "{$G| } *); | $" ~€

(24) 

(25) H:J  HK ‚. | " #P3| 5„G | †‡ 1+  /( ƒ4 f . Here two dimensionless numbers, the Reynolds number ˆŠ‰ , are defined by ~€. =?. VdRlQ/R  9 R. and. ˆŠ‰. ~€. (2.9). and the Froude number. V R =i. U Q/R;. (2.10). respectively. The Reynolds number indicates the ratio between inertia and viscous forces, whereas the quotient of the Reynolds number and the Froude number indicates the ratio between volume forces and viscous forces. Using the values in Table 2.1, we obtain ~€ m. ~€ @. ) S [. and. ˆŠ‰3m. @. ) ST] +. (2.11). From this we conclude that the viscous forces dominate in the Navier–Stokes equations. Thus the flow can be described by the following equations (we skip the tildes) D. P. "  ‹ 4 Œ F); "%$  F)+. (2.12). These equations are the dimensionless creeping flow or Stokes equations. In order to find a unique velocity  and a pressure  , at time , a set of boundary conditions has to be imposed (see Figure 2.1). Note that, in Figure 2.1 we have.

(26) CHAPTER 2. MATHEMATICAL MODELLING. 12. free boundary. plunger boundary mould boundary. Figure 2.1: The boundaries.  @Ž. . turned over the geometry by ) C for convenience. For simplicity we denote the glass domain by  at time , the boundary between the glass and the mould by   _ K Y‘e , the boundary between the glass and the plunger by Y , the free boundaries  by Y“’ (where the subscript ” means “atmosphere”). We first consider the free boundary. Surface tension, denoted by • , is a force acting on the free boundaries of the glass drop. One can find that for glass • approxa0– imately equals )+ p:o . The surface tension, as well as the plunger motion, plays its role in the system only in terms of boundary 9 R conditions. A typical velocity for a flow driven by surface tension is given by •—p . Thus, to investigate whether surface tension is of any significance, we have to compare this typical velocity with V,R the one induced by the9 plunger motion, e.g. . It is clear that the plunger velocity R is much larger than •—p during most of the pressing phase. From this observation one can derive that the surface tension will not be significant. Free boundaries are R in contact with the surrounding, thus we assume the atmospheric pressure, say  , to prevail everywhere. For the description of the atmospheric pressure in terms of boundary condition the stress vector B˜ has to be prescribed on the free boundaries. To be precise, the boundary condition on the free boundary is given by ™˜E ƒ4š. R ˜. . on Y“’M;. (2.13). where ˜E 5

(27) ›—L is the outward unit normal. Now we consider boundary conditions on the mould and the plunger. They depend.

(28) 2.1. MODELLING THE FLOW. 13. on the roughness of the mould and the plunger. We may impose a no-slip condition, i.e. the velocity of the flow equals the velocity of the boundary. The opposite of a no-slip condition is a no–friction condition, i.e. the normal component of the velocity of the flow equals the normal component of the velocity of the boundary and vanishing shear stresses. But it is also possible to impose a boundary condition describing a certain amount of friction. For example, one can impose the following boundary condition $. D . . $ ˜E *. g  œ ˜š; $ $ B˜. >4Ÿžœl

(29) - 4wgœ¡ ;. (2.14). where is the unit tangential direction, gœ is the velocity of the boundary and žœ is the slip–parameter indicating the amount of friction. This boundary condition means that the normal component of the velocity of the flow equals the normal component of the velocity of the boundary and the shear stress is proportional to the tangential velocity difference. Applying the above boundary conditions to the mould and the plunger yields $. D . B˜. and D. ˜E $I ¢). >4Ÿžeš. $. $. . . on Y‘eŸ;. $ _ ˜E *.  ˜ $ $ _ _ ™˜. 54Ÿž -

(30)  4£ .  on Y _ ;. (2.15). (2.16). where  _ is the velocity of the plunger. In reality, the plunger is actually driven by a given force exerted on it. Therefore _ is not known. We omit a further discussion, however, and assume  _ is given  throughout the pressing. Since at least some part of the boundary is moving, we use a quasi–static approach to describe the movement of a material fluid particle, i.e. D¤. .  s. ¥6

(31)

(32) ¦C; R s K s. when ¢)+ ¤t. (2.17). During the pressing, we are mainly interested in the movement of the free boundary. The modelling problem is then to predict the velocity and thus the position..

(33) CHAPTER 2. MATHEMATICAL MODELLING. 14. 2.2 Modelling the temperature Temperature plays an important role in the forming process, since only sufficiently 9 hot glass can be deformed. The most important quantity is the viscosity . By using dimension analysis, we show that during the pressing the temperature is preserved along a pathline (the line which a given fluid particle follows). Therefore a uniform temperature field remains uniform in the pressing. Consequently the viscosity could be viewed as a constant as we did in the previous section. During the dwell the flow is absent, the process is therefore a pure heat exchange problem.. 2.2.1 The pressing The equation describing the conservation of energy is given by  g§0¨. "©$. #ª. >46. "%$«. . 4. ;. (2.18). « §. where is the internal energy, is the heat fluxª due to heat conduction (we can ¨ ignore the radiation effect during the pressing), is the dissipation function. We assume that the conduction obeys Fourier’s law «. b ". >4. where X is the temperature and Therefore  §0¨. b. X¬;. (2.19). is the conductivity (we assume. . "%$. ƒ46. #ª­ . b ". U X¬+. b. is constant).. (2.20). §. For an incompressible flow, the dissipation function is given by 9. ª. 8

(34) LM¯® . I“® - U +. (2.21). The internal energy can be connected to the temperature by ¨. ^‘_. X¬;. (2.22).

