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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

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