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June  2014, 3(2): 331-348. doi: 10.3934/eect.2014.3.331

Shape optimization for non-Newtonian fluids in time-dependent domains

Jan Sokołowski 1, and  Jan Stebel 2,

1. 

Institut Élie Cartan Nancy, UMR 7502((Université Lorraine, CNRS, INRIA), Laboratoire de Mathématiques, Université de Lorraine, B.P.239, 54506 Vandoeuvre-lès-Nancy Cedex, France
2. 

Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 110 00 Praha 1, Czech Republic

Received  September 2013 Revised  February 2014 Published  May 2014

We study the model of an incompressible non-Newtonian fluid in a~moving domain. The domain is defined as a tube built by the velocity field $\mathbf{V}$ and described by the family of domains $\Omega_t$ parametrized by $t\in[0,T]$. A new shape optimization problem associated with the model is defined for a family of initial domains $\Omega_0$ and admissible velocity vector fields. It is shown that such shape optimization problems are well posed under the classical conditions on compactness of the admissible shapes [18]. For the state problem, we prove the existence of weak solutions and their continuity with respect to perturbations of the time-dependent boundary, provided that the power-law index $r\ge11/5$.
Keywords: Navier-Stokes equations., time-dependent domain, Shape optimization, incompressible viscous fluid.
Mathematics Subject Classification: Primary: 35Q30, 76D55; Secondary: 35R3.
Citation: Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations & Control Theory, 2014, 3 (2) : 331-348. doi: 10.3934/eect.2014.3.331
References:
[1]

N. Arada, Regularity of flows and optimal control of shear-thinning fluids,, Nonlinear Analysis: Theory, 89 (2013), 81.  doi: 10.1016/j.na.2013.04.015.  Google Scholar

[2]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006).  doi: 10.1090/memo/0852.  Google Scholar

[3]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization,, Second edition, (2011).  doi: 10.1137/1.9780898719826.  Google Scholar

[4]

M. C. Delfour and J. P. Zolésio, Oriented distance function and its evolution equation for initial sets with thin boundary,, SIAM Journal on Control and Optimization, 42 (2004), 2286.  doi: 10.1137/S0363012902411945.  Google Scholar

[5]

L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids,, Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 9 (2010), 1.   Google Scholar

[6]

R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical navier-stokes equations,, Journal of Convex Analysis, 6 (1999), 293.   Google Scholar

[7]

E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa and J. Stebel, Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time-dependent domains,, Journal of Differential Equations, 254 (2013), 125.  doi: 10.1016/j.jde.2012.08.019.  Google Scholar

[8]

E. Feireisl, J. Neustupa and J. Stebel, Convergence of a Brinkman-type penalization for compressible fluid flows,, Journal of Differential Equations, 250 (2011), 596.  doi: 10.1016/j.jde.2010.09.031.  Google Scholar

[9]

J. Frehse, J. Málek and M. Steinhauer, On existence results for fluids with shear dependent viscosity-unsteady flows,, Partial Differential Equations, 406 (2000), 121.   Google Scholar

[10]

J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the lipschitz truncation method,, SIAM Journal on Mathematical Analysis, 34 (2003), 1064.  doi: 10.1137/S0036141002410988.  Google Scholar

[11]

O. A. Ladyzhenskaya, New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems,, Trudy Mat. Inst. Steklov., 102 (1967), 85.   Google Scholar

[12]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Second English edition, (1969).   Google Scholar

[13]

O. A. Ladyzhenskaya, Initial-boundary problem for Navier-Stokes equations in domains with time-varying boundaries,, Zapiski Nauchnykh Seminarov LOMI, 11 (1968), 97.   Google Scholar

[14]

J.-L. Lions, Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires,, (French) Dunod; Gauthier-Villars, (1969).   Google Scholar

[15]

J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations,, in Evolutionary equations. Vol. II, (2005), 371.   Google Scholar

[16]

M. Moubachir and J.-P. Zolésio, Moving Shape Analysis and Control, vol. 277,, Chapman & Hall/CRC, (2006).  doi: 10.1201/9781420003246.  Google Scholar

[17]

J. Neustupa, Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method,, Mathematical Methods in the Applied Sciences, 32 (2009), 653.  doi: 10.1002/mma.1059.  Google Scholar

[18]

P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations, Theory and Shape Optimization,, Springer-Verlag, (2012).  doi: 10.1007/978-3-0348-0367-0.  Google Scholar

[19]

K. Rajagopal, Mechanics of non-Newtonian fluids,, in Recent Developments in Theoretical Fluid Mechanics (Winter School, (1992), 129.   Google Scholar

[20]

W. Schowalter, Mechanics of Non-Newtonian Fluids,, Pergamon Press, (1978).   Google Scholar

[21]

T. Slawig, Distributed control for a class of non-Newtonian fluids,, Journal of Differential Equations, 219 (2005), 116.  doi: 10.1016/j.jde.2005.03.009.  Google Scholar

[22]

J. Sokołowski and J. Stebel, Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids,, Control and Cybernetics, 40 (2011), 1077.   Google Scholar