(35) 2.2. MODELLING THE TEMPERATURE. 15. where ^`_ is the specific heat at constant pressure. Therefore, during the pressing the energy equation can be written as ^`_ . ª}. X §. b ". U X¬+. (2.23). §. Define the dimensionless temperature X°P 4hXZe. X f =?. F. ;. X. P. (2.24). where X±=? FXZY64hXZe is the temperature drop in the relevant area. By using this dimensionless temperature and other dimensionless quantities listed in in (2.8), we find that the temperature satisfies the following dimensionless equation (we skip tildes for ease of writing) §. X. @. ". r². 5³ ^ ª r² ;. U X. (2.25). §. where ³µ´ is the Eckhard number, defined as ³ ^ =?. and. r!. ^`_. V R P U X. +. (2.26). is the Peclet number, defined as r!. V,RlQ/R b  9 + R ^‘_ U. =?. (2.27). Simple calculation shows (see table 2.1 for typical values) @ r! ¢¶+. 8<· @. ) S [ ;. ³ ^ r!. @. 8¸· +. @. ) S [ +. (2.28). We conclude that the right hand side in (2.25) can be neglected. This results in X §. ¢)G+. (2.29). §. This means that the temperature is preserved along a pathline. Hence a uniform temperature field remains uniform. During the short process of the pressing, the glass gob can be viewed as having uniform temperature field except for a very thin thermal boundary layer. This implies that constant viscosity during the pressing does make sense..

(36) CHAPTER 2. MATHEMATICAL MODELLING. 16. 2.2.2 The dwell After the plunger has come to a standstill, ª the parison remains in the mould for some time. There is no flow any more,. ¹) , so the heat equation reads as (a special case of (2.23))  . ^‘_. ,X . b ". U X. in ºY:+. (2.30). Again we may ignore the radiation effect since the conduction dominates. We now analyse the boundary conditions as follows. We may assume that the glass makes perfect thermal contact with both the mould and the plunger. Therefore we have . X¥ ¢Xde»

(37) . and  . . _. X½ *X.

(38) . . on Y‘e¼;. (2.31).  on Y _ +. (2.32). we suppose that XZeŸ

(39)  and XdYM

(40)  are known. We may also generally specify Robin– type boundary conditions on the glass–metal interfaces (cf. [6]), i.e. b¾. . F¿Y‘eŸ

(41) Xde¼

(42) À4hÁB. bG¾. . on eÂ;. . _ _. ±¿Y 

(43) X

(44) À4hÁB.  on _ ;. (2.33). where ¿Y‘e ( ¿Y _ ) is the contact conductance between glass and the mould (plunger). As for the free boundary, noting that the flux from glass to air is much smaller than the fluxes from glass to the mould and the plunger, we use the following Neumann boundary condition   X ˜. *). . on Y“’:+. (2.34). Since we assume the uniform temperature field in the beginning, the initial condition for the temperature of the pressing is R Xà ¢X Y. . when F)G+. (2.35).

(45) 2.3. CHOOSING SOLUTION METHODS. 17. 2.3 Choosing solution methods In the previous two sections we derived mathematical models to describe the formation of the parison. We assumed that At the beginning of the pressing, the initial temperature field is uniform. Ä. During the pressing, the plunger is driven by a given velocity. Ä. Ä. Ä. During the dwell, the glass makes a perfect contact with both the mould and the plunger. We ignore the radiation effects since the conduction dominates during the dwell. The geometry is axisymmetric.. Under these assumptions, the flow during the pressing can be modelled as an time– dependent, axisymmetric Stokes equation with free boundaries. The flow is governed by an initial boundary value problem consisting of (2.12), (2.13), (2.15), (2.16) and (2.17). The heat exchange during the dwell can be modelled by a time–dependent, axisymmetric parabolic problem (2.30)–(2.35). Now we are in the position to solve the above problems. Although there several alternatives, we choose the boundary element method as our main discretisation method based on the following considerations Ä Ä. We are mainly interested in the evolution of the free boundaries.. Ä. The flow may have large gradients inside the domain. We can imagine that during the pressing the velocity field near the plunger changes drastically. A domain method needs a very fine grid to capture the changes. On the contrary, this problem is much less serious for the boundary element method. We expect the boundary element method is easier and cheaper under this circumstance. The boundary element method has been mature for inhomogeneous problems and time dependent problems..

(46) 18. CHAPTER 2. MATHEMATICAL MODELLING.

(47) Chapter 3 BIEs and their uniqueness In order to solve a boundary value problem (BVP) by the boundary element method (BEM), one needs to translate the BVP into a boundary integral equation (BIE). There are basically two methods to derive BIEs from BVPs: the indirect method and the direct method. In the direct method, the unknowns of the BIEs are the physical variables, for instance the potentials for potential problems or the velocities for Stokes problems. In the indirect method, the unknowns are density functions (cf. [23]), and the physical variables are computed based on the density functions afterwards. Obviously the physical variables in the direct method are more attractive to users than the density functions in the indirect method. Furthermore, the direct method is somewhat more general – as long as the fundamental solution and Green’s identity (reciprocity principle) with respect to the BVP under question are available, we can derive the corresponding BIE. Hence we adopt the direct method. Since the solution of the BVP is also a solution of the corresponding BIE, the solution existence of the BIE is always guaranteed. However, in general the BVP and the corresponding BIE are not exactly equivalent in the sense that the BIE may have solution(s) not satisfying the original BVP. In some cases, although the solution of the original BVP is unique, the corresponding BIE may have more than one solution. In these cases, we need extra condition(s) to single out the desired solution. We discuss the uniqueness problem of BIEs in some detail for the potential problem; for the Stokes problem we point out that a similar situation may happen as well. For axisymmetric problems, the derivation of the BIEs requires more mathematical effort than their two– or three–dimensional counterparts (cf. [3]). There exist two approaches.

