Page 503 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 503
MATHEMATICAL ANALYSIS: THE INTERACTION OF FLUIDS/ VISCOELASTIC
MATERIALS AND SOLIDS (MS-36)
Optimal boundary control for steady motions of a self-propelled body in a
Navier-Stokes liquid
Ana Leonor Silvestre, ana.silvestre@math.tecnico.ulisboa.pt
Instituto Superior Técnico, Universidade de Lisboa, Portugal
Consider a rigid body S ⊂ R3 immersed in a Navier-Stokes liquid and the motion of the body-
fluid interaction system described from a reference frame attached to S. We are interested in
steady motions of this coupled system, where the region occupied by the fluid is the exterior
domain Ω = R3 \ S. An important question that arises in this context is: How can a self-
propelled motion of S with a target velocity V (x) := ξ + ω × x be generated in such a way that
the drag about S is minimized?
We solve this problem using boundary controls v∗, acting on the whole ∂Ω or just on a
portion Γ of ∂Ω. Firstly, an appropriate drag functional is derived from the energy equation of
the fluid and the problem is formulated as an optimal control problem.
The drag minimization problem is solved for localized controls, such that supp v∗ ⊂ Γ, and
for tangential controls, i.e, v∗ · n|∂Ω = 0, where n is the outward unit normal to ∂Ω. Under
smallness restrictions on the objectives |ξ| and |ω| and on the boundary controls, we prove
the existence of optimal solutions, justify the Gâteaux derivative of the control-to-state map,
establish the well-posedness of the corresponding adjoint equations and, finally, derive the first
order optimality conditions.
This is joint work with Toshiaki Hishida (Nagoya University, Japan) and Takéo Takahashi
(Université de Lorraine, CNRS, Inria, IECL, Nancy, France).
An energetic variational approach for wormlike micelle solutions: Coarse
graining and dynamic stability
Yiwei Wang, ywang487@iit.edu
Illinois Institute of Technology, United States
Coauthors: Chun Liu, Teng-Fei Zhang
Wormlike micelles are self-assemblies of polymer chains that can break and recombine re-
versibly. In this talk, we present a thermodynamically consistent two-species micro-macro
model of wormlike micellar solutions by employing an energetic variational approach. The
model incorporates a breakage and combination process of polymer chains into the classical
micro-macro dumbbell model of polymeric fluids in a unified variational framework. The mod-
eling approach can be applied to other reactive or active complex fluids. Different maximum
entropy closure approximations to the new model will be discussed. By imposing a proper dis-
sipation in the coarse-grained level, the closure model, obtained by “closure-then-variation”,
preserves the thermodynamical structure of both mechanical and chemical parts of the original
system. The resulting model is an Oldroyd-B type system coupled with a chemical reaction.
We’ll also present the dynamic stability analysis on the micro-macro model. In particular, we
show the global existence of classical solutions near the global equilibrium, which indicates the
consistency between the detailed balance conditions in a chemical reaction and the global equi-
librium state of each species. The is joint work with Prof. Chun Liu (IIT) and Prof. Teng-Fei
Zhang (CUG).
501
MATERIALS AND SOLIDS (MS-36)
Optimal boundary control for steady motions of a self-propelled body in a
Navier-Stokes liquid
Ana Leonor Silvestre, ana.silvestre@math.tecnico.ulisboa.pt
Instituto Superior Técnico, Universidade de Lisboa, Portugal
Consider a rigid body S ⊂ R3 immersed in a Navier-Stokes liquid and the motion of the body-
fluid interaction system described from a reference frame attached to S. We are interested in
steady motions of this coupled system, where the region occupied by the fluid is the exterior
domain Ω = R3 \ S. An important question that arises in this context is: How can a self-
propelled motion of S with a target velocity V (x) := ξ + ω × x be generated in such a way that
the drag about S is minimized?
We solve this problem using boundary controls v∗, acting on the whole ∂Ω or just on a
portion Γ of ∂Ω. Firstly, an appropriate drag functional is derived from the energy equation of
the fluid and the problem is formulated as an optimal control problem.
The drag minimization problem is solved for localized controls, such that supp v∗ ⊂ Γ, and
for tangential controls, i.e, v∗ · n|∂Ω = 0, where n is the outward unit normal to ∂Ω. Under
smallness restrictions on the objectives |ξ| and |ω| and on the boundary controls, we prove
the existence of optimal solutions, justify the Gâteaux derivative of the control-to-state map,
establish the well-posedness of the corresponding adjoint equations and, finally, derive the first
order optimality conditions.
This is joint work with Toshiaki Hishida (Nagoya University, Japan) and Takéo Takahashi
(Université de Lorraine, CNRS, Inria, IECL, Nancy, France).
An energetic variational approach for wormlike micelle solutions: Coarse
graining and dynamic stability
Yiwei Wang, ywang487@iit.edu
Illinois Institute of Technology, United States
Coauthors: Chun Liu, Teng-Fei Zhang
Wormlike micelles are self-assemblies of polymer chains that can break and recombine re-
versibly. In this talk, we present a thermodynamically consistent two-species micro-macro
model of wormlike micellar solutions by employing an energetic variational approach. The
model incorporates a breakage and combination process of polymer chains into the classical
micro-macro dumbbell model of polymeric fluids in a unified variational framework. The mod-
eling approach can be applied to other reactive or active complex fluids. Different maximum
entropy closure approximations to the new model will be discussed. By imposing a proper dis-
sipation in the coarse-grained level, the closure model, obtained by “closure-then-variation”,
preserves the thermodynamical structure of both mechanical and chemical parts of the original
system. The resulting model is an Oldroyd-B type system coupled with a chemical reaction.
We’ll also present the dynamic stability analysis on the micro-macro model. In particular, we
show the global existence of classical solutions near the global equilibrium, which indicates the
consistency between the detailed balance conditions in a chemical reaction and the global equi-
librium state of each species. The is joint work with Prof. Chun Liu (IIT) and Prof. Teng-Fei
Zhang (CUG).
501
