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THE METHOD OF REGULARIZED STOKESLETS∗

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THE METHOD OF REGULARIZED STOKESLETS∗RICARDO CORTEZ†SIAM J. SCI. COMPUT.c"2001 Society for Industrial and Applied MathematicsVol. 23, No. 4, pp. 1204–1225Abstract. A numerical method for computing Stokes flows in the presence of immersed bound-aries and obstacles is presented. The method is based on the smoothing of the forces, leading toregularized Stokeslets. The resulting expressions provide the pressure and velo city field as functionsof the forcing. The latter expression can also be inverted to find the forces that impose a givenvelocity boundary condition. The numerical examples presented demonstrate the wide applicabilityof the method and its properties. Solutions converge with second-order accuracy when forces areexerted along smooth boundaries. Examples of segmented boundaries and forcing at random pointsare also presented.Key words. Stokes flow, immersed b oundariesAMS subject classifications. 76D07, 65M99, 65D32PII. S106482750038146X1. Introduction. The numerical method presented here is for the computationof two- and three-dimensional Stokes flows driven by external forcing. The forces areapplied on volumes or along boundaries, which may be curves, segments, or sets ofdisconnected points. In this way, the method applies to moving interfaces, elasticmembranes interacting with a fluid, or flows through an array of fixed obstacles.Stokes flows are of interest in many physical applications, particularly those inwhich the relevant length scales are extremely small or the fluid is extremely viscous.Many such applications emerge from biology, including the locomotion of bacteriaand other cells, flagellated microorganisms, and flows in small capillaries. Otherapplications are the study of free surfaces, such as bubble motion, and flows aroundimpurities or through a porous medium, which may be modeled as a collection ofpoint obstacles for dilute cases.The steady Stokes equations in two or three dimensions areµ∆u = ∇p − F,(1)∇· u =0,(2)where µ is the fluid viscosity, p is the pressure, u is the velocity, and F is force. Afundamental solution of these equations is called a Stokeslet, and it represents thevelocity due to a concentrated external force acting on the fluid at a single point [32,34, 1, 25]. Other, more singular solutions can also be derived from the Stokeslet bydifferentiation. Many important models have been created from the superposition ofthese fundamental solutions. Examples are analyses of flagellar motions [17, 16, 24];the beating motion of cilia [4]; flows between plates, inside cylinders, or in periodicgeometries [26, 27, 15, 33]; and slender body theories [6, 20, 23, 13]. Many of theseanalyses make use of images to enforce boundary conditions on the surface of spheresor planes.∗Received by the editors November 21, 2000; accepted for publication (in revised form) April 9,2001; published electronically November 7, 2001. This work was supported in part by NSF grantDMS-9816951.http://www.siam.org/journals/sisc/23-4/38146.html†Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118([email protected]).1204THE METHOD OF REGULARIZED STOKESLETS 1205The singularity in the Stokeslet is proportional to 1/r in three dimensions andlog(r) in two dimensions. Consequently, when forces are concentrated on surfacesembedded in R3(curves in R2), the expression for velocity is integrable and the flowis bounded in the vicinity of the surfaces. Most of the works cited above rely on thisfact and the smoothness of the surfaces to get final expressions for the fluid velocity.When the forces are concentrated along curves in R3(points in R2) or when sur-faces are not smooth, the situation is more difficult because the velocity formula is sin-gular. One technique for dealing with this problem is to desingularize the expressionsby introducing a small cutoff parameter in the kernels in order to regularize them. Thisapproach has been extremely useful for the modeling of vortex motion [7, 3, 21, 14],interface motion in inviscid fluids [5, 35, 10, 8, 36], and other processes [11, 9, 12]. Inall of these applications, the errors involved have b een analyzed extensively and arewell understood.Our approach here is to consider the forces to be applied over a small ball, wherethey vary smoothly from a maximum value at the center to zero on its surface, ratherthan being concentrated at points (as Dirac measures). The radius of the support ofthe forces is a numerical parameter that can be controlled independently from anyboundary discretization. Sometimes the radius of the ball is infinite, but the forcesdecay fast away from the center. From this starting point, expressions for the pressureand velocity due to this regularized force are derived. These expressions are boundedin any bounded set and differ only from the standard Stokeslet near the points wherethe forces are exerted. The resulting method is applicable to any situation in whichforces drive the motion, whether they are concentrated along interfaces or points. Thisgives the method wide applicability. The derivation of the expressions is presented insection 2 for radially symmetric regularizations and a specific example is presentedin detail. Section 3 contains several numerical examples with particular focus on theperformance of the method. The case of a smooth closed boundary in R2is furtherdeveloped in section 4, where the velocity expressions can be written in terms ofsingle and double layer potentials. Recent work by Beale and Lai [2] is used to achievesecond-order accuracy everywhere. The final section contains concluding remarks andfuture directions.2. Equations. We first consider the generic situation in which the forces arespread over a small ball centered at the points x0. The force is given byF(x)=f0φ!(x − x0),(3)where φ!is a radially symmetric smooth function with the property that!φ!(x)dx =1. Examples of these functions, called blobs or cutoffs, arein R2: φ!(x)=1π#2e−|x|2/!2and φ!(x)=3#32π(|x|2+ #2)5/2,in R3: φ!(x)=34π#3e−|x|3/!3and φ!(x)=15#48π(r2+ #2)7/2.A typical blob is displayed in Figure 1. The graph shows the same blob with twodifferent values of the parameter #, which controls the width, or spreading. A tighterfunction (smaller #) must necessarily be taller for the total integral to be 1. In thelimit # → 0, the blob approaches a Dirac delta.1206 RICARDO CORTEZ–3 –2 –1 0 1 2 300.10.20.30.40.50.60.7rFig. 1. Typical blob for two different


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