• Previous Article
    Combining robust state estimation with nonlinear model predictive control to regulate the acute inflammatory response to pathogen
  • MBE Home
  • This Issue
  • Next Article
    Global stability of a multi-group model with vaccination age, distributed delay and random perturbation
2015, 12(5): 1107-1126. doi: 10.3934/mbe.2015.12.1107

Cilium height difference between strokes is more effective in driving fluid transport in mucociliary clearance: A numerical study

1. 

Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, United States

2. 

Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, United States

Received  January 2015 Revised  March 2015 Published  June 2015

Mucociliary clearance is the first line of defense in our airway. The purpose of this study is to identify and study key factors in the cilia motion that influence the transport ability of the mucociliary system. Using a rod-propel-fluid model, we examine the effects of cilia density, beating frequency, metachronal wavelength, and the extending height of the beating cilia. We first verify that asymmetry in the cilia motion is key to developing transport in the mucus flow. Next, two types of asymmetries between the effective and recovery strokes of the cilia motion are considered, the cilium beating velocity difference and the cilium height difference. We show that the cilium height difference is more efficient in driving the transport, and the more bend the cilium during the recovery stroke is, the more effective the transport would be. It is found that the transport capacity of the mucociliary system increases with cilia density and cilia beating frequency, but saturates above by a threshold value in both density and frequency. The metachronal wave that results from the phase lag among cilia does not contribute much to the mucus transport, which is consistent with the experimental observation of Sleigh (1989). We also test the effect of mucus viscosity, whose value is found to be inversely proportional to the transport ability. While multiple parts have to interplay and coordinate to allow for most effective mucociliary clearance, our findings from a simple model move us closer to understanding the effects of the cilia motion on the efficiency of this clearance system.
Citation: Ling Xu, Yi Jiang. Cilium height difference between strokes is more effective in driving fluid transport in mucociliary clearance: A numerical study. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1107-1126. doi: 10.3934/mbe.2015.12.1107
References:
[1]

B. A. Afzelius, Ultrastructural basis for ciliary motility, Eur J Respir Dis Suppl., 128 (1983), 280-286.

[2]

S. A. Baba, Regular steps in bending cilia during the effective stroke, Nature, 282 (1979), 717-720. doi: 10.1038/282717a0.

[3]

C. Barton and S. Raynor, Analytical investigation of cilia induced mucous flow, Bull Math Biophys., 29 (1967), 419-428. doi: 10.1007/BF02476581.

[4]

J. R. Blake, A note on the image system for a stokeslet in a no slip boundary, Proc. Gamb. Phil. Soc., 70 (1971), 303-310. doi: 10.1017/S0305004100049902.

[5]

J. R. Blake, A model for the micro-structure in ciliated organisms, J. Fluid Mech., 55 (1972), 1-23.

[6]

B. Button, L.-H. Cai, C. Ehre, M. Kesimer, D. B. Hill, J. K. Sheehan, R. C. Boucher and M. Rubinstein, A periciliary brush promotes the lung health by separating the mucus layer from airway epithelia, Science, 337 (2012), 937-941. doi: 10.1126/science.1223012.

[7]

A. Braiman and Z. Priel, Efficient mucociliary transport relies on efficient regulation of ciliary beating, Respir Physiol Neurobiol., 163 (2008), 202-207.

[8]

D. L. Brown, R. Cortez and M. L. Minion, Accurate projection methods for the incompressible Navier Stokes equations, J. Comput Phys., 168 (2001), 464-499. doi: 10.1006/jcph.2001.6715.

[9]

M. A. Chilvers and C. O'Callaghan, Analysis of ciliary beat pattern and beat frequency using digital high speed imaging: comparison with the photomultiplier and photodiode methods, Thorax, 55 (2000), 314-317. doi: 10.1136/thorax.55.4.314.

[10]

M. A. Chilvers, A. Rutman and C. O'Callaghan, Ciliary beat pattern is associated with specific ultrastructural defects in primary ciliary dyskinesia, J Allergy Clin Immunol., 112 (2003), 518-524. doi: 10.1016/S0091-6749(03)01799-8.

[11]

J. R. Clamp, Chemical aspects of mucus. General considerations, Br Med Bull., 34 (1978), 25-27.

[12]

R. Cortez, The method of regularized Stokeslets, SIAM J. Sci. Comput., 23 (2001), 1204-1225. doi: 10.1137/S106482750038146X.

[13]

R. Cortez, L. Fauci and A. Medovikov, The method of regularized Stokeslets in three dimensions: Analysis, validation, and application to helical swimming, Phys. Fluids, 17 (2005), 031504, 21pp. doi: 10.1063/1.1830486.

[14]

R. H. Dillon, L Fauci, C. Omoto and X. Yang, Fluid dynamic models of flagellar and ciliary beating, Ann N Y Acad Sci., 1101 (2007), 494-505.

[15]

S. H. Donaldson, W. D. Bennett, K. L. Zeman, M. R. Knowles, R. Tarran and R. C. Boucher, Mucus clearance and lung function in cystic fibrosis with hypertonic saline, N. ENGL. J. MED., 354 (2006), 241-250.

[16]

J. Elgeti and G. Gompper, Emergence of metachronal waves in cilia arrays, PNAS, 110 (2013), 4470-4475. doi: 10.1073/pnas.1218869110.

[17]

D. Eshel and Z. Priel, Characterization of metachronal wave of beating cilia on forg's palate epithelium in tissue culture, J. Physiol., 388 (1987), 1-8.

[18]

L. Fauci, C. Peskin, A Computational Model of Aquatic Animal Locomotion, J. Comput Phys., 77 (1988), 85-108.

[19]

H. Flores, E. Lobaton, S. Mendez-Diez, S. Tlupova and R. Cortez, A study of bacterial flagellar bundling, Bull Math Biol., 67 (2005), 137-168. doi: 10.1016/j.bulm.2004.06.006.

[20]

W. M. Foster, E. Langenback and E. H. Bergofsky, Measurement to tracheal and bronchial mucus velocities in man: relation to lung clearance, J Appl Physiol Respir Environ Exerc Physiol., 48 (1980), 965-971.

[21]

M. Friedman, R. Dougherty, S. R. Nelson, R. P. White, M. A. Sackner and A. Wanner, Acute effects of an aerosol hair spray on tracheal mucociliary transport, Am Rev Respir Dis., 116 (1977), 281-286.

[22]

L. Gheber, A. Korngreen and Z. Priel, Effect of viscosity on metachrony in mucus propelling cilia, Cell Motil Cytoskeleton, 39 (1998), 9-20.

[23]

J. Kim and P. Moin, Application of a fractional step method to incompressible Navier-Stokes equations, J. Comput Phys., 59 (1985), 308-323. doi: 10.1016/0021-9991(85)90148-2.

[24]

M. R. Knowles and R. C. Boucher, Mucus clearance as a primary innate defense mechanism for mammalian airways, J Clin Invest., 109 (2002), 571-577.

[25]

M. C. Lai and C. S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput Phys., 160 (2000), 705-719. doi: 10.1006/jcph.2000.6483.

[26]

R. L. Leopold, M. J. O'Mahony, X. J. Lian, A. E. Tilley, B. G. Harvey and R. G. Crystal, Smoking is associated with shortened airway cilia, PloS One, 4 (2009), e8157. doi: 10.1371/journal.pone.0008157.

[27]

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady and Time Dependent Problems, {SIAM}, 2007. doi: 10.1137/1.9780898717839.

[28]

A. Livraghi and S. H. Randell, Cystic fibrosis and other respiratory diseases of impaired mucus clearance, Toxicol Pathol, 35 (2007), 116-129. doi: 10.1080/01926230601060025.

[29]

P. M. Low, C. L. Luk, M. J. Dulfano and R. J. Finch, Ciliary beat frequency of human respiratory tract by different sampling techniques, Am Rev Respir Dis., 130 (1984), 497-498.

[30]

M. R. Marino and E. Aiello, Cinemicrographic analysis of beat dynamics of human respiratory cilia, Cell Motility, 2 (1982), 35-39. doi: 10.1002/cm.970020709.

[31]

H. Matsui, S. H. Randell, S. W. Peretti, C. W. Davis and R. C. Boucher, Coordinated clearance of periciliary liquid and mucus from airway surfaces, J Clin Invest., 102 (1998), 1125-1131. doi: 10.1172/JCI2687.

[32]

M. Salathe, Cilia and Mucus, from Development to Respiratory Defense, CRC Press, 2001.

[33]

S. M. Mitran, Metachronal wave formation in a model of pulmonary cilia, Comput Struct., 85 (2007), 763-774. doi: 10.1016/j.compstruc.2007.01.015.

[34]

P. G. Noone, M. W. Leigh, A. Sannuti, S. L. Minnix, J. L. Carson, M. Hazucha, M. A. Zariwala and M. R. Knowles, Primary ciliary dyskinesia: Diagnostic and phenotypic features, Am J Respir Crit Care Med, 169 (2004), 459-467. doi: 10.1164/rccm.200303-365OC.

