---
_id: '14042'
abstract:
- lang: eng
  text: Long-time and large-data existence of weak solutions for initial- and boundary-value
    problems concerning three-dimensional flows of incompressible fluids is nowadays
    available not only for Navier–Stokes fluids but also for various fluid models
    where the relation between the Cauchy stress tensor and the symmetric part of
    the velocity gradient is nonlinear. The majority of such studies however concerns
    models where such a dependence is explicit (the stress is a function of the velocity
    gradient), which makes the class of studied models unduly restrictive. The same
    concerns boundary conditions, or more precisely the slipping mechanisms on the
    boundary, where the no-slip is still the most preferred condition considered in
    the literature. Our main objective is to develop a robust mathematical theory
    for unsteady internal flows of implicitly constituted incompressible fluids with
    implicit relations between the tangential projections of the velocity and the
    normal traction on the boundary. The theory covers numerous rheological models
    used in chemistry, biorheology, polymer and food industry as well as in geomechanics.
    It also includes, as special cases, nonlinear slip as well as stick–slip boundary
    conditions. Unlike earlier studies, the conditions characterizing admissible classes
    of constitutive equations are expressed by means of tools of elementary calculus.
    In addition, a fully constructive proof (approximation scheme) is incorporated.
    Finally, we focus on the question of uniqueness of such weak solutions.
acknowledgement: "M. Bulíček and J. Málek acknowledge the support of the project No.
  20-11027X financed by the Czech Science foundation (GAČR). M. Bulíček and J. Málek
  are members of the Nečas Center for Mathematical Modelling.\r\nOpen access publishing
  supported by the National Technical Library in Prague."
article_number: '72'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Miroslav
  full_name: Bulíček, Miroslav
  last_name: Bulíček
- first_name: Josef
  full_name: Málek, Josef
  last_name: Málek
- first_name: Erika
  full_name: Maringová, Erika
  id: dbabca31-66eb-11eb-963a-fb9c22c880b4
  last_name: Maringová
citation:
  ama: Bulíček M, Málek J, Maringová E. On unsteady internal flows of incompressible
    fluids characterized by implicit constitutive equations in the bulk and on the
    boundary. <i>Journal of Mathematical Fluid Mechanics</i>. 2023;25(3). doi:<a href="https://doi.org/10.1007/s00021-023-00803-w">10.1007/s00021-023-00803-w</a>
  apa: Bulíček, M., Málek, J., &#38; Maringová, E. (2023). On unsteady internal flows
    of incompressible fluids characterized by implicit constitutive equations in the
    bulk and on the boundary. <i>Journal of Mathematical Fluid Mechanics</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00021-023-00803-w">https://doi.org/10.1007/s00021-023-00803-w</a>
  chicago: Bulíček, Miroslav, Josef Málek, and Erika Maringová. “On Unsteady Internal
    Flows of Incompressible Fluids Characterized by Implicit Constitutive Equations
    in the Bulk and on the Boundary.” <i>Journal of Mathematical Fluid Mechanics</i>.
    Springer Nature, 2023. <a href="https://doi.org/10.1007/s00021-023-00803-w">https://doi.org/10.1007/s00021-023-00803-w</a>.
  ieee: M. Bulíček, J. Málek, and E. Maringová, “On unsteady internal flows of incompressible
    fluids characterized by implicit constitutive equations in the bulk and on the
    boundary,” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 25, no. 3. Springer
    Nature, 2023.
  ista: Bulíček M, Málek J, Maringová E. 2023. On unsteady internal flows of incompressible
    fluids characterized by implicit constitutive equations in the bulk and on the
    boundary. Journal of Mathematical Fluid Mechanics. 25(3), 72.
  mla: Bulíček, Miroslav, et al. “On Unsteady Internal Flows of Incompressible Fluids
    Characterized by Implicit Constitutive Equations in the Bulk and on the Boundary.”
    <i>Journal of Mathematical Fluid Mechanics</i>, vol. 25, no. 3, 72, Springer Nature,
    2023, doi:<a href="https://doi.org/10.1007/s00021-023-00803-w">10.1007/s00021-023-00803-w</a>.
  short: M. Bulíček, J. Málek, E. Maringová, Journal of Mathematical Fluid Mechanics
    25 (2023).