(48) CHAPTER 3. BIES AND THEIR UNIQUENESS. 20. to axisymmetric BIEs: the first is based on axisymmetric fundamental solutions, while the second uses three–dimensional BIEs and performs an integration step with respect to the angular direction. Both approaches finally lead to the same formulation. We employ the second approach to derive axisymmetric BIEs since the three–dimensional BIEs for the Laplace equation and the Stokes equation are widely available in literature.. 3.1 The BIE for the Laplace equation We start with potential problems, which provide us with a template of how the BIEs  are derived. Let us seek a function Á which is harmonic in an (open) domain  and   satisfies some kind of boundary condition on the boundary =?. ". D. ¤ s s£Å *Æ¥Ç ; U Á1

(49) N F); È (3.1) ÁÉ ¢ d; È given function; stands for a generic boundary condition (trace opÈ. where is a erator, for simplicity we assume that is a linear operator). To clearly specify boundary conditions, let us define the characteristic function Ê'

(50) s  for a subset       @ : Ê'

(51) s (. if s­Å ; Ê'

(52) s N F) if s£Å . Æ =i. 4 W. W. U. W. The boundary condition in the problem (3.1) could be one of the following È 1. Dirichlet boundary condition: Á‹=i ¢ÁšË È ¾ ËÌÍ =i HCÎ 2. Neumann boundary condition: Á =i. È @ HÏ Í ¾

(53) 4OÊN 3. mixed boundary condition: Á‹=i *ÊÐÁšË ËÌ Í È ¾ 4. Robin boundary condition: Á¢=i ÒÑÐÁšË ž ; where Ñ and ž are given funcËÓÌ tions. As we said, to transfer the Laplace equation into a BIE, we need Green’s identity and the fundamental solution of the Laplacian. Theorem 3.1 (Green’s identity) For two functions ÁÐ;y ÅÕÔ U

(54) ¼NÖ Ô W

(55) :Ÿ ׏  , there   ØTÙ Ø holds ".

(56) Á. U !4w. ". U ÁB §. Õ.

(57) LÁ Ì.   ›. 4w.  Á ›.  §.  +. (3.2). The fundamental solution for the Laplacian is defined by ". ÚU ÁZÛ

(58) s ;¦Ü/N ƒ4ŸÝT

(59) s. 4wÜ/y;. (3.3).

(60) 3.1. THE BIE FOR THE LAPLACE EQUATION. 21. where Ý is the Dirac Ý function. It is well known that Á Û is given by ÁZÛ

(61). ßà s. ;¡Ü'N. ä. W U á¬â¯ã Þ. t S. 8. W Ú ä ;. for. W ä W Ú ä ; [ á t S. ;. §. for. a. §. (3.4) +. ¾ The normal derivative of Á Û

(62) s ¡; Ü' at a point Ü is denoted by Û . Simple computation gives  ¾. Û

(63). s. ßà. s Á Û

(64) ¦; Ü/. ˜ Ú. ;¡Ü'š=?. ® Ïä ç t Ú I W U á6æ t S ® ÏIä ç t Ú W [ á æ t S Þå å. 8. Ú&è S ä-é ;. for. Ú&è S äëê ;. for. $. Here ˜ Ú is the outward unit normal at point Ü , and ì product.. $dí. ;. §. §. a. (3.5) + Q. ;. stands for the U inner. In Green’s identity, identifying Á as the solution of problem (3.1), and replacing s  by the fundamental solution Á Û

(65) ¦; Ü/ (as a function of Ü , in the distributional sense), we obtain ØGÙ. Ø. ÝT

(66). s. 4wÜ/.Á1

(67) -Ü/ §. Ø.  Ú. s. ÁZÛ

(68). ¾. ;¦Ü/. .

(69) LÜ/ §. Ì. ¾. Ú 4. Û

(70). s. ;¦Ü/`ÁÀ

(71) LÜ/. Ì. . Ú +. §. (3.6). The left hand side is a domain integral which can be directly evaluated: Ø Ù. s. Ýî

(72). 4£Ü/`ÁÀ

(73) LÜ/. ^.  Ú. §. W.

(74). s. `ÁÀ

(75) ÙGû û. where ^ W.

(76). s. ßà 6=i. Þ. ); ^ s

(77) N. @ ;. C;. (3.7). sÃÅ ï. R/ö÷ñøTù`ú ö÷ñø ù. âñð¯òôó`õ. s. ;. » ׏ ; . s£Å s£Å. ;. (3.8). »;. and ü is a circle (or a sphere in the three dimensional case) around s with a small $ radiusó , ý<

(78)  is the area (or volume in the three dimensional case) function (cf. ¨ Figure 3.1). Therefore (3.6) becomes ^. Ø. W.

(79). s. .Á1

(80). s. Ø. ¾  Ì. Û

(81). s. ;¦Ü/`ÁÀ

(82) LÜ/ §. . Ú. Ì. Á—Ûþ

(83). s. ;¦Ü/. ¾.

(84) -Ü' §. . Ú ;. s­Å Ç. ¤ +. (3.9).

(85) CHAPTER 3. BIES AND THEIR UNIQUENESS. 22. Γ. x θ. Ω. Figure 3.1: Geometric Definition of the Problem.. We are interested in the following two cases:  If s£Å , we haveØ Ø ¾. ^ s s

(86) `ÁÀ

(87) . otherwise if ÁÀ

(88). s. Û

(89). s. . ;¡Ü'.ÁÀ

(90) LÜ/ §. Ì. s­Å Ø . s. Û

(91). s. ÁZÛþ

(92). ¾. ;¦Ü/. .

(93) LÜ/. Ú ;. §. Ì. , we have ¾. . Ú. . s­Å. ;. (3.10). Ø . ;¡Ü'.ÁÀ

(94) LÜ/. Ì. §. Ú. ÁZÛ

(95). s. ;¦Ü/. ¾.

