doi: 10.3934/eect.2021043
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Local null controllability of a class of non-Newtonian incompressible viscous fluids

1. 

Departamento de Matemática, Universidade Estadual do Piauí, Teresina, PI, 64002-150, Brasil

2. 

Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, RJ, 24210-200, Brasil

3. 

Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, RJ, Niterói, RJ, 24210-200, Brasil

* Corresponding author: Juan Límaco (jlimaco@id.uff.br)

Received  October 2020 Revised  June 2021 Early access August 2021

Fund Project: Funding: This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001

We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.

Citation: Pitágoras Pinheiro de Carvalho, Juan Límaco, Denilson Menezes, Yuri Thamsten. Local null controllability of a class of non-Newtonian incompressible viscous fluids. Evolution Equations and Control Theory, doi: 10.3934/eect.2021043
References:
[1]

V. Alekseev, V. Tikhomirov and S. Fomin, Optimal Control, Contemporary Soviet Mathematics. Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.

[2]

H. Beirão da Veiga, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or non-slip boundary conditions, Comm. Pure Appl. Math., 58 (2005), 552-577.  doi: 10.1002/cpa.20036.

[3]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[4]

P. Carreau, Rheological Equations of State from Molecular Network Theories, PhD thesis, PhD thesis, University of Wisconsin, Madison, 1968.

[5]

N. Carreno and S. Guerrero, Local null controllability of the n-dimensional Navier–Stokes system with n- 1 scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2.

[6]

F. W. Chaves-Silva and S. Guerrero, A uniform controllability result for the keller–segel system, Asympto. Anal., 92 (2015), 313-338.  doi: 10.3233/ASY-141282.

[7]

Y. Cho and K. Kensey, Effects of The Non-Newtonian Viscosity of Blood on Hemodynamics of Diiseased Arterial Flows, vol. 1, Prat, 1989.

[8]

Y. I. Cho and K. R. Kensey, Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: Steady flows, Biorheology, 28 (1991), 241-262.  doi: 10.3233/BIR-1991-283-415.

[9]

E. Christiansen and S. Kelsey, Isothermal and nonisothermal, laminar, inelastic, non-newtonian tube-entrance flow following a contraction, Chemical Engineering Science, 28 (1973), 1099-1113.  doi: 10.1016/0009-2509(73)80013-2.

[10]

G. R. CokeletE. MerrillE. GillilandH. ShinA. Britten and R. Wells Jr, The rheology of human blood measurement near and at zero shear rate, Transactions of the Society of Rheology, 7 (1963), 303-317.  doi: 10.1122/1.548959.

[11]

J.-M. Coron and S. Guerrero, Null controllability of the n-dimensional Stokes system with n- 1 scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019.

[12]

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier–Stokes system with a distributed control having two vanishing components, Inven. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.

[13]

J.-M. CoronF. Marbach and F. Sueur, Small-time global exact controllability of the navier-stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc. (JEMS), 22 (2020), 1625-1673.  doi: 10.4171/JEMS/952.

[14]

M. M. Cross, Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems, Journal of Colloid Science, 20 (1965), 417-437.  doi: 10.1016/0095-8522(65)90022-X.

[15]

P. Davies, A. Mazher, D. Giddens, C. Zarins and S. Glagov, Effects of nonnewtonian fluid behavior on wall shear in a separated flow region, Proc. 1st World Conf. of Biomech, 1 (1990), 301.

[16]

Q. Du and M. D. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Ana. Appl., 155 (1991), 21-45.  doi: 10.1016/0022-247X(91)90024-T.

[17]

A. el GibalyO. A. El-BassiounyO. DiaaA. I. ShehataT. Hassan and K. M. Saqr, Effects of non-newtonian viscosity on the hemodynamics of cerebral aneurysms, Applied Mechanics and Materials, 819 (2016), 366-370.  doi: 10.4028/www.scientific.net/AMM.819.366.

[18]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[19]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl., 83 (2004), 1501-1542.  doi: 10.1016/j.matpur.2004.02.010.

[20]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Some controllability results for the n-dimensional Navier–Stokes and Boussinesq systems with n-1 scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X.

[21]

E. Fernández-CaraJ. Limaco and S. de Menezes, Theoretical and numerical local null controllability of a Ladyzhenskaya–Smagorinsky model of turbulence, J. Math. Fluid Mech., 17 (2015), 669-698.  doi: 10.1007/s00021-015-0232-7.

