Skip to main content
SpringerLink
Log in

Validation of a Galerkin technique on a boundary integral equation for creeping flow around a torus

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

A validation of the numerical solution for the steady and axisymmetric creeping flow around a three-dimensional torus is presented. This solution is obtained by means of the boundary element method. Both a Galerkin weighting technique and collocation to the centroid of the elements are employed. The curve of the viscous drag force as a function of the diameter of the torus relative to its thickness is compared against a semi-analytical solution and laboratory experimental measurements taken from the literature. The semi-analytical solution, as it is known for this kind of geometry, involves the Legendre functions of first and second kind of order one and semi-integer degrees, also called toroidal harmonics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Amarakoon AMD, Hussey RG, Good BJ, Grimsal EG (1982) Drag measurements for axisymmetric motion of a torus at low Reynolds number. Phys Fluids 25(9):1495–1501

    Article  Google Scholar 

  • Berli C, Cardona A (2009) On the calculation of viscous damping of microbeam resonators in air. J Sound Vib 327(1–2):249–253

    Article  Google Scholar 

  • Dargush GF, Grigoriev MM (2005) Fast and accurate solutions of steady Stokes flows using multilevel boundary element methods. J Fluids Eng Trans ASME 127(4):640–646

    Article  MathSciNet  Google Scholar 

  • D’Elía J, Battaglia L, Cardona A, Franck G (2012) Galerkin boundary elements for a computation of the surface tractions in exterior Stokes flows. Technical report, CIMEC-INTEC, UNL-CONICET

    Google Scholar 

  • D’Elía J, Battaglia L, Cardona A, Storti M (2011) Full numerical quadrature of weakly singular double surface integrals in Galerkin boundary element methods. Int J Numer Methods Biomed Eng 27(2):314–334. doi:10.1002/cnm.1309

    Article  MATH  MathSciNet  Google Scholar 

  • D’Elía J, Battaglia L, Storti M, Cardona A (2008) Galerkin boundary integral equations applied to three dimensional Stokes flows. In: Cardona A, Storti M, Zuppa C (eds) Mecánica Computacional, vol XXVII. San Luis, pp 2397–2410

  • Dorrepaal JM, Majumdar SR, O’Neill ME, Ranger K (1976) A closed torus in Stokes flow. Q J Mech Appl Math 29(381)

  • Duffy MG (1982) Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J Numer Anal 19(6):1260–1262

    Article  MATH  MathSciNet  Google Scholar 

  • Fang Z, Mammoli AA, Ingber MS (2001) Analyzing irreversibilities in Stokes flows containing suspensed particles using the traction boundary integral equation method. Eng Anal Bound Elements 25:249–257

    Article  MATH  Google Scholar 

  • Galvis E, Yarusevych S, Culham JR (2012) Incompressible laminar developing flow in microchannels. J Fluids Eng Trans ASME 134(1): 014, 503

    Google Scholar 

  • Gonzalez O (2009) On stable, complete, and singularity-free boundary integral formulations of exterior Stokes flow. SIAM J Appl Math 69:933–958

    Article  MATH  MathSciNet  Google Scholar 

  • Hebeker FK (1986) Efficient boundary element methods for three-dimensional exterior viscous flow. Numer Methods PDE 2(4):273–297

    Article  MATH  MathSciNet  Google Scholar 

  • Hsiao GC, Wendland WL (2008) Boundary integral equations. Springer, Berlin

    Book  MATH  Google Scholar 

  • Ingber MS, Mammoli AA (1999) A comparison of integral formulations for the analysis of low Reynolds number flows. Eng Anal Bound Elements 23:307–315

    Article  MATH  Google Scholar 

  • Ingber MS, Mondy LA (1993) Direct second kind boundary integral formulation for Stokes flow problems. Comput Mech 11:11–27

    Article  MATH  MathSciNet  Google Scholar 

  • Kim S, Karrila SJ (1991) Microhydrodynamics: principles and selected applications. Butterwoth-Heinemann, UK

    Google Scholar 

  • Ladyzhenskaya OA (1969) The mathematical theory of viscous incompressible flow, 2nd edn. Gordon and Breach Science Publishers, New York

    MATH  Google Scholar 

  • Lepchev D, Weihs D (2010) Low Reynolds number flow in spiral microchannels. J Fluids Eng Trans ASME 132(7):071202

    Article  Google Scholar 

  • Majumdar SR, O’Neill ME (1977) On axisymmetric Stokes flow past a torus. Z Angew Math Phys 28(4):541–550