(96) LÜ/. Ì. . s}Å. Ú ;. §. »+. (3.11). The equation (3.10) is referred to as the boundary integral equation (BIE). The boundary data can be obtained by solving this BIE together with the boundary condition(s) (cf. chapter 4). Once the boundary data is available, one may compute Á inside the domain  by using (3.11). To find the boundary data we need to combine the BIE together with the boundary condition(s), i.e. D. ^ s s

(97) `ÁÀ

(98)  È ÁÉ ½ Z+. ¥ÿ. Ì. ¾. Û

(99). s. ;¦Ü/`ÁÀ

(100) -Ü'.  §. Ú. ÿ. Á Û

(101) Ì. s. ;¦Ü/. ¾.

(102) LÜ/.  §. Ú ;. s}Å.  ;. (3.12). From the above procedure, we can see that the problem (3.12) is always solvable if the original BVP (3.1) is solvable, but the uniqueness of the BIE is not usually implied by the uniqueness of the original BVP. For further development and ease of writing we (loosely) define the single layer potential and the double layer potential, generated by a density function  , by Ø

(103) “Gl

(104). s. 6=i. Ì. ÁZÛ

(105). s. ;¦Ü/‘,

(106) -Ü/ §. . Ú ;. (3.13).

(107) 3.2. AN INSTRUCTIVE EXAMPLE. 23. Ø. and ¤.

(108) . s. Gl

(109). ¾. º=?. Û

(110). s. ;¦Ü/‘,

(111) LÜ/. Ì.  §. Ú +. (3.14). Then the BIE (3.10) can be written as ¤. ^

(112). `Á. ¾. . ;. (3.15). where  is the identity operator. Before going on to investigate the solution uniqueness of the BIE, we first take a look at a simple example.. 3.2 An instructive example To show that a BIE corresponding to a given BVP may have more than one solu tion, we consider a particular example in two dimensions. Let be a circle with ‰ radius . The boundary integral equation reads

(113) 8. @. ¤ . ¾. .ÁÉ . ;. (3.16). since the boundary is smooth. To investigate the solvability¤ of the above equation, Q  let us analyse eigenvalues and eigenfunctions of  and in U

(114)  . Using polar coordinates 8  

(115) 

(116)  6; 'AÐ; )G; ; ðã ð¯ã ¾ we find that, the integral kernels Á Û and Û are simple: @ @ @ 8 8 8 ‰ s  8  ÁZÛ

(117) ;¡Ü'N.

(118)

(119) 4.   

(120) »4'¦¡C; 5. 4  s 4wܵj âã j âã âã s. ‰.

(121) 

(122)  G;. .AÐ;. . ‰. Ü}. and ¾. Û

(123). s. 8. @. ;¦Ü/N. s ìØ Ú ; 4£Ü s j 4 ܵj U w . Then the operators read Ø ¡². ÁZÛ

(124) 4 . ̉ . s. ì. ‰ â¯ã. (3.18). @. 54 . ‰. . +. (3.19). Ø. ;¦Ü/‘,

(125) LÜ/ 8. í. (3.17). âã.  §. ‰.

(126). 8. R. Uá. ÁZÛ

(127). s.

(128) îy;¦Ü6

(129) !'¡‘,

(130) LÜ6

(131) '¦. 4. 8. í.   þ

(132) »4"'¡C;C,

(133) -Ü6

(134) !'¡. ;. §. . (3.20).

(135) CHAPTER 3. BIES AND THEIR UNIQUENESS. 24 and Ø ¤. Ø ¾. <. ;¦Ü/‘,

(136) LÜ/. Ì @ 4. s. Û

(137). . . @ ì.  §. ‰. Uá ¾ R. Û

(138). s.

(139)  C;¦Ü6

(140) '¦`,

(141) LÜ6

(142) !'¡. í. §. ;C,

(143) -ܺ

(144) '¦. . (3.21). +. These allow us to determine eigenvalues and eigenfunctions of operators ¤ .. . and. 8 8 8 8'& ‰ Å Q

(145) 4.

(146)  

(147) Â4#'¡ as a function of µ4( , and we Note that U $ ); % â¯ã âã can find its Fourier series: 8 8 8 ‰. â¯ã 8. ‰.

(148). âã. â¯ã. 8. â¯ã. +-,. Thus @ . ß Þåå. W. ›. 54.

(149) 

(150)  

(151) L›' .   

(152) L›/'. ‰. ‰. à.

(153)

(154)  

(155) L›'î¡(.   åå.

(156)  . S S.

(157) L›' ¦(. ð‰ ¯ã ‰. U. U. âã. ‡ +. ‡ +. ‰ $ @ ‰. >4. â¯ã. åå åå. (3.22). W 8. * ). ‰.   þ

(158) »4"'¡.   d›N

(159) »4"'. ›. +-,. 8. 4 * ).

(160) ›0îy;. ðã.

(161) L›/'¡C+. ðã. ;.   þ

(162) ›0îC;.

(163) ›0î.. ðã. ›. 8 @. ›‹. @. $$$. ;. ; 8. ;. (3.23). $$$. ;. ;. +. So we conclude‡ that 4 is an eigenvalue of  and the corresponding eigen@ function is ; S + are alsoâ ãeigenvalues 8 $$$ of  and the corresponding eigenfunctions @ U ; ; are   

(164) ›'  and

(165) L›'  , ›. . ¤. For. ð¯ã. , it is easy to verify ¤ @. ß Þåå åå. ¤ åå. ¤. S,W. U. S,W U. $ @. ; $.

(166) 

(167)  

(168) L›' ¦( F)Š F). à åå.   

(169) ›0îC;. $

(170) . ð¯ã.

(171) ›' ¦( F)Š F).

