Validation of a Galerkin technique on a boundary integral equation for creeping flow around a torus
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Validation of a Galerkin technique on a boundary integral equation for creeping flow around a torus Sofía Sarraf · Ezequiel López · Gustavo Ríos Rodríguez · Jorge D’Elía
Received: 13 December 2012 / Accepted: 23 April 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
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. Keywords Creeping flow · Steady flow · Boundary element method · Collocation technique · Galerkin weighting · Toroidal harmonics Mathematics Subject Classification (2000)
65N38 · 76D07
Communicated by Gustavo Buscaglia. S. Sarraf · E. López Departamento de Mecánica Aplicada, Fac. de Ing., Univ. Nac. del Comahue, CONICET, Buenos Aires 1400, 8300 Neuquén, Argentina e-mail: [email protected] E. López e-mail: [email protected] G. Ríos Rodríguez · J. D’Elía (B) 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 e-mail: [email protected] URL: http://www.cimec.org.ar G. Ríos Rodríguez e-mail: [email protected]
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1 Introduction The computation of steady Stokes flows around closed rigid bodies can be of interest in fluid and biomedical engineering. Possible applications are multilevel boundary element method (BEM) for steady Stokes flows in irregular two-dimensional domains (Dargush and Grigoriev 2005); low Reynolds number flow of an incompressible fluid in spiral microchannels that are used in DNA identifying lab-on-a-chip devices (Lepchev and Weihs 2010); creeping flow regime in oscillatory-flow mixers with flexible chambers (Shipman et al. 2007); laminar flow in compact heat exchangers and microcoolers in electronics packaging (Galvis 2012); laminar fully developed flow in micro-/minichannels with non-circular cross sections (Tamayol and Bahrami 2010); as well as micro-electro-mechanical systems (MEMS) (Berli and Cardona 2009; Méndez et al. 2008; Wang 2002). Indirect formulations in this flow case are commonly related to hydrodynamic doubleand single-layer potentials (Ladyzhenskaya 1969; Pozrikidis 1996). An indirect Boundary Integral Equation (BIE) uses as a starting point the potentials produced by surface layers of (fictitious) singularities (Sauter and Schwab 2011). These surface singularity layers generat
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