A new mesh relaxation approach and automatic time‐step control method for boundary integral simulations of a viscous drop

Siqueira, Ivan R.; Rebouças, R.B.; Oliveira, Taygoara Felamingo de; Cunha, Francisco Ricardo

doi:10.1002/fld.4346

Cited by 8 publications

(7 citation statements)

References 52 publications

(134 reference statements)

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“…where a 1 ¼ 0:95, a 2 ¼ 48:75, and a 3 % À143:3778 are constants. The numerical computation of D T is related to the evaluation of a Cauchy-Green shape tensor for the drop surface, and more details can be found in the original paper of Oliveira and Cunha [16] and in the subsequent work of Siqueira et al [74]. The mesh extrapolation procedure discussed so far was applied to study the dynamics of a drop with k ¼ 30 at Ca k ¼ 10, where Ca k ¼ kCa is the capillary number based on the drop viscosity used in the case of high-viscosity drops.…”

Section: Extrapolation To Mesh Independent Resultsmentioning

confidence: 99%

“…Then, a variation of the Stokes' Theorem is used to compute both nðxÞ and jðxÞ. These methods of discretization and evaluation of geometric quantities of the mesh are widespread in BI simulations of emulsion drops, and more details can be found elsewhere [16,26,29,73,74].…”

Section: Methodsmentioning

confidence: 99%

“…Nevertheless, the Lagrangian character of this equation might lead to a huge mesh distortion, even when the drop shape reaches a stable configuration. To overcome this problem, we have used the temporal evolution equation and the mesh relaxation method recently proposed by Siqueira et al [74]. The method considers the modified evolution equation dx 0 =dt ¼ ½uðx 0 Þ Á nðxÞnðxÞ þ Pðx 0 Þ, where Pðx 0 Þ is an arbitrary, non-physical velocity tangential to the drop surface, and uses additional iterations of pure relaxation in the form dx 0 =dt Ã ¼ Pðx 0 Þ in a virtual temporal march t Ã between each time step of the physical temporal march t. The relaxation velocity is defined as follows [74]:…”

Section: Methodsmentioning

confidence: 99%

“…This new formulation remains non-singular even when the singularity is exactly on the drop surface, since it is not on the path of integration. However, the nonsingular formulation is much more expensive in terms of computational cost when compared to the standard singular BI formulation, as discussed by Siqueira et al [74]. In addition to that because of the Lagrangian nature of the grid points, BI simulations deal with the continuous distortion of the computational mesh regardless of the BI formulation used.…”

Section: Introductionmentioning

confidence: 99%

“…In addition to that because of the Lagrangian nature of the grid points, BI simulations deal with the continuous distortion of the computational mesh regardless of the BI formulation used. To overcome this problem, several mesh relaxation and/or stabilization techniques [26,29,73,74] and topological transformations [24,75,76] have been proposed.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the volume conservation of emulsion drops in boundary integral simulations

Siqueira

Rebouças

Cunha

et al. 2017

J Braz. Soc. Mech. Sci. Eng.

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Boundary Integral simulations are very common to study the microhydrodynamics of viscous drops and predict the rheology of emulsions. The standard boundary integral formulation for the velocity field at the drop surface is given by surface integrals of singular Green's functions in space and depends on geometric quantities of the mesh, as the local mean curvature and the unitary exterior normal vector. Typically, the drops deform and rotate under the flow action, which leads to a great distortion of the computational mesh and, therefore, might produce several numerical errors in the surface velocity calculation. In this work, we investigate the numerical errors in the calculation of the surface velocity in boundary integral simulations of a viscous drop in simple shear flows. Assuming that the flow is incompressible, such that the total volume of the drops must remain constant, we use the net flow rate across the drop surface as a measure of the numerical errors in the simulations. For both small and moderate regimes of drop surface deformation, we show that the net mass flow rate across the drop surface is positive, and strongly depends on the drop-to-basis fluid viscosity ratio, capillary number of the flow, and mesh refinement. These results indicate that the volume of the drops increases continuously during the simulations. To remove the spurious mass flow rate from the simulations, we present a mesh convergence study based on an extrapolation procedure to an almost continuous mesh, so that the results become independent of mesh refinement and recover the theoretical prediction of a null net flow rate and constant drop volume in incompressible flows. The extrapolation method proposed here can be straightforwardly implemented to improve the accuracy of BI simulations used to study drop microhydrodynamics and emulsion rheology. As a matter of example, we applied the method to predict the surface deformation of a high-viscosity emulsion drop, and the result is compared in very good agreement with asymptotic theories available in the related literature.

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Section: Extrapolation To Mesh Independent Resultsmentioning

confidence: 99%