(172) L›'îy; ðã ¤. ›. 8 @. ›3. @. $$$. ;. ;. ;. 8. ;. (3.24). ;. $$$ +. So we conclude that S,W is an eigenvalue of and the corresponding eigenfunction ¤ @ U is ; ) is also an eigenvalue and the corresponding eigenfunctions are   

(173) ›0î 8 $$$ of @ and

(174) L›'  , ›. . ; ; ð¯ã.

(175) 3.2. AN INSTRUCTIVE EXAMPLE. 25. Now we consider the uniqueness of the BIE (3.16) on the unit circle. Since  has ¤ @ has eigenvalue S,W and eigenvalue ) and the corresponding eigenfunction is ; @ U the corresponding eigenfunction is , we obtain )! 5

(176) 8. @. 21 . 1. ¤ Ô . Î. . Ô43 ;. (3.25). ¾ where Ô and Ô43 are constant. The equality (3.25) implies that

(177) LÁÀ; / ©

(178) Ô ; Ô43  Î Î solves the BIE (3.16). ¾. ï Ô Ô43 ) , then

(179) LÁÀ; '

(180) ;  can’t be the boundary data of a harObviously, if Ô43ô > ¾ Î of a harmonic function should satisfy monic function since the normal derivative Ø ¾. F);. § Ì. but. . (3.26). Ø . Ô43 Ì. Ô43. §. oE

(181). . ï  ¢)G+. (3.27). The consequence of the above observation is significant: For the Dirichlet bound¾   ary condition Á Ô ; the mixed boundary condition ÁÉ Ô on , Ô43 on ; ¾ W U Î Ô Ô43 Î and the Robin boundary condition ÑÀÁ ž. FÑ ž , the corresponding BIE Î (3.16) all assume more than one solution, although the original BVP is uniquely solvable. To select the real solution from the set of solutions of the BIE we need extra condition(s). It can be shown that the only condition we need is actually (3.26) (see Section 3.4). ¤. As for the ¤ Neumann problem, note that the operator W  ¤ has eigenvalues ) and  U W since has eigenvalues S,W and ) . Therefore W is not generally invertQ  U U U ible in U

(182)  . This is not a surprise because the corresponding BVP has the same ¾ behaviour, i.e. Á can differ up to a constant for a given . we may single out the solution as follows. Define ³ R =?. where ³. R7. span 5. @6 ;. is the orthogonal complement of ³ Q. ³ R. W. ³R 7 ;. Q. (3.28). . in U

(183)  .. . Note that ³ is a closed subspace of U

(184)  , i.e. ³ is a Hilbert space on its own. It ¤ W W  is injective and surjective from ³ to itself. Therefore is easy to verify that W U. W.

(185) CHAPTER 3. BIES AND THEIR UNIQUENESS. 26 ¤. has bounded inverse from ³ to itself. Since, as an operator in ³ , W W W U U is self–adjoint and has only eigenvalue W , we find that W. . jM

(186) 8. @. ¤ . U. ¤ . 8.  S,W j98;:'. +. (3.29). To investigate the uniqueness issue for general Laplace problems we need some knowledge about the classic potential theory.. 3.3 Some results from classic potential theory To make this chapter self–contained, we cite some useful results from classic potential theory without proof; for details, we refer to [17] and [23]. The interior and exterior of  are denoted by  and =< respectively. We have the S following theorem which characterises the properties of single and double layer potentials (cf. [17]). . Theorem 3.2 (Jump relation) Let  Å*Ô ¤

(187)  . Then the single layer potential with density  is continuous throughout Ç . On the boundary there holds Ø

(188). > . s. ( %

(189)  . Gl

(190). s. Á Û

(191). (. s.  Ú. s­Å. s ,

(192) 8 . s­Å. ;¦Ü/‘,

(193) -Ü' §. Ì . and. > ?. s. ˜.

(194). ;. (3.30).  Ø. . . > . s Á Û

(195) ;¡Ü'. (. › Ì. ,

(196) LÜ/. t. §. . ÚA@.  ;. (3.31). where the integrals exist as improper integrals. ¤. The double layer potential > with density  can be continuously extended from =< ׏ < , and from  to = to ׏ with limiting values Ø S. ¤. > ?.

(197). s. ( ƒ

(198) . ¤. S. GC

(199). s. s ,

(200) 8 . /B. ¾. Û

(201). s. ;¡Ü'`,

(202) LÜ/. Ì.  §. Ú B. s ,

(203) 8 . s­Å.  +. (3.32). Furthermore, in the  sense of . ¤. > ?. ˜.

(204). s. 6=i. C. âñð¯ò õ. <. RED. " >. ¤.F s. B#¿,˜6

(205). s. HG0;C˜š

(206). s. JI!;. (3.33).

(207) 3.3. SOME RESULTS FROM CLASSIC POTENTIAL THEORY we have  . > ?. ˜.  Ø. ¤. Ì. ¾  s Û

(208) ;¦Ü/ › t. ,

(209) LÜ/ §. . s£Å Ú.  +. 27. (3.34). where the integral is to be understood as the Hadamard integrals. In addition to this, we recall the uniqueness theorem for classic Dirichlet problems, Neumann problems, Robin problems and mixed problems. These classic boundary value problems are: Ä. Interior Dirichlet problem Find a function Á ÅEÔ U

(210)   which is harmonic  S and satisfies the boundary condition Á Á × on . Ä. Exterior Dirichlet problem Find a function Á ÅÔ U

(211) =<1 which is harmonic  and satisfies the boundary condition Á Á × on , for j s jLK M it is required @ @ that ÁÉ ONŒ

(212)  in 2D or Á PT

(213)  in 3D. Ä. Interior Neumann problem Find a function Á ÅEÔ U

(214)   which is harmonic ¾ ¾  S and satisfies the boundary condition × on . Ä. Å Ô U

(215) =<1 which is harmonic Exterior Neumann problem Find a function Á  ¾ ¾  and satisfies the boundary condition × on , for j s jQK M it is required @ that ÁÉ PT

(216)  . Ä. Å Ô Interior Robin problem Find a function Á   which is harmonic and U

(217)  ¾ S satisfies the boundary condition ÑÀÁ ž. *• on . Ä. Interior mixed problem Find a function Á ÅEÔ U

(218)   which is harmonic and ¾S ¾     =?. 4 satisfies the boundary condition Á Á × on and × on . W. U. W. For these problems we have the following uniqueness theorem: Theorem 3.3 (Uniqueness theorem) Both the interior and the exterior Dirichlet problem have at most one solution; two solutions of the interior Neumann problem can differ only by a constant, the exterior Neumann problem has at most one solution. Both interior Robin problem and mixed problem have at most one solution..

