Efficient and accurate time adaptive multigrid simulations of droplet spreading
An efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non‐linear lubrication equations. The algorithm is fully implicit and has embedded within it an adaptive time‐stepping scheme that enables the same to be...
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| Published in | International journal for numerical methods in fluids Vol. 45; no. 11; pp. 1161 - 1186 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Chichester, UK
John Wiley & Sons, Ltd
20.08.2004
Wiley |
| Subjects | |
| Online Access | Get full text |
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| Abstract | An efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non‐linear lubrication equations. The algorithm is fully implicit and has embedded within it an adaptive time‐stepping scheme that enables the same to be optimized in a controlled manner subject to a specific error tolerance. The method is first validated against a range of analytical and existing numerical predictions commensurate with droplet spreading and then used to simulate a series of new, three‐dimensional flows consisting of droplet motion on substrates containing topographic and wetting heterogeneities. The latter are of particular interest and reveal how droplets can be made to spread preferentially on substrates owing to an interplay between different topographic and surface wetting characteristics. Copyright © 2004 John Wiley & Sons, Ltd. |
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| AbstractList | An efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non‐linear lubrication equations. The algorithm is fully implicit and has embedded within it an adaptive time‐stepping scheme that enables the same to be optimized in a controlled manner subject to a specific error tolerance. The method is first validated against a range of analytical and existing numerical predictions commensurate with droplet spreading and then used to simulate a series of new, three‐dimensional flows consisting of droplet motion on substrates containing topographic and wetting heterogeneities. The latter are of particular interest and reveal how droplets can be made to spread preferentially on substrates owing to an interplay between different topographic and surface wetting characteristics. Copyright © 2004 John Wiley & Sons, Ltd. An efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non-linear lubrication equations. The algorithm is fully implicit and has embedded within it an adaptive time-stepping scheme that enables the same to be optimized in a controlled manner subject to a specific error tolerance. The method is first validated against a range of analytical and existing numerical predictions commensurate with droplet spreading and then used to simulate a series of new, three-dimensional flows consisting of droplet motion on substrates containing topographic and wetting heterogeneities. The latter are of particular interest and reveal how droplets can be made to spread preferentially on substrates owing to an interplay between different topographic and surface wetting characteristics. |
| Author | Jimack, P. K. Sellier, M. Gaskell, P. H. Thompson, H. M. |
| Author_xml | – sequence: 1 givenname: P. H. surname: Gaskell fullname: Gaskell, P. H. email: p.h.gaskell@efm.leeds.ac.uk organization: Engineering Fluid Mechanics Research Group, School of Mechanical Engineering, The University of Leeds, Leeds LS2 9JT, U.K – sequence: 2 givenname: P. K. surname: Jimack fullname: Jimack, P. K. organization: School of Computing, The University of Leeds, Leeds LS2 9JT, U.K – sequence: 3 givenname: M. surname: Sellier fullname: Sellier, M. organization: Engineering Fluid Mechanics Research Group, School of Mechanical Engineering, The University of Leeds, Leeds LS2 9JT, U.K – sequence: 4 givenname: H. M. surname: Thompson fullname: Thompson, H. M. organization: Engineering Fluid Mechanics Research Group, School of Mechanical Engineering, The University of Leeds, Leeds LS2 9JT, U.K |