[23]

J. Sokołowski and J. Stebel, Shape sensitivity analysis of incompressible non-Newtonian fluids,, in System Modeling and Optimization, (2013), 427.   Google Scholar

[24]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis,, Springer Series in Computational Mathematics, (1992).   Google Scholar

[25]

C. Truesdell, W. Noll and S. Antman, The Non-linear Field Theories Of Mechanics,, Springer Verlag, (2004).  doi: 10.1007/978-3-662-10388-3.  Google Scholar

[26]

D. Wachsmuth and T. Roubíček, Optimal control of planar flow of incompressible non-Newtonian fluids,, Z. Anal. Anwend., 29 (2010), 351.  doi: 10.4171/ZAA/1412.  Google Scholar

show all references

References:
[1]

N. Arada, Regularity of flows and optimal control of shear-thinning fluids,, Nonlinear Analysis: Theory, 89 (2013), 81.  doi: 10.1016/j.na.2013.04.015.  Google Scholar

[2]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006).  doi: 10.1090/memo/0852.  Google Scholar

[3]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization,, Second edition, (2011).  doi: 10.1137/1.9780898719826.  Google Scholar

[4]

M. C. Delfour and J. P. Zolésio, Oriented distance function and its evolution equation for initial sets with thin boundary,, SIAM Journal on Control and Optimization, 42 (2004), 2286.  doi: 10.1137/S0363012902411945.  Google Scholar

[5]

L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids,, Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 9 (2010), 1.   Google Scholar

[6]

R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical navier-stokes equations,, Journal of Convex Analysis, 6 (1999), 293.   Google Scholar

[7]

E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa and J. Stebel, Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time-dependent domains,, Journal of Differential Equations, 254 (2013), 125.  doi: 10.1016/j.jde.2012.08.019.  Google Scholar

[8]

E. Feireisl, J. Neustupa and J. Stebel, Convergence of a Brinkman-type penalization for compressible fluid flows,, Journal of Differential Equations, 250 (2011), 596.  doi: 10.1016/j.jde.2010.09.031.  Google Scholar

[9]

J. Frehse, J. Málek and M. Steinhauer, On existence results for fluids with shear dependent viscosity-unsteady flows,, Partial Differential Equations, 406 (2000), 121.   Google Scholar

[10]

J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the lipschitz truncation method,, SIAM Journal on Mathematical Analysis, 34 (2003), 1064.  doi: 10.1137/S0036141002410988.  Google Scholar

[11]

O. A. Ladyzhenskaya, New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems,, Trudy Mat. Inst. Steklov., 102 (1967), 85.   Google Scholar

[12]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Second English edition, (1969).   Google Scholar

[13]

O. A. Ladyzhenskaya, Initial-boundary problem for Navier-Stokes equations in domains with time-varying boundaries,, Zapiski Nauchnykh Seminarov LOMI, 11 (1968), 97.   Google Scholar

[14]

J.-L. Lions, Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires,, (French) Dunod; Gauthier-Villars, (1969).   Google Scholar

[15]

J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations,, in Evolutionary equations. Vol. II, (2005), 371.   Google Scholar

[16]

M. Moubachir and J.-P. Zolésio, Moving Shape Analysis and Control, vol. 277,, Chapman & Hall/CRC, (2006).  doi: 10.1201/9781420003246.  Google Scholar

[17]

J. Neustupa, Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method,, Mathematical Methods in the Applied Sciences, 32 (2009), 653.  doi: 10.1002/mma.1059.  Google Scholar

[18]

P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations, Theory and Shape Optimization,, Springer-Verlag, (2012).  doi: 10.1007/978-3-0348-0367-0.  Google Scholar

[19]

K. Rajagopal, Mechanics of non-Newtonian fluids,, in Recent Developments in Theoretical Fluid Mechanics (Winter School, (1992), 129.   Google Scholar

[20]

W. Schowalter, Mechanics of Non-Newtonian Fluids,, Pergamon Press, (1978).   Google Scholar

[21]

T. Slawig, Distributed control for a class of non-Newtonian fluids,, Journal of Differential Equations, 219 (2005), 116.  doi: 10.1016/j.jde.2005.03.009.  Google Scholar

[22]

J. Sokołowski and J. Stebel, Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids,, Control and Cybernetics, 40 (2011), 1077.   Google Scholar

[23]

J. Sokołowski and J. Stebel, Shape sensitivity analysis of incompressible non-Newtonian fluids,, in System Modeling and Optimization, (2013), 427.   Google Scholar

[24]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis,, Springer Series in Computational Mathematics, (1992).   Google Scholar

[25]

C. Truesdell, W. Noll and S. Antman, The Non-linear Field Theories Of Mechanics,, Springer Verlag, (2004).  doi: 10.1007/978-3-662-10388-3.  Google Scholar

[26]

D. Wachsmuth and T. Roubíček, Optimal control of planar flow of incompressible non-Newtonian fluids,, Z. Anal. Anwend., 29 (2010), 351.  doi: 10.4171/ZAA/1412.  Google Scholar

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