(219) CHAPTER 3. BIES AND THEIR UNIQUENESS. 28. 3.4 Uniqueness of BIEs As shown before, the solution existence of a BIE is guaranteed by the solution existence of the related BVP, so we only need to consider uniqueness. We suppose the boundary is smooth; therefore the coefficient ^

(220) s ¬ W . The underlying function  space in this section is the continuous function space Ô U

(221)  .. 3.4.1 Dirichlet boundary condition ¾. Theorem 3.4 For a given Á , there is at most one function which satisfies the BIE Ø @ 8 Á. ¤ Á. . ¾ . ¾. and the extra condition.  §. Ì. To show this, we prove that Á F) implies. ¾. F) ¾. > =? . F)+. (3.35). . Define a single layer potential. ;. (3.36). by the jump relation theorem, we have. ><. ÿ. Because Ì. ¾ §. . ¢). @ 8 Á. ¾. . ¤ Á. . s­Å. F)G;. +. (3.37). , we have (omitting constants) R. R >. R . ¾. Ø . R. R. Ø. Ì â¯ã. R.

(222). Ø Ì. K. ). âã Ì.

(223).

(224). j. ¾. 4wÜÂj s.

(225) j. âã. R. . s.

(226) lj. s. R. §. 4£ÜµjÀ4. s. as j. . 4£Üµj  s j j jQK. R. âã. $ R¾. s j. j. ¾ . R. (3.38). §.  R. § M¥y+. Since the exterior Dirichlet problem has at most one solution we have >3 *) s­Å =< . On the other hand, >. S. . ¾. @ 8 Á. ¤ Á. F)G;. s­Å.  +. for. (3.39).

(227) 3.4. UNIQUENESS OF BIES. 29. Since the interior Dirichlet problem has at most one solution we obtain > ±) for s£Å  . S. Therefore HS

(228) T. HS-U HÏ. HIÏ. F). . Using the jump relation theorem again we conclude  ¾. . >. ˜. S.  4. >V<. *)+. ˜. (3.40). 3.4.2 Mixed boundary condition The boundary conditions are Ë. Á Ë Ë. ËÌ. :. ½Á. ¾ Ë ;. W. Ë. ¾. Ë é ËÌ. . U. ;. .  W. + U. (3.41). What we are looking for is ¾. ¾ Ë. =i. W. Ë Ë. ËÌ. Ø. Ø. ¾. 8 Á Ì. :. ÛyÁ. ¾ . W § Ì. é. . Û¦Á. +. (3.42) Ø. ÁZÛ :. Ì. The extra condition becomesØ. Ë é ËÌ. Ø. U §. =? ¢Á Ë U. :. The corresponding BIE can be written as @. Ë. Á. and. ¾  W §. Ì. é. ÁZÛ. ¾  U §. ;. (3.43). Ø ¾ Ì. :. . ¾. W §. ƒ4. +. U § é. Ì. . (3.44). We have the following ¾. . ¾. . Theorem 3.5 For a given pair Á on and on , there is at most one pair   W W U U W on and Á on , which satisfies the BIE (3.43) and the extra condition (3.44). W. U. U. ¾. ¾.  W implies

(229) ;¦Á N W . We need to prove that

(230) Á ; ( ¾ W U ¾ U ¾ @ @ W

(231) 4hÊN`Á , ?= FÊ

(232) 4OÊ( Define Á‹=? ¢ÊgÁ and a potential W. U. W. > =i. . ¾. 4. ¤. U. ÁÀ+. (3.45).

(233) CHAPTER 3. BIES AND THEIR UNIQUENESS. 30. ¤. ¾. )+ Omitting By the jump relation theorem, we have >< X. 4Y Á­4 Î.  U s constants,, the behaviour of > when j jQK ) can be characterised by R >. R. R. ¾.  Ø. . R. R. Ø .

(234). Ø Ì.  K. ). Ì. R. j

(235). â¯ã. R. Á. Ø. 4wܵj s.

(236) lj. â¯ã.

(237). s.

(238) j. Ì âã. R. ¤ R. ¾ . R. §. 4wÜÂjg4. s s. as j. 4£Üµj  s j j jZK. $ R¾ R. s j. â¯ã. Ì j. ¾ . R. ا R. §. j. Ì. M¥C+. s.  Á. R. §. @. s j. Ì@.  R. í. s ì՛ Ú ; 4­Ü s Ø j 4wµ Ü j U. R. R. 4wܵj. j64±jCܵj. R R. (3.46). § . R. Á.  R. Á. §. Since the exterior Dirichlet problem has unique solution, we obtain s­Å. >É ¢)G;. Consequently HS + T F)G;  but from  (3.45) H. . ><. . ›. ›[. . . ¾. =<(+ . ¾ 8 4. >. ›. S. . . ›V[.  4.  . (3.47). ›V[. ÁÐ;. (3.48). ÁÐ+. (3.49).  ¾. ¾. ¤. 8 4.  ¤ ›[. From the above three equations we have   >. ›. S. ¾. . if s­Å. *);. U. ;. (3.50). while >. S. . ¾. 4. ¤ Á. Á8. . if s£Å. F);. W. +. (3.51). Therefore >É ¢)G;. s­Å . S. +. Using the jump relation theorem again we finally obtain

(239). (3.52) ¾ W. ;yÁ U. N W. ..

