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

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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 - A, 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

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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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