# American Institute of Mathematical Sciences

June  2016, 21(4): 1027-1050. doi: 10.3934/dcdsb.2016.21.1027

## Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization

 1 Dipartimento di Matematica, Università di Pisa, I-56127 Pisa, Italy 2 Department of Civil Infrastructure and Environmental Engineering, Khalifa University of Science, Technology & Research (KUSTAR), Abu Dhabi, United Arab Emirates 3 Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, United States

Received  May 2015 Revised  January 2016 Published  March 2016

We study mathematical and physical properties of a family of recently introduced, reduced-order approximate deconvolution models. We first show a connection between these models and the Navier-Stokes-Voigt model, and also that Navier-Stokes-Voigt can be re-derived in the approximate deconvolution framework. We then study the energy balance and spectra of the model, and provide results of some turbulent-flow computations that backs up the theory. Analysis of global attractors for the model is also provided, as is a detailed analysis of the Voigt model's treatment of pulsatile flow.
Citation: Luigi C. Berselli, Tae-Yeon Kim, Leo G. Rebholz. Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027
##### References:
 [1] N. A. Adams and S. Stolz, Deconvolution Methods for Subgrid-Scale Approximation in Large Eddy Simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, 2001. [2] N. A. Adams and S. Stolz, A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys., 178 (2002), 391-426. doi: 10.1006/jcph.2002.7034. [3] H. Beirão da Veiga, Time periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains-Leray's problem for periodic flows, Arch. Ration. Mech. Anal., 178 (2005), 301-325. ibidem 198 (2010), 1095 doi: 10.1007/s00205-005-0376-3. [4] L. C. Berselli, Analysis of a Large Eddy Simulation model based on anisotropic filtering, J. Math. Anal. Appl., 386 (2012), 149-170. doi: 10.1016/j.jmaa.2011.07.044. [5] L. C. Berselli, Towards fluid equations by approximate deconvolution models, in Mathematical Aspects of Fluid Mechanics, vol. 402 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2012, 1-22. [6] L. C. Berselli and L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130. doi: 10.1016/j.na.2011.08.011. [7] L. C. Berselli, F. Guerra, B. Mazzolai and E. Sinibaldi, Pulsatile viscous flows in elliptical vessels and annuli: solution to the inverse problem, with application to blood and cerebrospinal fluid flow, SIAM J. Appl. Math., 74 (2014), 40-59. doi: 10.1137/120903385. [8] L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer-Verlag, Berlin, 2006. doi: 10.1007/b137408. [9] L. C. Berselli and R. Lewandowski, Convergence of approximate deconvolution models to the mean Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 171-198. doi: 10.1016/j.anihpc.2011.10.001. [10] L. C. Berselli and S. Spirito, On the Construction of Suitable Weak Solutions to the 3D Navier-Stokes Equations in a Bounded Domain by an Artificial Compressibility Method, Technical Report 1504.07800, arXiv, 2015, URL http://arxiv.org/abs/1504.07800. To appear in Commun. Contemp. Math. [11] A. L. Bowers, T.-Y. Kim, M. Neda, L. G. Rebholz and E. Fried, The Leray-$\alpha\beta$-deconvolution model: Energy analysis and numerical algorithms, Appl. Math. Model., 37 (2013), 1225-1241. doi: 10.1016/j.apm.2012.03.040. [12] Y. Cao, E. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848, \urlprefixhttp://projecteuclid.org/euclid.cms/1175797613. doi: 10.4310/CMS.2006.v4.n4.a8. [13] R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976, Mathematics in Science and Engineering, Vol. 127. [14] T. Chacón 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. [15] S. Chen, G. Doolen, R. Kraichnan and Z. She, On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. Fluids A, 5 (1993), 458-463. doi: 10.1063/1.858897. [16] S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Phys. D, 133 (1999), 66-83, Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998). doi: 10.1016/S0167-2789(99)00099-8. [17] A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629-649. doi: 10.1098/rspa.2004.1373. [18] A. Dunca and Y. Epshteyn, On the Stolz-Adams deconvolution model for the large-eddy simulation of turbulent flows, SIAM J. Math. Anal., 37 (2006), 1890-1902 (electronic). doi: 10.1137/S0036141003436302. [19] A. A. Dunca, A two-level multiscale deconvolution method for the large eddy simulation of turbulent flows, Math. Models Methods Appl. Sci., 22 (2012), 1250001, 30pp. doi: 10.1142/S0218202512500017. [20] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519, Advances in nonlinear mathematics and science. doi: 10.1016/S0167-2789(01)00191-9. [21] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, vol. 83 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754. [22] U. Frisch, Turbulence, The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995. [23] Y. C. Fung, Biomechanics. Circulation, Springer-Verlag, 1997. [24] K. J. Galvin, L. G. Rebholz and C. Trenchea, Efficient, unconditionally stable, and optimally accurate FE algorithms for approximate deconvolution models, SIAM J. Numer. Anal., 52 (2014), 678-707. doi: 10.1137/120887412. [25] D. F. Hinz, T.-Y. Kim and E. Fried, Statistics of the Navier-Stokes-alpha-beta regularization model for fluid turbulence, J. Phys. A, 47 (2014), 055501, 21pp. doi: 10.1088/1751-8113/47/5/055501. [26] K. J. K. Galvin, Advancements in Finite Element Methods for Newtonian and Non-Newtonian Flows, PhD thesis, Clemson University, 2013. [27] V. K. Kalantarov, B. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152. doi: 10.1007/s00332-008-9029-7. [28] V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714. doi: 10.1007/s11401-009-0205-3. [29] T.-Y. Kim, M. Cassiani, J. D. Albertson, J. E. Dolbow, E. Fried and M. E. Gurtin, Impact of the inherent separation of scales in the Navier-Stokes-$\alpha\beta$ equations, Phys. Rev. E (3), 79 (2009), 045307, 4pp. doi: 10.1103/PhysRevE.79.045307. [30] T.-Y. Kim, L. G. Rebholz and E. Fried, A deconvolution enhancement of the Navier-Stokes-$\alpha\beta$ model, J. Comput. Phys., 231 (2012), 4015-4027. doi: 10.1016/j.jcp.2011.12.003. [31] P. Kuberry, A. Larios, L. G. Rebholz and N. E. Wilson, Numerical approximation of the Voigt regularization for incompressible Navier-Stokes and magnetohydrodynamic flows, Comput. Math. Appl., 64 (2012), 2647-2662. doi: 10.1016/j.camwa.2012.07.010. [32] A. Larios, The Inviscid Voigt-Regularization for Hydrodynamic Models: Global Regularity, Boundary Conditions, and Blow-Up Phenomena, PhD thesis, Univ. California, Irvine, 2011. [33] A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627. doi: 10.3934/dcdsb.2010.14.603. [34] W. J. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128 (electronic). doi: 10.3934/dcdsb.2006.6.111. [35] W. J. Layton and L. Rebholz, Approximate Deconvolution Models of Turbulence Approximate Deconvolution Models of Turbulence, vol. 2042 of Lecture Notes in Mathematics, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-24409-4. [36] B. Levant, F. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14. [37] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 98-136, Boundary value problems of mathematical physics and related questions in the theory of functions, 7. [38] A. P. Oskolkov, On the theory of unsteady flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 115 (1982), 191-202, 310, Boundary value problems of mathematical physics and related questions in the theory of functions, 14. [39] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Handbook of numerical analysis. Vol. XII, Handb. Numer. Anal., XII, North-Holland, Amsterdam, 2004, 3-127. doi: 10.1016/S1570-8659(03)12001-7. [40] L. G. Rebholz, Well-posedness of a reduced order approximate deconvolution turbulence model, J. Math. Anal. Appl., 405 (2013), 738-741. doi: 10.1016/j.jmaa.2013.04.036. [41] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Scientific Computation, Springer-Verlag, Berlin, 2001, An introduction, With an introduction by Marcel Lesieur, Translated from the 1998 French original by the author. doi: 10.1007/978-3-662-04416-2. [42] Z. S. She, E. Jackson and S. A. Orszag, Statistical aspects of vortex dynamics in turbulence, in New perspectives in turbulence (Newport, RI, 1989), Springer, New York, 1991, 315-328. doi: 10.1007/978-1-4612-3156-1_12. [43] I. Stanculescu and C. C. Manica, Numerical analysis of Leray-Tikhonov deconvolution models of fluid motion, Comput. Math. Appl., 60 (2010), 1440-1456. doi: 10.1016/j.camwa.2010.06.026. [44] S. Stolz and N. A. Adams, An approximate deconvolution procedure for large-eddy simulation, Phys. Fluids, 11 (1999), 1699-1701. doi: 10.1063/1.869867. [45] S. Stolz, N. A. Adams and L. Kleiser, An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows, Phys. Fluids, 13 (2001), 997-1015. doi: 10.1063/1.1350896. [46] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [47] J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127 (1955), 553-563. doi: 10.1113/jphysiol.1955.sp005276.

