A spectral boundary element approach to three-dimensional Stokes flow

Muldowney, Gregory P.; Higdon, J. J. L.

doi:10.1017/s0022112095003260

Cited by 74 publications

(82 citation statements)

References 29 publications

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“…On any boundary element we spread collocation points as explained in [8] in such a way that the resulting nodes distribution on the entire (truncated) contour L is symmetric with respect to the (O, e z ) axis. Isoparametric interpolations of both the velocity u and the prescribed traction σ · n on L are employed and the integration of the kernels B αβ (x, x 0 ) and C αβ (x, x 0 ) on each boundary element are performed by regularizing the weakly-singular terms B αα (x, x 0 ) when the node x 0 belongs to the selected boundary element and using the iterative treatment of [9] to accurately calculate each encoutered regular integration.…”

Section: Methodsmentioning

confidence: 99%

Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface

Pigeonneau

Sellier

2011

Physics of Fluids

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Résumé :On examine l'ascension de bulles au voisinage de Abstract : The gravity-driven migration of N ≥ 1 bubble(s) near a free surface is addressed within the assumption of negligible inertial effects by solving a boundary-integral equation at each time and assuming axisymmetric free surface and bubble(s) with axis of revolution aligned with the gravity. The implemented boundary element method permits one to accurately invert at a reasonable cpu time cost the encountered boundary integration on the liquid boundary and to determine

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Section: Methodsmentioning

confidence: 99%

Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface

Pigeonneau

Sellier

2011

Physics of Fluids

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“…The numerical solution of the boundary integral equation of § 2.2 is achieved through the spectral boundary element method (Occhialini et al 1992;Muldowney & Higdon 1995). Briefly, each boundary is divided into a small number of surface elements which are parameterized by a variable ξ on the interval [−1, 1].…”

Section: Methodsmentioning

confidence: 99%

“…The geometry and physical † As an alternative to Step 2, the surface stress may be evaluated directly as an integral of ∆f and u over the boundary surfaces. The kernels required for this integration are the two dimensional versions of T and Q as given in Muldowney & Higdon (1995). These kernels require additional time for numerical quadratures, and the preferred choice will be dictated by a balance between quadrature effort and matrix inversion time.…”

Section: Methodsmentioning

confidence: 99%

“…All numerical derivatives are evaluated using the so-called collocation derivative, that is using analytical differentiation of the high-order polynomials which comprise the basis functions for the spectral element expansions. See Muldowney & Higdon (1995) A comment may be warranted concerning the constant-volume constraint in Step 4. While the solution of the boundary integral equation yields a velocity field which is divergence free, the boundary perturbation represented by d admits a broad class of surfaces which may include different volumes of fluid.…”

Section: Implementation Of Newton Iterationmentioning

confidence: 99%

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Displacement of fluid droplets from solid surfaces in low-Reynolds-number shear flows

Dimitrakopoulos

Higdon

1997

J. Fluid Mech.

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The yield conditions for the displacement of fluid droplets from solid boundaries are studied through a series of numerical computations. The study includes gravitational and interfacial forces, but is restricted to two-dimensional droplets and low-Reynoldsnumber flow. A comprehensive study is conducted, covering a wide range of viscosity ratio λ, Bond number B d , capillary number Ca and contact angles θ A and θ R . The yield conditions for drop displacement are calculated and the critical shear rates are presented as functions Ca(λ, B d , θ A , ∆θ) where ∆θ = θ A − θ R is the contact angle hysteresis. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surface profiles. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While excellent agreement is found in the joint asymptotic limits ∆θ θ A 1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets. Gravitational forces normal to the solid boundary have a significant effect on the displacement process, reducing the critical shear rate for viscous drops and increasing the rate for inviscid droplets. The low-viscosity limit λ → 0 is shown to be a singular limit in the lubrication theory, and the proper scaling for Ca at small λ is identified.

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“…By taking the Laplacian with respect to u(x) in Equation (9), the vorticity function is derived as follows:…”

Section: Bie For the Domain Pointmentioning

confidence: 99%

A new method for Stokes problems with circular boundaries using degenerate kernel and Fourier series

Chen

Hsiao

Leu

2007

Numerical Meth Engineering

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SUMMARYThis study is concerned with the Stokes flow of an incompressible fluid of constant density and viscosity with circular boundaries. To fully capture the circular boundary, the boundary densities in the direct and indirect boundary integral equations (BIEs) are expanded in terms of Fourier series. The kernel functions in either the direct BIE or the indirect BIE are expanded to degenerate kernels by using the separation of field and source points. Consequently, the improper integrals are transformed to series sum and are easily calculated. The linear algebraic system can be established by matching the boundary conditions at the collocation points. Then, the unknown Fourier coefficients can be easily determined. Finally, several examples including circular and eccentric domains are presented to demonstrate the validity of the present method. Five gains were obtained: (1) meshless approach; (2) free of boundary-layer effect; (3) singularity free; (4) exponential convergence; and (5) well-posed model.

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A spectral boundary element approach to three-dimensional Stokes flow

Cited by 74 publications

References 29 publications

Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface

Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface

Displacement of fluid droplets from solid surfaces in low-Reynolds-number shear flows

A new method for Stokes problems with circular boundaries using degenerate kernel and Fourier series

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