• 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

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 $h ≥ 0$.

Citation: Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217
References:
[1]

J. W. BarrettH. 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.

[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.

[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.

[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.

[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.

[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.

[7]

M. ChugunovaM. 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.

[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.

[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.

[10]

M. ChugunovaJ. 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.

[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.

[12]

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

[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.

[14]

S. JachalskiG. 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.

[15]

A. MavromoustakiL. WangJ. 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.

[16]

N. MurisicB. PausaderD. 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.

[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.

[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)

[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.

[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.

[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.

show all references

References:
[1]

J. W. BarrettH. 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.

[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.

[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.

[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.

[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.

[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.

[7]

M. ChugunovaM. 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.

[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.

[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.

[10]

M. ChugunovaJ. 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.

[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.

[12]

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

[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.

[14]

S. JachalskiG. 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.

[15]

A. MavromoustakiL. WangJ. 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.

[16]

N. MurisicB. PausaderD. 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.

[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.

[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)

[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.

[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.

[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.

[1]

Yang Liu, Wenke Li. A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4367-4381. doi: 10.3934/dcdss.2021112

[2]

Marco Donatelli, Luca Vilasi. Existence of multiple solutions for a fourth-order problem with variable exponent. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2471-2481. 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 and Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110

[4]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete and 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 and Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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 and Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691

[19]

Zhen-Zhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022080

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (232)
  • HTML views (105)
  • Cited by (0)

Other articles
by authors

[Back to Top]