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Erschienen in:
The Work of Yoshihiro Shibata Kapitel lesen Erstes Kapitel lesen

2016 | OriginalPaper | Buchkapitel

On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on the Impermeable Boundary

verfasst von : Miroslav Bulíček, Josef Málek

Erschienen in: Recent Developments of Mathematical Fluid Mechanics

Verlag: Springer Basel

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Abstract

In the analysis of weak solutions relevant to evolutionary flows of incompressible fluids with non-constant viscosity or with non-linear constitutive equation, it is in general an open question whether a globally integrable pressure exists if the flows are subject to no-slip boundary conditions. Here we overcome this deficiency by considering threshold boundary conditions stating that the fluid adheres to the boundary until certain critical value for the wall shear stress is reached. Once the wall shear stress exceeds this critical value, the fluid slips. The main ingredient in our approach is to look at this type of activated, stick-slip, boundary condition as an implicit constitutive equation on the boundary.
We prove the long-time and large-data existence of weak solutions, with integrable pressure, to unsteady internal flows of Bingham and Navier-Stokes fluids subject to such threshold slip boundary conditions.

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Vorheriges Kapitel Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles in an Incompressible Solvent
Nächstes Kapitel Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity
Fußnoten
1
By weak solution, we mean here a broad class of generalized solutions such as suitable weak solution, renormalized solution, entropy solution, dissipative solution, etc.
 
2
The reason why only \(\mathbb{S}\) appears in the constitutive (1)3 is due to the fact that the fluid is incompressible.
 
3
We refer to standard interpolation inequality \(\|z\|_{\frac{2(d+2)} {d} } \leq \| z\|_{2}^{ \frac{2} {d+2} }\|z\|_{ \frac{2d} {d-2} }^{ \frac{d} {d+2} }\) valid if d ≥ 3, and \(\|z\|_{\frac{2(d+2)} {d} } \leq c\|z\|_{2}^{ \frac{2} {d+2} }\|\nabla z\|_{2}^{ \frac{d} {d+2} }\) that holds even if d ≥ 2.
 
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Metadaten
Titel
On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on the Impermeable Boundary
verfasst von
Miroslav Bulíček
Josef Málek
Copyright-Jahr
2016
Verlag
Springer Basel
Buch
Recent Developments of Mathematical Fluid Mechanics

Print ISBN: 978-3-0348-0938-2

Electronic ISBN: 978-3-0348-0939-9

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-0348-0939-9_8