American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Selection of calibrated subaction when temperature goes to zero in the discounted problem
  • DCDS Home
  • This Issue
  • Next Article
    A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations
October  2018, 38(10): 4979-4996. doi: 10.3934/dcds.2018217

Existence of weak solutions for particle-laden flow with surface tension

Roman M. Taranets 1,2,3,, and  Jeffrey T. Wong 2,4,

1. 

Institute of Applied Mathematics and Mechanics of the NASU, 1, Dobrovol'skogo Str., 84100, Sloviansk, Ukraine
2. 

Department of Mathematics, University of California, Los Angeles, California 90095-1555, USA
3. 

Vasyl' Stus Donetsk National University, 21, 600-richya Str., 21021, Vinnytsia, Ukraine
4. 

Department of Mathematics, Duke University, Durham, North Carolina, 27708-0320, USA

* Corresponding author: Roman M. Taranets

Received  October 2017 Revised  May 2018 Published  July 2018

Fund Project: This work was supported in part by NSF grant DMS-1312543, and by a grant from Ministry of Education and Science of Ukraine (0118U003138 to Roman Taranets).

We prove the existence of solutions for a coupled system modeling the flow of a suspension of fluid and negatively buoyant non-colloidal particles in the thin film limit. The equations take the form of a fourth-order non-linear degenerate parabolic equation for the film height $h$ coupled to a second-order degenerate parabolic equation for the particle density $ψ$. We prove the existence of physically relevant solutions, which satisfy the uniform bounds $0 ≤ ψ/h ≤ 1$ and $h ≥ 0$.

Keywords: Fourth-order degenerate parabolic equations, existence of weak solutions, thin liquid films, suspensions, multiphase flows.
Mathematics Subject Classification: Primary: 35K65, 35K55; Secondary: 76D08.
Citation: Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217
References:
[1]

J. W. Barrett, H. Garcke and R. Nürnberg, Finite element approximation of surfactant spreading on a thin film, SIAM Journal on Numerical Analysis, 41 (2003), 1427-1464.  doi: 10.1137/S003614290139799X.  Google Scholar

[2]

S. Berres, R. Bürger and E. Tory, Mixed-type systems of convection-diffusion equations modeling polydisperse sedimentation, in Analysis and Simulation of Multifield Problems (eds. W. Wendland and M. Efendiev), Springer Nature (2003), 257-262. doi: 10.1007/978-3-540-36527-3_30.  Google Scholar

[3]

A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions, Communications on Pure and Applied Mathematics, 49 (1996), 85-123.  doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.  Google Scholar

[4]

A. L. Bertozzi and M. Pugh, Long-wave instabilities and saturation in thin film equations, Communications on Pure and Applied Mathematics, 51 (1998), 625-661.  doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9.  Google Scholar

[5]

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Higher Order Nonlinear Degenerate Parabolic Equations, 83 (1990), 179-206.  doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar

[6]

F. Boyer, E. Guazelli and O. Pouliquen Unifying suspension and granular rheology Physical Review Letters, 107 (2011), 188301. doi: 10.1103/PhysRevLett.107.188301.  Google Scholar

[7]

M. Chugunova, M. C. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM Journal on Mathematical Analysis, 42 (2010), 1826-1853.  doi: 10.1137/090777062.  Google Scholar

[8]

M. Chugunova and R. M. Taranets, Nonnegative weak solutions for a degenerate system modeling the spreading of surfactant on thin films, Applied Mathematics Research eXpress, 2013 (2013), 102-126.  doi: 10.1093/amrx/abs014.  Google Scholar

[9]

M. Chugunova and R. M. Taranets, Blow-up with mass concentration for the long-wave unstable thin-film equation, Applicable Analysis, 95 (2016), 944-962.  doi: 10.1080/00036811.2015.1047829.  Google Scholar

[10]

M. Chugunova, J. R. King and R. M. Taranets, The interface dynamics of a surfactant drop on a thin viscous film, European Journal of Applied Mathematics, 28 (2017), 656-686.  doi: 10.1017/S0956792516000474.  Google Scholar

[11]

R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Reviews of Modern Physics, 81 (2009), 1131. doi: 10.1103/RevModPhys.81.1131.  Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[13]

H. Garcke and S. Wieland, Surfactant spreading on thin viscous films: Nonnegative solutions of a coupled degenerate system, SIAM Journal on Mathematical Analysis, 37 (2006), 2025-2048.  doi: 10.1137/040617017.  Google Scholar

[14]

S. Jachalski, G. Kitavtsev and R. M. Taranets, Weak solutions to lubrication systems describing the evolution of bilayer thin films, Communications in Mathematical Sciences, 12 (2014), 527-544.  doi: 10.4310/CMS.2014.v12.n3.a7.  Google Scholar

[15]

A. Mavromoustaki, L. Wang, J. Wong and A. L. Bertozzi, Surface tension effects for particle settling and resuspension in viscous thin films, Nonlinearity, 31 (2018), 3151-3171.  doi: 10.1088/1361-6544/aab91d.  Google Scholar

[16]

N. Murisic, B. Pausader, D. Peschka and A. L. Bertozzi, Dynamics of particle settling and resuspension in viscous liquid films, Journal of Fluid Mechanics, 717 (2013), 203-231.  doi: 10.1017/jfm.2012.567.  Google Scholar

[17]

A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Reviews of Modern Physics, 69 (1997), 931. doi: 10.1103/RevModPhys.69.931.  Google Scholar

[18]

A. E. Shishkov and R. M. Taranets, On the thin-film equation with nonlinear convection in multidimensional domains, Ukr. Math. Bull, 1 (2004), 407{450. (Russian: http://dspace.nbuv.gov.ua/handle/123456789/124625)  Google Scholar

[19]

J. Wong, Modeling and Analysis of Thin-Film Incline Flow: Bidensity Suspensions and Surface, Tension Effects, Ph. D thesis, University of California, Los Angeles, 2017.  Google Scholar

[20]

L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM Journal on Numerical Analysis, 37 (1999), 523-555.  doi: 10.1137/S0036142998335698.  Google Scholar

[21]

J. Zhou, B. Dupuy, A. L. Bertozzi and A. E. Hosoi, Theory for shock dynamics in particle-laden thin films, Physical Review Letters, 94 (2005), 117803. doi: 10.1103/PhysRevLett.94.117803.  Google Scholar

show all references

References:
[1]

J. W. Barrett, H. Garcke and R. Nürnberg, Finite element approximation of surfactant spreading on a thin film, SIAM Journal on Numerical Analysis, 41 (2003), 1427-1464.  doi: 10.1137/S003614290139799X.  Google Scholar

[2]

S. Berres, R. Bürger and E. Tory, Mixed-type systems of convection-diffusion equations modeling polydisperse sedimentation, in Analysis and Simulation of Multifield Problems (eds. W. Wendland and M. Efendiev), Springer Nature (2003), 257-262. doi: 10.1007/978-3-540-36527-3_30.  Google Scholar

[3]

A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions, Communications on Pure and Applied Mathematics, 49 (1996), 85-123.  doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.  Google Scholar

[4]

A. L. Bertozzi and M. Pugh, Long-wave instabilities and saturation in thin film equations, Communications on Pure and Applied Mathematics, 51 (1998), 625-661.  doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9.  Google Scholar

[5]

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Higher Order Nonlinear Degenerate Parabolic Equations, 83 (1990), 179-206.  doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar

[6]

F. Boyer, E. Guazelli and O. Pouliquen Unifying suspension and granular rheology Physical Review Letters, 107 (2011), 188301. doi: 10.1103/PhysRevLett.107.188301.  Google Scholar

[7]

M. Chugunova, M. C. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM Journal on Mathematical Analysis, 42 (2010), 1826-1853.  doi: 10.1137/090777062.  Google Scholar

[8]