    Article  MATH  Google Scholar 

  • Méndez C, Paquay S, Klapka I, Raskin JP (2008) Effect of geometrical nonlinearity on MEMS thermoelastic damping. Nonlinear Anal R World Appl 10(3):1579–1588

    Article  Google Scholar 

  • Polimeridis AG, Mosig JR (2010) Complete semi-analytical treatment of weakly singular integrals on planar triangles via the direct evaluation method. Int J Numer Methods Eng 83(12):1625–1650

    Article  MATH  MathSciNet  Google Scholar 

  • Power H, Miranda G (1987) Second kind integral equation formulation of Stokes flows past a particle of arbitrary shape. SIAM J Appl Math 47(4):689–698

    Article  MATH  MathSciNet  Google Scholar 

  • Pozrikidis C (1996) Introduction to theoretical and computational fluid dynamics. Oxford University Press, Oxford

    Google Scholar 

  • Pozrikidis C (1997) Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, Cambridge

    Google Scholar 

  • Sarraf S, López E, Ríos Rodríguez G, D’Elía J (2012) Simulación del flujo reptante exterior a un toro tridimensional mediante el método de elementos de borde. In: Cardona A, Kohan PH, Quinteros RD, Storti MA (eds) Mecánica Computacional, vol XXXI. Salta, Argentina, pp 321–332

    Google Scholar 

  • Sauter SA, Schwab C (2011) Boundary element methods. Springer, Berlin

    Book  MATH  Google Scholar 

  • Shipman TN, Prasad AK, Davidson SL, Cohee DR (2007) Particle image velocimetry evaluation of a novel oscillatory-flow flexible chamber mixer. J Fluids Eng Trans ASME 129(2):179–187

    Article  Google Scholar 

  • Tamayol A, Bahrami M (2010) Laminar flow in microchannels with noncircular cross section. J Fluids Eng Trans ASME 132(11):111, 201

    Google Scholar 

  • Taylor DJ (2003a) Accurate and efficient numerical integration of weakly singulars integrals in Galerkin EFIE solutions. IEEE Trans Antennas Propag 51(7):1630–1637

    Article  Google Scholar 

  • Taylor DJ (2003b) Errata to “Accurate and efficient numerical integration of weakly singulars integrals in Galerkin EFIE solutions”. IEEE Trans Antennas Propag 51(9):2543–2543

    Article  Google Scholar 

  • Wang X (2002) Fast Stokes: a fast 3-D fluid simulation program for micro-electro-mechanical systems. PhD thesis. MIT, Cambridge

  • Younggren GK, Acrivos A (1975) Stokes flow past a particle of arbitrary shape: a numerical method of solutions. J Fluid Mech 69(2):377–403

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work has received financial support from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET, Argentina, grant PIP 112-20111-00978), Universidad Nacional del Litoral (UNL, Argentina, Grant CAI+D 2009-III-4-2), Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT, Argentina, grants PICT 2010-2492/16, PICT 2009-1141/07, PICT-PRH 2009-0147), EU-IRSES (PIRSES-GA-2009-246977), and was performed with the Free Software Foundation /GNU-Project resources such as GNU-Linux-OS, GNU-GFortran, GNU-Octave, GNU-Git, GNU-Doxygen, and GNU-GIMP, as well as other Open Source resources as NETGEN, ParaView, OpenDX, Xfig and LATEX.

Author information

Authors and Affiliations

  1. Departamento de Mecánica Aplicada, Fac. de Ing., Univ. Nac. del Comahue, CONICET, Buenos Aires 1400, 8300 , Neuquén, Argentina

    Sofía Sarraf & Ezequiel López

  2. Centro Internacional de Métodos Computacionales en Ingeniería (CIMEC), Instituto de Desarrollo Tecnológico para la Industria Química (INTEC), Univ. Nac. del Litoral, CONICET, Güemes 3450, 3000 , Santa Fe, Argentina

    Gustavo Ríos Rodríguez & Jorge D’Elía

Authors
  1. Sofía Sarraf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ezequiel López
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gustavo Ríos Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jorge D’Elía
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jorge D’Elía.

Additional information

Communicated by Gustavo Buscaglia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sarraf, S., López, E., Ríos Rodríguez, G. et al. Validation of a Galerkin technique on a boundary integral equation for creeping flow around a torus. Comp. Appl. Math. 33, 63–80 (2014). https://doi.org/10.1007/s40314-013-0043-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-013-0043-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation