Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid (2024)

Iglesias Martinez, J. A., Mercier, G. (2020). Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid. Applied Mathematics and Optimization: an international journal with applications to stochastics, 82(2), 399-432. https://doi.org/10.1007/s00245-018-9531-8

Iglesias Martinez, Jose Alberto ; Mercier, Gwenael ; Scherzer, Otmar. / Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid. In: Applied Mathematics and Optimization: an international journal with applications to stochastics. 2020 ; Vol. 82, No. 2. pp. 399-432.

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title = "Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid",

abstract = "We consider the flow of multiple particles in a Bingham fluid in an anti-plane shear flow configuration. The limiting situation in which the internal and applied forces balance and the fluid and particles stop flowing, that is, when the flow settles, is formulated as finding the optimal ratio between the total variation functional and a linear functional. The minimal value for this quotient is referred to as the critical yield number or, in analogy to Rayleigh quotients, generalized eigenvalue. This minimum value can in general only be attained by discontinuous, hence not physical, velocities. However, we prove that these generalized eigenfunctions, whose jumps we refer to as limiting yield surfaces, appear as rescaled limits of the physical velocities. Then, we show the existence of geometrically simple minimizers. Furthermore, a numerical method for the minimization is then considered. It is based on a nonlinear finite difference discretization, whose consistency is proven, and a standard primal-dual descent scheme. Finally, numerical examples show a variety of geometric solutions exhibiting the properties discussed in the theoretical sections.",

keywords = "Bingham fluid, CHEEGER SETS, Critical yield number, Exchange flow, Piecewise constant solutions, Settling, Total variation, UNIAXIAL EXCHANGE FLOWS",

author = "{Iglesias Martinez}, {Jose Alberto} and Gwenael Mercier and Otmar Scherzer",

note = "Publisher Copyright: {\textcopyright} 2018, The Author(s).",

year = "2020",

month = oct,

doi = "10.1007/s00245-018-9531-8",

language = "English",

volume = "82",

pages = "399--432",

journal = "Applied Mathematics and Optimization: an international journal with applications to stochastics",

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Iglesias Martinez, JA, Mercier, G 2020, 'Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid', Applied Mathematics and Optimization: an international journal with applications to stochastics, vol. 82, no. 2, pp. 399-432. https://doi.org/10.1007/s00245-018-9531-8

Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid. / Iglesias Martinez, Jose Alberto (Corresponding author); Mercier, Gwenael; Scherzer, Otmar.

In: Applied Mathematics and Optimization: an international journal with applications to stochastics, Vol. 82, No. 2, 10.2020, p. 399-432.

Publications: Contribution to journal › Article › Peer Reviewed

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AU - Iglesias Martinez, Jose Alberto

AU - Mercier, Gwenael

AU - Scherzer, Otmar

N1 - Publisher Copyright:© 2018, The Author(s).

PY - 2020/10

Y1 - 2020/10

N2 - We consider the flow of multiple particles in a Bingham fluid in an anti-plane shear flow configuration. The limiting situation in which the internal and applied forces balance and the fluid and particles stop flowing, that is, when the flow settles, is formulated as finding the optimal ratio between the total variation functional and a linear functional. The minimal value for this quotient is referred to as the critical yield number or, in analogy to Rayleigh quotients, generalized eigenvalue. This minimum value can in general only be attained by discontinuous, hence not physical, velocities. However, we prove that these generalized eigenfunctions, whose jumps we refer to as limiting yield surfaces, appear as rescaled limits of the physical velocities. Then, we show the existence of geometrically simple minimizers. Furthermore, a numerical method for the minimization is then considered. It is based on a nonlinear finite difference discretization, whose consistency is proven, and a standard primal-dual descent scheme. Finally, numerical examples show a variety of geometric solutions exhibiting the properties discussed in the theoretical sections.

AB - We consider the flow of multiple particles in a Bingham fluid in an anti-plane shear flow configuration. The limiting situation in which the internal and applied forces balance and the fluid and particles stop flowing, that is, when the flow settles, is formulated as finding the optimal ratio between the total variation functional and a linear functional. The minimal value for this quotient is referred to as the critical yield number or, in analogy to Rayleigh quotients, generalized eigenvalue. This minimum value can in general only be attained by discontinuous, hence not physical, velocities. However, we prove that these generalized eigenfunctions, whose jumps we refer to as limiting yield surfaces, appear as rescaled limits of the physical velocities. Then, we show the existence of geometrically simple minimizers. Furthermore, a numerical method for the minimization is then considered. It is based on a nonlinear finite difference discretization, whose consistency is proven, and a standard primal-dual descent scheme. Finally, numerical examples show a variety of geometric solutions exhibiting the properties discussed in the theoretical sections.

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Iglesias Martinez JA, Mercier G, Scherzer O. Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid. Applied Mathematics and Optimization: an international journal with applications to stochastics. 2020 Oct;82(2):399-432. Epub 2018 Oct. doi: 10.1007/s00245-018-9531-8