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| Keywords | Algorithms Three dimensional flow Computational fluid dynamics Multigrid Spreading Digital simulation Drops Adaptive method |
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| References | Trottenberg U. In Multigrid, Trottenberg U, Oostorbe CW, Schuller A (eds) (Guest Contribution by Brandt A, Ooswald P, Stuber K). San Diego Conference. Academic Press: London, 2001. Lelah MD, Marmur A. Spreading kinetics of drops on glass. Journal of Colloid and Interface Science 1981; 82:518-525. Stillwagon LE, Larson RG. Fundamentals of topographic surface levelling. Journal of Applied Physics 1988; 63:5251. Spitaleri RM, Corinaldesi L. A multigrid semi-implicit finite difference method for the two-dimensional shallow water equations. International Journal for Numerical Methods in Fluids 1997; 25:1229-1240. Christov CI, Pontes J, Walgraef MG, Velarde MG. Implicit time splitting for fourth-order parabolic equations. Computer Methods in Applied Mechanics and Engineering 1997; 148:209-224. Tanner L. The spreading of silicon drops on horizontal surfaces. Journal of Physics D 1979; 12:1473-1484. Weidner DE, Schwartz LW, Eley RR. Role of surface tension gradients in correcting coating defects in corners. Journal of Colloid and Interface Science 1996; 179:66-75. Hackbusch W. Multi-Grid Methods and Applications. Springer: Berlin, 1985. Eres MH, Schwartz LW, Roy RV. Fingering phenomena for driven coating films. Physics of Fluids 2000; 12(6):1278-1295. Thompson CP, Lezeau P. Application of the full approximation storage method to the numerical simulation of two-dimensional steady incompressible viscous multiphase flows. International Journal for Numerical Methods in Fluids 1998; 28:1217-1239. Zhornitskaya L, Bertozzi AL. Positivity-preserving numerical schemes for lubrication-type equations. SIAM Journal of Numerical Analysis 2000; 37(2):523-555. Schwartz LW, Eley RR. Simulation of droplet motion on low-energy and heterogeneous surfaces. Journal of Colloid and Interface Science 1998; 202:173-188. Weidner DE, Schwartz LW, Eres MH. Simulation of coating layer evolution and drop formation on horizontal cylinders. Journal of Colloid and Interface Science 1997; 187(1):243-258. Dormand JR. Numerical Methods for Differential Equations-a Computational Approach. CRC Press: Boca Raton, 1996. De Gennes PG. Wetting: statics and dynamics. Reviews of Modern Physics 1985; 57:827. Chapra SC, Canale RP. Numerical Methods for Engineers. McGraw-Hill: New York, 1998. Oron A, Davis SH, Bankoff SG. Long-scale evolution of thin liquid films. Reviews of Modern Physics 1997; 69(3):931-980. Liao S-J, Mashayek F. A multigrid approach for steady state laminar viscous flows. International Journal for Numerical Methods in Fluids 2001; 37:107-123. Bertozzi A. The mathematics of moving contact lines in thin liquid films. Notices of the AMS 1998; 45(6):689-697. Schwartz LW. Hysteretic effects in droplet motion on heterogeneous substrates: direct numerical simulation. Langmuir 1998; 14(12):3440-3453. Diez JA, Kondic L, Bertozzi A. Global models for moving contact lines. Physics Reviews E 2000; 63:011208. Chou M-H. A multigrid difference approach to steady flow between eccentric rotating cylinders. International Journal for Numerical Methods in Fluids 2000; 34:479-494. Mazouchi A, Homsy GM. Free surface Stokes flow over topography. Physics of Fluids 2001; 13(10):2751-2761. Gaskell PH, Jimack PK, Sellier M, Thompson HM, Wilson MCT. Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography. Journal of Fluid Mechanics 2004; 509:253-280. Wesseling P. Introduction to Multigrid Methods. Wiley: New York, 1992. Hocking LM. Rival contact-angle models and the spreading of drops. Journal of Fluid Mechanics 1992; 239:671-681. Mitlin VS. On dewetting conditions. Colloids Surfaces A: Physicochemical Engineering Aspects 1994; 89:97-101. Peurrung LM, Graves DB. Spin coating over topography. IEEE Transactions on Semiconductor Manufacturing 1993; 6(1):72-76. Kondic L, Diez J. Pattern formation in the flow of thin film down an inclined plane: constant flux configuration. Physics of Fluids 2001; 13(11):3168-3184. Decré MJ, Baret J-C. Gravity driven flows of low viscosity liquids over two-dimensional topographies. Journal of Fluid Mechanics 2003; 487:147-166. Nakaya C. Spread of fluid drops over a horizontal plane. Journal of Physics Society Japan 1974; 37:539. Orchard SE. On surface levelling in viscous liquids and gels. Applied Scientific Research A 1962; 11:451. Starov VM, Kalinin VV, Chen T-D. Advances in Colloid Interface Science 1994; 50:187. 1974; 37 1998; 28 1979; 12 1997; 69 1997; 25 2000; 63 1998 1994; 89 1996 1992 1962; 11 1981; 82 1998; 45 2004; 509 1993; 6 1997; 148 2001 2000; 37 2000; 12 2000; 34 1997; 187 1992; 239 1985 2001; 37 1998; 202 1982 1988; 63 1996; 179 1994; 50 2001; 13 2003; 487 1985; 57 1998; 14 Trottenberg U (e_1_2_1_29_2) 2001 Chapra SC (e_1_2_1_20_2) 1998 e_1_2_1_22_2 e_1_2_1_23_2 Bertozzi A (e_1_2_1_5_2) 1998; 45 e_1_2_1_26_2 e_1_2_1_27_2 e_1_2_1_24_2 e_1_2_1_25_2 Wesseling P (e_1_2_1_14_2) 1992 e_1_2_1_6_2 e_1_2_1_30_2 e_1_2_1_7_2 Orchard SE (e_1_2_1_21_2) 1962; 11 e_1_2_1_4_2 e_1_2_1_2_2 e_1_2_1_11_2 e_1_2_1_34_2 e_1_2_1_3_2 e_1_2_1_12_2 e_1_2_1_33_2 e_1_2_1_32_2 e_1_2_1_10_2 e_1_2_1_31_2 e_1_2_1_15_2 e_1_2_1_16_2 e_1_2_1_13_2 e_1_2_1_35_2 e_1_2_1_19_2 Dormand JR (e_1_2_1_28_2) 1996 e_1_2_1_8_2 e_1_2_1_17_2 e_1_2_1_9_2 e_1_2_1_18_2 |
| References_xml | – reference: De Gennes PG. Wetting: statics and dynamics. Reviews of Modern Physics 1985; 57:827. – reference: Peurrung LM, Graves DB. Spin coating over topography. IEEE Transactions on Semiconductor Manufacturing 1993; 6(1):72-76. – reference: Bertozzi A. The mathematics of moving contact lines in thin liquid films. Notices of the AMS 1998; 45(6):689-697. – reference: Zhornitskaya L, Bertozzi AL. Positivity-preserving numerical schemes for lubrication-type equations. SIAM Journal of Numerical Analysis 2000; 37(2):523-555. – reference: Eres MH, Schwartz LW, Roy RV. Fingering phenomena for driven coating films. Physics of Fluids 2000; 12(6):1278-1295. – reference: Tanner L. The spreading of silicon drops on horizontal surfaces. Journal of Physics D 1979; 12:1473-1484. – reference: Mazouchi A, Homsy GM. Free surface Stokes flow over topography. Physics of Fluids 2001; 13(10):2751-2761. – reference: Chapra SC, Canale RP. Numerical Methods for Engineers. McGraw-Hill: New York, 1998. – reference: Hocking LM. Rival contact-angle models and the spreading of drops. Journal of Fluid Mechanics 1992; 239:671-681. – reference: Kondic L, Diez J. Pattern formation in the flow of thin film down an inclined plane: constant flux configuration. Physics of Fluids 2001; 13(11):3168-3184. – reference: Liao S-J, Mashayek F. A multigrid approach for steady state laminar viscous flows. International Journal for Numerical Methods in Fluids 2001; 37:107-123. – reference: Spitaleri RM, Corinaldesi L. A multigrid semi-implicit finite difference method for the two-dimensional shallow water equations. International Journal for Numerical Methods in Fluids 1997; 25:1229-1240. – reference: Stillwagon LE, Larson RG. Fundamentals of topographic surface levelling. Journal of Applied Physics 1988; 63:5251. – reference: Oron A, Davis SH, Bankoff SG. Long-scale evolution of thin liquid films. Reviews of Modern Physics 1997; 69(3):931-980. – reference: Lelah MD, Marmur A. Spreading kinetics of drops on glass. Journal of Colloid and Interface Science 1981; 82:518-525. – reference: Hackbusch W. Multi-Grid Methods and Applications. Springer: Berlin, 1985. – reference: Nakaya C. Spread of fluid drops over a horizontal plane. Journal of Physics Society Japan 1974; 37:539. – reference: Orchard SE. On surface levelling in viscous liquids and gels. Applied Scientific Research A 1962; 11:451. – reference: Diez JA, Kondic L, Bertozzi A. Global models for moving contact lines. Physics Reviews E 2000; 63:011208. – reference: Schwartz LW. Hysteretic effects in droplet motion on heterogeneous substrates: direct numerical simulation. Langmuir 1998; 14(12):3440-3453. – reference: Wesseling P. Introduction to Multigrid Methods. Wiley: New York, 1992. – reference: Mitlin VS. On dewetting conditions. Colloids Surfaces A: Physicochemical Engineering Aspects 1994; 89:97-101. – reference: Weidner DE, Schwartz LW, Eley RR. Role of surface tension gradients in correcting coating defects in corners. Journal of Colloid and Interface Science 1996; 179:66-75. – reference: Chou M-H. A multigrid difference approach to steady flow between eccentric rotating cylinders. International Journal for Numerical Methods in Fluids 2000; 34:479-494. – reference: Christov CI, Pontes J, Walgraef MG, Velarde MG. Implicit time splitting for fourth-order parabolic equations. Computer Methods in Applied Mechanics and Engineering 1997; 148:209-224. – reference: Weidner DE, Schwartz LW, Eres MH. Simulation of coating layer evolution and drop formation on horizontal cylinders. Journal of Colloid and Interface Science 1997; 187(1):243-258. – reference: Dormand JR. Numerical Methods for Differential Equations-a Computational Approach. CRC Press: Boca Raton, 1996. – reference: Starov VM, Kalinin VV, Chen T-D. Advances in Colloid Interface Science 1994; 50:187. – reference: Gaskell PH, Jimack PK, Sellier M, Thompson HM, Wilson MCT. Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography. Journal of Fluid Mechanics 2004; 509:253-280. – reference: Trottenberg U. In Multigrid, Trottenberg U, Oostorbe CW, Schuller A (eds) (Guest Contribution by Brandt A, Ooswald P, Stuber K). San Diego Conference. Academic Press: London, 2001. – reference: Decré MJ, Baret J-C. Gravity driven flows of low viscosity liquids over two-dimensional topographies. Journal of Fluid Mechanics 2003; 487:147-166. – reference: Schwartz LW, Eley RR. Simulation of droplet motion on low-energy and heterogeneous surfaces. Journal of Colloid and Interface Science 1998; 202:173-188. – reference: Thompson CP, Lezeau P. Application of the full approximation storage method to the numerical simulation of two-dimensional steady incompressible viscous multiphase flows. International Journal for Numerical Methods in Fluids 1998; 28:1217-1239. – year: 1985 – volume: 6 start-page: 72 issue: 1 year: 1993 end-page: 76 article-title: Spin coating over topography publication-title: IEEE Transactions on Semiconductor Manufacturing – volume: 69 start-page: 931 issue: 3 year: 1997 end-page: 980 article-title: Long‐scale evolution of thin liquid films publication-title: Reviews of Modern Physics – volume: 13 start-page: 3168 issue: 11 year: 2001 end-page: 3184 article-title: Pattern formation in the flow of thin film down an inclined plane: constant flux configuration publication-title: Physics of Fluids – volume: 34 start-page: 479 year: 2000 end-page: 494 article-title: A multigrid difference approach to steady flow between eccentric rotating cylinders publication-title: International Journal for Numerical Methods in Fluids – volume: 37 start-page: 539 year: 1974 article-title: Spread of fluid drops over a horizontal plane publication-title: Journal of 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| Snippet | An efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non‐linear... An efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non-linear... |
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| SubjectTerms | adaptive time-stepping Computational methods in fluid dynamics droplets Drops and bubbles Exact sciences and technology Fluid dynamics fully implicit Fundamental areas of phenomenology (including applications) lubrication approximation multigrid Nonhomogeneous flows Physics spreading |
| Title | Efficient and accurate time adaptive multigrid simulations of droplet spreading |
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