(240) 3.4. UNIQUENESS OF BIES. 31. 3.4.3 Robin boundary condition The boundary condition is ÑÀÁ. ¾ ž. *•À+. (3.53). Here we consider only the situation that Ñ and ž are both non–zero constants. Plugging (3.53) into the BIE, we have Á8. @. ¤.

(241) . Ñ. .Á . ž. •À+ . ž. (3.54). The extra condition we need becomes Ø .

(242) •Œ4hÑÀÁB. *)+. §. Ì. (3.55). We have the following Theorem 3.6 For a given function • , there is at most one function Á which satisfies (3.54) and (3.55). We need to prove that •É *) implies Á F) . Define a potential ¤. >=? 5

(243) . Ñ. y`ÁÐ;. ž. (3.56). by the jump relation theorem, we have Á8. ><‹. ÿ. Since R >.  Á. Ì. F). §. R . R. Ñ Ë.

(244). . Ë ž ËØ Ñ Ë. Ì. Ë. Ë ž Ë ).

(245). R. Ì. `Á . ž. R. as j. s.

(246). j. Á. . s£Å. F);. +. (3.57). R. Ø . 4£Üµj`Á s.

(247) j. âã âã. s.

(248) j. Ì âã. R. Ë. ¤. R. “Á. Ë. Ñ. (see 3.55), we have (omitting constants). Ë ž Ë Ø Ñ R Ë Ë Ë ž Ë Ø. . K. Ñ Ë. ¤.

(249) . s. jLK. R. §. 4£ÜµjÀ4 4wÜÂj  s j j M¥y+. R. $. Á. s j. âã R. Ì. s ì#› Ú ; 4wÜ s Ø j 4 ܵj U w. j`Á. s. R.  ا. Ì@.  R. §. R. Ì. R. j. s. j. í  Á. R. §. @. R. 4wܵj. jš4±jCܵj. R. R. Á. § . R. §.  R. Á. (3.58).

(250) CHAPTER 3. BIES AND THEIR UNIQUENESS. 32. Since the exterior Dirichlet problem has at most one solution we have h > ) for s­Å =< . On the other hand, >. S. >4. ¤. Á8. Ñ Á ž. “Á. By differentiating we  have . >V<.  ¤ ›. . and. Ñ Á ž. “Án4wÁÉ >4¼ÁÐ+. (3.59). . ¢)!. ›. ¤. Á8. Ñ Á ž. “Á. t. 4. Ñ Á8 ;. (3.60). ž. .  >. ›. S. . ¤ ›. Ñ Á. . ž. t. Á. Ñ Á8 + ž. (3.61). . Therefore.  >. ›. S. Ñ. ÁÐ+ ž. (3.62). From (3.59) and (3.62), the following boundary condition holds for >  Ñ\> S. ž.  >. ›. S. S. ¢)G+. (3.63). >. Since the interior Robin problem has at most one solution we obtain ƒ s­Å  .. ). for. S. Again, from (3.59), we see Á ¢) .. 3.4.4 Neumann boundary condition The non–uniqueness of the Neumann problem is well understood. Nonetheless, to make this section complete we give the following ¾. Theorem 3.7 For a given , there is at most one Á which satisfies @ 8 Á ¤ Á. . ¾. R and the extra condition ÁÀ

(251) s ( ¢);. s R Å.  +. (3.64).

(252) 3.5. THE BIE FOR THE STOKES EQUATION We need to prove that. ¾. F). 33. implies Á ¢) . Define a potential ¤. >‹=i . ÁÐ+. (3.65). Using the jump relation theorem, we have. >V<. we can easily show that. R. R >. @ 8 Á. ) ; K. >. ˜ Ô. Á. *);. as j^ ]/jZK . and >£. have. ¤. S. . (a constant) for s*Å . Therefore > F) for s£Å =< . So. )+. >V<. (3.66). F);. ˜. (3.67). . Using the jump relation theorem again, we S. ÁÉ >V<. 4"> S. Ô. 54. +. (3.68). R Combining this with the extra condition ÁÀ

(253) s ( *) gives Á F) _. s£Å. . .. 3.5 The BIE for the Stokes equation Now we shift our attention to Stokes equation in an (open) domain  with bound ary ". D. ". s. "%U^$ `

(254). À4 g

(255) s

(256) ( F). `. s. N F). s­Å. ¤. ¢Æ¥Ç. ;. (3.69). where ` and  are the velocity and pressure respectively. Similarly to the Laplace equation, we need Green’s identity for Stokes equation and the fundamental solution of Stokes equation. It is easier to use Einstein’s convention. Theorem 3.8 (Green’s identity for Stokes equation) if ` and  are divergence free, ¾ Ù and  and are smooth scalars then ÿ. ". $ Á,.

(257). 3. U d4. ÿ. where M

(258) ` .; —N 54šdÝl . Ì. $ Á,ëM

(259) Lš;. Á,¯® . ¾. H. H [a À4wM`

(260). ". U Ádd4. `›64w-M

(261). ÁT“® . H. H. ;.—.› `. ¾. and M

(262) Lš; N ƒ4. _. &. . [a &.  § ¾.  §. (3.70) ;. ÝC. ¯® . þ‘® +.