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##### References:
 [1] N. A. Adams and S. Stolz, Deconvolution Methods for Subgrid-Scale Approximation in Large Eddy Simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, 2001. [2] N. A. Adams and S. Stolz, A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys., 178 (2002), 391-426. doi: 10.1006/jcph.2002.7034. [3] H. Beirão da Veiga, Time periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains-Leray's problem for periodic flows, Arch. Ration. Mech. Anal., 178 (2005), 301-325. ibidem 198 (2010), 1095 doi: 10.1007/s00205-005-0376-3. [4] L. C. Berselli, Analysis of a Large Eddy Simulation model based on anisotropic filtering, J. Math. Anal. Appl., 386 (2012), 149-170. doi: 10.1016/j.jmaa.2011.07.044. [5] L. C. Berselli, Towards fluid equations by approximate deconvolution models, in Mathematical Aspects of Fluid Mechanics, vol. 402 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2012, 1-22. [6] L. C. Berselli and L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130. doi: 10.1016/j.na.2011.08.011. [7] L. C. Berselli, F. Guerra, B. Mazzolai and E. Sinibaldi, Pulsatile viscous flows in elliptical vessels and annuli: solution to the inverse problem, with application to blood and cerebrospinal fluid flow, SIAM J. Appl. Math., 74 (2014), 40-59. doi: 10.1137/120903385. [8] L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer-Verlag, Berlin, 2006. doi: 10.1007/b137408. [9] L. C. Berselli and R. Lewandowski, Convergence of approximate deconvolution models to the mean Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 171-198. doi: 10.1016/j.anihpc.2011.10.001. [10] L. C. Berselli and S. Spirito, On the Construction of Suitable Weak Solutions to the 3D Navier-Stokes Equations in a Bounded Domain by an Artificial Compressibility Method, Technical Report 1504.07800, arXiv, 2015, URL http://arxiv.org/abs/1504.07800. To appear in Commun. Contemp. Math. [11] A. L. Bowers, T.-Y. Kim, M. Neda, L. G. Rebholz and E. Fried, The Leray-$\alpha\beta$-deconvolution model: Energy analysis and numerical algorithms, Appl. Math. Model., 37 (2013), 1225-1241. doi: 10.1016/j.apm.2012.03.040. [12] Y. Cao, E. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848, \urlprefixhttp://projecteuclid.org/euclid.cms/1175797613. doi: 10.4310/CMS.2006.v4.n4.a8. [13] R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976, Mathematics in Science and Engineering, Vol. 127. [14] T. Chacón 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. [15] S. Chen, G. Doolen, R. Kraichnan and Z. She, On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. Fluids A, 5 (1993), 458-463. doi: 10.1063/1.858897. [16] S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Phys. D, 133 (1999), 66-83, Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998). doi: 10.1016/S0167-2789(99)00099-8. [17] A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629-649. doi: 10.1098/rspa.2004.1373. [18] A. Dunca and Y. Epshteyn, On the Stolz-Adams deconvolution model for the large-eddy simulation of turbulent flows, SIAM J. Math. Anal., 37 (2006), 1890-1902 (electronic). doi: 10.1137/S0036141003436302. [19] A. A. Dunca, A two-level multiscale deconvolution method for the large eddy simulation of turbulent flows, Math. Models Methods Appl. Sci., 22 (2012), 1250001, 30pp. doi: 10.1142/S0218202512500017. [20] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519, Advances in nonlinear mathematics and science. doi: 10.1016/S0167-2789(01)00191-9. [21] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, vol. 83 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754. [22] U. Frisch, Turbulence, The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995. [23] Y. C. Fung, Biomechanics. Circulation, Springer-Verlag, 1997. [24] K. J. Galvin, L. G. Rebholz and C. Trenchea, Efficient, unconditionally stable, and optimally accurate FE algorithms for approximate deconvolution models, SIAM J. Numer. Anal., 52 (2014), 678-707. doi: 10.1137/120887412. [25] D. F. Hinz, T.