Page 512 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 512
MULTISCALE MODELING AND METHODS: APPLICATION IN ENGINEERING,
BIOLOGY AND MEDICINE (MS-80)
Steady state non-Newtonian flow with strain rate dependent viscosity in
thin tube structure with no slip boundary condition
Grigory Panasenko, grigory.panasenko@univ-st-etienne.fr
Université Jean Monnet, France, and Vilnius University, Lithuania
Coauthors: Bogdan Vernescu, Konstantinas Pileckas
Thin tube structures are finite unions of thin cylinders depending on the small parameter, ratio
of the diameter of the cross section to the length of the cylinder. Flows in such domains model
blood flow in a network of vessels. The asymptotic expansion of the solution of the steady
Stokes and Navier-Stokes equations in these domains with no slip boundary condition was con-
structed in the papers [1], [2] and the book [3]. However, the blood exhibits a non-Newtonian
rheology, when the viscosity depends on the strain rate. In the present talk we consider such
rheology. Applying the Banach fixed point theorem we prove the existence and uniqueness of
a solution and its regularity. An asymptotic approximation is constructed and justified by an
error estimate. The first and the second authors are supported by the European Social Fund
(project No 09.3.3-LMT-K-712-01-0012) under grant agreement with the Research Council of
Lithuania (LMTLT).
References
[1] G. Panasenko, Asymptotic expansion of the solution of Navier-Stokes equation in a tube
structure, C.R. Acad. Sci. Paris, 326, Série IIb, 1998, 867-872.
[2] F. Blanc, O. Gipouloux, G. Panasenko, A.M. Zine, Asymptotic analysis and partial asymp-
totic decomposition of the domain for Stokes Equation in tube structure, Mathematical
Models and Methods in Applied Sciences, 1999, Vol. 9, 9, 1351-1378.
[3] G. Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht,
2005.
Unsteady micropolar fluid flow in a thick domain with multiscale
oscillating roughness and a subdifferential boundary condition
Laetitia Paoli, laetitia.paoli@univ-st-etienne.fr
Université de Saint-Etienne, France
Motivated by lubrication problems involving complex fluids we consider an unsteady micropo-
lar fluid flow in a two dimentional thick domain Ωε. Following [2] the problem is thus described
by a non-linear coupled variational system for the fluid velocity vε, the pressure pε and the an-
gular micro-rotation field Zε. We assume moreover a fluid-solid interface law of friction type
modelled by a subdifferential condition (see [3]).
Existence, uniqueness and uniform estimates for (vε, pε, Zε) are derived. Then we assume
that the thickness and roughness of Ωε are described by multiple separated scales of periodic
oscillations i.e.
Ωε = (z1, z2) : 0 < z1 < L, 0 < z2 < εmhε(z1)
with hε(z1) = h(z1, z1 , ,z1 · ·· , z1 ), 0 < ε << 1 and m ≥ 2.
ε εm
ε2
510
BIOLOGY AND MEDICINE (MS-80)
Steady state non-Newtonian flow with strain rate dependent viscosity in
thin tube structure with no slip boundary condition
Grigory Panasenko, grigory.panasenko@univ-st-etienne.fr
Université Jean Monnet, France, and Vilnius University, Lithuania
Coauthors: Bogdan Vernescu, Konstantinas Pileckas
Thin tube structures are finite unions of thin cylinders depending on the small parameter, ratio
of the diameter of the cross section to the length of the cylinder. Flows in such domains model
blood flow in a network of vessels. The asymptotic expansion of the solution of the steady
Stokes and Navier-Stokes equations in these domains with no slip boundary condition was con-
structed in the papers [1], [2] and the book [3]. However, the blood exhibits a non-Newtonian
rheology, when the viscosity depends on the strain rate. In the present talk we consider such
rheology. Applying the Banach fixed point theorem we prove the existence and uniqueness of
a solution and its regularity. An asymptotic approximation is constructed and justified by an
error estimate. The first and the second authors are supported by the European Social Fund
(project No 09.3.3-LMT-K-712-01-0012) under grant agreement with the Research Council of
Lithuania (LMTLT).
References
[1] G. Panasenko, Asymptotic expansion of the solution of Navier-Stokes equation in a tube
structure, C.R. Acad. Sci. Paris, 326, Série IIb, 1998, 867-872.
[2] F. Blanc, O. Gipouloux, G. Panasenko, A.M. Zine, Asymptotic analysis and partial asymp-
totic decomposition of the domain for Stokes Equation in tube structure, Mathematical
Models and Methods in Applied Sciences, 1999, Vol. 9, 9, 1351-1378.
[3] G. Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht,
2005.
Unsteady micropolar fluid flow in a thick domain with multiscale
oscillating roughness and a subdifferential boundary condition
Laetitia Paoli, laetitia.paoli@univ-st-etienne.fr
Université de Saint-Etienne, France
Motivated by lubrication problems involving complex fluids we consider an unsteady micropo-
lar fluid flow in a two dimentional thick domain Ωε. Following [2] the problem is thus described
by a non-linear coupled variational system for the fluid velocity vε, the pressure pε and the an-
gular micro-rotation field Zε. We assume moreover a fluid-solid interface law of friction type
modelled by a subdifferential condition (see [3]).
Existence, uniqueness and uniform estimates for (vε, pε, Zε) are derived. Then we assume
that the thickness and roughness of Ωε are described by multiple separated scales of periodic
oscillations i.e.
Ωε = (z1, z2) : 0 < z1 < L, 0 < z2 < εmhε(z1)
with hε(z1) = h(z1, z1 , ,z1 · ·· , z1 ), 0 < ε << 1 and m ≥ 2.
ε εm
ε2
510