[22]

E. Fernández-CaraA. Münch and D. A. Souza, On the numerical controllability of the two-dimensional heat, Stokes and Navier–Stokes equations, J. Sci. Comput., 70 (2017), 819-858.  doi: 10.1007/s10915-016-0266-x.

[23]

E. Fernández-CaraD. Nina-HuamánM. R. Nuñez-Chávez and F. B. Vieira, On the theoretical and numerical control of a one-dimensional nonlinear parabolic partial differential equation, J. Optim. Theory Appl., 175 (2017), 652-682.  doi: 10.1007/s10957-017-1190-4.

[24]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[25]

B. KjartansonD. ShieldsL. Domaschuk and C.-S. Man, The creep of ice measured with the pressuremeter, Canadian Geotechnical Journal, 25 (1988), 250-261.  doi: 10.1139/t88-029.

[26]

O. Ladyzhenskaya, New equations for the description of the viscoue incompressible fluids and solvability in the large of the boundary value problems for them, Boundary Value Problem of Mathematical Physics V, Amer. Math. Soc.: Providence.

[27]

O. Ladyzhenskaya, Modification of the Navier–Stokes equations for large velocity gradients, Seminars in Mathematics VA Stheklov Mathematical Institute, 7 (1970).

[28]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[29]

W. Layton, Energy dissipation in the smagorinsky model of turbulence, Appl. Math. Lett., 59 (2016), 56-59.  doi: 10.1016/j.aml.2016.03.008.

[30]

M. Lesieur, Turbulence in Fluids: Stochastic And Numerical Modelling, Mechanics of Fluids and Transport Processes, Martinus Nijhoff Publishers, Dordrecht, 1987. doi: 10.1007/978-94-009-3545-7.

[31]

J.-L. Lions, Quelques Méthodes de Résolution des Problemes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris 1969.

[32]

X. Liu and X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations, SIAM J. Control Opti., 50 (2012), 2046-2064.  doi: 10.1137/110851808.

[33]

J. MálekJ. Nečas and M. Rŭžička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci., 3 (1993), 35-63.  doi: 10.1142/S0218202593000047.

[34]

J. MálekJ. Necas and M. Ruzicka, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case $ p \geq 2$, Adv. Differential Equations, 6 (2001), 257-302. 

[35]

J. MálekK. R. Rajagopal and M. Rŭžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci., 5 (1995), 789-812.  doi: 10.1142/S0218202595000449.

[36]

A. V. Malevsky and D. A. Yuen, Strongly chaotic non-newtonian mantle convection, Geophysical & Astrophysical Fluid Dynamics, 65 (1992), 149-171.  doi: 10.1080/03091929208225244.

[37]

A. Metzner, Non-newtonian technology: Fluid mechanics, mixing, and heat transfer, Advances in Chemical Engineering, 1 (1956), 77-153.  doi: 10.1016/S0065-2377(08)60311-7.

[38]

S. Micu and T. Takahashi, Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity, J. Differential Equations, 264 (2018), 3664-3703.  doi: 10.1016/j.jde.2017.11.029.

[39]

M. Nakamura and T. Sawada, Numerical study on the flow of a non-newtonian fluid through an axisymmetric stenosis, J. Biomech. Eng., 110 (1988), 137-143.  doi: 10.1115/1.3108418.

[40]

M. Pokornỳ, Cauchy problem for the non-newtonian viscous incompressible fluid, Appl. Math., 41 (1996), 169-201. 

[41]

R. E. Powell and H. Eyring, Mechanisms for the relaxation theory of viscosity, Nature, 154 (1944), 427-428.  doi: 10.1038/154427a0.

[42]

L. Prandtl, Guide à Travers La mécanique des Fluides, Dunod, 1952.

[43]

T. C. Rebollo and R. Lewandowski, Mathematical and Numerical Foundations of Turbulence Models and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2014. doi: 10.1007/978-1-4939-0455-6.

[44]

F. ReeT. Ree and H. Eyring, Relaxation theory of transport problems in condensed systems, Industrial & Engineering Chemistry, 50 (1958), 1036-1040. 

[45]

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science & Business Media, 2000.