(263) CHAPTER 3. BIES AND THEIR UNIQUENESS. 34. The fundamental solution

(264) `  Û ;  Û  is defined by ". ÚU. ßà Þ ". Û

(265) `. s. ". $ Ú. `. Ú   Û

(266) s ¦; Ü/N 54ŸÝT

(267) s. ;¦Ü/À4. (3.71) s. Û

(268). 4wÜ'!bM;. ;¦Ü/N F);. where bM is the c th unit vector of a Cartesian coordinateû system. û It can be shown that (cf. [26]) Þå. for. ;¦Ü/. Ýl . [ áLd. ä. â¯ã. [Hg. ÷. t S. W Ú ä. [a. Sfä e. t S. ÷ [Hg Ú ä é Sfe g. ; h. (3.72). g. W ä SfÚ e ä ; Uá t S. a. û. û. ; and. §. Þå.  Û

(269) å. a. §. s. Á Û 

(270). ßà. for. W. ;¦Ü/Ò. s.  Û

(271) å. 8. s. Á Û 

(272). ßà. s. ;¦Ü/Ò. ;¡Ü'. i. W. ag. ä. áLdkt j S [Hg. ÷. Ú ä. [a. Sfä e. a. t S. ÷ [Hg Ú ä ê Sfe g. h. ;. (3.73). g. ä f S eÚ ä ê ; [ á t S. . ¾. ¾. Replacing  and by the fundamental solutions `  Û and  , identifying ` and  with the solution to the continuity equation and momentum equation, and letting the source point approach the boundary, we obtain the following boundary integral equation (cf. [4], [26] and [41]) Ø ^. 

(273). s. s. .ÁTM

(274). Ø ¾.  Ì. ¾. Û 

(275). s. ;¦Ü/.þ §. . Ú. Ì. Á—Û 

(276). s. ;¦Ü/Jl‘ §. . Ú ;. (3.74). where the kernels Û  are equal to: ¾. s. Û 

(277). ;¦Ü/N. 8. for §.

(278) m],Z4nMLC

(279) m ]Tº4nþC

(280) m]o 4no.›'o ; s j 4wܵj [. (3.75).

(281) m],d4nMC

(282) m]T º4nþþl

(283) m]o¬4no.›'o Rs Rp ;  4£Ü. (3.76). ; ¾. Û 

(284). s. a ;¦Ü/N.

(285) 3.6. UNIQUENESS OF THE STOKES EQUATION for §. a. 35. .. The coefficients ^  depend on the position of the point s , but in the BEM, we will show that it is not necessary to know the analytical expressions of ^  although they are available (cf. [4]). ·. «. ¾. Define matrices ` Û and Û whose

(286) q¦;!c entries are Á Û  and Û  respectively. ¤ § § Thus we can also define hydrodynamical potentials  and as follows Ø w=?. . s. Û

(287) `. . ;¦Ü/º

(288) -Ü/ §. Ì Ø. and ¤. «. w=i. s. Û

(289). . ;¡Ü'6

(290) LÜ/. Ì. §. Ú ;. (3.77). Ú +. (3.78). Then the integral equation becomes ¤ .

(291) r. .  `. Js. +. (3.79). 3.6 Uniqueness of the Stokes equation In this section we show that, for the Stokes equation, the uniqueness of the BIE exhibits the same phenomenon as the potential problem. Taking a circle with radius ‰ @ @ as an example in the two dimensional case, let #

(292) ;  A and t =i u   , we have v Ø —. Á—Û  I Ø. Ì Ô. Ô. Ø Ø. d. Ì. Ô d. Ì. § @. Ýl  d. Ì. .  . s j. â¯ã j. â¯ã. j. s. â¯ã. 4£Üµj @. 4wܵj @ s. 4wܵj. 

(293) m],Z4nMC

(294) m]Tº4nþ h þ s § j 4wܵj U

(295) m],d 4 n:Llw

(296) ]Tº 4 nþ.þ  h s j 4£Üµj U § s 

(297) m],d 4 n:Ll

(298) 4­Ü6;¡' h + s § j 4£Üµj U. Using polar coordinates s. ‰.

(299) 

(300)  G;. ðã. . .AÐ;. Ü}. ‰. (3.80). 8

(301) 

(302)  6;. ð¯ã. 'AÐ;. )G;. ;. (3.81).

(303) CHAPTER 3. BIES AND THEIR UNIQUENESS. 36 Ø. results inv. @. Ô. W. j. Ø ÌLx â ã. Ô ‰. s. Uá. R.

(304) m]. 4wÜÂj ‰. 4. j. @ 8. 8 4. â¯ã. x. 4n W.

(305). s W 8. 4. âã. C

(306). s. 4wܵj U. Ô ‰. Uá R. ‰ 4. 8 4. â¯ã. 4 @. 8.   

(307) »4"/¦. »4. ð¯ã.   

(308) »4'. 4. Ø Ô ‰. Ô ‰. So, if. ‰. Yz. . Uá R. @ 8

(309) @. »4. ðã.

(310) “4. ‰ 4. âã. x. 8. 'C8

(311)    8 »4   . ð¯ã. 8 4. ‰. 8. 8.

(312). 'À4. 4. ðã. . . § y. (3.82). y.  §. »4. ð¯ã.   

(313) »4'. âã.   

(314)  . @. 4. '.   

(315) »4"/¦. x @ 8

(316) @  4   þ

(317)  /À4

(318)  '¦  § ð ã y 8  ‰  Ô ‰.

(319) “4 C ; â¯ã and similarly v Ø @ s w

(320) ] 4 n l

(321) 4wÜ6;¡'  Ô U U. s s U j 4wܵj j 4 ܵj U £ Ø= Ì xâã @ 8 8 Uá ‰ Ô ‰. 4 4 8

(322) 4.   þ

(323) »4"'¡ R â ã â ã x.

(324) m. ðã. 8.

(325). âã. § y.

(326) m   »4

(327)  'l8

(328) m   8 »4

(329)   Ø. . 4wܚ;¦/. ðã. ' y. §. . (3.83).   þ

(330) »4"'¡.

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