[35]

C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput Phys., 25 (1977), 220-252. doi: 10.1016/0021-9991(77)90100-0.

[36]

E. M. Purcell, Life at low Reynolds number, AIP Conf. Proc., 28 (1976), p49. doi: 10.1063/1.30370.

[37]

W. S. Sale and P. Satir, Direction of active sliding of microtubules in Tetrahymena cilia, PNAS, 74 (1977), 2045-2049. doi: 10.1073/pnas.74.5.2045.

[38]

M. J. Sanderson and M. A. Sleigh, Ciliary activity of cultured rabbit tracheal epithelium: Beat pattern and metachrony, J Cell Sci., 47 (1981), 331-347.

[39]

P. Satir, Studies on cilia: II. examination of the distal region of the ciliary shaft and the role of the filaments in motility, J Cell Biol., 26 (1965), 805-834. doi: 10.1083/jcb.26.3.805.

[40]

A. Schmid and M. Salathe, Ciliary beat co-ordination by calcium, Biol Cell, 103 (2011), 159-169. doi: 10.1042/BC20100120.

[41]

P. R. Sears, K. Thompson, M. R. Knowles and C. W. Davis, Human airway ciliary dynamics, Am J Physiol Lung Cell Mol Phyiol., 304 (2013), L170-L183. doi: 10.1152/ajplung.00105.2012.

[42]

M. A. Sleigh, Ciliary function in transport of mucus, Eur J Respir Dis. Suppl., 128 (1983), 287-292.

[43]

M. A. Sleigh, Adaptations of ciliary systems for the propulsion of water and mucus, Comp Biochem Physiol A Comp Physiol., 94 (1989), 359-364. doi: 10.1016/0300-9629(89)90559-8.

[44]

G. Taylor, Analysis of the Swimming of Microscopic Organisms, Proc. R. Soc. Lond. A, 209(1951), 447-461.

[45]

E. O. Tuck, A note on a swimming problem, J. Fluid Mech., 31 (1968), 305-308.

[46]

E. Tuomanen, The surface of mammalian respiratory cilia: interactions between cilia and respiratory pathogens, Ciliary and Flagellar Membranes, (Ed: Bloodgood RA) (1990), 363-388. Springer US. doi: 10.1007/978-1-4613-0515-6_14.

[47]

X. Yang, R. H. Dillon and L. J. Fauci, An integrative computational model of multiciliary beating, Bull Math Biol., 70 (2008), 1192-1215. doi: 10.1007/s11538-008-9296-3.

show all references

References:
[1]

B. A. Afzelius, Ultrastructural basis for ciliary motility, Eur J Respir Dis Suppl., 128 (1983), 280-286.

[2]

S. A. Baba, Regular steps in bending cilia during the effective stroke, Nature, 282 (1979), 717-720. doi: 10.1038/282717a0.

[3]

C. Barton and S. Raynor, Analytical investigation of cilia induced mucous flow, Bull Math Biophys., 29 (1967), 419-428. doi: 10.1007/BF02476581.

[4]

J. R. Blake, A note on the image system for a stokeslet in a no slip boundary, Proc. Gamb. Phil. Soc., 70 (1971), 303-310. doi: 10.1017/S0305004100049902.

[5]

J. R. Blake, A model for the micro-structure in ciliated organisms, J. Fluid Mech., 55 (1972), 1-23.

[6]

B. Button, L.-H. Cai, C. Ehre, M. Kesimer, D. B. Hill, J. K. Sheehan, R. C. Boucher and M. Rubinstein, A periciliary brush promotes the lung health by separating the mucus layer from airway epithelia, Science, 337 (2012), 937-941. doi: 10.1126/science.1223012.

[7]

A. Braiman and Z. Priel, Efficient mucociliary transport relies on efficient regulation of ciliary beating, Respir Physiol Neurobiol., 163 (2008), 202-207.

[8]

D. L. Brown, R. Cortez and M. L. Minion, Accurate projection methods for the incompressible Navier Stokes equations, J. Comput Phys., 168 (2001), 464-499. doi: 10.1006/jcph.2001.6715.

[9]

M. A. Chilvers and C. O'Callaghan, Analysis of ciliary beat pattern and beat frequency using digital high speed imaging: comparison with the photomultiplier and photodiode methods, Thorax, 55 (2000), 314-317. doi: 10.1136/thorax.55.4.314.