[46]

J. Smagorinsky, General circulation experiments with the primitive equations: I. the basic experiment, Monthly Weather Review, 91 (1963), 99-164.  doi: 10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.

[47]

H. Steffan, W. Brandstätter, G. Bachler and R. Pucher, Comparison of newtonian and non-newtonian blood flow in stenotic vessels using numerical simulation, Biofluid Mechanics, Springer, (1990), 479–485. doi: 10.1007/978-3-642-52338-0_61.

[48]

J. Sutterby, Laminar converging flow of dilute polymer solutions in conical sections: Part i. viscosity data, new viscosity model, tube flow solution, AIChE Journal, 12 (1966), 63-68.  doi: 10.1002/aic.690120114.

[49]

J. L. Sutterby, Laminar converging flow of dilute polymer solutions in conical sections. II, Transactions of the Society of Rheology, 9 (1965), 227-241.  doi: 10.1122/1.549024.

[50]

L. Tartar, An Introduction to Navier-Stokes Equation and Oceanography, Lecture Notes of the Unione Matematica Italiana, 1. Springer-Verlag, Berlin; UMI, Bologna, 2006. doi: 10.1007/3-540-36545-1.

[51]

R. Temam and A. Chorin, Navier Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[52]

R. M. Turian, The critical stress in frictionally heated non-Newtonian plane Couette flow, Chemical Engineering Science, 24 (1969), 1581-1587.  doi: 10.1016/0009-2509(69)80097-7.

[53]

C. Van Der Veen and I. Whillans, New and improved determinations of the velocity of ice stream-b and ice stream-c, West Antartica J. Glaciology, 36 (1990), 324-339. 

show all references

References:
[1]

V. Alekseev, V. Tikhomirov and S. Fomin, Optimal Control, Contemporary Soviet Mathematics. Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.

[2]

H. Beirão da Veiga, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or non-slip boundary conditions, Comm. Pure Appl. Math., 58 (2005), 552-577.  doi: 10.1002/cpa.20036.

[3]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[4]

P. Carreau, Rheological Equations of State from Molecular Network Theories, PhD thesis, PhD thesis, University of Wisconsin, Madison, 1968.

[5]

N. Carreno and S. Guerrero, Local null controllability of the n-dimensional Navier–Stokes system with n- 1 scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2.

[6]

F. W. Chaves-Silva and S. Guerrero, A uniform controllability result for the keller–segel system, Asympto. Anal., 92 (2015), 313-338.  doi: 10.3233/ASY-141282.

[7]

Y. Cho and K. Kensey, Effects of The Non-Newtonian Viscosity of Blood on Hemodynamics of Diiseased Arterial Flows, vol. 1, Prat, 1989.

[8]

Y. I. Cho and K. R. Kensey, Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: Steady flows, Biorheology, 28 (1991), 241-262.  doi: 10.3233/BIR-1991-283-415.

[9]

E. Christiansen and S. Kelsey, Isothermal and nonisothermal, laminar, inelastic, non-newtonian tube-entrance flow following a contraction, Chemical Engineering Science, 28 (1973), 1099-1113.  doi: 10.1016/0009-2509(73)80013-2.

[10]

G. R. CokeletE. MerrillE. GillilandH. ShinA. Britten and R. Wells Jr, The rheology of human blood measurement near and at zero shear rate, Transactions of the Society of Rheology, 7 (1963), 303-317.  doi: 10.1122/1.548959.

[11]

J.-M. Coron and S. Guerrero, Null controllability of the n-dimensional Stokes system with n- 1 scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019.

[12]

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier–Stokes system with a distributed control having two vanishing components, Inven. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.

[13]

J.-M. CoronF. Marbach and F. Sueur, Small-time global exact controllability of the navier-stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc. (JEMS), 22 (2020), 1625-1673.  doi: 10.4171/JEMS/952.

[14]

M. M. Cross, Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems, Journal of Colloid Science, 20 (1965), 417-437.  doi: 10.1016/0095-8522(65)90022-X.

[15]

P. Davies, A. Mazher, D. Giddens, C. Zarins and S. Glagov, Effects of nonnewtonian fluid behavior on wall shear in a separated flow region, Proc. 1st World Conf. of Biomech, 1 (1990), 301.

[16]

Q. Du and M. D. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Ana. Appl., 155 (1991), 21-45.  doi: 10.1016/0022-247X(91)90024-T.