-Y. Kim and E. Fried, Statistics of the Navier-Stokes-alpha-beta regularization model for fluid turbulence, J. Phys. A, 47 (2014), 055501, 21pp. doi: 10.1088/1751-8113/47/5/055501. [26] K. J. K. Galvin, Advancements in Finite Element Methods for Newtonian and Non-Newtonian Flows, PhD thesis, Clemson University, 2013. [27] V. K. Kalantarov, B. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152. doi: 10.1007/s00332-008-9029-7. [28] V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714. doi: 10.1007/s11401-009-0205-3. [29] T.-Y. Kim, M. Cassiani, J. D. Albertson, J. E. Dolbow, E. Fried and M. E. Gurtin, Impact of the inherent separation of scales in the Navier-Stokes-$\alpha\beta$ equations, Phys. Rev. E (3), 79 (2009), 045307, 4pp. doi: 10.1103/PhysRevE.79.045307. [30] T.-Y. Kim, L. G. Rebholz and E. Fried, A deconvolution enhancement of the Navier-Stokes-$\alpha\beta$ model, J. Comput. Phys., 231 (2012), 4015-4027. doi: 10.1016/j.jcp.2011.12.003. [31] P. Kuberry, A. Larios, L. G. Rebholz and N. E. Wilson, Numerical approximation of the Voigt regularization for incompressible Navier-Stokes and magnetohydrodynamic flows, Comput. Math. Appl., 64 (2012), 2647-2662. doi: 10.1016/j.camwa.2012.07.010. [32] A. Larios, The Inviscid Voigt-Regularization for Hydrodynamic Models: Global Regularity, Boundary Conditions, and Blow-Up Phenomena, PhD thesis, Univ. California, Irvine, 2011. [33] A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627. doi: 10.3934/dcdsb.2010.14.603. [34] W. J. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128 (electronic). doi: 10.3934/dcdsb.2006.6.111. [35] W. J. Layton and L. Rebholz, Approximate Deconvolution Models of Turbulence Approximate Deconvolution Models of Turbulence, vol. 2042 of Lecture Notes in Mathematics, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-24409-4. [36] B. Levant, F. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14. [37] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 98-136, Boundary value problems of mathematical physics and related questions in the theory of functions, 7. [38] A. P. Oskolkov, On the theory of unsteady flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 115 (1982), 191-202, 310, Boundary value problems of mathematical physics and related questions in the theory of functions, 14. [39] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Handbook of numerical analysis. Vol. XII, Handb. Numer. Anal., XII, North-Holland, Amsterdam, 2004, 3-127. doi: 10.1016/S1570-8659(03)12001-7. [40] L. G. Rebholz, Well-posedness of a reduced order approximate deconvolution turbulence model, J. Math. Anal. Appl., 405 (2013), 738-741. doi: 10.1016/j.jmaa.2013.04.036. [41] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Scientific Computation, Springer-Verlag, Berlin, 2001, An introduction, With an introduction by Marcel Lesieur, Translated from the 1998 French original by the author. doi: 10.1007/978-3-662-04416-2. [42] Z. S. She, E. Jackson and S. A. Orszag, Statistical aspects of vortex dynamics in turbulence, in New perspectives in turbulence (Newport, RI, 1989), Springer, New York, 1991, 315-328. doi: 10.1007/978-1-4612-3156-1_12. [43] I. Stanculescu and C. C. Manica, Numerical analysis of Leray-Tikhonov deconvolution models of fluid motion, Comput. Math. Appl., 60 (2010), 1440-1456. doi: 10.1016/j.camwa.2010.06.026. [44] S. Stolz and N. A. Adams, An approximate deconvolution procedure for large-eddy simulation, Phys. Fluids, 11 (1999), 1699-1701. doi: 10.1063/1.869867. [45] S. Stolz, N. A. Adams and L. Kleiser, An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows, Phys. Fluids, 13 (2001), 997-1015. doi: 10.1063/1.1350896. [46] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [47] J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127 (1955), 553-563. doi: 10.1113/jphysiol.1955.sp005276.
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