[10]

M. A. Chilvers, A. Rutman and C. O'Callaghan, Ciliary beat pattern is associated with specific ultrastructural defects in primary ciliary dyskinesia, J Allergy Clin Immunol., 112 (2003), 518-524. doi: 10.1016/S0091-6749(03)01799-8.

[11]

J. R. Clamp, Chemical aspects of mucus. General considerations, Br Med Bull., 34 (1978), 25-27.

[12]

R. Cortez, The method of regularized Stokeslets, SIAM J. Sci. Comput., 23 (2001), 1204-1225. doi: 10.1137/S106482750038146X.

[13]

R. Cortez, L. Fauci and A. Medovikov, The method of regularized Stokeslets in three dimensions: Analysis, validation, and application to helical swimming, Phys. Fluids, 17 (2005), 031504, 21pp. doi: 10.1063/1.1830486.

[14]

R. H. Dillon, L Fauci, C. Omoto and X. Yang, Fluid dynamic models of flagellar and ciliary beating, Ann N Y Acad Sci., 1101 (2007), 494-505.

[15]

S. H. Donaldson, W. D. Bennett, K. L. Zeman, M. R. Knowles, R. Tarran and R. C. Boucher, Mucus clearance and lung function in cystic fibrosis with hypertonic saline, N. ENGL. J. MED., 354 (2006), 241-250.

[16]

J. Elgeti and G. Gompper, Emergence of metachronal waves in cilia arrays, PNAS, 110 (2013), 4470-4475. doi: 10.1073/pnas.1218869110.

[17]

D. Eshel and Z. Priel, Characterization of metachronal wave of beating cilia on forg's palate epithelium in tissue culture, J. Physiol., 388 (1987), 1-8.

[18]

L. Fauci, C. Peskin, A Computational Model of Aquatic Animal Locomotion, J. Comput Phys., 77 (1988), 85-108.

[19]

H. Flores, E. Lobaton, S. Mendez-Diez, S. Tlupova and R. Cortez, A study of bacterial flagellar bundling, Bull Math Biol., 67 (2005), 137-168. doi: 10.1016/j.bulm.2004.06.006.

[20]

W. M. Foster, E. Langenback and E. H. Bergofsky, Measurement to tracheal and bronchial mucus velocities in man: relation to lung clearance, J Appl Physiol Respir Environ Exerc Physiol., 48 (1980), 965-971.

[21]

M. Friedman, R. Dougherty, S. R. Nelson, R. P. White, M. A. Sackner and A. Wanner, Acute effects of an aerosol hair spray on tracheal mucociliary transport, Am Rev Respir Dis., 116 (1977), 281-286.

[22]

L. Gheber, A. Korngreen and Z. Priel, Effect of viscosity on metachrony in mucus propelling cilia, Cell Motil Cytoskeleton, 39 (1998), 9-20.

[23]

J. Kim and P. Moin, Application of a fractional step method to incompressible Navier-Stokes equations, J. Comput Phys., 59 (1985), 308-323. doi: 10.1016/0021-9991(85)90148-2.

[24]

M. R. Knowles and R. C. Boucher, Mucus clearance as a primary innate defense mechanism for mammalian airways, J Clin Invest., 109 (2002), 571-577.

[25]

M. C. Lai and C. S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput Phys., 160 (2000), 705-719. doi: 10.1006/jcph.2000.6483.

[26]

R. L. Leopold, M. J. O'Mahony, X. J. Lian, A. E. Tilley, B. G. Harvey and R. G. Crystal, Smoking is associated with shortened airway cilia, PloS One, 4 (2009), e8157. doi: 10.1371/journal.pone.0008157.

[27]

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady and Time Dependent Problems, {SIAM}, 2007. doi: 10.1137/1.9780898717839.

[28]

A. Livraghi and S. H. Randell, Cystic fibrosis and other respiratory diseases of impaired mucus clearance, Toxicol Pathol, 35 (2007), 116-129. doi: 10.1080/01926230601060025.

[29]

P. M. Low, C. L. Luk, M. J. Dulfano and R. J. Finch, Ciliary beat frequency of human respiratory tract by different sampling techniques, Am Rev Respir Dis., 130 (1984), 497-498.

[30]

M. R. Marino and E. Aiello, Cinemicrographic analysis of beat dynamics of human respiratory cilia, Cell Motility, 2 (1982), 35-39. doi: 10.1002/cm.970020709.