M. Chugunova and R. M. Taranets, Nonnegative weak solutions for a degenerate system modeling the spreading of surfactant on thin films, Applied Mathematics Research eXpress, 2013 (2013), 102-126.  doi: 10.1093/amrx/abs014.  Google Scholar

[9]

M. Chugunova and R. M. Taranets, Blow-up with mass concentration for the long-wave unstable thin-film equation, Applicable Analysis, 95 (2016), 944-962.  doi: 10.1080/00036811.2015.1047829.  Google Scholar

[10]

M. Chugunova, J. R. King and R. M. Taranets, The interface dynamics of a surfactant drop on a thin viscous film, European Journal of Applied Mathematics, 28 (2017), 656-686.  doi: 10.1017/S0956792516000474.  Google Scholar

[11]

R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Reviews of Modern Physics, 81 (2009), 1131. doi: 10.1103/RevModPhys.81.1131.  Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[13]

H. Garcke and S. Wieland, Surfactant spreading on thin viscous films: Nonnegative solutions of a coupled degenerate system, SIAM Journal on Mathematical Analysis, 37 (2006), 2025-2048.  doi: 10.1137/040617017.  Google Scholar

[14]

S. Jachalski, G. Kitavtsev and R. M. Taranets, Weak solutions to lubrication systems describing the evolution of bilayer thin films, Communications in Mathematical Sciences, 12 (2014), 527-544.  doi: 10.4310/CMS.2014.v12.n3.a7.  Google Scholar

[15]

A. Mavromoustaki, L. Wang, J. Wong and A. L. Bertozzi, Surface tension effects for particle settling and resuspension in viscous thin films, Nonlinearity, 31 (2018), 3151-3171.  doi: 10.1088/1361-6544/aab91d.  Google Scholar

[16]

N. Murisic, B. Pausader, D. Peschka and A. L. Bertozzi, Dynamics of particle settling and resuspension in viscous liquid films, Journal of Fluid Mechanics, 717 (2013), 203-231.  doi: 10.1017/jfm.2012.567.  Google Scholar

[17]

A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Reviews of Modern Physics, 69 (1997), 931. doi: 10.1103/RevModPhys.69.931.  Google Scholar

[18]

A. E. Shishkov and R. M. Taranets, On the thin-film equation with nonlinear convection in multidimensional domains, Ukr. Math. Bull, 1 (2004), 407{450. (Russian: http://dspace.nbuv.gov.ua/handle/123456789/124625)  Google Scholar

[19]

J. Wong, Modeling and Analysis of Thin-Film Incline Flow: Bidensity Suspensions and Surface, Tension Effects, Ph. D thesis, University of California, Los Angeles, 2017.  Google Scholar

[20]

L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM Journal on Numerical Analysis, 37 (1999), 523-555.  doi: 10.1137/S0036142998335698.  Google Scholar

[21]

J. Zhou, B. Dupuy, A. L. Bertozzi and A. E. Hosoi, Theory for shock dynamics in particle-laden thin films, Physical Review Letters, 94 (2005), 117803. doi: 10.1103/PhysRevLett.94.117803.  Google Scholar

[1]

Yang Liu, Wenke Li. A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021112
[2]

Marco Donatelli, Luca Vilasi. Existence of multiple solutions for a fourth-order problem with variable exponent. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021141
[3]

Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021110
[4]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234
[5]

Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667
[6]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1
[7]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617
[8]

Kristian Bredies. Weak solutions of linear degenerate parabolic equations and an application in image processing. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1203-1229. doi: 10.3934/cpaa.2009.8.1203
[9]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225
[10]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465
[11]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851
[12]

Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564
[13]

Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393-402. doi: 10.3934/proc.2003.2003.393
[14]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729
[15]

M. Ben Ayed, K. El Mehdi, M. Hammami. Nonexistence of bounded energy solutions for a fourth order equation on thin annuli. Communications on Pure & Applied Analysis, 2004, 3 (4) : 557-580. doi: 10.3934/cpaa.2004.3.557
[16]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[17]

Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005
[18]

Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic & Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691
[19]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
[20]

Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (167)
  • HTML views (103)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences