# American Institute of Mathematical Sciences

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

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

 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 $r\ge11/5$.
Citation: Jan Sokołowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains. Evolution Equations and 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, Methods & Applications, 89 (2013), 81-94. doi: 10.1016/j.na.2013.04.015. [2] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), x+128pp. doi: 10.1090/memo/0852. [3] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second edition, Advances in Design and Control, 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719826. [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), 2286-2304. doi: 10.1137/S0363012902411945. [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), 1-46. [6] R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical navier-stokes equations, Journal of Convex Analysis, 6 (1999), 293-318. [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), 125-140. doi: 10.1016/j.jde.2012.08.019. [8] E. Feireisl, J. Neustupa and J. Stebel, Convergence of a Brinkman-type penalization for compressible fluid flows, Journal of Differential Equations, 250 (2011), 596-606. doi: 10.1016/j.jde.2010.09.031. [9] J. Frehse, J. Málek and M. Steinhauer, On existence results for fluids with shear dependent viscosity-unsteady flows, Partial Differential Equations, Theory and Numerical Solution, 406 (2000), 121-129. [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), 1064-1083. doi: 10.1137/S0036141002410988. [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), 85-104. [12] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged, Translated from the Russian by Richard A. Silverman and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York, 1969. [13] O. A. Ladyzhenskaya, Initial-boundary problem for Navier-Stokes equations in domains with time-varying boundaries, Zapiski Nauchnykh Seminarov LOMI, 11 (1968), 97-128. [14] J.-L. Lions, Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires, (French) Dunod; Gauthier-Villars, 1969. [15] J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Evolutionary equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2005), 371-459. [16] M. Moubachir and J.-P. Zolésio, Moving Shape Analysis and Control, vol. 277, Chapman & Hall/CRC, Boca Raton, FL, 2006. doi: 10.1201/9781420003246. [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), 653-683. doi: 10.1002/mma.1059. [18] P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations, Theory and Shape Optimization, Springer-Verlag, Basel, 2012. doi: 10.1007/978-3-0348-0367-0. [19] K. Rajagopal, Mechanics of non-Newtonian fluids, in Recent Developments in Theoretical Fluid Mechanics (Winter School, Paseky, 1992), Pitman Res. Notes Math. Ser., 291, Longman Sci. Tech., Harlow, 1993, 129-162. [20] W. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, 1978. [21] T. Slawig, Distributed control for a class of non-Newtonian fluids, Journal of Differential Equations, 219 (2005), 116-143. doi: 10.1016/j.jde.2005.03.009. [22] J. Sokołowski and J. Stebel, Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids, Control and Cybernetics, 40 (2011), 1077-1097. [23] J. Sokołowski and J. Stebel, Shape sensitivity analysis of incompressible non-Newtonian fluids, in System Modeling and Optimization, Springer, 2013, 427-436. [24] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. [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. [26] D. Wachsmuth and T. Roubíček, Optimal control of planar flow of incompressible non-Newtonian fluids, Z. Anal. Anwend., 29 (2010), 351-376. doi: 10.4171/ZAA/1412.

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##### References:
 [1] N. Arada, Regularity of flows and optimal control of shear-thinning fluids, Nonlinear Analysis: Theory, Methods & Applications, 89 (2013), 81-94. doi: 10.1016/j.na.2013.04.015. [2] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), x+128pp. doi: 10.1090/memo/0852. [3] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second edition, Advances in Design and Control, 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719826. [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), 2286-2304. doi: 10.1137/S0363012902411945. [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), 1-46. [6] R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical navier-stokes equations, Journal of Convex Analysis, 6 (1999), 293-318. [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), 125-140. doi: 10.1016/j.jde.2012.08.019. [8] E. Feireisl, J. Neustupa and J. Stebel, Convergence of a Brinkman-type penalization for compressible fluid flows, Journal of Differential Equations, 250 (2011), 596-606. doi: 10.1016/j.jde.2010.09.031. [9] J. Frehse, J. Málek and M. Steinhauer, On existence results for fluids with shear dependent viscosity-unsteady flows, Partial Differential Equations, Theory and Numerical Solution, 406 (2000), 121-129. [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), 1064-1083. doi: 10.1137/S0036141002410988. [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), 85-104. [12] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged, Translated from the Russian by Richard A. Silverman and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York, 1969. [13] O. A. Ladyzhenskaya, Initial-boundary problem for Navier-Stokes equations in domains with time-varying boundaries, Zapiski Nauchnykh Seminarov LOMI, 11 (1968), 97-128. [14] J.-L. Lions, Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires, (French) Dunod; Gauthier-Villars, 1969. [15] J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Evolutionary equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2005), 371-459. [16] M. Moubachir and J.-P. Zolésio, Moving Shape Analysis and Control, vol. 277, Chapman & Hall/CRC, Boca Raton, FL, 2006. doi: 10.1201/9781420003246. [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), 653-683. doi: 10.1002/mma.1059. [18] P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations, Theory and Shape Optimization, Springer-Verlag, Basel, 2012. doi: 10.1007/978-3-0348-0367-0. [19] K. Rajagopal, Mechanics of non-Newtonian fluids, in Recent Developments in Theoretical Fluid Mechanics (Winter School, Paseky, 1992), Pitman Res. Notes Math. Ser., 291, Longman Sci. Tech., Harlow, 1993, 129-162. [20] W. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, 1978. [21] T. Slawig, Distributed control for a class of non-Newtonian fluids, Journal of Differential Equations, 219 (2005), 116-143. doi: 10.1016/j.jde.2005.03.009. [22] J. Sokołowski and J. Stebel, Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids, Control and Cybernetics, 40 (2011), 1077-1097. [23] J. Sokołowski and J. Stebel, Shape sensitivity analysis of incompressible non-Newtonian fluids, in System Modeling and Optimization, Springer, 2013, 427-436. [24] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. [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. [26] D. Wachsmuth and T. Roubíček, Optimal control of planar flow of incompressible non-Newtonian fluids, Z. Anal. Anwend., 29 (2010), 351-376. doi: 10.4171/ZAA/1412.
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