[31]

H. Matsui, S. H. Randell, S. W. Peretti, C. W. Davis and R. C. Boucher, Coordinated clearance of periciliary liquid and mucus from airway surfaces, J Clin Invest., 102 (1998), 1125-1131. doi: 10.1172/JCI2687.

[32]

M. Salathe, Cilia and Mucus, from Development to Respiratory Defense, CRC Press, 2001.

[33]

S. M. Mitran, Metachronal wave formation in a model of pulmonary cilia, Comput Struct., 85 (2007), 763-774. doi: 10.1016/j.compstruc.2007.01.015.

[34]

P. G. Noone, M. W. Leigh, A. Sannuti, S. L. Minnix, J. L. Carson, M. Hazucha, M. A. Zariwala and M. R. Knowles, Primary ciliary dyskinesia: Diagnostic and phenotypic features, Am J Respir Crit Care Med, 169 (2004), 459-467. doi: 10.1164/rccm.200303-365OC.

[35]

C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput Phys., 25 (1977), 220-252. doi: 10.1016/0021-9991(77)90100-0.

[36]

E. M. Purcell, Life at low Reynolds number, AIP Conf. Proc., 28 (1976), p49. doi: 10.1063/1.30370.

[37]

W. S. Sale and P. Satir, Direction of active sliding of microtubules in Tetrahymena cilia, PNAS, 74 (1977), 2045-2049. doi: 10.1073/pnas.74.5.2045.

[38]

M. J. Sanderson and M. A. Sleigh, Ciliary activity of cultured rabbit tracheal epithelium: Beat pattern and metachrony, J Cell Sci., 47 (1981), 331-347.

[39]

P. Satir, Studies on cilia: II. examination of the distal region of the ciliary shaft and the role of the filaments in motility, J Cell Biol., 26 (1965), 805-834. doi: 10.1083/jcb.26.3.805.

[40]

A. Schmid and M. Salathe, Ciliary beat co-ordination by calcium, Biol Cell, 103 (2011), 159-169. doi: 10.1042/BC20100120.

[41]

P. R. Sears, K. Thompson, M. R. Knowles and C. W. Davis, Human airway ciliary dynamics, Am J Physiol Lung Cell Mol Phyiol., 304 (2013), L170-L183. doi: 10.1152/ajplung.00105.2012.

[42]

M. A. Sleigh, Ciliary function in transport of mucus, Eur J Respir Dis. Suppl., 128 (1983), 287-292.

[43]

M. A. Sleigh, Adaptations of ciliary systems for the propulsion of water and mucus, Comp Biochem Physiol A Comp Physiol., 94 (1989), 359-364. doi: 10.1016/0300-9629(89)90559-8.

[44]

G. Taylor, Analysis of the Swimming of Microscopic Organisms, Proc. R. Soc. Lond. A, 209(1951), 447-461.

[45]

E. O. Tuck, A note on a swimming problem, J. Fluid Mech., 31 (1968), 305-308.

[46]

E. Tuomanen, The surface of mammalian respiratory cilia: interactions between cilia and respiratory pathogens, Ciliary and Flagellar Membranes, (Ed: Bloodgood RA) (1990), 363-388. Springer US. doi: 10.1007/978-1-4613-0515-6_14.

[47]

X. Yang, R. H. Dillon and L. J. Fauci, An integrative computational model of multiciliary beating, Bull Math Biol., 70 (2008), 1192-1215. doi: 10.1007/s11538-008-9296-3.

[1]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic and Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004

[2]

Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165

[3]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[4]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[5]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153

[6]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073

[7]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

[8]

D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1251-1259. doi: 10.3934/dcdsb.2015.20.1251

[9]

Mustafa A. H. Al-Jaboori, D. Wirosoetisno. Navier--Stokes equations on the $\beta$-plane. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 687-701. doi: 10.3934/dcdsb.2011.16.687

[10]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433

[11]

T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067

[12]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

[13]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403

[14]

Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189

[15]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

[16]

Tong Tang, Hongjun Gao. On the compressible Navier-Stokes-Korteweg equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2745-2766. doi: 10.3934/dcdsb.2016071

[17]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717

[18]

Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269

[19]

Vena Pearl Bongolan-walsh, David Cheban, Jinqiao Duan. Recurrent motions in the nonautonomous Navier-Stokes system. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 255-262. doi: 10.3934/dcdsb.2003.3.255

[20]

Andrea Giorgini, Roger Temam. Attractors for the Navier-Stokes-Cahn-Hilliard system. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2249-2274. doi: 10.3934/dcdss.2022118

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (231)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]