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A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations
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 |
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 $h ≥ 0$.
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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. |
| [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. |
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