[17]

A. el GibalyO. A. El-BassiounyO. DiaaA. I. ShehataT. Hassan and K. M. Saqr, Effects of non-newtonian viscosity on the hemodynamics of cerebral aneurysms, Applied Mechanics and Materials, 819 (2016), 366-370.  doi: 10.4028/www.scientific.net/AMM.819.366.

[18]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[19]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl., 83 (2004), 1501-1542.  doi: 10.1016/j.matpur.2004.02.010.

[20]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Some controllability results for the n-dimensional Navier–Stokes and Boussinesq systems with n-1 scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X.

[21]

E. Fernández-CaraJ. Limaco and S. de Menezes, Theoretical and numerical local null controllability of a Ladyzhenskaya–Smagorinsky model of turbulence, J. Math. Fluid Mech., 17 (2015), 669-698.  doi: 10.1007/s00021-015-0232-7.

[22]

E. Fernández-CaraA. Münch and D. A. Souza, On the numerical controllability of the two-dimensional heat, Stokes and Navier–Stokes equations, J. Sci. Comput., 70 (2017), 819-858.  doi: 10.1007/s10915-016-0266-x.

[23]

E. Fernández-CaraD. Nina-HuamánM. R. Nuñez-Chávez and F. B. Vieira, On the theoretical and numerical control of a one-dimensional nonlinear parabolic partial differential equation, J. Optim. Theory Appl., 175 (2017), 652-682.  doi: 10.1007/s10957-017-1190-4.

[24]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[25]

B. KjartansonD. ShieldsL. Domaschuk and C.-S. Man, The creep of ice measured with the pressuremeter, Canadian Geotechnical Journal, 25 (1988), 250-261.  doi: 10.1139/t88-029.

[26]

O. Ladyzhenskaya, New equations for the description of the viscoue incompressible fluids and solvability in the large of the boundary value problems for them, Boundary Value Problem of Mathematical Physics V, Amer. Math. Soc.: Providence.

[27]

O. Ladyzhenskaya, Modification of the Navier–Stokes equations for large velocity gradients, Seminars in Mathematics VA Stheklov Mathematical Institute, 7 (1970).

[28]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

[29]

W. Layton, Energy dissipation in the smagorinsky model of turbulence, Appl. Math. Lett., 59 (2016), 56-59.  doi: 10.1016/j.aml.2016.03.008.

[30]

M. Lesieur, Turbulence in Fluids: Stochastic And Numerical Modelling, Mechanics of Fluids and Transport Processes, Martinus Nijhoff Publishers, Dordrecht, 1987. doi: 10.1007/978-94-009-3545-7.

[31]

J.-L. Lions, Quelques Méthodes de Résolution des Problemes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris 1969.

[32]

X. Liu and X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations, SIAM J. Control Opti., 50 (2012), 2046-2064.  doi: 10.1137/110851808.

[33]

J. MálekJ. Nečas and M. Rŭžička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci., 3 (1993), 35-63.  doi: 10.1142/S0218202593000047.

[34]

J. MálekJ. Necas and M. Ruzicka, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case $ p \geq 2$, Adv. Differential Equations, 6 (2001), 257-302. 

[35]

J. MálekK. R. Rajagopal and M. Rŭžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci., 5 (1995), 789-812.  doi: 10.1142/S0218202595000449.

[36]

A. V. Malevsky and D. A. Yuen, Strongly chaotic non-newtonian mantle convection, Geophysical & Astrophysical Fluid Dynamics, 65 (1992), 149-171.  doi: 10.1080/03091929208225244.

[37]

A. Metzner, Non-newtonian technology: Fluid mechanics, mixing, and heat transfer, Advances in Chemical Engineering, 1 (1956), 77-153.  doi: 10.1016/S0065-2377(08)60311-7.

[38]

S. Micu and T. Takahashi, Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity, J. Differential Equations, 264 (2018), 3664-3703.  doi: 10.1016/j.jde.2017.11.029.

[39]

M. Nakamura and T. Sawada, Numerical study on the flow of a non-newtonian fluid through an axisymmetric stenosis, J. Biomech. Eng., 110 (1988), 137-143.  doi: 10.1115/1.3108418.

[40]

M. Pokornỳ, Cauchy problem for the non-newtonian viscous incompressible fluid, Appl. Math., 41 (1996), 169-201. 

[41]

R. E. Powell and H. Eyring, Mechanisms for the relaxation theory of viscosity, Nature, 154 (1944), 427-428.  doi: 10.1038/154427a0.

[42]

L. Prandtl, Guide à Travers La mécanique des Fluides, Dunod, 1952.

[43]

T. C. Rebollo and R. Lewandowski, Mathematical and Numerical Foundations of Turbulence Models and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2014. doi: 10.1007/978-1-4939-0455-6.

[44]

F. ReeT. Ree and H. Eyring, Relaxation theory of transport problems in condensed systems, Industrial & Engineering Chemistry, 50 (1958), 1036-1040. 

[45]

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science & Business Media, 2000.

[46]

J. Smagorinsky, General circulation experiments with the primitive equations: I. the basic experiment, Monthly Weather Review, 91 (1963), 99-164.  doi: 10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.

[47]

H. Steffan, W. Brandstätter, G. Bachler and R. Pucher, Comparison of newtonian and non-newtonian blood flow in stenotic vessels using numerical simulation, Biofluid Mechanics, Springer, (1990), 479–485. doi: 10.1007/978-3-642-52338-0_61.

[48]

J. Sutterby, Laminar converging flow of dilute polymer solutions in conical sections: Part i. viscosity data, new viscosity model, tube flow solution, AIChE Journal, 12 (1966), 63-68.  doi: 10.1002/aic.690120114.

[49]

J. L. Sutterby, Laminar converging flow of dilute polymer solutions in conical sections. II, Transactions of the Society of Rheology, 9 (1965), 227-241.  doi: 10.1122/1.549024.

[50]

L. Tartar, An Introduction to Navier-Stokes Equation and Oceanography, Lecture Notes of the Unione Matematica Italiana, 1. Springer-Verlag, Berlin; UMI, Bologna, 2006. doi: 10.1007/3-540-36545-1.

[51]

R. Temam and A. Chorin, Navier Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[52]

R. M. Turian, The critical stress in frictionally heated non-Newtonian plane Couette flow, Chemical Engineering Science, 24 (1969), 1581-1587.  doi: 10.1016/0009-2509(69)80097-7.

[53]

C. Van Der Veen and I. Whillans, New and improved determinations of the velocity of ice stream-b and ice stream-c, West Antartica J. Glaciology, 36 (1990), 324-339. 

Figure 1.  Meshes. The volume is nine, there are $ 8748 $ tetrahedra, $ 1810 $ vertices, and $ 1944 $ boundary triangles
Figure 2.  Initial condition $ y_0 $
Figure 3.  Cross sections $ x_1 = 0.9 $ of the control components
Figure 4.  Cross sections $ x_1 = 2.1 $ of the control components
Figure 5.  Cross sections $ x_1 = 0.9 $ of the state components
Figure 6.  Cross sections $ x_1 = 2.1 $ of the state components
Figure 7.  Time evolution of the control (left panel) and state (right panel) norms. We referred to the sum of the norms of the components as "Modulus"
Figure 8.  The control components at time zero
Figure 9.  Heat maps of the control components at $ t = 0.15. $
Figure 10.  Heat maps of the first state component at diverse times
Figure 11.  Heat maps of the second state component at diverse times
Table 1.  Data we used in the simulations. Above, we denote by $ \psi_0 $ the initial streamline function, which we fix as $ \psi_0(x_1, x_2) = (x_1x_2)^2(1-x_1)^2(1-x_2)^2. $
$ N $ $ \Omega $ $ \omega $ $ T $ $ y_0 $ $ \nu_0 $ $ \nu_1 $ $ r $
$ 2 $ $ \left]0, 3\right[^2 $ $ \left]0.5, 2.5\right[^2 $ $ 1 $ $ \nabla \times \psi_0 $ $ 100 $ $ 0.01 $ $ 2 $
$ N $ $ \Omega $ $ \omega $ $ T $ $ y_0 $ $ \nu_0 $ $ \nu_1 $ $ r $
$ 2 $ $ \left]0, 3\right[^2 $ $ \left]0.5, 2.5\right[^2 $ $ 1 $ $ \nabla \times \psi_0 $ $ 100 $ $ 0.01 $ $ 2 $
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