date_created: 2023-08-13T22:01:13Z
date_published: 2023-08-01T00:00:00Z
date_updated: 2023-12-13T12:08:08Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00021-023-00803-w
external_id:
  arxiv:
  - '2301.12834'
  isi:
  - '001040354900001'
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publication: Journal of Mathematical Fluid Mechanics
publication_identifier:
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  - 1422-6952
  issn:
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publication_status: published
publisher: Springer Nature
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title: On unsteady internal flows of incompressible fluids characterized by implicit
  constitutive equations in the bulk and on the boundary
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---
_id: '11842'
abstract:
- lang: eng
  text: We consider the flow of two viscous and incompressible fluids within a bounded
    domain modeled by means of a two-phase Navier–Stokes system. The two fluids are
    assumed to be immiscible, meaning that they are separated by an interface. With
    respect to the motion of the interface, we consider pure transport by the fluid
    flow. Along the boundary of the domain, a complete slip boundary condition for
    the fluid velocities and a constant ninety degree contact angle condition for
    the interface are assumed. In the present work, we devise for the resulting evolution
    problem a suitable weak solution concept based on the framework of varifolds and
    establish as the main result a weak-strong uniqueness principle in 2D. The proof
    is based on a relative entropy argument and requires a non-trivial further development
    of ideas from the recent work of Fischer and the first author (Arch. Ration. Mech.
    Anal. 236, 2020) to incorporate the contact angle condition. To focus on the effects
    of the necessarily singular geometry of the evolving fluid domains, we work for
    simplicity in the regime of same viscosities for the two fluids.
acknowledgement: The authors warmly thank their former resp. current PhD advisor Julian
  Fischer for the suggestion of this problem and for valuable initial discussions
  on the subjects of this paper. This project has received funding from the European
  Research Council (ERC) under the European Union’s Horizon 2020 research and innovation
  programme (grant agreement No 948819) , and from the Deutsche Forschungsgemeinschaft
  (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1
  – 390685813.
article_number: '93'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastian
  full_name: Hensel, Sebastian
  id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
  last_name: Hensel
  orcid: 0000-0001-7252-8072
- first_name: Alice
  full_name: Marveggio, Alice
  id: 25647992-AA84-11E9-9D75-8427E6697425
  last_name: Marveggio
citation:
  ama: Hensel S, Marveggio A. Weak-strong uniqueness for the Navier–Stokes equation
    for two fluids with ninety degree contact angle and same viscosities. <i>Journal
    of Mathematical Fluid Mechanics</i>. 2022;24(3). doi:<a href="https://doi.org/10.1007/s00021-022-00722-2">10.1007/s00021-022-00722-2</a>
  apa: Hensel, S., &#38; Marveggio, A. (2022). Weak-strong uniqueness for the Navier–Stokes
    equation for two fluids with ninety degree contact angle and same viscosities.
    <i>Journal of Mathematical Fluid Mechanics</i>. Springer Nature. <a href="https://doi.org/10.1007/s00021-022-00722-2">https://doi.org/10.1007/s00021-022-00722-2</a>
  chicago: Hensel, Sebastian, and Alice Marveggio. “Weak-Strong Uniqueness for the
    Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same
    Viscosities.” <i>Journal of Mathematical Fluid Mechanics</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1007/s00021-022-00722-2">https://doi.org/10.1007/s00021-022-00722-2</a>.
  ieee: S. Hensel and A. Marveggio, “Weak-strong uniqueness for the Navier–Stokes
    equation for two fluids with ninety degree contact angle and same viscosities,”
    <i>Journal of Mathematical Fluid Mechanics</i>, vol. 24, no. 3. Springer Nature,
    2022.
  ista: Hensel S, Marveggio A. 2022. Weak-strong uniqueness for the Navier–Stokes
    equation for two fluids with ninety degree contact angle and same viscosities.
    Journal of Mathematical Fluid Mechanics. 24(3), 93.
  mla: Hensel, Sebastian, and Alice Marveggio. “Weak-Strong Uniqueness for the Navier–Stokes
    Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities.”
    <i>Journal of Mathematical Fluid Mechanics</i>, vol. 24, no. 3, 93, Springer Nature,
    2022, doi:<a href="https://doi.org/10.1007/s00021-022-00722-2">10.1007/s00021-022-00722-2</a>.
  short: S. Hensel, A. Marveggio, Journal of Mathematical Fluid Mechanics 24 (2022).
date_created: 2022-08-14T22:01:45Z
date_published: 2022-08-01T00:00:00Z
date_updated: 2023-11-30T13:25:02Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00021-022-00722-2
ec_funded: 1
external_id:
  arxiv:
  - '2112.11154'
  isi:
  - '000834834300001'
file:
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has_accepted_license: '1'
intvolume: '        24'
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month: '08'
oa: 1
oa_version: Published Version
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
  call_identifier: H2020
  grant_number: '948819'
  name: Bridging Scales in Random Materials
publication: Journal of Mathematical Fluid Mechanics
publication_identifier:
  eissn:
  - 1422-6952
  issn:
  - 1422-6928
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
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scopus_import: '1'
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title: Weak-strong uniqueness for the Navier–Stokes equation for two fluids with ninety
  degree contact angle and same viscosities
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  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
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  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 24
year: '2022'
...